2eb21d5...master · laop

git.iko.soy/laop

93 files changed+2350−5957

since splitsnapshots

unifiedsplit

.ghc.environment.x86_64-linux-8.6.5deleted−17

@@ -1,17 +0,0 @@

1−-- This is a GHC environment file written by cabal. This means you can

2−-- run ghc or ghci and get the environment of the project as a whole.

3−-- But you still need to use cabal repl $target to get the environment

4−-- of specific components (libs, exes, tests etc) because each one can

5−-- have its own source dirs, cpp flags etc.

6−--

7−clear-package-db

8−global-package-db

9−package-db /home/bolt/.cabal/store/ghc-8.6.5/package.db

10−package-db packagedb/ghc-8.6.5

11−package-id laop-0.1.0.0-inplace

12−package-id base-4.12.0.0

13−package-id ghc-prim-0.5.3

14−package-id rts

15−package-id integer-gmp-1.0.2.0

16−package-id deepseq-1.4.4.0

17−package-id array-0.5.3.0

.github/workflows/ci.ymlnew file+43

@@ -0,0 +1,43 @@

1+name: CI

3+# Trigger the workflow on push or pull request, but only for the master branch

4+on:

5+ pull_request:

6+ push:

7+ branches: [master]

9+jobs:

10+ build:

11+ name: ghc ${{ matrix.ghc }}

12+ runs-on: ubuntu-16.04

13+ strategy:

14+ matrix:

15+ cabal: ["3.0"]

16+ ghc:

17+ - "8.6.5"

18+ - "8.8.3"

19+

20+ steps:

21+ - uses: actions/checkout@v2

22+ if: github.event.action == 'opened' || github.event.action == 'synchronize' || github.event.ref == 'refs/heads/master'

23+

24+ - uses: actions/setup-haskell@v1

25+ name: Setup Haskell

26+ with:

27+ ghc-version: ${{ matrix.ghc }}

28+ cabal-version: ${{ matrix.cabal }}

29+

30+ - uses: actions/cache@v1

31+ name: Cache ~/.cabal/store

32+ with:

33+ path: ~/.cabal/store

34+ key: ${{ runner.os }}-${{ matrix.ghc }}-cabal

35+

36+ - name: Build

37+ run: |

38+ cabal v2-update

39+ cabal v2-build --enable-tests --enable-benchmarks

40+

41+ - name: Test

42+ run: |

43+ cabal v2-test --enable-tests

.gitignore+54−2

@@ -1,3 +1,55 @@

1+### Haskell

2+dist

3+dist-*

4+cabal-dev

5+*.o

6+*.hi

7+*.chi

8+*.chs.h

9+*.dyn_o

10+*.dyn_hi

11+*.prof

12+*.aux

13+*.hp

14+*.eventlog

15+.virtualenv

16+.hsenv

17+.hpc

18+.cabal-sandbox/

19+cabal.sandbox.config

20+cabal.config

21+cabal.project.local

22+.ghc.environment.*

23+.HTF/

24+# Stack

125.stack-work/

2−laop.cabal

3−*~

26+stack.yaml.lock

27+

28+### IDE/support

29+# Vim

30+[._]*.s[a-v][a-z]

31+[._]*.sw[a-p]

32+[._]s[a-v][a-z]

33+[._]sw[a-p]

34+*~

35+tags

36+

37+# IntellijIDEA

38+.idea/

39+.ideaHaskellLib/

40+*.iml

41+

42+# Atom

43+.haskell-ghc-mod.json

44+

45+# VS

46+.vscode/

47+

48+# Emacs

49+*#

50+.dir-locals.el

51+TAGS

52+

53+# other

54+.DS_Store

55+hackage.sh

.travis.ymlnew file+47

@@ -0,0 +1,47 @@

1+sudo: true

2+language: haskell

4+git:

5+ depth: 5

7+cabal: "3.0"

9+cache:

10+ directories:

11+ - "$HOME/.cabal/store"

12+ - "$HOME/.stack"

13+ - "$TRAVIS_BUILD_DIR/.stack-work"

14+

15+matrix:

16+ include:

17+ - ghc: 8.6.5

18+ - ghc: 8.8.3

19+

20+ - ghc: 8.8.3

21+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack-8.8.3.yaml"

22+

23+ - ghc: 8.6.5

24+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack.yaml"

25+

26+

27+install:

28+ - |

29+ if [ -z "$STACK_YAML" ]; then

30+ cabal v2-update

31+ cabal v2-build --enable-tests --enable-benchmarks

32+ else

33+ curl -sSL https://get.haskellstack.org/ | sh

34+ stack --version

35+ stack build --system-ghc --test --bench --no-run-tests --no-run-benchmarks

36+ fi

37+

38+script:

39+ - |

40+ if [ -z "$STACK_YAML" ]; then

41+ cabal v2-test --enable-tests

42+ else

43+ stack test --system-ghc

44+ fi

45+

46+notifications:

47+ email: false

CHANGELOG.mdnew file+17

@@ -0,0 +1,17 @@

1+# Changelog

3+`laop` uses [PVP Versioning][1].

4+The changelog is available [on GitHub][2].

6+## 0.1.1.1

8+* Bump base version to work with cabal

10+## 0.1.1.0

11+

12+* Package is more organized and now has tests and benchmarks separated

13+* Repository is more organized and clean

14+* Added CI

15+

16+[1]: https://pvp.haskell.org

17+[2]: https://github.com/bolt12/laop2/releases

ChangeLog.mddeleted−3

@@ -1,3 +0,0 @@

1−# Changelog for laop

2−

3−## Unreleased changes

LICENSE+17−26

@@ -1,30 +1,21 @@

1+MIT License

5−Redistribution and use in source and binary forms, with or without

6−modification, are permitted provided that the following conditions are met:

5+Permission is hereby granted, free of charge, to any person obtaining a copy

6+of this software and associated documentation files (the "Software"), to deal

7+in the Software without restriction, including without limitation the rights

8+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell

9+copies of the Software, and to permit persons to whom the Software is

10+furnished to do so, subject to the following conditions:

711

8− * Redistributions of source code must retain the above copyright

9− notice, this list of conditions and the following disclaimer.

12+The above copyright notice and this permission notice shall be included in all

13+copies or substantial portions of the Software.

1014

11− * Redistributions in binary form must reproduce the above

12− copyright notice, this list of conditions and the following

13− disclaimer in the documentation and/or other materials provided

14− with the distribution.

15−

16− * Neither the name of Armando Santos nor the names of other

17− contributors may be used to endorse or promote products derived

18− from this software without specific prior written permission.

19−

20−THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS

21−"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT

22−LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR

23−A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT

24−OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,

25−SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT

26−LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,

27−DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY

28−THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT

29−(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE

30−OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

15+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR

16+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

17+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE

18+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER

19+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,

20+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE

21+SOFTWARE.

README.md+107−24

@@ -1,5 +1,11 @@

11# laop - Linear Algebra of Programming library

3+[![GitHub CI](https://github.com/bolt12/laop/workflows/CI/badge.svg)](https://github.com/bolt12/laop/actions)

4+[![Build status](https://img.shields.io/travis/bolt12/laop.svg?logo=travis)](https://travis-ci.org/bolt12/laop)

5+[![Hackage](https://img.shields.io/hackage/v/laop.svg?logo=haskell)](https://hackage.haskell.org/package/laop)

6+[![Stackage Lts](http://stackage.org/package/laop/badge/lts)](http://stackage.org/lts/package/laop)

7+[![Stackage Nightly](http://stackage.org/package/laop/badge/nightly)](http://stackage.org/nightly/package/laop)

39The LAoP discipline generalises relations and functions treating them as

410Boolean matrices and in turn consider these as arrows.

511

@@ -10,33 +16,38 @@ motivation behind the library, the underlying theory, and implementation details

1016This module offers many of the combinators mentioned in the work of

1117[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).

1218

13−See the candidate package in hackage [here](https://hackage.haskell.org/package/laop-0.1.0.0/candidate)

19+See the package in hackage [here](https://hackage.haskell.org/package/laop-0.1.1.0)

20+

21+A Functional Pearl has been written and can be regarded as the [reference document](https://github.com/bolt12/tymfgg-pearl) for this library.

1422

1523## Features

1624

17−This library offers:

18−

19−- An internal representation that uses normal datatypes (`Void`, `()`, `Either`) for matrix dimensions and two

20− type families `FromNat` and `Count` to make it easier to work with these type of matrices.

21−- A module that uses generalised types for the dimensions and is just a newtype wrapper

22− around the internal matrix data type.

23−- Other that uses type level naturals for the dimensions and is just a newtype wrapper

24− around the internal matrix data type.

25−- A very simple Probabilistic Programming module and several functions and data types to

26− make it easier to deal with sample space.

27−

28−Given this, this new matrix formulation compared to other libraries one has much more advantages, such as:

29−

30−- Has an inductive definition enabling writing matrix manipulation functions in a much more elegant and calculational way;

31−- Statically typed dimensions;

32−- Polymorphic data type dimensions;

33−- Polymorphic matrix content;

34−- Awesome dimension type inference;

35−- Fast type natural conversion via `FromNat` type family;

36−- Matrix `Junc` and `Split`-ing in O(1);

37−- Matrix composition takes advantage of divide-and-conquer and fusion laws.

38−

39−Unfortunately, this approach does not solve the issue of type dimensions being in some way constrainted making it impossible to write Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed.

25+This library offers 3 main matrix programming modules:

26+

27+- One in which matrices are typed with type level natural numbers;

28+- One in which matrices are typed with arbitrary generic data types;

29+- One in which matrices are regarded as relations, i.e. boolean matrices;

30+

31+There's also an experimental module that uses matrices to represent probability

32+distributions.

33+

34+The most interesting feature is that matrices are represented as an inductive data type.

35+Given this formulation, matrix algorithms can be expressed in a much more elegant and

36+calculational style than the traditional vector of vectors representation. This data type

37+guarantees that a matrix will always have valid dimensions. It can also express block matrix

38+computations naturally, which leads to total, efficient and statically typed manipulation

39+and transformation functions.

40+

41+Like matrix multiplication, other common operations, such as matrix transposition, benefit from a block-oriented structure that leads to a simple and natural divide-and-conquer algorithmic solution. Performance wise, this means that without much effort we can obtain optimal cache-oblivious algorithms.

42+

43+Given this, this new matrix formulation compared to other libraries:

44+

45+- Is more compositional and polymorphic and does not have partial matrix manipulation functions (hence less chances for usage errors);

46+- Our implementation of matrices enables simple manipulation of submatrices, making it particularly suitable for formal verification and equation reasoning, using the mathematical framework defined by the linear algebra of programming. Furthermore, the data type constructors ensure that the matrices of this kind are sound, i.e. malformed matrices with incorrect dimensions of the sort, can not be constructed.

47+

48+## Known issues

49+

50+Unfortunately, due to the use of type-level programming features, this approach sometimes requires type dimensions to be constrainted, in some way, impossible to write idiomatic Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed. And not all kinds of programs can be modeled using matrices, namely programs that deal with arbitrary infinite data types such as lists or integers (although there are some workarounds).

4051

4152## Notes

4253

@@ -118,6 +129,73 @@ grass_wet = row [0,1] . kp1

118129rainning :: Matrix Double (Bool, (Bool, Bool)) One

119130rainning = row [0,1] . kp2 . kp2

120131

132+-- Alcuin Puzzle

133+

134+data Being = Farmer | Fox | Goose | Beans

135+ deriving (Bounded, Enum, Eq, Show, Generic)

136+

137+data Bank = LeftB | RightB

138+ deriving (Bounded, Enum, Eq, Show, Generic)

139+

140+eats :: Being -> Being -> Bool

141+eats Fox Goose = True

142+eats Goose Beans = True

143+eats _ _ = False

144+

145+eatsR :: R.Relation Being Being

146+eatsR = R.toRel eats

147+

148+cross :: Bank -> Bank

149+cross LeftB = RightB

150+cross RightB = LeftB

151+

152+crossR :: R.Relation Bank Bank

153+crossR = R.fromF' cross

154+

155+-- | Initial state, everyone in the left bank

156+locationLeft :: Being -> Bank

157+locationLeft _ = LeftB

158+

159+locationLeftR :: R.Relation Being Bank

160+locationLeftR = R.fromF' locationLeft

161+

162+-- | Initial state, everyone in the right bank

163+locationRight :: Being -> Bank

164+locationRight _ = RightB

165+

166+locationRightR :: R.Relation Being Bank

167+locationRightR = R.fromF' locationRight

168+

169+-- Properties

170+

171+-- Being at the same bank

172+sameBank :: R.Relation Being Bank -> R.Relation Being Being

173+sameBank = R.ker

174+

175+-- Risk of somebody eating somebody else

176+canEat :: R.Relation Being Bank -> R.Relation Being Being

177+canEat w = sameBank w `R.intersection` eatsR

178+

179+-- "Starvation" property.

180+inv :: R.Relation Being Bank -> Bool

181+inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)

182+ where

183+ farmer :: R.Relation Being Being

184+ farmer = R.fromF' (const Farmer)

185+

186+-- Arbitrary state

187+bankState :: Being -> Bank -> Bool

188+bankState Farmer LeftB = True

189+bankState Fox LeftB = True

190+bankState Goose RightB = True

191+bankState Beans RightB = True

192+bankState _ _ = False

193+

194+bankStateR :: R.Relation Being Bank

195+bankStateR = R.toRel bankState

196+

197+-- Main

198+

121199main :: IO ()

122200main = do

123201 putStrLn "Monty Hall Problem solution:"

@@ -132,6 +210,8 @@ main = do

132210 prettyPrint (grass_wet . state)

133211 putStrLn "\n Probability of rain:"

134212 prettyPrint (rainning . state)

213+ putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"

214+ print (inv bankStateR)

135215```

136216

137217```Shell

@@ -190,4 +270,7 @@ Monty Hall Problem solution:

190270┌ ┐

191271│ 0.2 │

192272└ ┘

273+

274+Is the arbitrary state a valid state? (Alcuin Puzzle)

275+False

193276```

Setup.hsdeleted−2

@@ -1,2 +0,0 @@

1−import Distribution.Simple

2−main = defaultMain

appveyor.ymlnew file+34

@@ -0,0 +1,34 @@

1+clone_folder: "c:\\WORK"

2+clone_depth: 5

4+# Do not build feature branch with open Pull Requests

5+skip_branch_with_pr: true

7+platform:

8+ - x86_64

10+cache:

11+ - "C:\\SR"

12+ - dist-newstyle

13+

14+environment:

15+ global:

16+ CABOPTS: --store-dir=C:\\SR

17+

18+ matrix:

19+ - GHCVER: 8.6.5

20+

21+install:

22+ - choco source add -n mistuke -s https://www.myget.org/F/mistuke/api/v2

23+ - choco install -y cabal --version 2.4.1.0

24+ - choco install -y ghc --version 8.6.3

25+ - refreshenv

26+

27+before_build:

28+ - cabal --version

29+ - ghc --version

30+ - cabal %CABOPTS% v2-update

31+

32+build_script:

33+ - cabal %CABOPTS% v2-build --enable-tests

34+ - cabal %CABOPTS% v2-test --enable-tests

benchmark/Main.hsnew file+175

@@ -0,0 +1,175 @@

1+{-# LANGUAGE KindSignatures #-}

2+{-# LANGUAGE GeneralizedNewtypeDeriving #-}

3+{-# LANGUAGE DerivingVia #-}

4+{-# LANGUAGE DataKinds #-}

5+{-# LANGUAGE BangPatterns #-}

6+{-# LANGUAGE TypeApplications #-}

7+{-# LANGUAGE FlexibleContexts #-}

8+{-# LANGUAGE AllowAmbiguousTypes #-}

9+{-# LANGUAGE ScopedTypeVariables #-}

10+

11+module Main (main) where

12+

13+import LAoP.Dist.Internal

14+import LAoP.Utils

15+import LAoP.Matrix.Type hiding (CountableN, Countable, FLN)

16+import qualified LAoP.Matrix.Type as T (FLN)

17+import Criterion.Main

18+import Control.DeepSeq

19+import Test.QuickCheck

20+import GHC.TypeLits

21+import Data.Proxy

22+import Prelude hiding (id, (.))

