-- |
-- Module      : Data.Manifold.WithBoundary
-- Copyright   : (c) Justus Sagemüller 2020
-- License     : GPL v3
-- 
-- Maintainer  : (@) jsag $ hvl.no
-- Stability   : experimental
-- Portability : portable
-- 

{-# LANGUAGE FlexibleInstances        #-}
{-# LANGUAGE UndecidableInstances     #-}
{-# LANGUAGE TypeFamilies             #-}
{-# LANGUAGE MultiParamTypeClasses    #-}
{-# LANGUAGE FlexibleContexts         #-}
{-# LANGUAGE GADTs                    #-}
{-# LANGUAGE DefaultSignatures        #-}
{-# LANGUAGE DeriveGeneric            #-}
{-# LANGUAGE StandaloneDeriving       #-}
{-# LANGUAGE ConstraintKinds          #-}
{-# LANGUAGE UnicodeSyntax            #-}
{-# LANGUAGE ScopedTypeVariables      #-}
{-# LANGUAGE AllowAmbiguousTypes      #-}
{-# LANGUAGE TypeApplications         #-}
{-# LANGUAGE EmptyCase                #-}
{-# LANGUAGE LambdaCase               #-}
{-# LANGUAGE TypeOperators            #-}
{-# LANGUAGE TypeInType               #-}
{-# LANGUAGE NoStarIsType             #-}
{-# LANGUAGE CPP                      #-}


module Data.Manifold.WithBoundary
        ( SemimanifoldWithBoundary(..), PseudoAffineWithBoundary(..), ProjectableBoundary(..)
        , SmfdWBoundWitness(..)
        , AdditiveMonoid(..), HalfSpace(..)
        ) where

import Data.Manifold.WithBoundary.Class

import Data.VectorSpace
import Data.AffineSpace
import Data.Basis

import Math.Manifold.Core.PseudoAffine
import Data.Manifold.PseudoAffine
import Math.Manifold.Core.Types
import Data.Manifold.Types.Primitive
import Math.Manifold.VectorSpace.ZeroDimensional
import Math.LinearMap.Category ( Tensor(..), TensorSpace(..)
                               , LinearMap(..), LinearFunction(..), LinearSpace(..)
                               , Num', closedScalarWitness, ClosedScalarWitness(..)
                               , DualSpaceWitness(..), ScalarSpaceWitness(..)
                               , LinearManifoldWitness(..) )
#if MIN_VERSION_linearmap_category(0,6,0)
import Math.VectorSpace.DimensionAware
                ( DimensionAware(..), Dimensional(..), DimensionalityWitness(..)
                , dimensionality, DimensionalityCases(..)
                , dimensionalitySing )
import GHC.TypeLits
#if MIN_VERSION_singletons(3,0,0)
import Prelude.Singletons (SNum(..))
import GHC.TypeLits.Singletons (withKnownNat, SNat(..))
#else
import Data.Singletons.Prelude.Num (SNum(..), SNat(..))
import Data.Singletons.TypeLits (withKnownNat)
#endif
#endif
import Math.VectorSpace.Dual
import Math.VectorSpace.MiscUtil.MultiConstraints (SameScalar)
import Data.Monoid.Additive
import Data.Void
import Linear (V0, V1, V2, V3, V4)
import qualified Linear.Affine as LinAff

import Control.Applicative
import Control.Arrow

import qualified GHC.Generics as Gnrx
import GHC.Generics (Generic, (:*:)(..))
import Data.Kind (Type)
import Proof.Propositional (Empty(..))

import Data.CallStack (HasCallStack)



interiorSemimanifoldWitness :: ∀ a . SemimanifoldWithBoundary a
          => SemimanifoldWitness (Interior a)
interiorSemimanifoldWitness :: forall a.
SemimanifoldWithBoundary a =>
SemimanifoldWitness (Interior a)
interiorSemimanifoldWitness = case forall m. SemimanifoldWithBoundary m => SmfdWBoundWitness m
smfdWBoundWitness @a of
  SmfdWBoundWitness a
OpenManifoldWitness -> forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness
  SmfdWBoundWitness a
SmfdWBoundWitness -> forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness

#define VectorSpaceSansBoundary(v, s)                         \
instance (Num' (s), Eq (s), OpenManifold (s), ProjectableBoundary (s)) \
                   => SemimanifoldWithBoundary (v) where {      \
  type Interior (v) = v;                                 \
  type Boundary (v) = EmptyMfd (ZeroDim s);               \
  type HalfNeedle (v) = ℝay;                             \
  smfdWBoundWitness = OpenManifoldWitness;                \
  fromInterior = id;                                     \
  fromBoundary b = case b of {};                          \
  separateInterior = Right;                              \
  p|+^_ = case p of {};                                   \
  a.+^|b = Right $ a^+^b;                                \
  extendToBoundary _ _ = Nothing };                       \
instance (Num' (s), Eq (s), OpenManifold (s), ProjectableBoundary (s)) \
                  => PseudoAffineWithBoundary (v) where {\
  _!-|p = case p of {};                                   \
  (.--!) = (-) };                                        \
instance (Num' (s), Eq (s), OpenManifold (s), ProjectableBoundary (s)) \
                => ProjectableBoundary (v) where {              \
  projectToBoundary _ p = case p of {};                  \
  marginFromBoundary p = case p of {} }

VectorSpaceSansBoundary(ℝ,ℝ)
VectorSpaceSansBoundary(V0 s, s)
VectorSpaceSansBoundary(V1 s, s)
VectorSpaceSansBoundary(V2 s, s)
VectorSpaceSansBoundary(V3 s, s)
VectorSpaceSansBoundary(V4 s, s)

data ProductBoundary a b
  = BoundOfL !(Boundary a) !(Interior b)
  | BoundOfR !(Interior a) !(Boundary b)

data ProductBoundaryNeedleT (dn :: Dualness) a b v
  = ZeroProductBoundaryNeedle
  | NBoundOfL !(dn`Space`Needle (Boundary a)) !(dn`Space`Needle (Interior b)) !v
  | NBoundOfR !(dn`Space`Needle (Interior a)) !(dn`Space`Needle (Boundary b)) !v
type ProductBoundaryNeedle a b = ProductBoundaryNeedleT Vector a b
                                     (Scalar (Needle (Interior a)))

instance ( AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Interior b))
         , AdditiveGroup (dn`Space`Needle (Interior a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , AdditiveGroup v
         , ValidDualness dn )
    => AffineSpace (ProductBoundaryNeedleT dn a b v) where
  type Diff (ProductBoundaryNeedleT dn a b v) = ProductBoundaryNeedleT dn a b v
  ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle .+^ :: ProductBoundaryNeedleT dn a b v
-> Diff (ProductBoundaryNeedleT dn a b v)
-> ProductBoundaryNeedleT dn a b v
.+^ Diff (ProductBoundaryNeedleT dn a b v)
n = Diff (ProductBoundaryNeedleT dn a b v)
n
  ProductBoundaryNeedleT dn a b v
n .+^ Diff (ProductBoundaryNeedleT dn a b v)
ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle = ProductBoundaryNeedleT dn a b v
n
  NBoundOfL Space dn (Needle (Boundary a))
x Space dn (Needle (Interior b))
y v
v .+^ NBoundOfL Space dn (Needle (Boundary a))
ξ Space dn (Needle (Interior b))
υ v
β = forall (dn :: Dualness) a b v.
Space dn (Needle (Boundary a))
-> Space dn (Needle (Interior b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfL (Space dn (Needle (Boundary a))
xforall v. AdditiveGroup v => v -> v -> v
^+^Space dn (Needle (Boundary a))
ξ) (Space dn (Needle (Interior b))
yforall v. AdditiveGroup v => v -> v -> v
^+^Space dn (Needle (Interior b))
υ) (v
vforall v. AdditiveGroup v => v -> v -> v
^+^v
β)
  NBoundOfR Space dn (Needle (Interior a))
x Space dn (Needle (Boundary b))
y v
v .+^ NBoundOfR Space dn (Needle (Interior a))
ξ Space dn (Needle (Boundary b))
υ v
β = forall (dn :: Dualness) a b v.
Space dn (Needle (Interior a))
-> Space dn (Needle (Boundary b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfR (Space dn (Needle (Interior a))
xforall v. AdditiveGroup v => v -> v -> v
^+^Space dn (Needle (Interior a))
ξ) (Space dn (Needle (Boundary b))
yforall v. AdditiveGroup v => v -> v -> v
^+^Space dn (Needle (Boundary b))
υ) (v
vforall v. AdditiveGroup v => v -> v -> v
^+^v
β)
  ProductBoundaryNeedleT dn a b v
n .-. :: ProductBoundaryNeedleT dn a b v
-> ProductBoundaryNeedleT dn a b v
-> Diff (ProductBoundaryNeedleT dn a b v)
.-. ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle = ProductBoundaryNeedleT dn a b v
n
  NBoundOfL Space dn (Needle (Boundary a))
x Space dn (Needle (Interior b))
y v
v .-. NBoundOfL Space dn (Needle (Boundary a))
ξ Space dn (Needle (Interior b))
υ v
β = forall (dn :: Dualness) a b v.
Space dn (Needle (Boundary a))
-> Space dn (Needle (Interior b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfL (Space dn (Needle (Boundary a))
xforall v. AdditiveGroup v => v -> v -> v
^-^Space dn (Needle (Boundary a))
ξ) (Space dn (Needle (Interior b))
yforall v. AdditiveGroup v => v -> v -> v
^-^Space dn (Needle (Interior b))
υ) (v
vforall v. AdditiveGroup v => v -> v -> v
^-^v
β)
  NBoundOfR Space dn (Needle (Interior a))
x Space dn (Needle (Boundary b))
y v
v .-. NBoundOfR Space dn (Needle (Interior a))
ξ Space dn (Needle (Boundary b))
υ v
β = forall (dn :: Dualness) a b v.
Space dn (Needle (Interior a))
-> Space dn (Needle (Boundary b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfR (Space dn (Needle (Interior a))
xforall v. AdditiveGroup v => v -> v -> v
^-^Space dn (Needle (Interior a))
ξ) (Space dn (Needle (Boundary b))
yforall v. AdditiveGroup v => v -> v -> v
^-^Space dn (Needle (Boundary b))
υ) (v
vforall v. AdditiveGroup v => v -> v -> v
^-^v
β)

instance ( AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Interior b))
         , AdditiveGroup (dn`Space`Needle (Interior a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , AdditiveGroup v
         , ValidDualness dn )
    => AdditiveGroup (ProductBoundaryNeedleT dn a b v) where
  zeroV :: ProductBoundaryNeedleT dn a b v
zeroV = forall (dn :: Dualness) a b v. ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle
  ^+^ :: ProductBoundaryNeedleT dn a b v
-> ProductBoundaryNeedleT dn a b v
-> ProductBoundaryNeedleT dn a b v
(^+^) = forall p. AffineSpace p => p -> Diff p -> p
(.+^)
  negateV :: ProductBoundaryNeedleT dn a b v -> ProductBoundaryNeedleT dn a b v
negateV ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle = forall (dn :: Dualness) a b v. ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle
  negateV (NBoundOfL Space dn (Needle (Boundary a))
x Space dn (Needle (Interior b))
y v
v) = forall (dn :: Dualness) a b v.
Space dn (Needle (Boundary a))
-> Space dn (Needle (Interior b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfL (forall v. AdditiveGroup v => v -> v
negateV Space dn (Needle (Boundary a))
x) (forall v. AdditiveGroup v => v -> v
negateV Space dn (Needle (Interior b))
y) (forall v. AdditiveGroup v => v -> v
negateV v
v)
  negateV (NBoundOfR Space dn (Needle (Interior a))
x Space dn (Needle (Boundary b))
y v
v) = forall (dn :: Dualness) a b v.
Space dn (Needle (Interior a))
-> Space dn (Needle (Boundary b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfR (forall v. AdditiveGroup v => v -> v
negateV Space dn (Needle (Interior a))
x) (forall v. AdditiveGroup v => v -> v
negateV Space dn (Needle (Boundary b))
y) (forall v. AdditiveGroup v => v -> v
negateV v
v)

instance ∀ a b v dn .
         ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar VectorSpace
           '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , ValidDualness dn )
    => VectorSpace (ProductBoundaryNeedleT dn a b v) where
  type Scalar (ProductBoundaryNeedleT dn a b v) = Scalar v
  *^ :: Scalar (ProductBoundaryNeedleT dn a b v)
-> ProductBoundaryNeedleT dn a b v
-> ProductBoundaryNeedleT dn a b v
(*^) = forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @a (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @b (
            case (forall (dn :: Dualness). ValidDualness dn => DualnessSingletons dn
decideDualness @dn, forall m. SemimanifoldWithBoundary m => SmfdWBoundWitness m
smfdWBoundWitness @a, forall m. SemimanifoldWithBoundary m => SmfdWBoundWitness m
smfdWBoundWitness @b) of
     (DualnessSingletons dn
VectorWitness, SmfdWBoundWitness a
_, SmfdWBoundWitness b
_) -> \Scalar (Space 'Vector (Needle (Boundary a)))
μ -> \case
        ProductBoundaryNeedleT 'Vector a b v
ZeroProductBoundaryNeedle -> forall (dn :: Dualness) a b v. ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle
        NBoundOfL Space 'Vector (Needle (Boundary a))
x Space 'Vector (Needle (Interior b))
y v
v -> forall (dn :: Dualness) a b v.
Space dn (Needle (Boundary a))
-> Space dn (Needle (Interior b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfL (Scalar (Space 'Vector (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Vector (Needle (Boundary a))
x) (Scalar (Space 'Vector (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Vector (Needle (Interior b))
y) (Scalar (Space 'Vector (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^v
v)
        NBoundOfR Space 'Vector (Needle (Interior a))
x Space 'Vector (Needle (Boundary b))
y v
v -> forall (dn :: Dualness) a b v.
Space dn (Needle (Interior a))
-> Space dn (Needle (Boundary b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfR (Scalar (Space 'Vector (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Vector (Needle (Interior a))
x) (Scalar (Space 'Vector (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Vector (Needle (Boundary b))
y) (Scalar (Space 'Vector (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^v
v)
     (DualnessSingletons dn
FunctionalWitness, SmfdWBoundWitness a
SmfdWBoundWitness, SmfdWBoundWitness b
SmfdWBoundWitness)
                       -> case ( forall v. LinearSpace v => DualSpaceWitness v
dualSpaceWitness @(Needle (Interior a))
                               , forall v. LinearSpace v => DualSpaceWitness v
dualSpaceWitness @(Needle (Boundary a))
                               , forall v. LinearSpace v => DualSpaceWitness v
dualSpaceWitness @(Needle (Interior b))
                               , forall v. LinearSpace v => DualSpaceWitness v
dualSpaceWitness @(Needle (Boundary b)) ) of
       (DualSpaceWitness (Needle (Interior a))
DualSpaceWitness, DualSpaceWitness (Needle (Boundary a))
DualSpaceWitness, DualSpaceWitness (Needle (Interior b))
DualSpaceWitness, DualSpaceWitness (Needle (Boundary b))
DualSpaceWitness)
            -> \Scalar (Space 'Functional (Needle (Boundary a)))
μ -> \case
        ProductBoundaryNeedleT 'Functional a b v
ZeroProductBoundaryNeedle -> forall (dn :: Dualness) a b v. ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle
        NBoundOfL Space 'Functional (Needle (Boundary a))
x Space 'Functional (Needle (Interior b))
y v
v -> forall (dn :: Dualness) a b v.
Space dn (Needle (Boundary a))
-> Space dn (Needle (Interior b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfL (Scalar (Space 'Functional (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Functional (Needle (Boundary a))
x) (Scalar (Space 'Functional (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Functional (Needle (Interior b))
y) (Scalar (Space 'Functional (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^v
v)
        NBoundOfR Space 'Functional (Needle (Interior a))
x Space 'Functional (Needle (Boundary b))
y v
v -> forall (dn :: Dualness) a b v.
Space dn (Needle (Interior a))
-> Space dn (Needle (Boundary b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfR (Scalar (Space 'Functional (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Functional (Needle (Interior a))
x) (Scalar (Space 'Functional (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Space 'Functional (Needle (Boundary b))
y) (Scalar (Space 'Functional (Needle (Boundary a)))
μforall v. VectorSpace v => Scalar v -> v -> v
*^v
v)
    ))

#if MIN_VERSION_linearmap_category(0,6,0)
type family ProductBoundaryNeedleTDimension dna dnb dv where
  ProductBoundaryNeedleTDimension 
    ('Just n) ('Just m) ('Just o) = 'Just (n*m*o - 1)
  ProductBoundaryNeedleTDimension _ _ _ = 'Nothing

instance ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar LinearSpace
           '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , ValidDualness dn )
    => DimensionAware (ProductBoundaryNeedleT dn a b v) where
  type StaticDimension (ProductBoundaryNeedleT dn a b v)
          = ProductBoundaryNeedleTDimension (StaticDimension (dn`Space`Needle (Interior a)))
                                            (StaticDimension (dn`Space`Needle (Interior b)))
                                            (StaticDimension v)
  dimensionalityWitness :: DimensionalityWitness (ProductBoundaryNeedleT dn a b v)
dimensionalityWitness = case ( forall a.
SemimanifoldWithBoundary a =>
SemimanifoldWitness (Interior a)
interiorSemimanifoldWitness @a
                               , forall a.
SemimanifoldWithBoundary a =>
SemimanifoldWitness (Interior a)
interiorSemimanifoldWitness @b ) of
      (SemimanifoldWitness (Interior a)
SemimanifoldWitness, SemimanifoldWitness (Interior b)
SemimanifoldWitness)
                -> case ( forall v. DimensionAware v => DimensionalityCases v
dimensionality @(dn`Space`Needle (Interior a))
                        , forall v. DimensionAware v => DimensionalityCases v
dimensionality @(dn`Space`Needle (Interior b))
                        , forall v. DimensionAware v => DimensionalityCases v
dimensionality @v ) of
       (DimensionalityCases (Space dn (Needle (Interior a)))
StaticDimensionalCase, DimensionalityCases (Space dn (Needle (Interior b)))
StaticDimensionalCase, DimensionalityCases v
StaticDimensionalCase)
            -> forall (n :: Nat) r. Sing n -> (KnownNat n => r) -> r
withKnownNat (forall v (n :: Nat). Dimensional n v => Sing n
dimensionalitySing @(dn`Space`Needle (Interior a))
                               forall a (t1 :: a) (t2 :: a).
SNum a =>
Sing t1 -> Sing t2 -> Sing (Apply (Apply (*@#@$) t1) t2)
%* forall v (n :: Nat). Dimensional n v => Sing n
dimensionalitySing @(dn`Space`Needle (Interior b))
                               forall a (t1 :: a) (t2 :: a).
SNum a =>
Sing t1 -> Sing t2 -> Sing (Apply (Apply (*@#@$) t1) t2)
%* forall v (n :: Nat). Dimensional n v => Sing n
dimensionalitySing @v
                               forall a (t1 :: a) (t2 :: a).
SNum a =>
Sing t1 -> Sing t2 -> Sing (Apply (Apply (-@#@$) t1) t2)
%- forall (n :: Nat). KnownNat n => SNat n
SNat @1
                            )
                forall (n :: Nat) v. Dimensional n v => DimensionalityWitness v
IsStaticDimensional
instance ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar LinearSpace
           '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , n`Dimensional`(dn`Space`Needle (Interior a))
         , m`Dimensional`(dn`Space`Needle (Interior b))
         , o`Dimensional`v
         , KnownNat d, d ~ (n*m*o-1)
         , ValidDualness dn )
    => d`Dimensional`(ProductBoundaryNeedleT dn a b v) where
#endif