23+

24+selectM2 ::

25+ ( Num e,

26+ T.FLN b b,

27+ CountableN b

28+ ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

29+selectM2 m y = join y iden `comp` m

30+

31+selectD2 ::

32+ ( FLN b b,

33+ CountableN b

34+ ) => Dist (Either a b) -> Matrix Prob a b -> Dist b

35+selectD2 (D d) m = D (join m iden `comp` d)

36+

37+-- Composition with iden

38+compId m = comp iden m

39+

40+randomDist :: forall a . (Countable a, FLN () a) => Gen (Dist a)

41+randomDist = do

42+ let size = fromInteger (natVal (Proxy :: Proxy (Count a)))

43+ l <- vectorOf size (arbitrary :: Gen Prob)

44+ let ln = normalize l

45+ lr = map (:[]) ln

46+ m = fromLists lr

47+ return (D m)

48+

49+randomMatrix :: forall a b . (CountableDims a b, FLN a b) => Gen (Matrix Prob a b)

50+randomMatrix = do

51+ let cols = fromInteger (natVal (Proxy :: Proxy (Count a)))

52+ rows = fromInteger (natVal (Proxy :: Proxy (Count b)))

53+ l <- vectorOf (cols * rows) arbitrary

54+ let lr = buildList l cols

55+ m = fromLists lr

56+ return m

57+

58+buildList [] _ = []

59+buildList l r = take r l : buildList (drop r l) r

60+

61+normalize :: [Prob] -> [Prob]

62+normalize l = let l' = map abs l

63+ s = sum l'

64+ in map (/ s) l'

65+

66+-- Probability

67+newtype Probability = P Double

68+ deriving (Num, Show, Fractional, NFData) via Double

69+newtype Dist' a = D' { unD :: [(a,Probability)] }

70+ deriving (Show, NFData)

71+

72+instance Functor Dist' where

73+ fmap f (D' d) = D' [(f x,p) | (x,p) <- d]

74+

75+instance Applicative Dist' where

76+ pure x = D' [(x, 1)]

77+ fm <*> m = D' [(f x,q*p) | (f,p) <- unD fm, (x,q) <- unD m]

78+

79+class Applicative f => Selective f where

80+ select :: f (Either a b) -> f (a -> b) -> f b

81+

82+instance Selective Dist' where

83+ select x y = (\e f -> either f id e) <$> x <*> y -- selectA

84+

85+instance Monad Dist' where

86+ return = pure

87+ (D' d) >>= f = D' [(y,q*p) | (x,p) <- d, (y,q) <- unD (f x)]

88+

89+normalize2 :: forall a . (Enum a, Bounded a) => [Probability] -> [(a, Probability)]

90+normalize2 l = let l' = map abs l

91+ as = [minBound .. maxBound]

92+ s = sum l'

93+ probs = map (/ s) l'

94+ in zip as probs

95+

96+randomDist2 :: forall a . (Countable a, Enum a, Bounded a) => Gen (Dist' a)

97+randomDist2 = do

98+ let size = fromInteger (natVal (Proxy :: Proxy (Count a)))

99+ l <- vectorOf size (arbitrary :: Gen Prob)

100+ let ln = normalize2 @a (map P l)

101+ return (D' ln)

102+

103+normalize3 :: forall a . [a] -> [Probability] -> [(a, Probability)]

104+normalize3 as l = let s = sum l

105+ probs = map (/ s) l

106+ in zip as probs

107+

108+randomDistF :: forall a b . (CountableDims a b, CoArbitrary a, Arbitrary b) => Gen (Dist' (a -> b))

109+randomDistF = do

110+ let a = fromInteger (natVal (Proxy :: Proxy (Count a)))

111+ b = fromInteger (natVal (Proxy :: Proxy (Count b)))

112+ size = a * b

113+ l <- vectorOf size (arbitrary :: Gen Prob)

114+ l2 <- vectorOf size (arbitrary :: Gen (a -> b))

115+ let !ln = normalize3 l2 (map P l)

116+ return (D' ln)

117+

118+instance CoArbitrary (Natural a b)

119+

120+instance forall (a :: Nat) (b :: Nat). (KnownNat a, KnownNat b) => Arbitrary (Natural a b)

121+ where

122+ arbitrary =

123+ let bottom = fromInteger (natVal (Proxy :: Proxy a))

124+ top = fromInteger (natVal (Proxy :: Proxy b))

125+ in do

126+ x <- arbitrary `suchThat` (\n -> n >= bottom && n <= top)

127+ return (nat x)

128+

129+setupEnv = do

130+ m11 <- generate (resize 1 (randomMatrix @(Natural 0 10) @(Natural 0 10) ))

131+ m12 <- generate (resize 1 (randomMatrix @(Natural 0 10) @(Natural 0 10) ))

132+ m21 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))

133+ m22 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))

134+ m31 <- generate (resize 1 (randomMatrix @(Natural 0 200) @(Natural 0 200) ))

135+ m32 <- generate (resize 1 (randomMatrix @(Natural 0 200) @(Natural 0 200) ))

136+ return (m11, m12, m21, m22, m31, m32)

137+

138+setupEnv2 = do

139+ m21 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))

140+ m40 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Either (Natural 0 100) (Natural 0 100) )))

141+ dist <- generate (resize 1 (randomDist @(Either (Natural 0 100) (Natural 0 100)) ))

142+ dist2 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))

143+ distList1 <- generate (resize 1 (randomDist2 @(Either (Natural 0 100) (Natural 0 100))))

144+ distList2 <- generate (resize 1 (randomDistF @(Natural 0 100) @(Natural 0 100)))

145+ return (m21, m40, dist, dist2, distList1, distList2)

146+

147+main :: IO ()

148+main = defaultMain [

149+ env setupEnv $ \ ~(m11, m12, m21, m22, m31, m32) -> bgroup "Matrix Composition" [

150+ bgroup "10x10" [

151+ bench "WHNF - 10x10" $ whnf (comp m11) m12

152+ , bench "NF - 10x10" $ nf (comp m11) m12

153+ ],

154+ bgroup "100x100" [

155+ bench "WHNF - 100x100" $ whnf (comp m21) m22

156+ , bench "NF - 100x100" $ nf (comp m21) m22

157+ ],

158+ bgroup "200x200" [

159+ bench "WHNF - 200x200" $ whnf (comp m31) m32

160+ , bench "NF - 200x200" $ nf (comp m31) m32

161+ ],

162+ bgroup "200x200 - RULES" [

163+ bench "No rules - 200x200" $ nf (comp m21) iden

164+ , bench "Rules - 200x200" $ nf compId m21

165+ ] ],

166+ env setupEnv2 $ \ ~(m21, m40, dist, dist2, dl1, dl2) -> bgroup "Matrix vs List - `select`" [

167+ bgroup "Distribution `select` - 100+100 / 100x100" [

168+ bench "List Distribution - Applicative version" $ nf (select dl1) dl2

169+ , bench "Matrix Distribution - Applicative version" $ nf (selectD2 dist) dist2

170+ , bench "Matrix Distribution - Selective version" $ nf (selectD dist) dist2

171+ ],

172+ bgroup "Matrix `select` - 100x(100+100) / 100x100" [

173+ bench "Applicative version" $ nf (selectM2 m40) m21

174+ , bench "Selective version" $ nf (selectM m40) m21

175+ ] ] ]

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/build/autogen/Paths_laop.hsdeleted−50

@@ -1,50 +0,0 @@

1−{-# LANGUAGE CPP #-}

2−{-# LANGUAGE NoRebindableSyntax #-}

3−{-# OPTIONS_GHC -fno-warn-missing-import-lists #-}

4−module Paths_laop (

5− version,

6− getBinDir, getLibDir, getDynLibDir, getDataDir, getLibexecDir,

7− getDataFileName, getSysconfDir

8− ) where

9−

10−import qualified Control.Exception as Exception

11−import Data.Version (Version(..))

12−import System.Environment (getEnv)

13−import Prelude

14−

15−#if defined(VERSION_base)

16−

17−#if MIN_VERSION_base(4,0,0)

18−catchIO :: IO a -> (Exception.IOException -> IO a) -> IO a

19−#else

20−catchIO :: IO a -> (Exception.Exception -> IO a) -> IO a

21−#endif

22−

23−#else

24−catchIO :: IO a -> (Exception.IOException -> IO a) -> IO a

25−#endif

26−catchIO = Exception.catch

27−

28−version :: Version

29−version = Version [0,1,0,0] []

30−bindir, libdir, dynlibdir, datadir, libexecdir, sysconfdir :: FilePath

31−

32−bindir = "/home/bolt/.cabal/bin"

33−libdir = "/home/bolt/.cabal/lib/x86_64-linux-ghc-8.6.5/laop-0.1.0.0-inplace"

34−dynlibdir = "/home/bolt/.cabal/lib/x86_64-linux-ghc-8.6.5"

35−datadir = "/home/bolt/.cabal/share/x86_64-linux-ghc-8.6.5/laop-0.1.0.0"

36−libexecdir = "/home/bolt/.cabal/libexec/x86_64-linux-ghc-8.6.5/laop-0.1.0.0"

37−sysconfdir = "/home/bolt/.cabal/etc"

38−

39−getBinDir, getLibDir, getDynLibDir, getDataDir, getLibexecDir, getSysconfDir :: IO FilePath

40−getBinDir = catchIO (getEnv "laop_bindir") (\_ -> return bindir)

41−getLibDir = catchIO (getEnv "laop_libdir") (\_ -> return libdir)

42−getDynLibDir = catchIO (getEnv "laop_dynlibdir") (\_ -> return dynlibdir)

43−getDataDir = catchIO (getEnv "laop_datadir") (\_ -> return datadir)

44−getLibexecDir = catchIO (getEnv "laop_libexecdir") (\_ -> return libexecdir)

45−getSysconfDir = catchIO (getEnv "laop_sysconfdir") (\_ -> return sysconfdir)

46−

47−getDataFileName :: FilePath -> IO FilePath

48−getDataFileName name = do

49− dir <- getDataDir

50− return (dir ++ "/" ++ name)

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/build/autogen/cabal_macros.hdeleted−143

@@ -1,143 +0,0 @@

1−/* DO NOT EDIT: This file is automatically generated by Cabal */

2−

3−/* package laop-0.1.0.0 */

4−#ifndef VERSION_laop

5−#define VERSION_laop "0.1.0.0"

6−#endif /* VERSION_laop */

7−#ifndef MIN_VERSION_laop

8−#define MIN_VERSION_laop(major1,major2,minor) (\

9− (major1) < 0 || \

10− (major1) == 0 && (major2) < 1 || \

11− (major1) == 0 && (major2) == 1 && (minor) <= 0)

12−#endif /* MIN_VERSION_laop */

13−

14−/* package base-4.12.0.0 */

15−#ifndef VERSION_base

16−#define VERSION_base "4.12.0.0"

17−#endif /* VERSION_base */

18−#ifndef MIN_VERSION_base

19−#define MIN_VERSION_base(major1,major2,minor) (\

20− (major1) < 4 || \

21− (major1) == 4 && (major2) < 12 || \

22− (major1) == 4 && (major2) == 12 && (minor) <= 0)

23−#endif /* MIN_VERSION_base */

24−

25−/* package deepseq-1.4.4.0 */

26−#ifndef VERSION_deepseq

27−#define VERSION_deepseq "1.4.4.0"

28−#endif /* VERSION_deepseq */

29−#ifndef MIN_VERSION_deepseq

30−#define MIN_VERSION_deepseq(major1,major2,minor) (\

31− (major1) < 1 || \

32− (major1) == 1 && (major2) < 4 || \

33− (major1) == 1 && (major2) == 4 && (minor) <= 4)

34−#endif /* MIN_VERSION_deepseq */

35−

36−/* tool gcc-9.2.0 */

37−#ifndef TOOL_VERSION_gcc

38−#define TOOL_VERSION_gcc "9.2.0"

39−#endif /* TOOL_VERSION_gcc */

40−#ifndef MIN_TOOL_VERSION_gcc

41−#define MIN_TOOL_VERSION_gcc(major1,major2,minor) (\

42− (major1) < 9 || \

43− (major1) == 9 && (major2) < 2 || \

44− (major1) == 9 && (major2) == 2 && (minor) <= 0)

45−#endif /* MIN_TOOL_VERSION_gcc */

46−

47−/* tool ghc-8.6.5 */

48−#ifndef TOOL_VERSION_ghc

49−#define TOOL_VERSION_ghc "8.6.5"

50−#endif /* TOOL_VERSION_ghc */

51−#ifndef MIN_TOOL_VERSION_ghc

52−#define MIN_TOOL_VERSION_ghc(major1,major2,minor) (\

53− (major1) < 8 || \

54− (major1) == 8 && (major2) < 6 || \

55− (major1) == 8 && (major2) == 6 && (minor) <= 5)

56−#endif /* MIN_TOOL_VERSION_ghc */

57−

58−/* tool ghc-pkg-8.6.5 */

59−#ifndef TOOL_VERSION_ghc_pkg

60−#define TOOL_VERSION_ghc_pkg "8.6.5"

61−#endif /* TOOL_VERSION_ghc_pkg */

62−#ifndef MIN_TOOL_VERSION_ghc_pkg

63−#define MIN_TOOL_VERSION_ghc_pkg(major1,major2,minor) (\

64− (major1) < 8 || \

65− (major1) == 8 && (major2) < 6 || \

66− (major1) == 8 && (major2) == 6 && (minor) <= 5)

67−#endif /* MIN_TOOL_VERSION_ghc_pkg */

68−

69−/* tool haddock-2.22.0 */

70−#ifndef TOOL_VERSION_haddock

71−#define TOOL_VERSION_haddock "2.22.0"

72−#endif /* TOOL_VERSION_haddock */

73−#ifndef MIN_TOOL_VERSION_haddock

74−#define MIN_TOOL_VERSION_haddock(major1,major2,minor) (\

75− (major1) < 2 || \

76− (major1) == 2 && (major2) < 22 || \

77− (major1) == 2 && (major2) == 22 && (minor) <= 0)

78−#endif /* MIN_TOOL_VERSION_haddock */

79−

80−/* tool hpc-0.67 */

81−#ifndef TOOL_VERSION_hpc

82−#define TOOL_VERSION_hpc "0.67"

83−#endif /* TOOL_VERSION_hpc */

84−#ifndef MIN_TOOL_VERSION_hpc

85−#define MIN_TOOL_VERSION_hpc(major1,major2,minor) (\

86− (major1) < 0 || \

87− (major1) == 0 && (major2) < 67 || \

88− (major1) == 0 && (major2) == 67 && (minor) <= 0)

89−#endif /* MIN_TOOL_VERSION_hpc */

90−

91−/* tool hsc2hs-0.68.5 */

92−#ifndef TOOL_VERSION_hsc2hs

93−#define TOOL_VERSION_hsc2hs "0.68.5"

94−#endif /* TOOL_VERSION_hsc2hs */

95−#ifndef MIN_TOOL_VERSION_hsc2hs

96−#define MIN_TOOL_VERSION_hsc2hs(major1,major2,minor) (\

97− (major1) < 0 || \

98− (major1) == 0 && (major2) < 68 || \

99− (major1) == 0 && (major2) == 68 && (minor) <= 5)

100−#endif /* MIN_TOOL_VERSION_hsc2hs */

101−

102−/* tool pkg-config-1.6.3 */

103−#ifndef TOOL_VERSION_pkg_config

104−#define TOOL_VERSION_pkg_config "1.6.3"

105−#endif /* TOOL_VERSION_pkg_config */

106−#ifndef MIN_TOOL_VERSION_pkg_config

107−#define MIN_TOOL_VERSION_pkg_config(major1,major2,minor) (\

108− (major1) < 1 || \

109− (major1) == 1 && (major2) < 6 || \

110− (major1) == 1 && (major2) == 6 && (minor) <= 3)

111−#endif /* MIN_TOOL_VERSION_pkg_config */

112−

113−/* tool runghc-8.6.5 */

114−#ifndef TOOL_VERSION_runghc

115−#define TOOL_VERSION_runghc "8.6.5"

116−#endif /* TOOL_VERSION_runghc */

117−#ifndef MIN_TOOL_VERSION_runghc

118−#define MIN_TOOL_VERSION_runghc(major1,major2,minor) (\

119− (major1) < 8 || \

120− (major1) == 8 && (major2) < 6 || \

121− (major1) == 8 && (major2) == 6 && (minor) <= 5)

122−#endif /* MIN_TOOL_VERSION_runghc */

123−

124−/* tool strip-2.33 */

125−#ifndef TOOL_VERSION_strip

126−#define TOOL_VERSION_strip "2.33"

127−#endif /* TOOL_VERSION_strip */

128−#ifndef MIN_TOOL_VERSION_strip

129−#define MIN_TOOL_VERSION_strip(major1,major2,minor) (\

130− (major1) < 2 || \

131− (major1) == 2 && (major2) < 33 || \

132− (major1) == 2 && (major2) == 33 && (minor) <= 0)

133−#endif /* MIN_TOOL_VERSION_strip */

134−

135−#ifndef CURRENT_PACKAGE_KEY

136−#define CURRENT_PACKAGE_KEY "laop-0.1.0.0-inplace"

137−#endif /* CURRENT_PACKAGE_KEY */

138−#ifndef CURRENT_COMPONENT_ID

139−#define CURRENT_COMPONENT_ID "laop-0.1.0.0-inplace"

140−#endif /* CURRENT_COMPONENT_ID */

141−#ifndef CURRENT_PACKAGE_VERSION

142−#define CURRENT_PACKAGE_VERSION "0.1.0.0"

143−#endif /* CURRENT_PACKAGE_VERSION */

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/cache/configdeleted

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build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/Dist.htmldeleted−15

@@ -1,15 +0,0 @@

2− in Haskell. See <a href="https://github.com/bolt12/master-thesis">my Msc Thesis</a> for the

3− motivation behind the library, the underlying theory, and implementation details.This module exports a type synonym <code><a href="Dist.html#t:Dist" title="Dist">Dist</a></code>

4− that represents a stochastic distribution

5− matrix and several distribution construction

6− functions.</div></div><div id="synopsis"><details id="syn"><summary>Synopsis</summary><ul class="details-toggle" data-details-id="syn"><li class="src short">newtype <a href="#t:Dist">Dist</a> a = <a href="#v:D">D</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () a)</li><li class="src short">type <a href="#t:Prob">Prob</a> = <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Double" title="Prelude">Double</a></li><li class="src short"><a href="#v:fmapD">fmapD</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => (a -> b) -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b</li><li class="src short"><a href="#v:unitD">unitD</a> :: <a href="Dist.html#t:Dist" title="Dist">Dist</a> ()</li><li class="src short"><a href="#v:multD">multD</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> (a, b)</li><li class="src short"><a href="#v:selectD">selectD</a> :: (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b))) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> a b -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b</li><li class="src short"><a href="#v:returnD">returnD</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a)) => a -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a</li><li class="src short"><a href="#v:bindD">bindD</a> :: <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> a b -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b</li><li class="src short"><a href="#v:choose">choose</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => <a href="Dist.html#t:Prob" title="Dist">Prob</a> -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a</li><li class="src short"><a href="#v:shape">shape</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => (<a href="Dist.html#t:Prob" title="Dist">Prob</a> -> <a href="Dist.html#t:Prob" title="Dist">Prob</a>) -> [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a</li><li class="src short"><a href="#v:linear">linear</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a</li><li class="src short"><a href="#v:uniform">uniform</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a</li><li class="src short"><a href="#v:negExp">negExp</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a</li><li class="src short"><a href="#v:normal">normal</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a</li><li class="src short"><a href="#v:toValues">toValues</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> [(a, <a href="Dist.html#t:Prob" title="Dist">Prob</a>)]</li><li class="src short"><a href="#v:prettyDist">prettyDist</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a></li><li class="src short"><a href="#v:prettyPrintDist">prettyPrintDist</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="/package/base-4.12.0.0/docs/System-IO.html#t:IO" title="System.IO">IO</a> ()</li></ul></details></div><div id="interface"><h1>Documentation</h1><div class="doc">If the sum of the rows of a column matrix

7− is equal to 1 then this stochastic matrix

8− can be seen as a probability distribution.This module is still experimental but it's

9− already possible to model probabilistic programming

10− problems with it. Import <code><a href="Matrix.html#v:Nat" title="Matrix">Nat</a></code> or <code><a href="Matrix.html#v:Type" title="Matrix">Type</a></code>

11− to access LAoP matrix combinators and then all you have

12− to do is to define your sample space, either by creating a new data

13− type or by abstracting it out via <code>Natural</code>.Write manipulation functions and promote them to matrices via

14− <code>fromF</code> or <code>fromF'</code> and you're good to go!</div><a href="#g:1" id="g:1"><h1><code><a href="Dist.html#t:Dist" title="Dist">Dist</a></code> and <code><a href="Dist.html#t:Prob" title="Dist">Prob</a></code> type synonyms</h1></a><div class="top">newtype <a id="t:Dist" class="def">Dist</a> a <a href="src/Dist.Internal.html#Dist" class="link">Source</a> <a href="#t:Dist" class="selflink">#</a><div class="doc">Type synonym for column vector matrices. This represents a probability

15− distribution.</div><div class="subs constructors">Constructors<table><tr><td class="src"><a id="v:D" class="def">D</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () a)</td><td class="doc empty"> </td></tr></table></div></div><div class="top">type <a id="t:Prob" class="def">Prob</a> = <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Double" title="Prelude">Double</a> <a href="src/Dist.Internal.html#Prob" class="link">Source</a> <a href="#t:Prob" class="selflink">#</a><div class="doc">Type synonym for probability value</div></div><a href="#g:2" id="g:2"><h1>Functor instance equivalent functions</h1></a><div class="top"><a id="v:fmapD" class="def">fmapD</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => (a -> b) -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b <a href="src/Dist.Internal.html#fmapD" class="link">Source</a> <a href="#v:fmapD" class="selflink">#</a><div class="doc">Functor instance</div></div><a href="#g:3" id="g:3"><h1>Applicative equivalent functions</h1></a><div class="top"><a id="v:unitD" class="def">unitD</a> :: <a href="Dist.html#t:Dist" title="Dist">Dist</a> () <a href="src/Dist.Internal.html#unitD" class="link">Source</a> <a href="#v:unitD" class="selflink">#</a><div class="doc">Applicative/Monoidal instance <code>unit</code> function</div></div><div class="top"><a id="v:multD" class="def">multD</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> (a, b) <a href="src/Dist.Internal.html#multD" class="link">Source</a> <a href="#v:multD" class="selflink">#</a><div class="doc">Applicative/Monoidal instance <code>mult</code> function</div></div><a href="#g:4" id="g:4"><h1>Selective equivalent functions</h1></a><div class="top"><a id="v:selectD" class="def">selectD</a> :: (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b))) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> a b -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b <a href="src/Dist.Internal.html#selectD" class="link">Source</a> <a href="#v:selectD" class="selflink">#</a><div class="doc">Selective instance function</div></div><a href="#g:5" id="g:5"><h1>Monad equivalent functions</h1></a><div class="top"><a id="v:returnD" class="def">returnD</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a)) => a -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a <a href="src/Dist.Internal.html#returnD" class="link">Source</a> <a href="#v:returnD" class="selflink">#</a><div class="doc">Monad instance <code><a href="/package/base-4.12.0.0/docs/Control-Monad.html#v:return" title="Control.Monad">return</a></code> function</div></div><div class="top"><a id="v:bindD" class="def">bindD</a> :: <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> a b -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> b <a href="src/Dist.Internal.html#bindD" class="link">Source</a> <a href="#v:bindD" class="selflink">#</a><div class="doc">Monad instance '(>>=)' function</div></div><a href="#g:6" id="g:6"><h1>Distribution construction functions</h1></a><div class="top"><a id="v:choose" class="def">choose</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => <a href="Dist.html#t:Prob" title="Dist">Prob</a> -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a <a href="src/Dist.Internal.html#choose" class="link">Source</a> <a href="#v:choose" class="selflink">#</a><div class="doc">Constructs a Bernoulli distribution</div></div><div class="top"><a id="v:shape" class="def">shape</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => (<a href="Dist.html#t:Prob" title="Dist">Prob</a> -> <a href="Dist.html#t:Prob" title="Dist">Prob</a>) -> [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a <a href="src/Dist.Internal.html#shape" class="link">Source</a> <a href="#v:shape" class="selflink">#</a><div class="doc">Creates a distribution given a shape function</div></div><div class="top"><a id="v:linear" class="def">linear</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a <a href="src/Dist.Internal.html#linear" class="link">Source</a> <a href="#v:linear" class="selflink">#</a><div class="doc">Constructs a Linear distribution</div></div><div class="top"><a id="v:uniform" class="def">uniform</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a <a href="src/Dist.Internal.html#uniform" class="link">Source</a> <a href="#v:uniform" class="selflink">#</a><div class="doc">Constructs an Uniform distribution</div></div><div class="top"><a id="v:negExp" class="def">negExp</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a <a href="src/Dist.Internal.html#negExp" class="link">Source</a> <a href="#v:negExp" class="selflink">#</a><div class="doc">Constructs an Negative Exponential distribution</div></div><div class="top"><a id="v:normal" class="def">normal</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) => [a] -> <a href="Dist.html#t:Dist" title="Dist">Dist</a> a <a href="src/Dist.Internal.html#normal" class="link">Source</a> <a href="#v:normal" class="selflink">#</a><div class="doc">Constructs an Normal distribution</div></div><a href="#g:7" id="g:7"><h1>Converto to list of pairs</h1></a><div class="top"><a id="v:toValues" class="def">toValues</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> [(a, <a href="Dist.html#t:Prob" title="Dist">Prob</a>)] <a href="src/Dist.Internal.html#toValues" class="link">Source</a> <a href="#v:toValues" class="selflink">#</a><div class="doc">Transforms a <code><a href="Dist.html#t:Dist" title="Dist">Dist</a></code> into a list of pairs.</div></div><a href="#g:8" id="g:8"><h1>Pretty print distribution</h1></a><div class="top"><a id="v:prettyDist" class="def">prettyDist</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a> <a href="src/Dist.Internal.html#prettyDist" class="link">Source</a> <a href="#v:prettyDist" class="selflink">#</a><div class="doc">Pretty a distribution</div></div><div class="top"><a id="v:prettyPrintDist" class="def">prettyPrintDist</a> :: forall a. (<a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> <a href="Dist.html#t:Prob" title="Dist">Prob</a> () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => <a href="Dist.html#t:Dist" title="Dist">Dist</a> a -> <a href="/package/base-4.12.0.0/docs/System-IO.html#t:IO" title="System.IO">IO</a> () <a href="src/Dist.Internal.html#prettyPrintDist" class="link">Source</a> <a href="#v:prettyPrintDist" class="selflink">#</a><div class="doc">Pretty Print a distribution</div></div></div></div><div id="footer">Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.22.0</div></body></html>

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2− Boolean matrices and in turn consider these as arrows.LAoP is a library for algebraic (inductive) construction and manipulation of matrices

3− in Haskell. See <a href="https://github.com/bolt12/master-thesis">my Msc Thesis</a> for the

4− motivation behind the library, the underlying theory, and implementation details.This module offers a newtype wrapper around <code><a href="Matrix-Type.html#v:Matrix" title="Matrix.Type">Matrix</a></code> that

5− uses type level naturals instead of standard data types for the matrices

6− dimensions.</div></div><div id="synopsis"><details id="syn"><summary>Synopsis</summary><ul class="details-toggle" data-details-id="syn"><li class="src short">newtype <a href="#t:Matrix">Matrix</a> e (cols :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) (rows :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) = <a href="#v:M">M</a> (Matrix e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))</li><li class="src short"><a href="#v:empty">empty</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e 0 0</li><li class="src short"><a href="#v:one">one</a> :: e -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e 1 1</li><li class="src short"><a href="#v:junc">junc</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols1 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols2 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols3 rows</li><li class="src short"><a href="#v:split">split</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows1 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows2 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows3</li><li class="src short">type family <a href="#t:FromNat">FromNat</a> (n :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ...</li><li class="src short">type family <a href="#t:Count">Count</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a> where ...</li><li class="src short">type family <a href="#t:Normalize">Normalize</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ...</li><li class="src short">class <a href="#t:FromLists">FromLists</a> e cols rows</li><li class="src short"><a href="#v:fromLists">fromLists</a> :: <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) => [[e]] -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:toLists">toLists</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> [[e]]</li><li class="src short"><a href="#v:toList">toList</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> [e]</li><li class="src short"><a href="#v:matrixBuilder">matrixBuilder</a> :: (<a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => ((<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>, <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> e) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:row">row</a> :: <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) () => [e] -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols 1</li><li class="src short"><a href="#v:col">col</a> :: <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) => [e] -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e 1 rows</li><li class="src short"><a href="#v:zeros">zeros</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:ones">ones</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:bang">bang</a> :: forall e cols. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols 1</li><li class="src short"><a href="#v:constant">constant</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => e -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:columns">columns</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></li><li class="src short"><a href="#v:rows">rows</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></li><li class="src short"><a href="#v:tr">tr</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e rows cols</li><li class="src short"><a href="#v:select">select</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1)), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows2 ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1, <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols1 ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols2, <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols3) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols1 rows3 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols3 rows1 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols2 rows2</li><li class="src short"><a href="#v:cond">cond</a> :: (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols), <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e) => (a -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a>) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:abideJS">abideJS</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:abideSJ">abideSJ</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:-61--61--61-">(===)</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows1 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows2 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows3</li><li class="src short"><a href="#v:p1">p1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) m</li><li class="src short"><a href="#v:p2">p2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) n</li><li class="src short"><a href="#v:-124--124--124-">(|||)</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols1 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols2 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols3 rows</li><li class="src short"><a href="#v:i1">i1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (rows <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e rows (rows <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n)</li><li class="src short"><a href="#v:i2">i2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> rows) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e rows (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> rows)</li><li class="src short"><a href="#v:-45--124--45-">(-|-)</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (n <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> m) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (k <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> j) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e n k -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e m j -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (n <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> m) (k <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> j)</li><li class="src short"><a href="#v:-62--60-">(><)</a> :: forall e m p n q. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> p)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> q)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n))), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q))), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> p), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> q), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> p) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> q))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e m p -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e n q -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n) (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q)</li><li class="src short"><a href="#v:kp1">kp1</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k))), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) m</li><li class="src short"><a href="#v:kp2">kp2</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k))), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) k</li><li class="src short"><a href="#v:khatri">khatri</a> :: forall e cols a b. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b))), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> a), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> b), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> a) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> b))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols a -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols b -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b)</li><li class="src short"><a href="#v:identity">identity</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols cols</li><li class="src short"><a href="#v:comp">comp</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cr rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols cr -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:fromF">fromF</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)) => (a -> b) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows</li><li class="src short"><a href="#v:fromF-39-">fromF'</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> b)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> b) (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> a)) => (a -> b) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> a) (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> b)</li><li class="src short"><a href="#v:pretty">pretty</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a></li><li class="src short"><a href="#v:prettyPrint">prettyPrint</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/System-IO.html#t:IO" title="System.IO">IO</a> ()</li></ul></details></div><div id="interface"><h1>Documentation</h1><div class="doc">LAoP (Linear Algebra of Programming) Inductive Matrix definition.LAoP generalises relations and functions treating them as

7− Boolean matrices and in turn consider these as arrows.

8− This library offers many of the combinators mentioned in the work of

9− Macedo (2012) and Oliveira (2012).This definition is a wrapper around <code><a href="Matrix.html#v:Type" title="Matrix">Type</a></code> but

10− dimensions are type level Naturals. Type inference might not

11− be as desired.There exists two type families that make it easier to write

12− matrix dimensions: <code>FromNat</code> and <code>Count</code>. This approach

13− leads to a very straightforward implementation

14− of LAoP combinators. </div><a href="#g:1" id="g:1"><h1>Type safe matrix representation</h1></a><div class="top">newtype <a id="t:Matrix" class="def">Matrix</a> e (cols :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) (rows :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) <a href="src/Matrix.Nat.html#Matrix" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a><div class="subs constructors">Constructors<table><tr><td class="src"><a id="v:M" class="def">M</a> (Matrix e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))</td><td class="doc empty"> </td></tr></table></div><div class="subs instances"><details id="i:Matrix" open="open"><summary>Instances</summary><table><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="/package/base-4.12.0.0/docs/Control-Category.html#t:Category" title="Control.Category">Category</a> (<a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a> -> <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a> -> <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) <a href="src/Matrix.Nat.html#line-155" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc">It isn't possible to implement the <code><a href="/package/base-4.12.0.0/docs/Data-Function.html#v:id" title="Data.Function">id</a></code> function so it's

15− implementation is <code><a href="/package/base-4.12.0.0/docs/Prelude.html#v:undefined" title="Prelude">undefined</a></code>. However <code><a href="Matrix-Nat.html#v:comp" title="Matrix.Nat">comp</a></code> can be and this partial

16− class implementation exists just to make the code more readable.Please use <code><a href="Matrix-Nat.html#v:identity" title="Matrix.Nat">identity</a></code> instead.</td></tr><tr><td colspan="2"><details id="i:id:Matrix:Category:1"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Nat.html">Matrix.Nat</a> <div class="subs methods">Methods<a href="#v:id">id</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e a a <a href="#v:id" class="selflink">#</a><a href="#v:.">(.)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e b c -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e a b -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e a c <a href="#v:." class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> e => <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> (<a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows) <a href="src/Matrix.Nat.html#line-148" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Eq:2"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Nat.html">Matrix.Nat</a> <div class="subs methods">Methods<a href="#v:-61--61-">(==)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-61--61-" class="selflink">#</a><a href="#v:-47--61-">(/=)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-47--61-" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> (<a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows) <a href="src/Matrix.Nat.html#line-148" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Num:3"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Nat.html">Matrix.Nat</a> <div class="subs methods">Methods<a href="#v:-43-">(+)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:-43-" class="selflink">#</a><a href="#v:-45-">(-)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:-45-" class="selflink">#</a><a href="#v:-42-">(*)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:-42-" class="selflink">#</a><a href="#v:negate">negate</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:negate" class="selflink">#</a><a href="#v:abs">abs</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:abs" class="selflink">#</a><a href="#v:signum">signum</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:signum" class="selflink">#</a><a href="#v:fromInteger">fromInteger</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Integer" title="Prelude">Integer</a> -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:fromInteger" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e => <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> (<a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows) <a href="src/Matrix.Nat.html#line-148" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Ord:4"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Nat.html">Matrix.Nat</a> <div class="subs methods">Methods<a href="#v:compare">compare</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ordering" title="Data.Ord">Ordering</a> <a href="#v:compare" class="selflink">#</a><a href="#v:-60-">(<)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-60-" class="selflink">#</a><a href="#v:-60--61-">(<=)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-60--61-" class="selflink">#</a><a href="#v:-62-">(>)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-62-" class="selflink">#</a><a href="#v:-62--61-">(>=)</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-62--61-" class="selflink">#</a><a href="#v:max">max</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:max" class="selflink">#</a><a href="#v:min">min</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="#v:min" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e => <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> (<a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows) <a href="src/Matrix.Nat.html#line-148" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Show:5"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Nat.html">Matrix.Nat</a> <div class="subs methods">Methods<a href="#v:showsPrec">showsPrec</a> :: <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:ShowS" title="Text.Show">ShowS</a> <a href="#v:showsPrec" class="selflink">#</a><a href="#v:show">show</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a> <a href="#v:show" class="selflink">#</a><a href="#v:showList">showList</a> :: [<a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows] -> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:ShowS" title="Text.Show">ShowS</a> <a href="#v:showList" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/deepseq-1.4.4.0/docs/Control-DeepSeq.html#t:NFData" title="Control.DeepSeq">NFData</a> e => <a href="/package/deepseq-1.4.4.0/docs/Control-DeepSeq.html#t:NFData" title="Control.DeepSeq">NFData</a> (<a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows) <a href="src/Matrix.Nat.html#line-148" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:NFData:6"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Nat.html">Matrix.Nat</a> <div class="subs methods">Methods<a href="#v:rnf">rnf</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> () <a href="#v:rnf" class="selflink">#</a></div></details></td></tr></table></details></div></div><a href="#g:2" id="g:2"><h1>Primitives</h1></a><div class="top"><a id="v:empty" class="def">empty</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e 0 0 <a href="src/Matrix.Nat.html#empty" class="link">Source</a> <a href="#v:empty" class="selflink">#</a></div><div class="top"><a id="v:one" class="def">one</a> :: e -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e 1 1 <a href="src/Matrix.Nat.html#one" class="link">Source</a> <a href="#v:one" class="selflink">#</a></div><div class="top"><a id="v:junc" class="def">junc</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols1 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols2 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols3 rows <a href="src/Matrix.Nat.html#junc" class="link">Source</a> <a href="#v:junc" class="selflink">#</a></div><div class="top"><a id="v:split" class="def">split</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows1 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows2 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows3 <a href="src/Matrix.Nat.html#split" class="link">Source</a> <a href="#v:split" class="selflink">#</a></div><a href="#g:3" id="g:3"><h1>Auxiliary type families</h1></a><div class="top">type family <a id="t:FromNat" class="def">FromNat</a> (n :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ... <a href="src/Matrix.Internal.html#FromNat" class="link">Source</a> <a href="#t:FromNat" class="selflink">#</a><div class="doc">Type family that computes of a given type dimension from a given naturalThanks to Li-Yao Xia this type family is super fast.</div><div class="subs equations">Equations<table><tr><td class="src"><a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> 0 = <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> 1 = ()</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n = FromNat' (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Mod" title="GHC.TypeNats">Mod</a> n 2 <a href="/package/base-4.12.0.0/docs/Data-Type-Equality.html#t:-61--61-" title="Data.Type.Equality">==</a> 0) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Div" title="GHC.TypeNats">Div</a> n 2))</td><td class="doc empty"> </td></tr></table></div></div><div class="top">type family <a id="t:Count" class="def">Count</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a> where ... <a href="src/Matrix.Internal.html#Count" class="link">Source</a> <a href="#t:Count" class="selflink">#</a><div class="doc">Type family that computes the cardinality of a given type dimension.It can also count the cardinality of custom types that implement the