instance ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar LinearSpace
           '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , ValidDualness dn )
    => TensorSpace (ProductBoundaryNeedleT dn a b v) where
  type TensorProduct (ProductBoundaryNeedleT dn a b v) w
          = ProductBoundaryNeedleT dn a b (v⊗w)
  wellDefinedVector :: ProductBoundaryNeedleT dn a b v
-> Maybe (ProductBoundaryNeedleT dn a b v)
wellDefinedVector ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle = forall a. a -> Maybe a
Just forall (dn :: Dualness) a b v. ProductBoundaryNeedleT dn a b v
ZeroProductBoundaryNeedle
  wellDefinedTensor :: forall w.
(TensorSpace w,
 Scalar w ~ Scalar (ProductBoundaryNeedleT dn a b v)) =>
(ProductBoundaryNeedleT dn a b v ⊗ w)
-> Maybe (ProductBoundaryNeedleT dn a b v ⊗ w)
wellDefinedTensor t :: ProductBoundaryNeedleT dn a b v ⊗ w
t@(Tensor TensorProduct (ProductBoundaryNeedleT dn a b v) w
ProductBoundaryNeedleT dn a b (Tensor (Scalar v) v w)
ZeroProductBoundaryNeedle) = forall a. a -> Maybe a
Just ProductBoundaryNeedleT dn a b v ⊗ w
t
  
instance ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar LinearSpace
            '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , ValidDualness dn
         )
    => LinearSpace (ProductBoundaryNeedleT dn a b v) where
  type DualVector (ProductBoundaryNeedleT dn a b v)
         = ProductBoundaryNeedleT (Dual dn) a b (DualVector v)
  

instance ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar LinearSpace
            '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , ValidDualness dn
         )
    => Semimanifold (ProductBoundaryNeedleT dn a b v) where
  type Needle (ProductBoundaryNeedleT dn a b v) = ProductBoundaryNeedleT dn a b v
  .+~^ :: ProductBoundaryNeedleT dn a b v
-> Needle (ProductBoundaryNeedleT dn a b v)
-> ProductBoundaryNeedleT dn a b v
(.+~^) = forall v. AdditiveGroup v => v -> v -> v
(^+^)
  semimanifoldWitness :: SemimanifoldWitness (ProductBoundaryNeedleT dn a b v)
semimanifoldWitness = forall x.
(Semimanifold (Needle x), Needle (Needle x) ~ Needle x) =>
SemimanifoldWitness x
SemimanifoldWitness
  
instance ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar LinearSpace
            '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , ValidDualness dn
         )
    => PseudoAffine (ProductBoundaryNeedleT dn a b v) where
  ProductBoundaryNeedleT dn a b v
p.-~. :: ProductBoundaryNeedleT dn a b v
-> ProductBoundaryNeedleT dn a b v
-> Maybe (Needle (ProductBoundaryNeedleT dn a b v))
.-~.ProductBoundaryNeedleT dn a b v
q = forall (f :: Type -> Type) a. Applicative f => a -> f a
pure (ProductBoundaryNeedleT dn a b v
pforall v. AdditiveGroup v => v -> v -> v
^-^ProductBoundaryNeedleT dn a b v
q)
  .-~! :: HasCallStack =>
ProductBoundaryNeedleT dn a b v
-> ProductBoundaryNeedleT dn a b v
-> Needle (ProductBoundaryNeedleT dn a b v)
(.-~!) = forall v. AdditiveGroup v => v -> v -> v
(^-^)
  
instance ( SemimanifoldWithBoundary a, SemimanifoldWithBoundary b
         , SameScalar LinearSpace
            '[ v, dn`Space`Needle (Interior a), dn`Space`Needle (Interior b) ]
         , AdditiveGroup (dn`Space`Needle (Boundary a))
         , AdditiveGroup (dn`Space`Needle (Boundary b))
         , OpenManifold (Scalar v)
         , ValidDualness dn
         )
    => SemimanifoldWithBoundary (ProductBoundaryNeedleT dn a b v) where
  type Interior (ProductBoundaryNeedleT dn a b v) = ProductBoundaryNeedleT dn a b v
  type Boundary (ProductBoundaryNeedleT dn a b v) = EmptyMfd v
  type HalfNeedle (ProductBoundaryNeedleT dn a b v) = ℝay
  smfdWBoundWitness :: SmfdWBoundWitness (ProductBoundaryNeedleT dn a b v)
smfdWBoundWitness = forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness

instance ∀ a b . ( ProjectableBoundary a, ProjectableBoundary b
                 , SameScalar LinearSpace
                    '[ Needle (Interior a), Needle (Interior b) ]
                 , Num' (Scalar (Needle (Interior a)))
                 )
   => Semimanifold (ProductBoundary a b) where
  type Needle (ProductBoundary a b) = ProductBoundaryNeedle a b
--ProductBoundary x y.+~^(δx, δy)
--     = case (separateInterior x, separateInterior y) of
-- (Left bx, Right _) -> case y .+^| δy of
--            Right iy' -> undefined
  .+~^ :: ProductBoundary a b
-> Needle (ProductBoundary a b) -> ProductBoundary a b
(.+~^) = forall a. HasCallStack => a
undefined
  semimanifoldWitness :: SemimanifoldWitness (ProductBoundary a b)
semimanifoldWitness = case ( forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior a)
                             , forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior b) ) of
    (SemimanifoldWitness (Interior a)
SemimanifoldWitness, SemimanifoldWitness (Interior b)
SemimanifoldWitness)
       -> forall a. HasCallStack => a
undefined -- SemimanifoldWitness

instance ∀ a b . ( ProjectableBoundary a, ProjectableBoundary b
                 , SameScalar LinearSpace
                    '[ Needle (Interior a), Needle (Interior b) ]
                 , Num' (Scalar (Needle (Interior a)))
                 )
   => PseudoAffine (ProductBoundary a b) where
  ProductBoundary a b
p.-~! :: HasCallStack =>
ProductBoundary a b
-> ProductBoundary a b -> Needle (ProductBoundary a b)
.-~!ProductBoundary a b
q = case ProductBoundary a b
pforall x. PseudoAffine x => x -> x -> Maybe (Needle x)
.-~.ProductBoundary a b
q of
             Just Needle (ProductBoundary a b)
v -> Needle (ProductBoundary a b)
v
             Maybe (Needle (ProductBoundary a b))
Nothing -> forall a. HasCallStack => [Char] -> a
error [Char]
"No path found in product-space boundary."
  .-~. :: ProductBoundary a b
-> ProductBoundary a b -> Maybe (Needle (ProductBoundary a b))
(.-~.) = case ( forall x. PseudoAffine x => PseudoAffineWitness x
pseudoAffineWitness @(Interior a)
                , forall x. PseudoAffine x => PseudoAffineWitness x
pseudoAffineWitness @(Interior b) ) of
   (PseudoAffineWitness SemimanifoldWitness (Interior a)
SemimanifoldWitness, PseudoAffineWitness SemimanifoldWitness (Interior b)
SemimanifoldWitness)
    -> let BoundOfL Boundary a
bx Interior b
y − :: ProductBoundary a b
-> ProductBoundary a b
-> Maybe
     (ProductBoundaryNeedleT 'Vector a b (Scalar (Needle (Interior a))))
− BoundOfL Boundary a
bξ Interior b
υ
             = case (Boundary a
bxforall x. PseudoAffine x => x -> x -> Maybe (Needle x)
.-~.Boundary a
bξ, forall m. SemimanifoldWithBoundary m => Interior m -> m
fromInterior @b Interior b
yforall m.
PseudoAffineWithBoundary m =>
m -> m -> Maybe (Needle (Interior m))
.--.forall m. SemimanifoldWithBoundary m => Interior m -> m
fromInterior Interior b
υ) of
                 (Just Needle (Boundary a)
δbx, Just Needle (Interior b)
δy) -> forall a. a -> Maybe a
Just forall a b. (a -> b) -> a -> b
$ forall (dn :: Dualness) a b v.
Space dn (Needle (Boundary a))
-> Space dn (Needle (Interior b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfL Needle (Boundary a)
δbx Needle (Interior b)
δy Scalar (Needle (Interior a))
1
                 (Maybe (Needle (Boundary a))
_, Maybe (Needle (Interior b))
Nothing) -> forall a. Maybe a
Nothing
           BoundOfL Boundary a
bx Interior b
y − BoundOfR Interior a
ξ Boundary b
bυ
             = case ( forall m. SemimanifoldWithBoundary m => Boundary m -> m
fromBoundary @a Boundary a
bxforall m.
PseudoAffineWithBoundary m =>
m -> m -> Maybe (Needle (Interior m))
.--.forall m. SemimanifoldWithBoundary m => Interior m -> m
fromInterior Interior a
ξ
                    , forall m.
ProjectableBoundary m =>
m
-> Boundary m
-> Maybe (Needle (Boundary m), Scalar (Needle (Interior m)))
projectToBoundary (forall m. SemimanifoldWithBoundary m => Interior m -> m
fromInterior @b Interior b
y) Boundary b
bυ ) of
                 (Just Needle (Interior a)
δbx, Just (Needle (Boundary b)
δby, Scalar (Needle (Interior a))
dy))
                    -> forall a. a -> Maybe a
Just forall a b. (a -> b) -> a -> b
$ forall (dn :: Dualness) a b v.
Space dn (Needle (Interior a))
-> Space dn (Needle (Boundary b))
-> v
-> ProductBoundaryNeedleT dn a b v
NBoundOfR (Needle (Interior a)
δbxforall v s. (VectorSpace v, s ~ Scalar v) => v -> s -> v
^*(Scalar (Needle (Interior a))
1forall a. Num a => a -> a -> a
+Scalar (Needle (Interior a))
dy)) Needle (Boundary b)
δby Scalar (Needle (Interior a))
1
                 (Maybe (Needle (Interior a)),
 Maybe (Needle (Boundary b), Scalar (Needle (Interior a))))
_ -> forall a. Maybe a
Nothing
       in ProductBoundary a b
-> ProductBoundary a b
-> Maybe
     (ProductBoundaryNeedleT 'Vector a b (Scalar (Needle (Interior a))))
(−)
  pseudoAffineWitness :: PseudoAffineWitness (ProductBoundary a b)
pseudoAffineWitness = case ( forall x. PseudoAffine x => PseudoAffineWitness x
pseudoAffineWitness @(Interior a)
                             , forall x. PseudoAffine x => PseudoAffineWitness x
pseudoAffineWitness @(Interior b) ) of
    (PseudoAffineWitness SemimanifoldWitness (Interior a)
SemimanifoldWitness
     , PseudoAffineWitness SemimanifoldWitness (Interior b)
SemimanifoldWitness)
       -> forall a. HasCallStack => a
undefined {- PseudoAffineWitness SemimanifoldWitness -}

instance ∀ a b . ( ProjectableBoundary a, ProjectableBoundary b
                 , SameScalar LinearSpace
                    '[ Needle (Interior a), Needle (Interior b)
                     , FullSubspace (HalfNeedle a)
                     ]
                 , RealFrac'' (Scalar (Needle (Interior a)))
                 )
   => SemimanifoldWithBoundary (ProductBoundary a b) where
  type Interior (ProductBoundary a b) = ProductBoundary a b
  type Boundary (ProductBoundary a b) = EmptyMfd (Needle (Boundary a), Needle (Boundary b))
  type HalfNeedle (ProductBoundary a b) = (HalfNeedle a, Needle (Boundary b))
  Boundary (ProductBoundary a b)
q|+^ :: Boundary (ProductBoundary a b)
-> HalfNeedle (ProductBoundary a b) -> ProductBoundary a b
|+^HalfNeedle (ProductBoundary a b)
_ = case Boundary (ProductBoundary a b)
q of {}
  ProductBoundary a b
p.+^| :: ProductBoundary a b
-> Needle (Interior (ProductBoundary a b))
-> Either
     (Boundary (ProductBoundary a b),
      Scalar (Needle (Interior (ProductBoundary a b))))
     (Interior (ProductBoundary a b))
.+^|Needle (Interior (ProductBoundary a b))
q = forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ ProductBoundary a b
pforall x. Semimanifold x => x -> Needle x -> x
.+~^Needle (Interior (ProductBoundary a b))
q
  fromInterior :: Interior (ProductBoundary a b) -> ProductBoundary a b
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (ProductBoundary a b) -> ProductBoundary a b
fromBoundary Boundary (ProductBoundary a b)
q = case Boundary (ProductBoundary a b)
q of {}
  smfdWBoundWitness :: SmfdWBoundWitness (ProductBoundary a b)
smfdWBoundWitness = forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @a
     (case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @(Scalar (Needle (Interior a))) of
              ClosedScalarWitness (Scalar (Needle (Interior a)))
ClosedScalarWitness -> forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness)
  needleIsOpenMfd :: forall r.
(OpenManifold (Needle (Interior (ProductBoundary a b))) => r) -> r
needleIsOpenMfd OpenManifold (Needle (Interior (ProductBoundary a b))) => r
r = forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @a (forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @b
                        (case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @(Scalar (Needle (Interior a))) of
                           ClosedScalarWitness (Scalar (Needle (Interior a)))
ClosedScalarWitness -> OpenManifold (Needle (Interior (ProductBoundary a b))) => r
r))
  extendToBoundary :: Interior (ProductBoundary a b)
-> Needle (Interior (ProductBoundary a b))
-> Maybe (Boundary (ProductBoundary a b))
extendToBoundary Interior (ProductBoundary a b)
q = case Interior (ProductBoundary a b)
q of {}
  scalarIsOpenMfd :: forall r.
(OpenManifold (Scalar (Needle (Interior (ProductBoundary a b)))) =>
 r)
-> r
scalarIsOpenMfd OpenManifold (Scalar (Needle (Interior (ProductBoundary a b)))) =>
r
r = forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @a
     (case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @(Scalar (Needle (Interior a))) of
              ClosedScalarWitness (Scalar (Needle (Interior a)))
ClosedScalarWitness -> OpenManifold (Scalar (Needle (Interior (ProductBoundary a b)))) =>
r
r)
  boundaryHasSameScalar :: forall r.
((LinearSpace (Needle (Boundary (ProductBoundary a b))),
  Scalar (Needle (Boundary (ProductBoundary a b)))
  ~ Scalar (Needle (Interior (ProductBoundary a b)))) =>
 r)
-> r
boundaryHasSameScalar (LinearSpace (Needle (Boundary (ProductBoundary a b))),
 Scalar (Needle (Boundary (ProductBoundary a b)))
 ~ Scalar (Needle (Interior (ProductBoundary a b)))) =>
r
r = forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @a (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @b
     (case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @(Scalar (Needle (Interior a))) of
              ClosedScalarWitness (Scalar (Needle (Interior a)))
ClosedScalarWitness -> (LinearSpace (Needle (Boundary (ProductBoundary a b))),
 Scalar (Needle (Boundary (ProductBoundary a b)))
 ~ Scalar (Needle (Interior (ProductBoundary a b)))) =>
r
r))

instance (Empty (Boundary a), Empty (Boundary b)) => Empty (ProductBoundary a b) where
  eliminate :: forall x. ProductBoundary a b -> x
eliminate (BoundOfL Boundary a
ba Interior b
_) = forall a x. Empty a => a -> x
eliminate Boundary a
ba
  eliminate (BoundOfR Interior a
_ Boundary b
bb) = forall a x. Empty a => a -> x
eliminate Boundary b
bb

data ProductHalfNeedle a b
  = ProductHalfNeedle !(Needle (Interior a)) !(Needle (Interior b))

instance (AdditiveGroup (Needle (Interior a)), AdditiveGroup (Needle (Interior b)))
             => AdditiveMonoid (ProductHalfNeedle a b) where
  zeroHV :: ProductHalfNeedle a b
zeroHV = forall a b.
Needle (Interior a) -> Needle (Interior b) -> ProductHalfNeedle a b
ProductHalfNeedle forall v. AdditiveGroup v => v
zeroV forall v. AdditiveGroup v => v
zeroV
  addHVs :: ProductHalfNeedle a b
-> ProductHalfNeedle a b -> ProductHalfNeedle a b
addHVs (ProductHalfNeedle Needle (Interior a)
v Needle (Interior b)
w) (ProductHalfNeedle Needle (Interior a)
ϋ Needle (Interior b)
ĥ)
            = forall a b.
Needle (Interior a) -> Needle (Interior b) -> ProductHalfNeedle a b
ProductHalfNeedle (Needle (Interior a)
vforall v. AdditiveGroup v => v -> v -> v
^+^Needle (Interior a)
ϋ) (Needle (Interior b)
wforall v. AdditiveGroup v => v -> v -> v
^+^Needle (Interior b)
ĥ)
instance ( SemimanifoldWithBoundary a
         , SameScalar VectorSpace
            '[ Needle (Interior a), Needle (Interior b) ]
         , RealFrac'' (Scalar (Needle (Interior a)))
         ) => HalfSpace (ProductHalfNeedle a b) where
  type FullSubspace (ProductHalfNeedle a b) = ProductBoundaryNeedle a b
  type Ray (ProductHalfNeedle a b) = ℝay_ (Scalar (Needle (Interior a)))
  type MirrorJoin (ProductHalfNeedle a b) = (Needle (Interior a), Needle (Interior b))
  scaleNonNeg :: Ray (ProductHalfNeedle a b)
-> ProductHalfNeedle a b -> ProductHalfNeedle a b
scaleNonNeg = case forall m. SemimanifoldWithBoundary m => SmfdWBoundWitness m
smfdWBoundWitness @a of
     SmfdWBoundWitness a
SmfdWBoundWitness 
        -> forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @a (\(Cℝay Scalar (Needle (ZeroDim (Scalar (Needle (Interior a)))))
μ ZeroDim (Scalar (Needle (Interior a)))
Origin) (ProductHalfNeedle Needle (Interior a)
v Needle (Interior b)
w)
         -> forall a b.
Needle (Interior a) -> Needle (Interior b) -> ProductHalfNeedle a b
ProductHalfNeedle (Scalar (Needle (ZeroDim (Scalar (Needle (Interior a)))))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Needle (Interior a)
v) (Scalar (Needle (ZeroDim (Scalar (Needle (Interior a)))))
μforall v. VectorSpace v => Scalar v -> v -> v
*^Needle (Interior b)
w))
  fromFullSubspace :: FullSubspace (ProductHalfNeedle a b) -> ProductHalfNeedle a b
fromFullSubspace FullSubspace (ProductHalfNeedle a b)
ProductBoundaryNeedleT 'Vector a b (Scalar (Needle (Interior a)))
ZeroProductBoundaryNeedle = forall h. AdditiveMonoid h => h
zeroHV
  fullSubspaceIsVectorSpace :: forall r.
((VectorSpace (FullSubspace (ProductHalfNeedle a b)),
  ScalarSpace (Scalar (FullSubspace (ProductHalfNeedle a b))),
  Scalar (FullSubspace (ProductHalfNeedle a b))
  ~ MirrorJoin (Ray (ProductHalfNeedle a b))) =>
 r)
-> r
fullSubspaceIsVectorSpace (VectorSpace (FullSubspace (ProductHalfNeedle a b)),
 ScalarSpace (Scalar (FullSubspace (ProductHalfNeedle a b))),
 Scalar (FullSubspace (ProductHalfNeedle a b))
 ~ MirrorJoin (Ray (ProductHalfNeedle a b))) =>
r
q = forall a. HasCallStack => a
undefined
  projectToFullSubspace :: ProductHalfNeedle a b -> FullSubspace (ProductHalfNeedle a b)
projectToFullSubspace = forall a. HasCallStack => a
undefined
  rayIsHalfSpace :: forall r. (HalfSpace (Ray (ProductHalfNeedle a b)) => r) -> r
rayIsHalfSpace HalfSpace (Ray (ProductHalfNeedle a b)) => r
_ = forall a. HasCallStack => a
undefined
  fromPositiveHalf :: ProductHalfNeedle a b -> MirrorJoin (ProductHalfNeedle a b)
fromPositiveHalf = forall a. HasCallStack => a
undefined
  fromNegativeHalf :: ProductHalfNeedle a b -> MirrorJoin (ProductHalfNeedle a b)
fromNegativeHalf = forall a. HasCallStack => a
undefined