17− <code><a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:Generic" title="GHC.Generics">Generic</a></code> instance.</div><div class="subs equations">Equations<table><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m) = (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-45-" title="GHC.TypeNats">-</a> n) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> 1</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) = <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">(+)</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> a) (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> b)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (a, b) = <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> a) (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> b)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (a -> b) = <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-94-" title="GHC.TypeNats">(^)</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> b) (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:M1" title="GHC.Generics">M1</a> _ _ f p) = <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (f p)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:K1" title="GHC.Generics">K1</a> _ _ _) = 1</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:V1" title="GHC.Generics">V1</a> _) = 0</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:U1" title="GHC.Generics">U1</a> _) = 1</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t::-42-:" title="GHC.Generics">(:*:)</a> a b p) = <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (a p) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (b p)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t::-43-:" title="GHC.Generics">(:+:)</a> a b p) = <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (a p) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (b p)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> d = <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:Rep" title="GHC.Generics">Rep</a> d <a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:R" title="GHC.Generics">R</a>)</td><td class="doc empty"> </td></tr></table></div></div><div class="top">type family <a id="t:Normalize" class="def">Normalize</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ... <a href="src/Matrix.Internal.html#Normalize" class="link">Source</a> <a href="#t:Normalize" class="selflink">#</a><div class="doc">Type family that normalizes the representation of a given data

18− structure</div><div class="subs equations">Equations<table><tr><td class="src"><a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> d = <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> d)</td><td class="doc empty"> </td></tr></table></div></div><a href="#g:4" id="g:4"><h1>Matrix construction and conversion</h1></a><div class="top">class <a id="t:FromLists" class="def">FromLists</a> e cols rows <a href="src/Matrix.Internal.html#FromLists" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a><div class="doc">Type class for defining the <code>fromList</code> conversion function.Given that it is not possible to branch on types at the term level type

19− classes are needed bery much like an inductive definition but on types.</div><div class="subs minimal">Minimal complete definitionfromLists</div><div class="subs instances"><details id="i:FromLists" open="open"><summary>Instances</summary><table><tr><td class="src clearfix"> <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () () <a href="src/Matrix.Internal.html#line-311" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:1"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e () ()</div></details></td></tr><tr><td class="src clearfix"> <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a> <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a> <a href="src/Matrix.Internal.html#line-307" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:2"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a> <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a></div></details></td></tr><tr><td class="src clearfix"> (<a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () a, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () b, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> a)) => <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) <a href="src/Matrix.Internal.html#line-329" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:3"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b)</div></details></td></tr><tr><td class="src clearfix"> <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () rows => <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () rows) <a href="src/Matrix.Internal.html#line-325" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:4"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () rows)</div></details></td></tr><tr><td class="src clearfix"> (<a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e a (), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e b (), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> a)) => <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) () <a href="src/Matrix.Internal.html#line-319" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:5"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ()</div></details></td></tr><tr><td class="src clearfix"> <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e cols () => <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () cols) () <a href="src/Matrix.Internal.html#line-315" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:6"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () cols) ()</div></details></td></tr><tr><td class="src clearfix"> (<a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) c, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) d, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> c)) => <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> c d) <a href="src/Matrix.Internal.html#line-335" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:7"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> c d)</div></details></td></tr></table></details></div></div><div class="top"><a id="v:fromLists" class="def">fromLists</a> :: <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) => [[e]] -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#fromLists" class="link">Source</a> <a href="#v:fromLists" class="selflink">#</a></div><div class="top"><a id="v:toLists" class="def">toLists</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> [[e]] <a href="src/Matrix.Nat.html#toLists" class="link">Source</a> <a href="#v:toLists" class="selflink">#</a></div><div class="top"><a id="v:toList" class="def">toList</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> [e] <a href="src/Matrix.Nat.html#toList" class="link">Source</a> <a href="#v:toList" class="selflink">#</a></div><div class="top"><a id="v:matrixBuilder" class="def">matrixBuilder</a> :: (<a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => ((<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>, <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> e) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#matrixBuilder" class="link">Source</a> <a href="#v:matrixBuilder" class="selflink">#</a></div><div class="top"><a id="v:row" class="def">row</a> :: <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) () => [e] -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols 1 <a href="src/Matrix.Nat.html#row" class="link">Source</a> <a href="#v:row" class="selflink">#</a></div><div class="top"><a id="v:col" class="def">col</a> :: <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) => [e] -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e 1 rows <a href="src/Matrix.Nat.html#col" class="link">Source</a> <a href="#v:col" class="selflink">#</a></div><div class="top"><a id="v:zeros" class="def">zeros</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#zeros" class="link">Source</a> <a href="#v:zeros" class="selflink">#</a></div><div class="top"><a id="v:ones" class="def">ones</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#ones" class="link">Source</a> <a href="#v:ones" class="selflink">#</a></div><div class="top"><a id="v:bang" class="def">bang</a> :: forall e cols. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols 1 <a href="src/Matrix.Nat.html#bang" class="link">Source</a> <a href="#v:bang" class="selflink">#</a></div><div class="top"><a id="v:constant" class="def">constant</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows))) => e -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#constant" class="link">Source</a> <a href="#v:constant" class="selflink">#</a></div><a href="#g:5" id="g:5"><h1>Misc</h1></a><a href="#g:6" id="g:6"><h2>Get dimensions</h2></a><div class="top"><a id="v:columns" class="def">columns</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> <a href="src/Matrix.Nat.html#columns" class="link">Source</a> <a href="#v:columns" class="selflink">#</a></div><div class="top"><a id="v:rows" class="def">rows</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> <a href="src/Matrix.Nat.html#rows" class="link">Source</a> <a href="#v:rows" class="selflink">#</a></div><a href="#g:7" id="g:7"><h2>Matrix Transposition</h2></a><div class="top"><a id="v:tr" class="def">tr</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e rows cols <a href="src/Matrix.Nat.html#tr" class="link">Source</a> <a href="#v:tr" class="selflink">#</a></div><a href="#g:8" id="g:8"><h2>Selective operator</h2></a><div class="top"><a id="v:select" class="def">select</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1)), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows2 ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1, <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols1 ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols2, <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols3) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols1 rows3 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols3 rows1 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols2 rows2 <a href="src/Matrix.Nat.html#select" class="link">Source</a> <a href="#v:select" class="selflink">#</a></div><a href="#g:9" id="g:9"><h2>McCarthy's Conditional</h2></a><div class="top"><a id="v:cond" class="def">cond</a> :: (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e () (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols), <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e) => (a -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a>) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#cond" class="link">Source</a> <a href="#v:cond" class="selflink">#</a></div><a href="#g:10" id="g:10"><h2>Matrix "abiding"</h2></a><div class="top"><a id="v:abideJS" class="def">abideJS</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#abideJS" class="link">Source</a> <a href="#v:abideJS" class="selflink">#</a></div><div class="top"><a id="v:abideSJ" class="def">abideSJ</a> :: <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#abideSJ" class="link">Source</a> <a href="#v:abideSJ" class="selflink">#</a></div><a href="#g:11" id="g:11"><h1>Biproduct approach</h1></a><a href="#g:12" id="g:12"><h2>Split</h2></a><div class="top"><a id="v:-61--61--61-" class="def">(===)</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows1 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows2 -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows3 infixl 2 <a href="src/Matrix.Nat.html#%3D%3D%3D" class="link">Source</a> <a href="#v:-61--61--61-" class="selflink">#</a></div><a href="#g:13" id="g:13"><h3>Projections</h3></a><div class="top"><a id="v:p1" class="def">p1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) m <a href="src/Matrix.Nat.html#p1" class="link">Source</a> <a href="#v:p1" class="selflink">#</a></div><div class="top"><a id="v:p2" class="def">p2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) n <a href="src/Matrix.Nat.html#p2" class="link">Source</a> <a href="#v:p2" class="selflink">#</a></div><a href="#g:14" id="g:14"><h2>Junc</h2></a><div class="top"><a id="v:-124--124--124-" class="def">(|||)</a> :: <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols3 ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols1) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols2) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols1 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols2 rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols3 rows infixl 3 <a href="src/Matrix.Nat.html#%7C%7C%7C" class="link">Source</a> <a href="#v:-124--124--124-" class="selflink">#</a></div><a href="#g:15" id="g:15"><h3>Injections</h3></a><div class="top"><a id="v:i1" class="def">i1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (rows <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e rows (rows <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> n) <a href="src/Matrix.Nat.html#i1" class="link">Source</a> <a href="#v:i1" class="selflink">#</a></div><div class="top"><a id="v:i2" class="def">i2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> rows) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e rows (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> rows) <a href="src/Matrix.Nat.html#i2" class="link">Source</a> <a href="#v:i2" class="selflink">#</a></div><a href="#g:16" id="g:16"><h2>Bifunctors</h2></a><div class="top"><a id="v:-45--124--45-" class="def">(-|-)</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (n <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> m) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (k <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> j) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> j)) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e n k -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e m j -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (n <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> m) (k <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> j) infixl 5 <a href="src/Matrix.Nat.html#-%7C-" class="link">Source</a> <a href="#v:-45--124--45-" class="selflink">#</a></div><div class="top"><a id="v:-62--60-" class="def">(><)</a> :: forall e m p n q. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> p)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> q)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n))), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q))), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> p), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> q), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> n)), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> p) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> q))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e m p -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e n q -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> n) (p <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> q) infixl 4 <a href="src/Matrix.Nat.html#%3E%3C" class="link">Source</a> <a href="#v:-62--60-" class="selflink">#</a></div><a href="#g:17" id="g:17"><h2>Applicative matrix combinators</h2></a><div class="doc">Note that given the restrictions imposed it is not possible to

20− implement the standard type classes present in standard Haskell.

21− *** Matrix pairing projections</div><div class="top"><a id="v:kp1" class="def">kp1</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k))), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) m <a href="src/Matrix.Nat.html#kp1" class="link">Source</a> <a href="#v:kp1" class="selflink">#</a></div><div class="top"><a id="v:kp2" class="def">kp2</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k))), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> m) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> k))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> k) k <a href="src/Matrix.Nat.html#kp2" class="link">Source</a> <a href="#v:kp2" class="selflink">#</a></div><a href="#g:18" id="g:18"><h3>Matrix pairing</h3></a><div class="top"><a id="v:khatri" class="def">khatri</a> :: forall e cols a b. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b))), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> a), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b)) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> b), <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b) ~ <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> a) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> b))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols a -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols b -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols (a <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> b) <a href="src/Matrix.Nat.html#khatri" class="link">Source</a> <a href="#v:khatri" class="selflink">#</a></div><a href="#g:19" id="g:19"><h1>Matrix composition and lifting</h1></a><a href="#g:20" id="g:20"><h2>Arrow matrix combinators</h2></a><div class="doc">Note that given the restrictions imposed it is not possible to

22− implement the standard type classes present in standard Haskell.</div><div class="top"><a id="v:identity" class="def">identity</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols))) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols cols <a href="src/Matrix.Nat.html#identity" class="link">Source</a> <a href="#v:identity" class="selflink">#</a></div><div class="top"><a id="v:comp" class="def">comp</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cr rows -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols cr -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#comp" class="link">Source</a> <a href="#v:comp" class="selflink">#</a></div><div class="top"><a id="v:fromF" class="def">fromF</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> rows) (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)) => (a -> b) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows <a href="src/Matrix.Nat.html#fromF" class="link">Source</a> <a href="#v:fromF" class="selflink">#</a></div><div class="top"><a id="v:fromF-39-" class="def">fromF'</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> b)), <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">FromLists</a> e (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> b) (<a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Normalize</a> a)) => (a -> b) -> <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> a) (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> b) <a href="src/Matrix.Nat.html#fromF%27" class="link">Source</a> <a href="#v:fromF-39-" class="selflink">#</a></div><a href="#g:21" id="g:21"><h1>Matrix printing</h1></a><div class="top"><a id="v:pretty" class="def">pretty</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a> <a href="src/Matrix.Nat.html#pretty" class="link">Source</a> <a href="#v:pretty" class="selflink">#</a></div><div class="top"><a id="v:prettyPrint" class="def">prettyPrint</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Count</a> (<a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">FromNat</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/System-IO.html#t:IO" title="System.IO">IO</a> () <a href="src/Matrix.Nat.html#prettyPrint" class="link">Source</a> <a href="#v:prettyPrint" class="selflink">#</a></div></div></div><div id="footer">Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.22.0</div></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/Matrix-Type.htmldeleted−36

@@ -1,36 +0,0 @@

2− Boolean matrices and in turn consider these as arrows.LAoP is a library for algebraic (inductive) construction and manipulation of matrices

3− in Haskell. See <a href="https://github.com/bolt12/master-thesis">my Msc Thesis</a> for the

4− motivation behind the library, the underlying theory, and implementation details.This module offers a newtype wrapper around <code><a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a></code> that

5− uses arbitrary types instead of canonical data types for the matrices

6− dimensions.NOTE: If the types in the dimensions are custom they must need to

7− implement a <code>Generic</code> instance.</div></div><div id="synopsis"><details id="syn"><summary>Synopsis</summary><ul class="details-toggle" data-details-id="syn"><li class="src short">newtype <a href="#t:Matrix">Matrix</a> e (cols :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) (rows :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) = <a href="#v:M">M</a> (Matrix e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))</li><li class="src short">type <a href="#t:Zero">Zero</a> = <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a></li><li class="src short">type <a href="#t:One">One</a> = ()</li><li class="src short"><a href="#v:empty">empty</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:Zero" title="Matrix.Type">Zero</a> <a href="Matrix-Type.html#t:Zero" title="Matrix.Type">Zero</a></li><li class="src short"><a href="#v:one">one</a> :: e -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a></li><li class="src short"><a href="#v:junc">junc</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e b rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) rows</li><li class="src short"><a href="#v:split">split</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b)</li><li class="src short">type family <a href="#t:FromNat">FromNat</a> (n :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ...</li><li class="src short">type family <a href="#t:Count">Count</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a> where ...</li><li class="src short">type family <a href="#t:Normalize">Normalize</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ...</li><li class="src short">class <a href="#t:FromLists">FromLists</a> e cols rows</li><li class="src short"><a href="#v:fromLists">fromLists</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows) => [[e]] -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:toLists">toLists</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> [[e]]</li><li class="src short"><a href="#v:toList">toList</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> [e]</li><li class="src short"><a href="#v:matrixBuilder">matrixBuilder</a> :: (<a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => ((<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>, <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> e) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:row">row</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) () => [e] -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a></li><li class="src short"><a href="#v:col">col</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows) => [e] -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> rows</li><li class="src short"><a href="#v:zeros">zeros</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:ones">ones</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:bang">bang</a> :: forall e cols. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a></li><li class="src short"><a href="#v:constant">constant</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => e -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:fmapM">fmapM</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => (a -> b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c b</li><li class="src short"><a href="#v:unitM">unitM</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e () ()</li><li class="src short"><a href="#v:multM">multM</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b))), <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c (a, b)</li><li class="src short"><a href="#v:selectM">selectM</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b</li><li class="src short"><a href="#v:returnM">returnM</a> :: forall e a. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a)) => a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> a</li><li class="src short"><a href="#v:bindM">bindM</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e b c -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a c</li><li class="src short"><a href="#v:columns">columns</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></li><li class="src short"><a href="#v:rows">rows</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></li><li class="src short"><a href="#v:tr">tr</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e rows cols</li><li class="src short"><a href="#v:cond">cond</a> :: (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e) => (a -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a>) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b</li><li class="src short"><a href="#v:abideJS">abideJS</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:abideSJ">abideSJ</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:-61--61--61-">(===)</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b)</li><li class="src short"><a href="#v:p1">p1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) m</li><li class="src short"><a href="#v:p2">p2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) n</li><li class="src short"><a href="#v:-124--124--124-">(|||)</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e b rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) rows</li><li class="src short"><a href="#v:i1">i1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e m (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n)</li><li class="src short"><a href="#v:i2">i2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e n (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n)</li><li class="src short"><a href="#v:-45--124--45-">(-|-)</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> n m) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> k j) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e n k -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e m j -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> n m) (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> k j)</li><li class="src short"><a href="#v:-62--60-">(><)</a> :: forall e m p n q. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> p)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> q)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n))), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> p), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> q), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> p, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> q)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e m p -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e n q -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (m, n) (p, q)</li><li class="src short"><a href="#v:kp1">kp1</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (m, k) m</li><li class="src short"><a href="#v:kp2">kp2</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (m, k) k</li><li class="src short"><a href="#v:khatri">khatri</a> :: forall e cols a b. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (a, b)</li><li class="src short"><a href="#v:identity">identity</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a a</li><li class="src short"><a href="#v:comp">comp</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cr rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols cr -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:fromF">fromF</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)) => (a -> b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows</li><li class="src short"><a href="#v:fromF-39-">fromF'</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => (a -> b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b</li><li class="src short"><a href="#v:pretty">pretty</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a></li><li class="src short"><a href="#v:prettyPrint">prettyPrint</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/System-IO.html#t:IO" title="System.IO">IO</a> ()</li></ul></details></div><div id="interface"><h1>Documentation</h1><div class="doc">LAoP (Linear Algebra of Programming) Inductive Matrix definition.LAoP generalises relations and functions treating them as

8− Boolean matrices and in turn consider these as arrows.