instance ∀ a b .
         ( ProjectableBoundary a, ProjectableBoundary b
         , SameScalar LinearSpace
            '[ Needle (Interior a), Needle (Interior b)
             ]
         , RealFrac'' (Scalar (Needle (Interior a)))
         , ProjectableBoundary (Interior a), ProjectableBoundary (Interior b)
         ) => SemimanifoldWithBoundary (a,b) where
  type Interior (a,b) = (Interior a, Interior b)
  type Boundary (a,b) = ProductBoundary a b
  type HalfNeedle (a,b) = ProductHalfNeedle a b
  extendToBoundary :: Interior (a, b)
-> Needle (Interior (a, b)) -> Maybe (Boundary (a, b))
extendToBoundary = forall a. HasCallStack => a
undefined
  smfdWBoundWitness :: SmfdWBoundWitness (a, b)
smfdWBoundWitness = case (forall m. SemimanifoldWithBoundary m => SmfdWBoundWitness m
smfdWBoundWitness @a, forall m. SemimanifoldWithBoundary m => SmfdWBoundWitness m
smfdWBoundWitness @b) of
    (SmfdWBoundWitness a
OpenManifoldWitness, SmfdWBoundWitness b
OpenManifoldWitness)
        -> forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @a (forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @b (
             forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @(Needle a) (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @(Needle b)
               (case (forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior a), forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior b))
                 of (SemimanifoldWitness a
SemimanifoldWitness, SemimanifoldWitness b
SemimanifoldWitness)
                        -> forall m r.
ProjectableBoundary m =>
(ProjectableBoundary (Needle (Interior m)) => r) -> r
needleBoundaryIsTriviallyProjectible @a
                            (forall m r.
ProjectableBoundary m =>
(ProjectableBoundary (Needle (Interior m)) => r) -> r
needleBoundaryIsTriviallyProjectible @b forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness) )
            )))
    (SmfdWBoundWitness a
SmfdWBoundWitness, SmfdWBoundWitness b
SmfdWBoundWitness)
        -> forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @a
            (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @b
              (forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @(Interior a)
                (forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @(Interior b)
                  (case ( forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior a)
                        , forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior b)
                        , forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @(Scalar (Needle (Interior a)))
                        )
                 of (SemimanifoldWitness (Interior a)
SemimanifoldWitness, SemimanifoldWitness (Interior b)
SemimanifoldWitness, ClosedScalarWitness (Scalar (Needle (Interior a)))
ClosedScalarWitness)
                        -> forall m r.
ProjectableBoundary m =>
(ProjectableBoundary (Needle (Interior m)) => r) -> r
needleBoundaryIsTriviallyProjectible @a
                            (forall m r.
ProjectableBoundary m =>
(ProjectableBoundary (Needle (Interior m)) => r) -> r
needleBoundaryIsTriviallyProjectible @b
                              (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @(Needle (Interior a))
                                (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @(Needle (Interior b))
                                  forall m.
(OpenManifold (Interior m), OpenManifold (Boundary m),
 FullSubspace (HalfNeedle m) ~ Needle (Boundary m)) =>
SmfdWBoundWitness m
SmfdWBoundWitness)))))))
  boundaryHasSameScalar :: forall r.
((LinearSpace (Needle (Boundary (a, b))),
  Scalar (Needle (Boundary (a, b)))
  ~ Scalar (Needle (Interior (a, b)))) =>
 r)
-> r
boundaryHasSameScalar (LinearSpace (Needle (Boundary (a, b))),
 Scalar (Needle (Boundary (a, b)))
 ~ Scalar (Needle (Interior (a, b)))) =>
r
q = forall a. HasCallStack => a
undefined
  needleIsOpenMfd :: forall r. (OpenManifold (Needle (Interior (a, b))) => r) -> r
needleIsOpenMfd OpenManifold (Needle (Interior (a, b))) => r
_ = forall a. HasCallStack => a
undefined

instance ∀ a b .
         ( ProjectableBoundary a, ProjectableBoundary b
         , SameScalar LinearSpace
            '[ Needle (Interior a), Needle (Interior b)
             , Needle (Boundary a), Needle (Boundary b)
             ]
         , ProjectableBoundary (Interior a), ProjectableBoundary (Interior b)
         , RealFrac'' (Scalar (Needle (Interior a)))
         ) => PseudoAffineWithBoundary (a,b) where

instance ∀ a b .
         ( ProjectableBoundary a, ProjectableBoundary b
         , SameScalar LinearSpace
            '[ Needle (Interior a), Needle (Interior b)
             , Needle (Boundary a), Needle (Boundary b)
             ]
         , ProjectableBoundary (Interior a), ProjectableBoundary (Interior b)
         , RealFrac'' (Scalar (Needle (Interior a)))
         ) => ProjectableBoundary (a,b) where
  needleBoundaryIsTriviallyProjectible :: forall r.
(ProjectableBoundary (Needle (Interior (a, b))) => r) -> r
needleBoundaryIsTriviallyProjectible ProjectableBoundary (Needle (Interior (a, b))) => r
q
      = forall m r.
ProjectableBoundary m =>
(ProjectableBoundary (Needle (Interior m)) => r) -> r
needleBoundaryIsTriviallyProjectible @a
         (forall m r.
ProjectableBoundary m =>
(ProjectableBoundary (Needle (Interior m)) => r) -> r
needleBoundaryIsTriviallyProjectible @b
           (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @(Needle (Interior a))
             (forall m r.
SemimanifoldWithBoundary m =>
((LinearSpace (Needle (Boundary m)),
  Scalar (Needle (Boundary m)) ~ Scalar (Needle (Interior m))) =>
 r)
-> r
boundaryHasSameScalar @(Needle (Interior b))
               (forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @a
                 (forall m r.
SemimanifoldWithBoundary m =>
(OpenManifold (Needle (Interior m)) => r) -> r
needleIsOpenMfd @b
                   (case (forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior a), forall x. Semimanifold x => SemimanifoldWitness x
semimanifoldWitness @(Interior b))
                     of (SemimanifoldWitness (Interior a)
SemimanifoldWitness, SemimanifoldWitness (Interior b)
SemimanifoldWitness) -> ProjectableBoundary (Needle (Interior (a, b))) => r
q))))))

instance ∀ s . RealFloat'' s => SemimanifoldWithBoundary (S⁰_ s) where
  type Interior (S⁰_ s) = S⁰_ s
  type Boundary (S⁰_ s) = EmptyMfd (ZeroDim s)
  type HalfNeedle (S⁰_ s) = ZeroDim s
  fromInterior :: Interior (S⁰_ s) -> S⁰_ s
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (S⁰_ s) -> S⁰_ s
fromBoundary Boundary (S⁰_ s)
b = case Boundary (S⁰_ s)
b of {}
  separateInterior :: S⁰_ s -> Either (Boundary (S⁰_ s)) (Interior (S⁰_ s))
separateInterior = forall a b. b -> Either a b
Right
  Boundary (S⁰_ s)
p|+^ :: Boundary (S⁰_ s) -> HalfNeedle (S⁰_ s) -> S⁰_ s
|+^HalfNeedle (S⁰_ s)
_ = case Boundary (S⁰_ s)
p of {}
  S⁰_ s
NegativeHalfSphere .+^| :: S⁰_ s
-> Needle (Interior (S⁰_ s))
-> Either
     (Boundary (S⁰_ s), Scalar (Needle (Interior (S⁰_ s))))
     (Interior (S⁰_ s))
.+^| ZeroDim s
Needle (Interior (S⁰_ s))
Origin = forall a b. b -> Either a b
Right forall r. S⁰_ r
NegativeHalfSphere
  S⁰_ s
PositiveHalfSphere .+^| ZeroDim s
Needle (Interior (S⁰_ s))
Origin = forall a b. b -> Either a b
Right forall r. S⁰_ r
PositiveHalfSphere
  extendToBoundary :: Interior (S⁰_ s)
-> Needle (Interior (S⁰_ s)) -> Maybe (Boundary (S⁰_ s))
extendToBoundary Interior (S⁰_ s)
_ Needle (Interior (S⁰_ s))
_ = forall a. Maybe a
Nothing
  smfdWBoundWitness :: SmfdWBoundWitness (S⁰_ s)
smfdWBoundWitness = forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness

instance ∀ s . RealFloat'' s => SemimanifoldWithBoundary (S¹_ s) where
  type Interior (S¹_ s) = (S¹_ s)
  type Boundary (S¹_ s) = EmptyMfd (ZeroDim s)
  type HalfNeedle (S¹_ s) = ℝay_ s
  fromInterior :: Interior (S¹_ s) -> S¹_ s
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (S¹_ s) -> S¹_ s
fromBoundary Boundary (S¹_ s)
b = case Boundary (S¹_ s)
b of {}
  separateInterior :: S¹_ s -> Either (Boundary (S¹_ s)) (Interior (S¹_ s))
separateInterior = forall a b. b -> Either a b
Right
  Boundary (S¹_ s)
p|+^ :: Boundary (S¹_ s) -> HalfNeedle (S¹_ s) -> S¹_ s
|+^HalfNeedle (S¹_ s)
_ = case Boundary (S¹_ s)
p of {}
  S¹_ s
_ .+^| :: S¹_ s
-> Needle (Interior (S¹_ s))
-> Either
     (Boundary (S¹_ s), Scalar (Needle (Interior (S¹_ s))))
     (Interior (S¹_ s))
.+^| Needle (Interior (S¹_ s))
p = case Needle (Interior (S¹_ s))
p of {}
  extendToBoundary :: Interior (S¹_ s)
-> Needle (Interior (S¹_ s)) -> Maybe (Boundary (S¹_ s))
extendToBoundary Interior (S¹_ s)
_ Needle (Interior (S¹_ s))
_ = forall a. Maybe a
Nothing
  smfdWBoundWitness :: SmfdWBoundWitness (S¹_ s)
smfdWBoundWitness = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of ClosedScalarWitness s
ClosedScalarWitness -> forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness
  scalarIsOpenMfd :: forall r.
(OpenManifold (Scalar (Needle (Interior (S¹_ s)))) => r) -> r
scalarIsOpenMfd OpenManifold (Scalar (Needle (Interior (S¹_ s)))) => r
q = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of ClosedScalarWitness s
ClosedScalarWitness -> OpenManifold (Scalar (Needle (Interior (S¹_ s)))) => r
q
  boundaryHasSameScalar :: forall r.
((LinearSpace (Needle (Boundary (S¹_ s))),
  Scalar (Needle (Boundary (S¹_ s)))
  ~ Scalar (Needle (Interior (S¹_ s)))) =>
 r)
-> r
boundaryHasSameScalar (LinearSpace (Needle (Boundary (S¹_ s))),
 Scalar (Needle (Boundary (S¹_ s)))
 ~ Scalar (Needle (Interior (S¹_ s)))) =>
r
q = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of ClosedScalarWitness s
ClosedScalarWitness -> (LinearSpace (Needle (Boundary (S¹_ s))),
 Scalar (Needle (Boundary (S¹_ s)))
 ~ Scalar (Needle (Interior (S¹_ s)))) =>
r
q

instance ∀ s . RealFloat'' s => PseudoAffineWithBoundary (S¹_ s) where
  S¹_ s
_!-| :: S¹_ s -> Boundary (S¹_ s) -> HalfNeedle (S¹_ s)
!-|Boundary (S¹_ s)
p = case Boundary (S¹_ s)
p of {}
  .--! :: S¹_ s -> S¹_ s -> Needle (Interior (S¹_ s))
(.--!) = forall x. (PseudoAffine x, HasCallStack) => x -> x -> Needle x
(.-~!)