9− This library offers many of the combinators mentioned in the work of

10− Macedo (2012) and Oliveira (2012).This definition is a wrapper around <code><a href="Matrix.html#v:Type" title="Matrix">Type</a></code> but

11− dimensions are arbitrary data types. Type inference might not

12− be as desired.</div><a href="#g:1" id="g:1"><h1>Type safe matrix representation</h1></a><div class="top">newtype <a id="t:Matrix" class="def">Matrix</a> e (cols :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) (rows :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) <a href="src/Matrix.Type.html#Matrix" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a><div class="subs constructors">Constructors<table><tr><td class="src"><a id="v:M" class="def">M</a> (Matrix e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))</td><td class="doc empty"> </td></tr></table></div><div class="subs instances"><details id="i:Matrix" open="open"><summary>Instances</summary><table><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="/package/base-4.12.0.0/docs/Control-Category.html#t:Category" title="Control.Category">Category</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> -> <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> -> <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) <a href="src/Matrix.Type.html#line-174" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc">It isn't possible to implement the <code><a href="/package/base-4.12.0.0/docs/Data-Function.html#v:id" title="Data.Function">id</a></code> function so it's

13− implementation is <code><a href="/package/base-4.12.0.0/docs/Prelude.html#v:undefined" title="Prelude">undefined</a></code>. However <code><a href="Matrix-Type.html#v:comp" title="Matrix.Type">comp</a></code> can be and this partial

14− class implementation exists just to make the code more readable.Please use <code><a href="Matrix-Type.html#v:identity" title="Matrix.Type">identity</a></code> instead.</td></tr><tr><td colspan="2"><details id="i:id:Matrix:Category:1"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Type.html">Matrix.Type</a> <div class="subs methods">Methods<a href="#v:id">id</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a a <a href="#v:id" class="selflink">#</a><a href="#v:.">(.)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e b c -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a c <a href="#v:." class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> e => <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows) <a href="src/Matrix.Type.html#line-167" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Eq:2"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Type.html">Matrix.Type</a> <div class="subs methods">Methods<a href="#v:-61--61-">(==)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-61--61-" class="selflink">#</a><a href="#v:-47--61-">(/=)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-47--61-" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows) <a href="src/Matrix.Type.html#line-167" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Num:3"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Type.html">Matrix.Type</a> <div class="subs methods">Methods<a href="#v:-43-">(+)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:-43-" class="selflink">#</a><a href="#v:-45-">(-)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:-45-" class="selflink">#</a><a href="#v:-42-">(*)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:-42-" class="selflink">#</a><a href="#v:negate">negate</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:negate" class="selflink">#</a><a href="#v:abs">abs</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:abs" class="selflink">#</a><a href="#v:signum">signum</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:signum" class="selflink">#</a><a href="#v:fromInteger">fromInteger</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Integer" title="Prelude">Integer</a> -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:fromInteger" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e => <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows) <a href="src/Matrix.Type.html#line-167" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Ord:4"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Type.html">Matrix.Type</a> <div class="subs methods">Methods<a href="#v:compare">compare</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ordering" title="Data.Ord">Ordering</a> <a href="#v:compare" class="selflink">#</a><a href="#v:-60-">(<)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-60-" class="selflink">#</a><a href="#v:-60--61-">(<=)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-60--61-" class="selflink">#</a><a href="#v:-62-">(>)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-62-" class="selflink">#</a><a href="#v:-62--61-">(>=)</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-62--61-" class="selflink">#</a><a href="#v:max">max</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:max" class="selflink">#</a><a href="#v:min">min</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="#v:min" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e => <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows) <a href="src/Matrix.Type.html#line-167" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:Show:5"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Type.html">Matrix.Type</a> <div class="subs methods">Methods<a href="#v:showsPrec">showsPrec</a> :: <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:ShowS" title="Text.Show">ShowS</a> <a href="#v:showsPrec" class="selflink">#</a><a href="#v:show">show</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a> <a href="#v:show" class="selflink">#</a><a href="#v:showList">showList</a> :: [<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows] -> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:ShowS" title="Text.Show">ShowS</a> <a href="#v:showList" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/deepseq-1.4.4.0/docs/Control-DeepSeq.html#t:NFData" title="Control.DeepSeq">NFData</a> e => <a href="/package/deepseq-1.4.4.0/docs/Control-DeepSeq.html#t:NFData" title="Control.DeepSeq">NFData</a> (<a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows) <a href="src/Matrix.Type.html#line-167" class="link">Source</a> <a href="#t:Matrix" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Matrix:NFData:6"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Type.html">Matrix.Type</a> <div class="subs methods">Methods<a href="#v:rnf">rnf</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> () <a href="#v:rnf" class="selflink">#</a></div></details></td></tr></table></details></div></div><a href="#g:2" id="g:2"><h1>Type aliases</h1></a><div class="top">type <a id="t:Zero" class="def">Zero</a> = <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a> <a href="src/Matrix.Type.html#Zero" class="link">Source</a> <a href="#t:Zero" class="selflink">#</a><div class="doc">Zero type alias</div></div><div class="top">type <a id="t:One" class="def">One</a> = () <a href="src/Matrix.Type.html#One" class="link">Source</a> <a href="#t:One" class="selflink">#</a><div class="doc">One type alias</div></div><a href="#g:3" id="g:3"><h1>Primitives</h1></a><div class="top"><a id="v:empty" class="def">empty</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:Zero" title="Matrix.Type">Zero</a> <a href="Matrix-Type.html#t:Zero" title="Matrix.Type">Zero</a> <a href="src/Matrix.Type.html#empty" class="link">Source</a> <a href="#v:empty" class="selflink">#</a><div class="doc">Empty matrix constructor</div></div><div class="top"><a id="v:one" class="def">one</a> :: e -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> <a href="src/Matrix.Type.html#one" class="link">Source</a> <a href="#v:one" class="selflink">#</a><div class="doc">Unit matrix constructor</div></div><div class="top"><a id="v:junc" class="def">junc</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e b rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) rows <a href="src/Matrix.Type.html#junc" class="link">Source</a> <a href="#v:junc" class="selflink">#</a><div class="doc">Matrix <code>Junc</code> constructor</div></div><div class="top"><a id="v:split" class="def">split</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) <a href="src/Matrix.Type.html#split" class="link">Source</a> <a href="#v:split" class="selflink">#</a><div class="doc">Matrix <code>Split</code> constructor</div></div><a href="#g:4" id="g:4"><h1>Auxiliary type families</h1></a><div class="top">type family <a id="t:FromNat" class="def">FromNat</a> (n :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ... <a href="src/Matrix.Internal.html#FromNat" class="link">Source</a> <a href="#t:FromNat" class="selflink">#</a><div class="doc">Type family that computes of a given type dimension from a given naturalThanks to Li-Yao Xia this type family is super fast.</div><div class="subs equations">Equations<table><tr><td class="src"><a href="Matrix-Type.html#t:FromNat" title="Matrix.Type">FromNat</a> 0 = <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:FromNat" title="Matrix.Type">FromNat</a> 1 = ()</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:FromNat" title="Matrix.Type">FromNat</a> n = FromNat' (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Mod" title="GHC.TypeNats">Mod</a> n 2 <a href="/package/base-4.12.0.0/docs/Data-Type-Equality.html#t:-61--61-" title="Data.Type.Equality">==</a> 0) (<a href="Matrix-Type.html#t:FromNat" title="Matrix.Type">FromNat</a> (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Div" title="GHC.TypeNats">Div</a> n 2))</td><td class="doc empty"> </td></tr></table></div></div><div class="top">type family <a id="t:Count" class="def">Count</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a> where ... <a href="src/Matrix.Internal.html#Count" class="link">Source</a> <a href="#t:Count" class="selflink">#</a><div class="doc">Type family that computes the cardinality of a given type dimension.It can also count the cardinality of custom types that implement the

15− <code><a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:Generic" title="GHC.Generics">Generic</a></code> instance.</div><div class="subs equations">Equations<table><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m) = (m <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-45-" title="GHC.TypeNats">-</a> n) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> 1</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) = <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">(+)</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a) (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> b)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (a, b) = <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a) (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> b)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (a -> b) = <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-94-" title="GHC.TypeNats">(^)</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> b) (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:M1" title="GHC.Generics">M1</a> _ _ f p) = <a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (f p)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:K1" title="GHC.Generics">K1</a> _ _ _) = 1</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:V1" title="GHC.Generics">V1</a> _) = 0</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:U1" title="GHC.Generics">U1</a> _) = 1</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t::-42-:" title="GHC.Generics">(:*:)</a> a b p) = <a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (a p) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-" title="GHC.TypeNats">*</a> <a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (b p)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t::-43-:" title="GHC.Generics">(:+:)</a> a b p) = <a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (a p) <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-" title="GHC.TypeNats">+</a> <a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (b p)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> d = <a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:Rep" title="GHC.Generics">Rep</a> d <a href="/package/base-4.12.0.0/docs/GHC-Generics.html#t:R" title="GHC.Generics">R</a>)</td><td class="doc empty"> </td></tr></table></div></div><div class="top">type family <a id="t:Normalize" class="def">Normalize</a> (d :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a>) :: <a href="/package/base-4.12.0.0/docs/Data-Kind.html#t:Type" title="Data.Kind">Type</a> where ... <a href="src/Matrix.Internal.html#Normalize" class="link">Source</a> <a href="#t:Normalize" class="selflink">#</a><div class="doc">Type family that normalizes the representation of a given data

16− structure</div><div class="subs equations">Equations<table><tr><td class="src"><a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> d = <a href="Matrix-Type.html#t:FromNat" title="Matrix.Type">FromNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> d)</td><td class="doc empty"> </td></tr></table></div></div><a href="#g:5" id="g:5"><h1>Matrix construction and conversion</h1></a><div class="top">class <a id="t:FromLists" class="def">FromLists</a> e cols rows <a href="src/Matrix.Internal.html#FromLists" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a><div class="doc">Type class for defining the <code>fromList</code> conversion function.Given that it is not possible to branch on types at the term level type

17− classes are needed bery much like an inductive definition but on types.</div><div class="subs minimal">Minimal complete definitionfromLists</div><div class="subs instances"><details id="i:FromLists" open="open"><summary>Instances</summary><table><tr><td class="src clearfix"> <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () () <a href="src/Matrix.Internal.html#line-311" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:1"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e () ()</div></details></td></tr><tr><td class="src clearfix"> <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a> <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a> <a href="src/Matrix.Internal.html#line-307" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:2"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a> <a href="/package/base-4.12.0.0/docs/Data-Void.html#t:Void" title="Data.Void">Void</a></div></details></td></tr><tr><td class="src clearfix"> (<a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () a, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () b, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a)) => <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) <a href="src/Matrix.Internal.html#line-329" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:3"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b)</div></details></td></tr><tr><td class="src clearfix"> <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () rows => <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () rows) <a href="src/Matrix.Internal.html#line-325" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:4"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e () (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () rows)</div></details></td></tr><tr><td class="src clearfix"> (<a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e a (), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e b (), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a)) => <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) () <a href="src/Matrix.Internal.html#line-319" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:5"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ()</div></details></td></tr><tr><td class="src clearfix"> <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e cols () => <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () cols) () <a href="src/Matrix.Internal.html#line-315" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:6"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> () cols) ()</div></details></td></tr><tr><td class="src clearfix"> (<a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) c, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) d, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> c)) => <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> c d) <a href="src/Matrix.Internal.html#line-335" class="link">Source</a> <a href="#t:FromLists" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:ic:FromLists:FromLists:7"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Matrix-Internal.html">Matrix.Internal</a> <div class="subs methods">Methods<a href="#v:fromLists">fromLists</a> :: [[e]] -> Matrix e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> c d)</div></details></td></tr></table></details></div></div><div class="top"><a id="v:fromLists" class="def">fromLists</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows) => [[e]] -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#fromLists" class="link">Source</a> <a href="#v:fromLists" class="selflink">#</a><div class="doc">Build a matrix out of a list of list of elements. Throws a runtime

18− error if the dimensions do not match.</div></div><div class="top"><a id="v:toLists" class="def">toLists</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> [[e]] <a href="src/Matrix.Type.html#toLists" class="link">Source</a> <a href="#v:toLists" class="selflink">#</a><div class="doc">Converts a matrix to a list of lists of elements.</div></div><div class="top"><a id="v:toList" class="def">toList</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> [e] <a href="src/Matrix.Type.html#toList" class="link">Source</a> <a href="#v:toList" class="selflink">#</a><div class="doc">Converts a matrix to a list of elements.</div></div><div class="top"><a id="v:matrixBuilder" class="def">matrixBuilder</a> :: (<a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => ((<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>, <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> e) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#matrixBuilder" class="link">Source</a> <a href="#v:matrixBuilder" class="selflink">#</a><div class="doc">Matrix builder function. Constructs a matrix provided with

19− a construction function.</div></div><div class="top"><a id="v:row" class="def">row</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) () => [e] -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> <a href="src/Matrix.Type.html#row" class="link">Source</a> <a href="#v:row" class="selflink">#</a><div class="doc">Constructs a row vector matrix</div></div><div class="top"><a id="v:col" class="def">col</a> :: <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows) => [e] -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> rows <a href="src/Matrix.Type.html#col" class="link">Source</a> <a href="#v:col" class="selflink">#</a><div class="doc">Constructs a column vector matrix</div></div><div class="top"><a id="v:zeros" class="def">zeros</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#zeros" class="link">Source</a> <a href="#v:zeros" class="selflink">#</a><div class="doc">The zero matrix. A matrix wholly filled with zeros.</div></div><div class="top"><a id="v:ones" class="def">ones</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#ones" class="link">Source</a> <a href="#v:ones" class="selflink">#</a><div class="doc">The ones matrix. A matrix wholly filled with ones.Also known as T (Top) matrix.</div></div><div class="top"><a id="v:bang" class="def">bang</a> :: forall e cols. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> <a href="src/Matrix.Type.html#bang" class="link">Source</a> <a href="#v:bang" class="selflink">#</a><div class="doc">The T (Top) row vector matrix.</div></div><div class="top"><a id="v:constant" class="def">constant</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows))) => e -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#constant" class="link">Source</a> <a href="#v:constant" class="selflink">#</a><div class="doc">The constant matrix constructor. A matrix wholly filled with a given

20− value.</div></div><a href="#g:6" id="g:6"><h1>Functor instance equivalent function</h1></a><div class="top"><a id="v:fmapM" class="def">fmapM</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => (a -> b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c b <a href="src/Matrix.Type.html#fmapM" class="link">Source</a> <a href="#v:fmapM" class="selflink">#</a><div class="doc">Functor instance equivalent function</div></div><a href="#g:7" id="g:7"><h1>Applicative/Monoidal instance equivalent functions</h1></a><div class="top"><a id="v:unitM" class="def">unitM</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e () () <a href="src/Matrix.Type.html#unitM" class="link">Source</a> <a href="#v:unitM" class="selflink">#</a><div class="doc">Applicative instance equivalent <code>unit</code> function,</div></div><div class="top"><a id="v:multM" class="def">multM</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b))), <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e c (a, b) <a href="src/Matrix.Type.html#multM" class="link">Source</a> <a href="#v:multM" class="selflink">#</a><div class="doc">Applicative instance equivalent <code>unit</code> function,</div></div><a href="#g:8" id="g:8"><h1>Selective equivalent instance function</h1></a><div class="top"><a id="v:selectM" class="def">selectM</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b <a href="src/Matrix.Type.html#selectM" class="link">Source</a> <a href="#v:selectM" class="selflink">#</a><div class="doc">Selective functors <code>select</code> operator equivalent inspired by the

21− ArrowMonad solution presented in the paper.</div></div><a href="#g:9" id="g:9"><h1>Monad equivalent instance function</h1></a><div class="top"><a id="v:returnM" class="def">returnM</a> :: forall e a. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> e, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> a)) => a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e <a href="Matrix-Type.html#t:One" title="Matrix.Type">One</a> a <a href="src/Matrix.Type.html#returnM" class="link">Source</a> <a href="#v:returnM" class="selflink">#</a><div class="doc">Monad instance equivalent <code><a href="/package/base-4.12.0.0/docs/Control-Monad.html#v:return" title="Control.Monad">return</a></code> function,</div></div><div class="top"><a id="v:bindM" class="def">bindM</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e b c -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a c <a href="src/Matrix.Type.html#bindM" class="link">Source</a> <a href="#v:bindM" class="selflink">#</a><div class="doc">Monad instance equivalent '(>>=)' function,</div></div><a href="#g:10" id="g:10"><h1>Misc</h1></a><a href="#g:11" id="g:11"><h2>Get dimensions</h2></a><div class="top"><a id="v:columns" class="def">columns</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> <a href="src/Matrix.Type.html#columns" class="link">Source</a> <a href="#v:columns" class="selflink">#</a><div class="doc">Obtain the number of columns.NOTE: The <code><a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a></code> constaint is needed in order to obtain the

22− dimensions in constant time.TODO: A <code><a href="Matrix-Type.html#v:columns" title="Matrix.Type">columns</a></code> function that does not need the <code><a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a></code> constraint in

23− exchange for performance.</div></div><div class="top"><a id="v:rows" class="def">rows</a> :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> <a href="src/Matrix.Type.html#rows" class="link">Source</a> <a href="#v:rows" class="selflink">#</a><div class="doc">Obtain the number of rows.NOTE: The <code><a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a></code> constaint is needed in order to obtain the

24− dimensions in constant time.TODO: A <code><a href="Matrix-Type.html#v:rows" title="Matrix.Type">rows</a></code> function that does not need the <code><a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a></code> constraint in

25− exchange for performance.</div></div><a href="#g:12" id="g:12"><h2>Matrix Transposition</h2></a><div class="top"><a id="v:tr" class="def">tr</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e rows cols <a href="src/Matrix.Type.html#tr" class="link">Source</a> <a href="#v:tr" class="selflink">#</a><div class="doc">Matrix transposition.</div></div><a href="#g:13" id="g:13"><h2>McCarthy's Conditional</h2></a><div class="top"><a id="v:cond" class="def">cond</a> :: (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e () (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e) => (a -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a>) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b <a href="src/Matrix.Type.html#cond" class="link">Source</a> <a href="#v:cond" class="selflink">#</a><div class="doc">McCarthy's Conditional expresses probabilistic choice.</div></div><a href="#g:14" id="g:14"><h2>Matrix "abiding"</h2></a><div class="top"><a id="v:abideJS" class="def">abideJS</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#abideJS" class="link">Source</a> <a href="#v:abideJS" class="selflink">#</a><div class="doc">Matrix "abiding" followin the 'Junc'-'Split' abide law.Law:<pre><code>Junc</code> (<code>Split</code> a c) (<code>Split</code> b d) == <code>Split</code> (<code>Junc</code> a b) (<code>Junc</code> c d)

26−</pre></div></div><div class="top"><a id="v:abideSJ" class="def">abideSJ</a> :: <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#abideSJ" class="link">Source</a> <a href="#v:abideSJ" class="selflink">#</a><div class="doc">Matrix "abiding" followin the 'Split'-'Junc' abide law.Law:<pre><code>Split</code> (<code>Junc</code> a b) (<code>Junc</code> c d) == <code>Junc</code> (<code>Split</code> a c) (<code>Split</code> b d)

27−</pre></div></div><a href="#g:15" id="g:15"><h1>Biproduct approach</h1></a><a href="#g:16" id="g:16"><h2>Split</h2></a><div class="top"><a id="v:-61--61--61-" class="def">(===)</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) infixl 2 <a href="src/Matrix.Type.html#%3D%3D%3D" class="link">Source</a> <a href="#v:-61--61--61-" class="selflink">#</a><div class="doc">Matrix <code>Split</code> constructor</div></div><a href="#g:17" id="g:17"><h3>Projections</h3></a><div class="top"><a id="v:p1" class="def">p1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) m <a href="src/Matrix.Type.html#p1" class="link">Source</a> <a href="#v:p1" class="selflink">#</a><div class="doc">Biproduct first component projection</div></div><div class="top"><a id="v:p2" class="def">p2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) n <a href="src/Matrix.Type.html#p2" class="link">Source</a> <a href="#v:p2" class="selflink">#</a><div class="doc">Biproduct second component projection</div></div><a href="#g:18" id="g:18"><h2>Junc</h2></a><div class="top"><a id="v:-124--124--124-" class="def">(|||)</a> :: <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e b rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> a b) rows infixl 3 <a href="src/Matrix.Type.html#%7C%7C%7C" class="link">Source</a> <a href="#v:-124--124--124-" class="selflink">#</a><div class="doc">Matrix <code>Junc</code> constructor</div></div><a href="#g:19" id="g:19"><h3>Injections</h3></a><div class="top"><a id="v:i1" class="def">i1</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e m (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) <a href="src/Matrix.Type.html#i1" class="link">Source</a> <a href="#v:i1" class="selflink">#</a><div class="doc">Biproduct first component injection</div></div><div class="top"><a id="v:i2" class="def">i2</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e n (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> m n) <a href="src/Matrix.Type.html#i2" class="link">Source</a> <a href="#v:i2" class="selflink">#</a><div class="doc">Biproduct second component injection</div></div><a href="#g:20" id="g:20"><h2>Bifunctors</h2></a><div class="top"><a id="v:-45--124--45-" class="def">(-|-)</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> n m) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> k j) ~ <a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> j)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e n k -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e m j -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> n m) (<a href="/package/base-4.12.0.0/docs/Data-Either.html#t:Either" title="Data.Either">Either</a> k j) infixl 5 <a href="src/Matrix.Type.html#-%7C-" class="link">Source</a> <a href="#v:-45--124--45-" class="selflink">#</a><div class="doc">Matrix coproduct functor also known as matrix direct sum.</div></div><div class="top"><a id="v:-62--60-" class="def">(><)</a> :: forall e m p n q. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> p)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> q)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n))), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> p), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> q), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, n) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> n), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (p, q) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> p, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> q)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e m p -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e n q -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (m, n) (p, q) infixl 4 <a href="src/Matrix.Type.html#%3E%3C" class="link">Source</a> <a href="#v:-62--60-" class="selflink">#</a><div class="doc">Matrix product functor also known as kronecker product</div></div><a href="#g:21" id="g:21"><h2>Applicative matrix combinators</h2></a><div class="doc">Note that given the restrictions imposed it is not possible to

28− implement the standard type classes present in standard Haskell.

29− *** Matrix pairing projections</div><div class="top"><a id="v:kp1" class="def">kp1</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (m, k) m <a href="src/Matrix.Type.html#kp1" class="link">Source</a> <a href="#v:kp1" class="selflink">#</a><div class="doc">Khatri Rao product first component projection matrix.</div></div><div class="top"><a id="v:kp2" class="def">kp2</a> :: forall e m k. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (m, k) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> m, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> k)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e (m, k) k <a href="src/Matrix.Type.html#kp2" class="link">Source</a> <a href="#v:kp2" class="selflink">#</a><div class="doc">Khatri Rao product second component projection matrix.</div></div><a href="#g:22" id="g:22"><h3>Matrix pairing</h3></a><div class="top"><a id="v:khatri" class="def">khatri</a> :: forall e cols a b. (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b))), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b)) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b), <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (a, b) ~ <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a, <a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols a -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols b -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols (a, b) <a href="src/Matrix.Type.html#khatri" class="link">Source</a> <a href="#v:khatri" class="selflink">#</a><div class="doc">Khatri Rao Matrix product also known as matrix pairing.NOTE: That this is not a true categorical product, see for instance:<pre> | kp1 <code><a href="Matrix-Type.html#v:comp" title="Matrix.Type">comp</a></code> khatri a b == a

30−khatri a b ==> |

31− | kp2 <code><a href="Matrix-Type.html#v:comp" title="Matrix.Type">comp</a></code> khatri a b == b

32−</pre>Emphasis on the implication symbol.</div></div><a href="#g:23" id="g:23"><h1>Matrix composition and lifting</h1></a><a href="#g:24" id="g:24"><h2>Arrow matrix combinators</h2></a><div class="doc">Note that given the restrictions imposed it is not possible to

33− implement the standard type classes present in standard Haskell.</div><div class="top"><a id="v:identity" class="def">identity</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a))) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a a <a href="src/Matrix.Type.html#identity" class="link">Source</a> <a href="#v:identity" class="selflink">#</a><div class="doc">Identity matrix</div></div><div class="top"><a id="v:comp" class="def">comp</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cr rows -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols cr -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#comp" class="link">Source</a> <a href="#v:comp" class="selflink">#</a><div class="doc">Matrix composition. Equivalent to matrix-matrix multiplication.This definition takes advantage of divide-and-conquer and fusion laws

34− from LAoP.</div></div><div class="top"><a id="v:fromF" class="def">fromF</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> rows) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)) => (a -> b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows <a href="src/Matrix.Type.html#fromF" class="link">Source</a> <a href="#v:fromF" class="selflink">#</a><div class="doc">Lifts functions to matrices with arbitrary dimensions.NOTE: Be careful to not ask for a matrix bigger than the cardinality of

35− types <code>a</code> or <code>b</code> allows.</div></div><div class="top"><a id="v:fromF-39-" class="def">fromF'</a> :: (<a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> a, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> b, <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> b, <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> e, <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> e, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)), <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b)), <a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">FromLists</a> e (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> b) (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> a)) => (a -> b) -> <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e a b <a href="src/Matrix.Type.html#fromF%27" class="link">Source</a> <a href="#v:fromF-39-" class="selflink">#</a><div class="doc">Lifts functions to matrices with dimensions matching <code>a</code> and <code>b</code>

36− cardinality's.</div></div><a href="#g:25" id="g:25"><h1>Matrix printing</h1></a><div class="top"><a id="v:pretty" class="def">pretty</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a> <a href="src/Matrix.Type.html#pretty" class="link">Source</a> <a href="#v:pretty" class="selflink">#</a><div class="doc">Matrix pretty printer</div></div><div class="top"><a id="v:prettyPrint" class="def">prettyPrint</a> :: (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> (<a href="Matrix-Type.html#t:Count" title="Matrix.Type">Count</a> (<a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Normalize</a> cols)), <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> e) => <a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix</a> e cols rows -> <a href="/package/base-4.12.0.0/docs/System-IO.html#t:IO" title="System.IO">IO</a> () <a href="src/Matrix.Type.html#prettyPrint" class="link">Source</a> <a href="#v:prettyPrint" class="selflink">#</a><div class="doc">Matrix pretty printer</div></div></div></div><div id="footer">Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.22.0</div></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/Utils.htmldeleted−52

@@ -1,52 +0,0 @@

2− in Haskell. See <a href="https://github.com/bolt12/master-thesis">my Msc Thesis</a> for the

3− motivation behind the library, the underlying theory, and implementation details.This module provides the <code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code> data type.

4− The semantic associated with this data type is that

5− it's meant to be a restricted <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> value.</div></div><div id="synopsis"><details id="syn"><summary>Synopsis</summary><ul class="details-toggle" data-details-id="syn"><li class="src short">data <a href="#t:Natural">Natural</a> (start :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) (end :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>)</li><li class="src short"><a href="#v:nat">nat</a> :: forall n m. (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> n, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> m) => <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m</li><li class="src short"><a href="#v:coerceNat">coerceNat</a> :: (<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> a a' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> b b' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> c c'</li><li class="src short"><a href="#v:coerceNat2">coerceNat2</a> :: ((<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>, <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> a a', <a href="Utils.html#t:Natural" title="Utils">Natural</a> b b') -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> c c' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> d d'</li><li class="src short"><a href="#v:coerceNat3">coerceNat3</a> :: (<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> a) -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> b b' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> c c' -> a</li></ul></details></div><div id="interface"><h1>Documentation</h1><div class="doc">Utility module that provides the <code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code> data type.

6− The semantic associated with this data type is that

7− it's meant to be a restricted <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> value. For example

8− the type <code>Natural 1 6</code> can only be instanciated with <code>nat n</code>

9− where <code>1 <= n <= 6</code>. Why, You might ask, because with normal

10− <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code>s it is not possible to have a decent <code>Enum (Int, Int)</code>

11− instance. See the following probabilistic programming model as and

12− example:We want to calculate the probability of the sum of two dice throws.

13− To do this we start by defining the sample space: <pre>type SampleSpace = Int -- We think <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> are enough

14−

15−die :: Dist Int 6

16−die = unifrom [1..6]

17−

18−-- Promote <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> addition to a matrix

19−addM = fromF (uncurry (+)) -- Impossible

20−</pre>The last line is impossible because <code>(Int, Int)</code> does not have

21− a good <code><a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a></code> instance: <code>[(0, 1), (0, 2), .. (0, maxBound), (1, 0),

22− ..]</code>. And we'd like the addition matrix to be of 36 columns by 12

23− rows but limited to integers up to <code>6</code>!One way to solve this issue is by defining and auxilary data type to

24− represent the sample space:<pre>data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6

25− deriving (Show, Eq, Enum, Bounded) -- Enum and Bounded are

26− important

27−</pre>And write the sample space addition function:<pre>ssAdd :: SampleSpace -> SampleSpace -> Int

28−ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)

29−</pre>And then promote that function to matrix and everything is alright:<pre>ssAddM = fromF' (uncurry ssAdd)

30−

31−dieSumProb = ssAddM <code>comp</code> (khatri die die)

32−</pre>This is a nice solution for small sample spaces. But for larger ones

33− it is not feasible to write a data type with hundreds of constructors

34− and then write manipulation functions that need to deal with them.

35− To mitigate this limitation the <code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code> type comes a long way and

36− allows one to model the sample in an easier way. See for instance:<pre>ssAdd :: Natural 1 6 -> Natural 1 6 -> Natural 1 12

37−ssAdd = coerceNat (+)

38−

39−ssAddM = fromF' (uncurry sumSS)

40−

41−die :: Dist (Natural 1 6) 6

42−die = uniform [nat <code>1 </code>6 1 .. nat 6]

43−

44−dieSumProb = ssAddM <code>comp</code> (khatri die die)

45−</pre><ul><li><code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code> data type</li></ul></div><div class="top">data <a id="t:Natural" class="def">Natural</a> (start :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) (end :: <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat" title="GHC.TypeNats">Nat</a>) <a href="src/Utils.Internal.html#Natural" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a><div class="doc">Wrapper around <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code>s that have a restrictive semantic associated.