instance ∀ s . RealFloat'' s => ProjectableBoundary (S¹_ s) where
  scalarBoundaryIsTriviallyProjectible :: forall r.
(ProjectableBoundary (Scalar (Needle (Interior (S¹_ s)))) => r)
-> r
scalarBoundaryIsTriviallyProjectible ProjectableBoundary (Scalar (Needle (Interior (S¹_ s)))) => r
q = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of
     ClosedScalarWitness s
ClosedScalarWitness -> ProjectableBoundary (Scalar (Needle (Interior (S¹_ s)))) => r
q
  projectToBoundary :: S¹_ s
-> Boundary (S¹_ s)
-> Maybe
     (Needle (Boundary (S¹_ s)), Scalar (Needle (Interior (S¹_ s))))
projectToBoundary S¹_ s
_ Boundary (S¹_ s)
p = case Boundary (S¹_ s)
p of {}
  marginFromBoundary :: Boundary (S¹_ s) -> Scalar (Needle (Interior (S¹_ s))) -> S¹_ s
marginFromBoundary Boundary (S¹_ s)
p = case Boundary (S¹_ s)
p of {}

instance ∀ s . RealFloat'' s => SemimanifoldWithBoundary (S²_ s) where
  type Interior (S²_ s) = S²_ s
  type Boundary (S²_ s) = EmptyMfd s
  type HalfNeedle (S²_ s) = ℝay_ s
  fromInterior :: Interior (S²_ s) -> S²_ s
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (S²_ s) -> S²_ s
fromBoundary Boundary (S²_ s)
b = case Boundary (S²_ s)
b of {}
  separateInterior :: S²_ s -> Either (Boundary (S²_ s)) (Interior (S²_ s))
separateInterior = forall a b. b -> Either a b
Right
  Boundary (S²_ s)
p|+^ :: Boundary (S²_ s) -> HalfNeedle (S²_ s) -> S²_ s
|+^HalfNeedle (S²_ s)
_ = case Boundary (S²_ s)
p of {}
  S²_ s
_ .+^| :: S²_ s
-> Needle (Interior (S²_ s))
-> Either
     (Boundary (S²_ s), Scalar (Needle (Interior (S²_ s))))
     (Interior (S²_ s))
.+^| Needle (Interior (S²_ s))
p = case Needle (Interior (S²_ s))
p of {}
  extendToBoundary :: Interior (S²_ s)
-> Needle (Interior (S²_ s)) -> Maybe (Boundary (S²_ s))
extendToBoundary Interior (S²_ s)
_ Needle (Interior (S²_ s))
_ = forall a. Maybe a
Nothing
  smfdWBoundWitness :: SmfdWBoundWitness (S²_ s)
smfdWBoundWitness = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of ClosedScalarWitness s
ClosedScalarWitness -> forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness
  scalarIsOpenMfd :: forall r.
(OpenManifold (Scalar (Needle (Interior (S²_ s)))) => r) -> r
scalarIsOpenMfd OpenManifold (Scalar (Needle (Interior (S²_ s)))) => r
q = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of ClosedScalarWitness s
ClosedScalarWitness -> OpenManifold (Scalar (Needle (Interior (S²_ s)))) => r
q
  boundaryHasSameScalar :: forall r.
((LinearSpace (Needle (Boundary (S²_ s))),
  Scalar (Needle (Boundary (S²_ s)))
  ~ Scalar (Needle (Interior (S²_ s)))) =>
 r)
-> r
boundaryHasSameScalar (LinearSpace (Needle (Boundary (S²_ s))),
 Scalar (Needle (Boundary (S²_ s)))
 ~ Scalar (Needle (Interior (S²_ s)))) =>
r
q = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of ClosedScalarWitness s
ClosedScalarWitness -> (LinearSpace (Needle (Boundary (S²_ s))),
 Scalar (Needle (Boundary (S²_ s)))
 ~ Scalar (Needle (Interior (S²_ s)))) =>
r
q

instance ∀ s . RealFloat'' s => PseudoAffineWithBoundary (S²_ s) where
  S²_ s
_!-| :: S²_ s -> Boundary (S²_ s) -> HalfNeedle (S²_ s)
!-|Boundary (S²_ s)
p = case Boundary (S²_ s)
p of {}
  .--! :: S²_ s -> S²_ s -> Needle (Interior (S²_ s))
(.--!) = forall x. (PseudoAffine x, HasCallStack) => x -> x -> Needle x
(.-~!)

instance ∀ s . RealFloat'' s => ProjectableBoundary (S²_ s) where
  scalarBoundaryIsTriviallyProjectible :: forall r.
(ProjectableBoundary (Scalar (Needle (Interior (S²_ s)))) => r)
-> r
scalarBoundaryIsTriviallyProjectible ProjectableBoundary (Scalar (Needle (Interior (S²_ s)))) => r
q = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of
     ClosedScalarWitness s
ClosedScalarWitness -> ProjectableBoundary (Scalar (Needle (Interior (S²_ s)))) => r
q
  projectToBoundary :: S²_ s
-> Boundary (S²_ s)
-> Maybe
     (Needle (Boundary (S²_ s)), Scalar (Needle (Interior (S²_ s))))
projectToBoundary S²_ s
_ Boundary (S²_ s)
p = case Boundary (S²_ s)
p of {}
  marginFromBoundary :: Boundary (S²_ s) -> Scalar (Needle (Interior (S²_ s))) -> S²_ s
marginFromBoundary Boundary (S²_ s)
p = case Boundary (S²_ s)
p of {}