46− A value of type <code><code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code> n m</code> can only be instanciated with some <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code>

47− <code>i</code> that's <code>n <= i <= m</code>.</div><div class="subs instances"><details id="i:Natural" open="open"><summary>Instances</summary><table><tr><td class="src clearfix"> (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> n, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> m) => <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Bounded" title="Prelude">Bounded</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m) <a href="src/Utils.Internal.html#line-138" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Natural:Bounded:1"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:minBound">minBound</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:minBound" class="selflink">#</a><a href="#v:maxBound">maxBound</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:maxBound" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> n, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> m) => <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Enum" title="Prelude">Enum</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m) <a href="src/Utils.Internal.html#line-142" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Natural:Enum:2"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:succ">succ</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:succ" class="selflink">#</a><a href="#v:pred">pred</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:pred" class="selflink">#</a><a href="#v:toEnum">toEnum</a> :: <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:toEnum" class="selflink">#</a><a href="#v:fromEnum">fromEnum</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> <a href="#v:fromEnum" class="selflink">#</a><a href="#v:enumFrom">enumFrom</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> [<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m] <a href="#v:enumFrom" class="selflink">#</a><a href="#v:enumFromThen">enumFromThen</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> [<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m] <a href="#v:enumFromThen" class="selflink">#</a><a href="#v:enumFromTo">enumFromTo</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> [<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m] <a href="#v:enumFromTo" class="selflink">#</a><a href="#v:enumFromThenTo">enumFromThenTo</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> [<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m] <a href="#v:enumFromThenTo" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq" title="Data.Eq">Eq</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end) <a href="src/Utils.Internal.html#line-101" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Natural:Eq:3"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:-61--61-">(==)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-61--61-" class="selflink">#</a><a href="#v:-47--61-">(/=)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-47--61-" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> n, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> m) => <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Num" title="Prelude">Num</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> n m) <a href="src/Utils.Internal.html#line-105" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc">Throws a runtime error if any of the operations overflows or

48− underflows.</td></tr><tr><td colspan="2"><details id="i:id:Natural:Num:4"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:-43-">(+)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:-43-" class="selflink">#</a><a href="#v:-45-">(-)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:-45-" class="selflink">#</a><a href="#v:-42-">(*)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:-42-" class="selflink">#</a><a href="#v:negate">negate</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:negate" class="selflink">#</a><a href="#v:abs">abs</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:abs" class="selflink">#</a><a href="#v:signum">signum</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:signum" class="selflink">#</a><a href="#v:fromInteger">fromInteger</a> :: <a href="/package/base-4.12.0.0/docs/Prelude.html#t:Integer" title="Prelude">Integer</a> -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="#v:fromInteger" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord" title="Data.Ord">Ord</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end) <a href="src/Utils.Internal.html#line-101" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Natural:Ord:5"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:compare">compare</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-Ord.html#t:Ordering" title="Data.Ord">Ordering</a> <a href="#v:compare" class="selflink">#</a><a href="#v:-60-">(<)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-60-" class="selflink">#</a><a href="#v:-60--61-">(<=)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-60--61-" class="selflink">#</a><a href="#v:-62-">(>)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-62-" class="selflink">#</a><a href="#v:-62--61-">(>=)</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool" title="Data.Bool">Bool</a> <a href="#v:-62--61-" class="selflink">#</a><a href="#v:max">max</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end <a href="#v:max" class="selflink">#</a><a href="#v:min">min</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end <a href="#v:min" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Text-Read.html#t:Read" title="Text.Read">Read</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end) <a href="src/Utils.Internal.html#line-101" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Natural:Read:6"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:readsPrec">readsPrec</a> :: <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Text-ParserCombinators-ReadP.html#t:ReadS" title="Text.ParserCombinators.ReadP">ReadS</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end) <a href="#v:readsPrec" class="selflink">#</a><a href="#v:readList">readList</a> :: <a href="/package/base-4.12.0.0/docs/Text-ParserCombinators-ReadP.html#t:ReadS" title="Text.ParserCombinators.ReadP">ReadS</a> [<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end] <a href="#v:readList" class="selflink">#</a><a href="#v:readPrec">readPrec</a> :: <a href="/package/base-4.12.0.0/docs/Text-ParserCombinators-ReadPrec.html#t:ReadPrec" title="Text.ParserCombinators.ReadPrec">ReadPrec</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end) <a href="#v:readPrec" class="selflink">#</a><a href="#v:readListPrec">readListPrec</a> :: <a href="/package/base-4.12.0.0/docs/Text-ParserCombinators-ReadPrec.html#t:ReadPrec" title="Text.ParserCombinators.ReadPrec">ReadPrec</a> [<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end] <a href="#v:readListPrec" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:Show" title="Text.Show">Show</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end) <a href="src/Utils.Internal.html#line-101" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Natural:Show:7"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:showsPrec">showsPrec</a> :: <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:ShowS" title="Text.Show">ShowS</a> <a href="#v:showsPrec" class="selflink">#</a><a href="#v:show">show</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> <a href="/package/base-4.12.0.0/docs/Data-String.html#t:String" title="Data.String">String</a> <a href="#v:show" class="selflink">#</a><a href="#v:showList">showList</a> :: [<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end] -> <a href="/package/base-4.12.0.0/docs/Text-Show.html#t:ShowS" title="Text.Show">ShowS</a> <a href="#v:showList" class="selflink">#</a></div></details></td></tr><tr><td class="src clearfix"> <a href="/package/deepseq-1.4.4.0/docs/Control-DeepSeq.html#t:NFData" title="Control.DeepSeq">NFData</a> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> start end) <a href="src/Utils.Internal.html#line-101" class="link">Source</a> <a href="#t:Natural" class="selflink">#</a></td><td class="doc empty"> </td></tr><tr><td colspan="2"><details id="i:id:Natural:NFData:8"><summary class="hide-when-js-enabled">Instance details</summary>Defined in <a href="Utils-Internal.html">Utils.Internal</a> <div class="subs methods">Methods<a href="#v:rnf">rnf</a> :: <a href="Utils.html#t:Natural" title="Utils">Natural</a> start end -> () <a href="#v:rnf" class="selflink">#</a></div></details></td></tr></table></details></div></div><div class="top"><a id="v:nat" class="def">nat</a> :: forall n m. (<a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> n, <a href="/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat" title="GHC.TypeNats">KnownNat</a> m) => <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> n m <a href="src/Utils.Internal.html#nat" class="link">Source</a> <a href="#v:nat" class="selflink">#</a><div class="doc">Natural constructor function. Throws a runtime error if the <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code>

49− value is greater than the corresponding <code>m</code> or lower than <code>n</code> in the <code><code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code> n m</code> type.</div></div><a href="#g:1" id="g:1"><h1>Coerce auxiliar functions to help promote <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> typed functions to</h1></a><div class="top"><a id="v:coerceNat" class="def">coerceNat</a> :: (<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> a a' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> b b' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> c c' <a href="src/Utils.Internal.html#coerceNat" class="link">Source</a> <a href="#v:coerceNat" class="selflink">#</a><div class="doc">Auxiliary function that promotes binary <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> functions to <code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code>

50− binary functions.</div></div><div class="top"><a id="v:coerceNat2" class="def">coerceNat2</a> :: ((<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>, <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a>) -> (<a href="Utils.html#t:Natural" title="Utils">Natural</a> a a', <a href="Utils.html#t:Natural" title="Utils">Natural</a> b b') -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> c c' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> d d' <a href="src/Utils.Internal.html#coerceNat2" class="link">Source</a> <a href="#v:coerceNat2" class="selflink">#</a><div class="doc">Auxiliary function that promotes ternary (binary) <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> functions to <code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code>

51− functions.</div></div><div class="top"><a id="v:coerceNat3" class="def">coerceNat3</a> :: (<a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> <a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a> -> a) -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> b b' -> <a href="Utils.html#t:Natural" title="Utils">Natural</a> c c' -> a <a href="src/Utils.Internal.html#coerceNat3" class="link">Source</a> <a href="#v:coerceNat3" class="selflink">#</a><div class="doc">Auxiliary function that promotes ternary (binary) <code><a href="/package/base-4.12.0.0/docs/Data-Int.html#t:Int" title="Data.Int">Int</a></code> functions to <code><a href="Utils.html#t:Natural" title="Utils">Natural</a></code>

52− functions.</div></div></div></div><div id="footer">Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.22.0</div></body></html>

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1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /><title>laop-0.1.0.0: An inductive matrix definition library à la LAoP (Index)</title><link href="ocean.css" rel="stylesheet" type="text/css" title="Ocean" /><link rel="stylesheet" type="text/css" href="quick-jump.css" /><script src="haddock-bundle.min.js" async="async" type="text/javascript"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="/package/laop-0.1.0.0">Contents</a></li><li><a href="doc-index.html">Index</a></li></ul>laop-0.1.0.0: An inductive matrix definition library à la LAoP</div><div id="content"><div id="index">Index<table><tr><td class="src">-|-</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:-45--124--45-" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:-45--124--45-" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">===</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:-61--61--61-" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:-61--61--61-" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">><</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:-62--60-" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:-62--60-" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">abideJS</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:abideJS" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:abideJS" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">abideSJ</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:abideSJ" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:abideSJ" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">bang</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:bang" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:bang" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">bindD</td><td class="module"><a href="Dist.html#v:bindD" title="Dist">Dist</a></td></tr><tr><td class="src">bindM</td><td class="module"><a href="Matrix-Type.html#v:bindM" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">choose</td><td class="module"><a href="Dist.html#v:choose" title="Dist">Dist</a></td></tr><tr><td class="src">coerceNat</td><td class="module"><a href="Utils.html#v:coerceNat" title="Utils">Utils</a></td></tr><tr><td class="src">coerceNat2</td><td class="module"><a href="Utils.html#v:coerceNat2" title="Utils">Utils</a></td></tr><tr><td class="src">coerceNat3</td><td class="module"><a href="Utils.html#v:coerceNat3" title="Utils">Utils</a></td></tr><tr><td class="src">col</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:col" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:col" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">columns</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:columns" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:columns" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">comp</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:comp" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:comp" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">cond</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:cond" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:cond" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">constant</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:constant" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:constant" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">Count</td><td class="module"><a href="Matrix-Type.html#t:Count" title="Matrix.Type">Matrix.Type</a>, <a href="Matrix-Nat.html#t:Count" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">D</td><td class="module"><a href="Dist.html#v:D" title="Dist">Dist</a></td></tr><tr><td class="src">Dist</td><td class="module"><a href="Dist.html#t:Dist" title="Dist">Dist</a></td></tr><tr><td class="src">empty</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:empty" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:empty" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">fmapD</td><td class="module"><a href="Dist.html#v:fmapD" title="Dist">Dist</a></td></tr><tr><td class="src">fmapM</td><td class="module"><a href="Matrix-Type.html#v:fmapM" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">fromF</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:fromF" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:fromF" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">fromF'</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:fromF-39-" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:fromF-39-" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">FromLists</td><td class="module"><a href="Matrix-Type.html#t:FromLists" title="Matrix.Type">Matrix.Type</a>, <a href="Matrix-Nat.html#t:FromLists" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">fromLists</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:fromLists" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:fromLists" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">FromNat</td><td class="module"><a href="Matrix-Type.html#t:FromNat" title="Matrix.Type">Matrix.Type</a>, <a href="Matrix-Nat.html#t:FromNat" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">i1</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:i1" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:i1" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">i2</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:i2" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:i2" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">identity</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:identity" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:identity" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">junc</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:junc" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:junc" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">khatri</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:khatri" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:khatri" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">kp1</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:kp1" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:kp1" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">kp2</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:kp2" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:kp2" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">linear</td><td class="module"><a href="Dist.html#v:linear" title="Dist">Dist</a></td></tr><tr><td class="src">M</td><td> </td></tr><tr><td class="alt">1 (Data Constructor)</td><td class="module"><a href="Matrix-Type.html#v:M" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Data Constructor)</td><td class="module"><a href="Matrix-Nat.html#v:M" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">Matrix</td><td> </td></tr><tr><td class="alt">1 (Type/Class)</td><td class="module"><a href="Matrix-Type.html#t:Matrix" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Type/Class)</td><td class="module"><a href="Matrix-Nat.html#t:Matrix" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">matrixBuilder</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:matrixBuilder" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:matrixBuilder" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">multD</td><td class="module"><a href="Dist.html#v:multD" title="Dist">Dist</a></td></tr><tr><td class="src">multM</td><td class="module"><a href="Matrix-Type.html#v:multM" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">nat</td><td class="module"><a href="Utils.html#v:nat" title="Utils">Utils</a></td></tr><tr><td class="src">Natural</td><td class="module"><a href="Utils.html#t:Natural" title="Utils">Utils</a></td></tr><tr><td class="src">negExp</td><td class="module"><a href="Dist.html#v:negExp" title="Dist">Dist</a></td></tr><tr><td class="src">normal</td><td class="module"><a href="Dist.html#v:normal" title="Dist">Dist</a></td></tr><tr><td class="src">Normalize</td><td class="module"><a href="Matrix-Type.html#t:Normalize" title="Matrix.Type">Matrix.Type</a>, <a href="Matrix-Nat.html#t:Normalize" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">One</td><td class="module"><a href="Matrix-Type.html#t:One" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">one</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:one" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:one" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">ones</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:ones" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:ones" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">p1</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:p1" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:p1" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">p2</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:p2" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:p2" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">pretty</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:pretty" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:pretty" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">prettyDist</td><td class="module"><a href="Dist.html#v:prettyDist" title="Dist">Dist</a></td></tr><tr><td class="src">prettyPrint</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:prettyPrint" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:prettyPrint" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">prettyPrintDist</td><td class="module"><a href="Dist.html#v:prettyPrintDist" title="Dist">Dist</a></td></tr><tr><td class="src">Prob</td><td class="module"><a href="Dist.html#t:Prob" title="Dist">Dist</a></td></tr><tr><td class="src">returnD</td><td class="module"><a href="Dist.html#v:returnD" title="Dist">Dist</a></td></tr><tr><td class="src">returnM</td><td class="module"><a href="Matrix-Type.html#v:returnM" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">row</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:row" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:row" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">rows</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:rows" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:rows" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">select</td><td class="module"><a href="Matrix-Nat.html#v:select" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">selectD</td><td class="module"><a href="Dist.html#v:selectD" title="Dist">Dist</a></td></tr><tr><td class="src">selectM</td><td class="module"><a href="Matrix-Type.html#v:selectM" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">shape</td><td class="module"><a href="Dist.html#v:shape" title="Dist">Dist</a></td></tr><tr><td class="src">split</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:split" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:split" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">toList</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:toList" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:toList" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">toLists</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:toLists" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:toLists" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">toValues</td><td class="module"><a href="Dist.html#v:toValues" title="Dist">Dist</a></td></tr><tr><td class="src">tr</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:tr" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:tr" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">uniform</td><td class="module"><a href="Dist.html#v:uniform" title="Dist">Dist</a></td></tr><tr><td class="src">unitD</td><td class="module"><a href="Dist.html#v:unitD" title="Dist">Dist</a></td></tr><tr><td class="src">unitM</td><td class="module"><a href="Matrix-Type.html#v:unitM" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">Zero</td><td class="module"><a href="Matrix-Type.html#t:Zero" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="src">zeros</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:zeros" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:zeros" title="Matrix.Nat">Matrix.Nat</a></td></tr><tr><td class="src">|||</td><td> </td></tr><tr><td class="alt">1 (Function)</td><td class="module"><a href="Matrix-Type.html#v:-124--124--124-" title="Matrix.Type">Matrix.Type</a></td></tr><tr><td class="alt">2 (Function)</td><td class="module"><a href="Matrix-Nat.html#v:-124--124--124-" title="Matrix.Nat">Matrix.Nat</a></td></tr></table></div></div><div id="footer">Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.22.0</div></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/doc-index.jsondeleted−1