instance ∀ s . RealFloat'' s => SemimanifoldWithBoundary (D¹_ s) where
  type Interior (D¹_ s) = s
  type Boundary (D¹_ s) = (S⁰_ s)
  type HalfNeedle (D¹_ s) = ℝay_ s
  fromBoundary :: Boundary (D¹_ s) -> D¹_ s
fromBoundary S⁰_ s
Boundary (D¹_ s)
NegativeHalfSphere = forall r. r -> D¹_ r
D¹ (-s
1)
  fromBoundary S⁰_ s
Boundary (D¹_ s)
PositiveHalfSphere = forall r. r -> D¹_ r
D¹ s
1
  fromInterior :: Interior (D¹_ s) -> D¹_ s
fromInterior = forall r. r -> D¹_ r
D¹ forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Floating a => a -> a
tanh
  separateInterior :: D¹_ s -> Either (Boundary (D¹_ s)) (Interior (D¹_ s))
separateInterior (D¹ (-1)) = forall a b. a -> Either a b
Left forall r. S⁰_ r
NegativeHalfSphere
  separateInterior (D¹ s
1) = forall a b. a -> Either a b
Left forall r. S⁰_ r
PositiveHalfSphere
  separateInterior (D¹ s
x) = forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ forall a. Floating a => a -> a
atanh s
x
  S⁰_ s
Boundary (D¹_ s)
NegativeHalfSphere|+^ :: Boundary (D¹_ s) -> HalfNeedle (D¹_ s) -> D¹_ s
|+^Cℝay Scalar (Needle (ZeroDim s))
l ZeroDim s
Origin = forall r. r -> D¹_ r
D¹ forall a b. (a -> b) -> a -> b
$ s
1 forall a. Num a => a -> a -> a
- s
4forall a. Fractional a => a -> a -> a
/(Scalar (Needle (ZeroDim s))
lforall a. Num a => a -> a -> a
+s
2)
  S⁰_ s
Boundary (D¹_ s)
PositiveHalfSphere|+^Cℝay Scalar (Needle (ZeroDim s))
l ZeroDim s
Origin = forall r. r -> D¹_ r
D¹ forall a b. (a -> b) -> a -> b
$ s
4forall a. Fractional a => a -> a -> a
/(Scalar (Needle (ZeroDim s))
lforall a. Num a => a -> a -> a
+s
2) forall a. Num a => a -> a -> a
- s
1
  .+^| :: D¹_ s
-> Needle (Interior (D¹_ s))
-> Either
     (Boundary (D¹_ s), Scalar (Needle (Interior (D¹_ s))))
     (Interior (D¹_ s))
(.+^|) = case (forall v. TensorSpace v => LinearManifoldWitness v
linearManifoldWitness @s, forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s) of
   (LinearManifoldWitness s
LinearManifoldWitness, ClosedScalarWitness s
ClosedScalarWitness) ->
    let addBD¹ :: D¹_ b -> b -> Either (S⁰_ r, b) b
addBD¹ (D¹ b
p) b
l
          | b
p' forall a. Ord a => a -> a -> Bool
>= b
1    = forall a b. a -> Either a b
Left (forall r. S⁰_ r
PositiveHalfSphere, (b
p'forall a. Num a => a -> a -> a
-b
1) forall a. Fractional a => a -> a -> a
/ b
l)
          | b
p' forall a. Ord a => a -> a -> Bool
<= -b
1   = forall a b. a -> Either a b
Left (forall r. S⁰_ r
NegativeHalfSphere, (b
p'forall a. Num a => a -> a -> a
+b
1) forall a. Fractional a => a -> a -> a
/ b
l)
          | Bool
otherwise  = forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ forall a. Floating a => a -> a
atanh b
p'
         where p' :: b
p' = b
pforall a. Num a => a -> a -> a
+b
l
    in forall {b} {r}.
(Ord b, Floating b) =>
D¹_ b -> b -> Either (S⁰_ r, b) b
addBD¹
  extendToBoundary :: Interior (D¹_ s)
-> Needle (Interior (D¹_ s)) -> Maybe (Boundary (D¹_ s))
extendToBoundary = case (forall v. TensorSpace v => LinearManifoldWitness v
linearManifoldWitness @s, forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s) of
   (LinearManifoldWitness s
LinearManifoldWitness, ClosedScalarWitness s
ClosedScalarWitness) ->
    let e2b :: p -> a -> Maybe (S⁰_ r)
e2b p
_ a
dir
         | a
dir forall a. Ord a => a -> a -> Bool
> a
0    = forall a. a -> Maybe a
Just forall r. S⁰_ r
PositiveHalfSphere
         | a
dir forall a. Ord a => a -> a -> Bool
< a
0    = forall a. a -> Maybe a
Just forall r. S⁰_ r
NegativeHalfSphere
         | Bool
otherwise  = forall a. Maybe a
Nothing
    in forall {a} {p} {r}. (Ord a, Num a) => p -> a -> Maybe (S⁰_ r)
e2b
  smfdWBoundWitness :: SmfdWBoundWitness (D¹_ s)
smfdWBoundWitness = case forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s of ClosedScalarWitness s
ClosedScalarWitness -> forall m.
(OpenManifold (Interior m), OpenManifold (Boundary m),
 FullSubspace (HalfNeedle m) ~ Needle (Boundary m)) =>
SmfdWBoundWitness m
SmfdWBoundWitness
  scalarIsOpenMfd :: forall r.
(OpenManifold (Scalar (Needle (Interior (D¹_ s)))) => r) -> r
scalarIsOpenMfd OpenManifold (Scalar (Needle (Interior (D¹_ s)))) => r
q = case (forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s, forall v. TensorSpace v => LinearManifoldWitness v
linearManifoldWitness @s) of
   (ClosedScalarWitness s
ClosedScalarWitness, LinearManifoldWitness s
LinearManifoldWitness) -> OpenManifold (Scalar (Needle (Interior (D¹_ s)))) => r
q
  boundaryHasSameScalar :: forall r.
((LinearSpace (Needle (Boundary (D¹_ s))),
  Scalar (Needle (Boundary (D¹_ s)))
  ~ Scalar (Needle (Interior (D¹_ s)))) =>
 r)
-> r
boundaryHasSameScalar (LinearSpace (Needle (Boundary (D¹_ s))),
 Scalar (Needle (Boundary (D¹_ s)))
 ~ Scalar (Needle (Interior (D¹_ s)))) =>
r
q = case (forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s, forall v. TensorSpace v => LinearManifoldWitness v
linearManifoldWitness @s) of
   (ClosedScalarWitness s
ClosedScalarWitness, LinearManifoldWitness s
LinearManifoldWitness) -> (LinearSpace (Needle (Boundary (D¹_ s))),
 Scalar (Needle (Boundary (D¹_ s)))
 ~ Scalar (Needle (Interior (D¹_ s)))) =>
r
q
  needleIsOpenMfd :: forall r. (OpenManifold (Needle (Interior (D¹_ s))) => r) -> r
needleIsOpenMfd OpenManifold (Needle (Interior (D¹_ s))) => r
q = case (forall s. Num' s => ClosedScalarWitness s
closedScalarWitness @s, forall v. TensorSpace v => LinearManifoldWitness v
linearManifoldWitness @s) of
   (ClosedScalarWitness s
ClosedScalarWitness, LinearManifoldWitness s
LinearManifoldWitness) -> OpenManifold (Needle (Interior (D¹_ s))) => r
q


instance ( Num' n, OpenManifold n, LinearManifold (a n)
         , Scalar (a n) ~ n, Needle (a n) ~ a n )
            => SemimanifoldWithBoundary (LinAff.Point a n) where
  type Boundary (LinAff.Point a n) = EmptyMfd (ZeroDim n)
  type Interior (LinAff.Point a n) = LinAff.Point a n
  type HalfNeedle (LinAff.Point a n) = ℝay
  smfdWBoundWitness :: SmfdWBoundWitness (Point a n)
smfdWBoundWitness = forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness
  LinAff.P a n
p.+^| :: Point a n
-> Needle (Interior (Point a n))
-> Either
     (Boundary (Point a n), Scalar (Needle (Interior (Point a n))))
     (Interior (Point a n))
.+^|Needle (Interior (Point a n))
v = forall a b. b -> Either a b
Right forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (f :: Type -> Type) a. f a -> Point f a
LinAff.P forall a b. (a -> b) -> a -> b
$ a n
pforall v. AdditiveGroup v => v -> v -> v
^+^Needle (Interior (Point a n))
v
  fromInterior :: Interior (Point a n) -> Point a n
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (Point a n) -> Point a n
fromBoundary Boundary (Point a n)
b = case Boundary (Point a n)
b of {}
  Boundary (Point a n)
b|+^ :: Boundary (Point a n) -> HalfNeedle (Point a n) -> Point a n
|+^HalfNeedle (Point a n)
_ = case Boundary (Point a n)
b of {}

instance ( Num' n, OpenManifold n, LinearManifold (a n)
         , Scalar (a n) ~ n, Needle (a n) ~ a n )
            => PseudoAffineWithBoundary (LinAff.Point a n) where
  LinAff.P a n
p.--! :: Point a n -> Point a n -> Needle (Interior (Point a n))
.--!LinAff.P a n
q = a n
pforall v. AdditiveGroup v => v -> v -> v
^-^a n
q
  Point a n
_!-| :: Point a n -> Boundary (Point a n) -> HalfNeedle (Point a n)
!-|Boundary (Point a n)
b = case Boundary (Point a n)
b of {}

instance ∀ n a .  ( Num' n, OpenManifold n, LinearManifold (a n), ProjectableBoundary n
                  , Scalar (a n) ~ n, Needle (a n) ~ a n )
            => ProjectableBoundary (LinAff.Point a n) where
  projectToBoundary :: Point a n
-> Boundary (Point a n)
-> Maybe
     (Needle (Boundary (Point a n)),
      Scalar (Needle (Interior (Point a n))))
projectToBoundary Point a n
_ Boundary (Point a n)
b = case Boundary (Point a n)
b of {}
  marginFromBoundary :: Boundary (Point a n)
-> Scalar (Needle (Interior (Point a n))) -> Point a n
marginFromBoundary Boundary (Point a n)
b Scalar (Needle (Interior (Point a n)))
_ = case Boundary (Point a n)
b of {}

instance ( LinearSpace v, LinearSpace w
         , s ~ Scalar v, s ~ Scalar w
         , Num' s, OpenManifold s
         ) => SemimanifoldWithBoundary (Tensor s v w) where
  type Interior (Tensor s v w) = (Tensor s v w)
  type Boundary (Tensor s v w) = EmptyMfd (ZeroDim s)
  type HalfNeedle (Tensor s v w) = ℝay_ s
  smfdWBoundWitness :: SmfdWBoundWitness (Tensor s v w)
smfdWBoundWitness = forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness
  fromInterior :: Interior (Tensor s v w) -> Tensor s v w
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (Tensor s v w) -> Tensor s v w
fromBoundary Boundary (Tensor s v w)
b = case Boundary (Tensor s v w)
b of {}
  separateInterior :: Tensor s v w
-> Either (Boundary (Tensor s v w)) (Interior (Tensor s v w))
separateInterior = forall a b. b -> Either a b
Right
  Boundary (Tensor s v w)
p|+^ :: Boundary (Tensor s v w)
-> HalfNeedle (Tensor s v w) -> Tensor s v w
|+^HalfNeedle (Tensor s v w)
_ = case Boundary (Tensor s v w)
p of {}
  Tensor s v w
a.+^| :: Tensor s v w
-> Needle (Interior (Tensor s v w))
-> Either
     (Boundary (Tensor s v w),
      Scalar (Needle (Interior (Tensor s v w))))
     (Interior (Tensor s v w))
.+^|Needle (Interior (Tensor s v w))
b = forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ Tensor s v w
aforall v. AdditiveGroup v => v -> v -> v
^+^Needle (Interior (Tensor s v w))
b
  extendToBoundary :: Interior (Tensor s v w)
-> Needle (Interior (Tensor s v w))
-> Maybe (Boundary (Tensor s v w))
extendToBoundary Interior (Tensor s v w)
_ Needle (Interior (Tensor s v w))
_ = forall a. Maybe a
Nothing

instance ( LinearSpace v, LinearSpace w
         , s ~ Scalar v, s ~ Scalar w
         , Num' s, OpenManifold s
         ) => PseudoAffineWithBoundary (Tensor s v w) where
  Tensor s v w
_!-| :: Tensor s v w
-> Boundary (Tensor s v w) -> HalfNeedle (Tensor s v w)
!-|Boundary (Tensor s v w)
p = case Boundary (Tensor s v w)
p of {}
  .--! :: Tensor s v w -> Tensor s v w -> Needle (Interior (Tensor s v w))
(.--!) = forall v. AdditiveGroup v => v -> v -> v
(^-^)

instance ( LinearSpace v, LinearSpace w
         , s ~ Scalar v, s ~ Scalar w
         , Num' s, OpenManifold s
         ) => SemimanifoldWithBoundary (LinearMap s v w) where
  type Interior (LinearMap s v w) = (LinearMap s v w)
  type Boundary (LinearMap s v w) = EmptyMfd (ZeroDim s)
  type HalfNeedle (LinearMap s v w) = ℝay
  smfdWBoundWitness :: SmfdWBoundWitness (LinearMap s v w)
smfdWBoundWitness = forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness
  fromInterior :: Interior (LinearMap s v w) -> LinearMap s v w
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (LinearMap s v w) -> LinearMap s v w
fromBoundary Boundary (LinearMap s v w)
b = case Boundary (LinearMap s v w)
b of {}
  separateInterior :: LinearMap s v w
-> Either (Boundary (LinearMap s v w)) (Interior (LinearMap s v w))
separateInterior = forall a b. b -> Either a b
Right
  Boundary (LinearMap s v w)
p|+^ :: Boundary (LinearMap s v w)
-> HalfNeedle (LinearMap s v w) -> LinearMap s v w
|+^HalfNeedle (LinearMap s v w)
_ = case Boundary (LinearMap s v w)
p of {}
  LinearMap s v w
a.+^| :: LinearMap s v w
-> Needle (Interior (LinearMap s v w))
-> Either
     (Boundary (LinearMap s v w),
      Scalar (Needle (Interior (LinearMap s v w))))
     (Interior (LinearMap s v w))
.+^|Needle (Interior (LinearMap s v w))
b = forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ LinearMap s v w
aforall v. AdditiveGroup v => v -> v -> v
^+^Needle (Interior (LinearMap s v w))
b
  extendToBoundary :: Interior (LinearMap s v w)
-> Needle (Interior (LinearMap s v w))
-> Maybe (Boundary (LinearMap s v w))
extendToBoundary Interior (LinearMap s v w)
_ Needle (Interior (LinearMap s v w))
_ = forall a. Maybe a
Nothing

instance ( LinearSpace v, LinearSpace w
         , s ~ Scalar v, s ~ Scalar w
         , Num' s, OpenManifold s, ProjectableBoundary s
         ) => ProjectableBoundary (LinearMap s v w) where
  projectToBoundary :: LinearMap s v w
-> Boundary (LinearMap s v w)
-> Maybe
     (Needle (Boundary (LinearMap s v w)),
      Scalar (Needle (Interior (LinearMap s v w))))
projectToBoundary LinearMap s v w
_ Boundary (LinearMap s v w)
p = case Boundary (LinearMap s v w)
p of {}
  marginFromBoundary :: Boundary (LinearMap s v w)
-> Scalar (Needle (Interior (LinearMap s v w))) -> LinearMap s v w
marginFromBoundary Boundary (LinearMap s v w)
p = case Boundary (LinearMap s v w)
p of {}

instance ( LinearSpace v, LinearSpace w
         , s ~ Scalar v, s ~ Scalar w
         , Num' s, OpenManifold s
         ) => PseudoAffineWithBoundary (LinearMap s v w) where
  LinearMap s v w
_!-| :: LinearMap s v w
-> Boundary (LinearMap s v w) -> HalfNeedle (LinearMap s v w)
!-|Boundary (LinearMap s v w)
p = case Boundary (LinearMap s v w)
p of {}
  .--! :: LinearMap s v w
-> LinearMap s v w -> Needle (Interior (LinearMap s v w))
(.--!) = forall v. AdditiveGroup v => v -> v -> v
(^-^)

instance ( LinearSpace v, LinearSpace w
         , s ~ Scalar v, s ~ Scalar w
         , Num' s, OpenManifold s
         ) => SemimanifoldWithBoundary (LinearFunction s v w) where
  type Interior (LinearFunction s v w) = (LinearFunction s v w)
  type Boundary (LinearFunction s v w) = EmptyMfd (ZeroDim s)
  type HalfNeedle (LinearFunction s v w) = ℝay
  smfdWBoundWitness :: SmfdWBoundWitness (LinearFunction s v w)
smfdWBoundWitness = forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness
  fromInterior :: Interior (LinearFunction s v w) -> LinearFunction s v w
fromInterior = forall a. a -> a
id
  fromBoundary :: Boundary (LinearFunction s v w) -> LinearFunction s v w
fromBoundary Boundary (LinearFunction s v w)
b = case Boundary (LinearFunction s v w)
b of {}
  separateInterior :: LinearFunction s v w
-> Either
     (Boundary (LinearFunction s v w)) (Interior (LinearFunction s v w))
separateInterior = forall a b. b -> Either a b
Right
  Boundary (LinearFunction s v w)
p|+^ :: Boundary (LinearFunction s v w)
-> HalfNeedle (LinearFunction s v w) -> LinearFunction s v w
|+^HalfNeedle (LinearFunction s v w)
_ = case Boundary (LinearFunction s v w)
p of {}
  LinearFunction s v w
a.+^| :: LinearFunction s v w
-> Needle (Interior (LinearFunction s v w))
-> Either
     (Boundary (LinearFunction s v w),
      Scalar (Needle (Interior (LinearFunction s v w))))
     (Interior (LinearFunction s v w))
.+^|Needle (Interior (LinearFunction s v w))
b = forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ LinearFunction s v w
aforall v. AdditiveGroup v => v -> v -> v
^+^Needle (Interior (LinearFunction s v w))
b
  extendToBoundary :: Interior (LinearFunction s v w)
-> Needle (Interior (LinearFunction s v w))
-> Maybe (Boundary (LinearFunction s v w))
extendToBoundary Interior (LinearFunction s v w)
_ Needle (Interior (LinearFunction s v w))
_ = forall a. Maybe a
Nothing

instance ( LinearSpace v, LinearSpace w
         , s ~ Scalar v, s ~ Scalar w
         , Num' s, OpenManifold s
         ) => PseudoAffineWithBoundary (LinearFunction s v w) where
  LinearFunction s v w
_!-| :: LinearFunction s v w
-> Boundary (LinearFunction s v w)
-> HalfNeedle (LinearFunction s v w)
!-|Boundary (LinearFunction s v w)
p = case Boundary (LinearFunction s v w)
p of {}
  .--! :: LinearFunction s v w
-> LinearFunction s v w -> Needle (Interior (LinearFunction s v w))
(.--!) = forall v. AdditiveGroup v => v -> v -> v
(^-^)



instance ( Semimanifold a
         , Semimanifold (VRep a), Needle a ~ GenericNeedle a
         , OpenManifold (Scalar (Needle (Gnrx.Rep a Void)))
         , LinearSpace (Needle (Gnrx.Rep a Void))
         , Num' (Scalar (Needle (Gnrx.Rep a Void))) )
            => SemimanifoldWithBoundary (GenericNeedle a) where
  type Interior (GenericNeedle a) = GenericNeedle a
  type Boundary (GenericNeedle a) = EmptyMfd (ZeroDim (Scalar (Needle (Gnrx.Rep a Void))))
  type HalfNeedle (GenericNeedle a) = ℝay_ (Scalar (Needle (Gnrx.Rep a Void)))
  extendToBoundary :: Interior (GenericNeedle a)
-> Needle (Interior (GenericNeedle a))
-> Maybe (Boundary (GenericNeedle a))
extendToBoundary Interior (GenericNeedle a)
_ Needle (Interior (GenericNeedle a))
_ = forall a. Maybe a
Nothing
  smfdWBoundWitness :: SmfdWBoundWitness (GenericNeedle a)
smfdWBoundWitness = forall m. OpenManifold m => SmfdWBoundWitness m
OpenManifoldWitness
  needleIsOpenMfd :: forall r.
(OpenManifold (Needle (Interior (GenericNeedle a))) => r) -> r
needleIsOpenMfd OpenManifold (Needle (Interior (GenericNeedle a))) => r
q = OpenManifold (Needle (Interior (GenericNeedle a))) => r
q
  scalarIsOpenMfd :: forall r.
(OpenManifold (Scalar (Needle (Interior (GenericNeedle a)))) => r)
-> r
scalarIsOpenMfd OpenManifold (Scalar (Needle (Interior (GenericNeedle a)))) => r
q = OpenManifold (Scalar (Needle (Interior (GenericNeedle a)))) => r
q
  boundaryHasSameScalar :: forall r.
((LinearSpace (Needle (Boundary (GenericNeedle a))),
  Scalar (Needle (Boundary (GenericNeedle a)))
  ~ Scalar (Needle (Interior (GenericNeedle a)))) =>
 r)
-> r
boundaryHasSameScalar (LinearSpace (Needle (Boundary (GenericNeedle a))),
 Scalar (Needle (Boundary (GenericNeedle a)))
 ~ Scalar (Needle (Interior (GenericNeedle a)))) =>
r
q = (LinearSpace (Needle (Boundary (GenericNeedle a))),
 Scalar (Needle (Boundary (GenericNeedle a)))
 ~ Scalar (Needle (Interior (GenericNeedle a)))) =>
r
q
  Boundary (GenericNeedle a)
b|+^ :: Boundary (GenericNeedle a)
-> HalfNeedle (GenericNeedle a) -> GenericNeedle a
|+^HalfNeedle (GenericNeedle a)
_ = case Boundary (GenericNeedle a)
b of {}
  GenericNeedle a
p .+^| :: GenericNeedle a
-> Needle (Interior (GenericNeedle a))
-> Either
     (Boundary (GenericNeedle a),
      Scalar (Needle (Interior (GenericNeedle a))))
     (Interior (GenericNeedle a))
.+^| Needle (Interior (GenericNeedle a))
k = forall a b. b -> Either a b
Right forall a b. (a -> b) -> a -> b
$ GenericNeedle a
pforall v. AdditiveGroup v => v -> v -> v
^+^Needle (Interior (GenericNeedle a))
k
  fromBoundary :: Boundary (GenericNeedle a) -> GenericNeedle a
fromBoundary Boundary (GenericNeedle a)
b = case Boundary (GenericNeedle a)
b of {}


instance ( Semimanifold a
         , Semimanifold (VRep a), Needle a ~ GenericNeedle a
         , OpenManifold (Scalar (Needle (Gnrx.Rep a Void)))
         , LinearSpace (Needle (Gnrx.Rep a Void))
         , Num' (Scalar (Needle (Gnrx.Rep a Void))) )
            => PseudoAffineWithBoundary (GenericNeedle a) where
  GenericNeedle a
_ !-| :: GenericNeedle a
-> Boundary (GenericNeedle a) -> HalfNeedle (GenericNeedle a)
!-| Boundary (GenericNeedle a)
b = case Boundary (GenericNeedle a)
b of {}
  .--! :: GenericNeedle a
-> GenericNeedle a -> Needle (Interior (GenericNeedle a))
(.--!) = forall v. AdditiveGroup v => v -> v -> v
(^-^)