@@ -1 +0,0 @@

1−[{"display_html":"data <a href=\"#t:Natural\">Natural</a> (start :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a>) (end :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a>)","name":"Natural","module":"Utils","link":"Utils.html#t:Natural"},{"display_html":"<a href=\"#v:nat\">nat</a> :: forall n m. (<a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> n, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> m) => <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a> -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> n m","name":"nat","module":"Utils","link":"Utils.html#v:nat"},{"display_html":"<a href=\"#v:coerceNat\">coerceNat</a> :: (<a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a> -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a> -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>) -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> a a' -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> b b' -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> c c'","name":"coerceNat","module":"Utils","link":"Utils.html#v:coerceNat"},{"display_html":"<a href=\"#v:coerceNat2\">coerceNat2</a> :: ((<a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>, <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>) -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a> -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>) -> (<a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> a a', <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> b b') -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> c c' -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> d d'","name":"coerceNat2","module":"Utils","link":"Utils.html#v:coerceNat2"},{"display_html":"<a href=\"#v:coerceNat3\">coerceNat3</a> :: (<a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a> -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a> -> a) -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> b b' -> <a href=\"Utils.html#t:Natural\" title=\"Utils\">Natural</a> c c' -> a","name":"coerceNat3","module":"Utils","link":"Utils.html#v:coerceNat3"},{"display_html":"newtype <a href=\"#t:Matrix\">Matrix</a> e (cols :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a>) (rows :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a>) = <a href=\"#v:M\">M</a> (Matrix e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows))","name":"Matrix M","module":"Matrix.Type","link":"Matrix-Type.html#t:Matrix"},{"display_html":"type <a href=\"#t:Zero\">Zero</a> = <a href=\"/package/base-4.12.0.0/docs/Data-Void.html#t:Void\" title=\"Data.Void\">Void</a>","name":"Zero","module":"Matrix.Type","link":"Matrix-Type.html#t:Zero"},{"display_html":"type <a href=\"#t:One\">One</a> = ()","name":"One","module":"Matrix.Type","link":"Matrix-Type.html#t:One"},{"display_html":"<a href=\"#v:empty\">empty</a> :: <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e <a href=\"Matrix-Type.html#t:Zero\" title=\"Matrix.Type\">Zero</a> <a href=\"Matrix-Type.html#t:Zero\" title=\"Matrix.Type\">Zero</a>","name":"empty","module":"Matrix.Type","link":"Matrix-Type.html#v:empty"},{"display_html":"<a href=\"#v:one\">one</a> :: e -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e <a href=\"Matrix-Type.html#t:One\" title=\"Matrix.Type\">One</a> <a href=\"Matrix-Type.html#t:One\" title=\"Matrix.Type\">One</a>","name":"one","module":"Matrix.Type","link":"Matrix-Type.html#v:one"},{"display_html":"<a href=\"#v:junc\">junc</a> :: <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e b rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) rows","name":"junc","module":"Matrix.Type","link":"Matrix-Type.html#v:junc"},{"display_html":"<a href=\"#v:split\">split</a> :: <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols a -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b)","name":"split","module":"Matrix.Type","link":"Matrix-Type.html#v:split"},{"display_html":"type family <a href=\"#t:FromNat\">FromNat</a> (n :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a>) :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a> where ...","name":"FromNat","module":"Matrix.Type","link":"Matrix-Type.html#t:FromNat"},{"display_html":"type family <a href=\"#t:Count\">Count</a> (d :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a>) :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a> where ...","name":"Count","module":"Matrix.Type","link":"Matrix-Type.html#t:Count"},{"display_html":"type family <a href=\"#t:Normalize\">Normalize</a> (d :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a>) :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a> where ...","name":"Normalize","module":"Matrix.Type","link":"Matrix-Type.html#t:Normalize"},{"display_html":"class <a href=\"#t:FromLists\">FromLists</a> e cols rows","name":"FromLists","module":"Matrix.Type","link":"Matrix-Type.html#t:FromLists"},{"display_html":"<a href=\"#v:fromLists\">fromLists</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows) => [[e]] -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"fromLists","module":"Matrix.Type","link":"Matrix-Type.html#v:fromLists"},{"display_html":"<a href=\"#v:toLists\">toLists</a> :: <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> [[e]]","name":"toLists","module":"Matrix.Type","link":"Matrix-Type.html#v:toLists"},{"display_html":"<a href=\"#v:toList\">toList</a> :: <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> [e]","name":"toList","module":"Matrix.Type","link":"Matrix-Type.html#v:toList"},{"display_html":"<a href=\"#v:matrixBuilder\">matrixBuilder</a> :: (<a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows))) => ((<a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>, <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>) -> e) -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"matrixBuilder","module":"Matrix.Type","link":"Matrix-Type.html#v:matrixBuilder"},{"display_html":"<a href=\"#v:row\">row</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) () => [e] -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols <a href=\"Matrix-Type.html#t:One\" title=\"Matrix.Type\">One</a>","name":"row","module":"Matrix.Type","link":"Matrix-Type.html#v:row"},{"display_html":"<a href=\"#v:col\">col</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows) => [e] -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e <a href=\"Matrix-Type.html#t:One\" title=\"Matrix.Type\">One</a> rows","name":"col","module":"Matrix.Type","link":"Matrix-Type.html#v:col"},{"display_html":"<a href=\"#v:zeros\">zeros</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows))) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"zeros","module":"Matrix.Type","link":"Matrix-Type.html#v:zeros"},{"display_html":"<a href=\"#v:ones\">ones</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows))) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"ones","module":"Matrix.Type","link":"Matrix-Type.html#v:ones"},{"display_html":"<a href=\"#v:bang\">bang</a> :: forall e cols. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> e, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) (), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols))) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols <a href=\"Matrix-Type.html#t:One\" title=\"Matrix.Type\">One</a>","name":"bang","module":"Matrix.Type","link":"Matrix-Type.html#v:bang"},{"display_html":"<a href=\"#v:constant\">constant</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows))) => e -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"constant","module":"Matrix.Type","link":"Matrix-Type.html#v:constant"},{"display_html":"<a href=\"#v:fmapM\">fmapM</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> b, <a href=\"/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq\" title=\"Data.Eq\">Eq</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord\" title=\"Data.Ord\">Ord</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)) => (a -> b) -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e c a -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e c b","name":"fmapM","module":"Matrix.Type","link":"Matrix-Type.html#v:fmapM"},{"display_html":"<a href=\"#v:unitM\">unitM</a> :: <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e () ()","name":"unitM","module":"Matrix.Type","link":"Matrix-Type.html#v:unitM"},{"display_html":"<a href=\"#v:multM\">multM</a> :: (<a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b))), <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a, <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e c a -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e c b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e c (a, b)","name":"multM","module":"Matrix.Type","link":"Matrix-Type.html#v:multM"},{"display_html":"<a href=\"#v:selectM\">selectM</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols b","name":"selectM","module":"Matrix.Type","link":"Matrix-Type.html#v:selectM"},{"display_html":"<a href=\"#v:returnM\">returnM</a> :: forall e a. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> e, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> a)) => a -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e <a href=\"Matrix-Type.html#t:One\" title=\"Matrix.Type\">One</a> a","name":"returnM","module":"Matrix.Type","link":"Matrix-Type.html#v:returnM"},{"display_html":"<a href=\"#v:bindM\">bindM</a> :: <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e b c -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a c","name":"bindM","module":"Matrix.Type","link":"Matrix-Type.html#v:bindM"},{"display_html":"<a href=\"#v:columns\">columns</a> :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>","name":"columns","module":"Matrix.Type","link":"Matrix-Type.html#v:columns"},{"display_html":"<a href=\"#v:rows\">rows</a> :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>","name":"rows","module":"Matrix.Type","link":"Matrix-Type.html#v:rows"},{"display_html":"<a href=\"#v:tr\">tr</a> :: <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e rows cols","name":"tr","module":"Matrix.Type","link":"Matrix-Type.html#v:tr"},{"display_html":"<a href=\"#v:cond\">cond</a> :: (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord\" title=\"Data.Ord\">Ord</a> e) => (a -> <a href=\"/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool\" title=\"Data.Bool\">Bool</a>) -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a b","name":"cond","module":"Matrix.Type","link":"Matrix-Type.html#v:cond"},{"display_html":"<a href=\"#v:abideJS\">abideJS</a> :: <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"abideJS","module":"Matrix.Type","link":"Matrix-Type.html#v:abideJS"},{"display_html":"<a href=\"#v:abideSJ\">abideSJ</a> :: <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"abideSJ","module":"Matrix.Type","link":"Matrix-Type.html#v:abideSJ"},{"display_html":"<a href=\"#v:-61--61--61-\">(===)</a> :: <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols a -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b)","name":"===","module":"Matrix.Type","link":"Matrix-Type.html#v:-61--61--61-"},{"display_html":"<a href=\"#v:p1\">p1</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n) m","name":"p1","module":"Matrix.Type","link":"Matrix-Type.html#v:p1"},{"display_html":"<a href=\"#v:p2\">p2</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n) n","name":"p2","module":"Matrix.Type","link":"Matrix-Type.html#v:p2"},{"display_html":"<a href=\"#v:-124--124--124-\">(|||)</a> :: <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e b rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) rows","name":"|||","module":"Matrix.Type","link":"Matrix-Type.html#v:-124--124--124-"},{"display_html":"<a href=\"#v:i1\">i1</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e m (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n)","name":"i1","module":"Matrix.Type","link":"Matrix-Type.html#v:i1"},{"display_html":"<a href=\"#v:i2\">i2</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e n (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> m n)","name":"i2","module":"Matrix.Type","link":"Matrix-Type.html#v:i2"},{"display_html":"<a href=\"#v:-45--124--45-\">(-|-)</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> j)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> j) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> j), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> j) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> j), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> n m) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> k j) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> j)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e n k -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e m j -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> n m) (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> k j)","name":"-|-","module":"Matrix.Type","link":"Matrix-Type.html#v:-45--124--45-"},{"display_html":"<a href=\"#v:-62--60-\">(><)</a> :: forall e m p n q. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> p)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> q)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, n))), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (p, q))), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, n)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, n)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (p, q)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> p), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (p, q)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> q), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, n) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m, <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> n), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (p, q) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> p, <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> q)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e m p -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e n q -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (m, n) (p, q)","name":"><","module":"Matrix.Type","link":"Matrix-Type.html#v:-62--60-"},{"display_html":"<a href=\"#v:kp1\">kp1</a> :: forall e m k. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, k))), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, k)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, k) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m, <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (m, k) m","name":"kp1","module":"Matrix.Type","link":"Matrix-Type.html#v:kp1"},{"display_html":"<a href=\"#v:kp2\">kp2</a> :: forall e m k. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, k))), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, k)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (m, k) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> m, <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> k)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e (m, k) k","name":"kp2","module":"Matrix.Type","link":"Matrix-Type.html#v:kp2"},{"display_html":"<a href=\"#v:khatri\">khatri</a> :: forall e cols a b. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b))), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a, <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols a -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols b -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols (a, b)","name":"khatri","module":"Matrix.Type","link":"Matrix-Type.html#v:khatri"},{"display_html":"<a href=\"#v:identity\">identity</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a))) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a a","name":"identity","module":"Matrix.Type","link":"Matrix-Type.html#v:identity"},{"display_html":"<a href=\"#v:comp\">comp</a> :: <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cr rows -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols cr -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"comp","module":"Matrix.Type","link":"Matrix-Type.html#v:comp"},{"display_html":"<a href=\"#v:fromF\">fromF</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> b, <a href=\"/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq\" title=\"Data.Eq\">Eq</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord\" title=\"Data.Ord\">Ord</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> rows) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)) => (a -> b) -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows","name":"fromF","module":"Matrix.Type","link":"Matrix-Type.html#v:fromF"},{"display_html":"<a href=\"#v:fromF-39-\">fromF'</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> b, <a href=\"/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq\" title=\"Data.Eq\">Eq</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord\" title=\"Data.Ord\">Ord</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> e (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)) => (a -> b) -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e a b","name":"fromF'","module":"Matrix.Type","link":"Matrix-Type.html#v:fromF-39-"},{"display_html":"<a href=\"#v:pretty\">pretty</a> :: (<a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)), <a href=\"/package/base-4.12.0.0/docs/Text-Show.html#t:Show\" title=\"Text.Show\">Show</a> e) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/Data-String.html#t:String\" title=\"Data.String\">String</a>","name":"pretty","module":"Matrix.Type","link":"Matrix-Type.html#v:pretty"},{"display_html":"<a href=\"#v:prettyPrint\">prettyPrint</a> :: (<a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> cols)), <a href=\"/package/base-4.12.0.0/docs/Text-Show.html#t:Show\" title=\"Text.Show\">Show</a> e) => <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/System-IO.html#t:IO\" title=\"System.IO\">IO</a> ()","name":"prettyPrint","module":"Matrix.Type","link":"Matrix-Type.html#v:prettyPrint"},{"display_html":"newtype <a href=\"#t:Matrix\">Matrix</a> e (cols :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a>) (rows :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a>) = <a href=\"#v:M\">M</a> (Matrix e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows))","name":"Matrix M","module":"Matrix.Nat","link":"Matrix-Nat.html#t:Matrix"},{"display_html":"<a href=\"#v:empty\">empty</a> :: <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e 0 0","name":"empty","module":"Matrix.Nat","link":"Matrix-Nat.html#v:empty"},{"display_html":"<a href=\"#v:one\">one</a> :: e -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e 1 1","name":"one","module":"Matrix.Nat","link":"Matrix-Nat.html#v:one"},{"display_html":"<a href=\"#v:junc\">junc</a> :: <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols3 ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols1) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols2) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols1 rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols2 rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols3 rows","name":"junc","module":"Matrix.Nat","link":"Matrix-Nat.html#v:junc"},{"display_html":"<a href=\"#v:split\">split</a> :: <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows3 ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows1) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows2) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows1 -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows2 -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows3","name":"split","module":"Matrix.Nat","link":"Matrix-Nat.html#v:split"},{"display_html":"type family <a href=\"#t:FromNat\">FromNat</a> (n :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a>) :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a> where ...","name":"FromNat","module":"Matrix.Nat","link":"Matrix-Nat.html#t:FromNat"},{"display_html":"type family <a href=\"#t:Count\">Count</a> (d :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a>) :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:Nat\" title=\"GHC.TypeNats\">Nat</a> where ...","name":"Count","module":"Matrix.Nat","link":"Matrix-Nat.html#t:Count"},{"display_html":"type family <a href=\"#t:Normalize\">Normalize</a> (d :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a>) :: <a href=\"/package/base-4.12.0.0/docs/Data-Kind.html#t:Type\" title=\"Data.Kind\">Type</a> where ...","name":"Normalize","module":"Matrix.Nat","link":"Matrix-Nat.html#t:Normalize"},{"display_html":"class <a href=\"#t:FromLists\">FromLists</a> e cols rows","name":"FromLists","module":"Matrix.Nat","link":"Matrix-Nat.html#t:FromLists"},{"display_html":"<a href=\"#v:fromLists\">fromLists</a> :: <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows) => [[e]] -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"fromLists","module":"Matrix.Nat","link":"Matrix-Nat.html#v:fromLists"},{"display_html":"<a href=\"#v:toLists\">toLists</a> :: <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> [[e]]","name":"toLists","module":"Matrix.Nat","link":"Matrix-Nat.html#v:toLists"},{"display_html":"<a href=\"#v:toList\">toList</a> :: <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> [e]","name":"toList","module":"Matrix.Nat","link":"Matrix-Nat.html#v:toList"},{"display_html":"<a href=\"#v:matrixBuilder\">matrixBuilder</a> :: (<a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows))) => ((<a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>, <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>) -> e) -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"matrixBuilder","module":"Matrix.Nat","link":"Matrix-Nat.html#v:matrixBuilder"},{"display_html":"<a href=\"#v:row\">row</a> :: <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) () => [e] -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols 1","name":"row","module":"Matrix.Nat","link":"Matrix-Nat.html#v:row"},{"display_html":"<a href=\"#v:col\">col</a> :: <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e () (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows) => [e] -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e 1 rows","name":"col","module":"Matrix.Nat","link":"Matrix-Nat.html#v:col"},{"display_html":"<a href=\"#v:zeros\">zeros</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"zeros","module":"Matrix.Nat","link":"Matrix-Nat.html#v:zeros"},{"display_html":"<a href=\"#v:ones\">ones</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"ones","module":"Matrix.Nat","link":"Matrix-Nat.html#v:ones"},{"display_html":"<a href=\"#v:bang\">bang</a> :: forall e cols. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> e, <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols 1","name":"bang","module":"Matrix.Nat","link":"Matrix-Nat.html#v:bang"},{"display_html":"<a href=\"#v:constant\">constant</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows))) => e -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"constant","module":"Matrix.Nat","link":"Matrix-Nat.html#v:constant"},{"display_html":"<a href=\"#v:columns\">columns</a> :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>","name":"columns","module":"Matrix.Nat","link":"Matrix-Nat.html#v:columns"},{"display_html":"<a href=\"#v:rows\">rows</a> :: <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/Data-Int.html#t:Int\" title=\"Data.Int\">Int</a>","name":"rows","module":"Matrix.Nat","link":"Matrix-Nat.html#v:rows"},{"display_html":"<a href=\"#v:tr\">tr</a> :: <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e rows cols","name":"tr","module":"Matrix.Nat","link":"Matrix-Nat.html#v:tr"},{"display_html":"<a href=\"#v:select\">select</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows1) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows1), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows1)), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows2 ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows1, <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols1 ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols2, <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows3 ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols3) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows1)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols1 rows3 -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols3 rows1 -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols2 rows2","name":"select","module":"Matrix.Nat","link":"Matrix-Nat.html#v:select"},{"display_html":"<a href=\"#v:cond\">cond</a> :: (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)) ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e () (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols), <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord\" title=\"Data.Ord\">Ord</a> e) => (a -> <a href=\"/package/base-4.12.0.0/docs/Data-Bool.html#t:Bool\" title=\"Data.Bool\">Bool</a>) -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"cond","module":"Matrix.Nat","link":"Matrix-Nat.html#v:cond"},{"display_html":"<a href=\"#v:abideJS\">abideJS</a> :: <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"abideJS","module":"Matrix.Nat","link":"Matrix-Nat.html#v:abideJS"},{"display_html":"<a href=\"#v:abideSJ\">abideSJ</a> :: <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"abideSJ","module":"Matrix.Nat","link":"Matrix-Nat.html#v:abideSJ"},{"display_html":"<a href=\"#v:-61--61--61-\">(===)</a> :: <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows3 ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows1) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows2) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows1 -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows2 -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows3","name":"===","module":"Matrix.Nat","link":"Matrix-Nat.html#v:-61--61--61-"},{"display_html":"<a href=\"#v:p1\">p1</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> n) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> n) m","name":"p1","module":"Matrix.Nat","link":"Matrix-Nat.html#v:p1"},{"display_html":"<a href=\"#v:p2\">p2</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> n) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> n) n","name":"p2","module":"Matrix.Nat","link":"Matrix-Nat.html#v:p2"},{"display_html":"<a href=\"#v:-124--124--124-\">(|||)</a> :: <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols3 ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols1) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols2) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols1 rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols2 rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols3 rows","name":"|||","module":"Matrix.Nat","link":"Matrix-Nat.html#v:-124--124--124-"},{"display_html":"<a href=\"#v:i1\">i1</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (rows <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> n) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e rows (rows <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> n)","name":"i1","module":"Matrix.Nat","link":"Matrix-Nat.html#v:i1"},{"display_html":"<a href=\"#v:i2\">i2</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> rows) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e rows (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> rows)","name":"i2","module":"Matrix.Nat","link":"Matrix-Nat.html#v:i2"},{"display_html":"<a href=\"#v:-45--124--45-\">(-|-)</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> j)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> j) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> j), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> j) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> j), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (n <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> m) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (k <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> j) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> j)) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e n k -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e m j -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e (n <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> m) (k <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-43-\" title=\"GHC.TypeNats\">+</a> j)","name":"-|-","module":"Matrix.Nat","link":"Matrix-Nat.html#v:-45--124--45-"},{"display_html":"<a href=\"#v:-62--60-\">(><)</a> :: forall e m p n q. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> p)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> q)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> n))), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (p <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> q))), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> n)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> n)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (p <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> q)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> p), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (p <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> q)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> q), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> n) ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> <a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> n)), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (p <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> q) ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> p) <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> <a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> q))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e m p -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e n q -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> n) (p <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> q)","name":"><","module":"Matrix.Nat","link":"Matrix-Nat.html#v:-62--60-"},{"display_html":"<a href=\"#v:kp1\">kp1</a> :: forall e m k. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k))), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k) ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> <a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k) m","name":"kp1","module":"Matrix.Nat","link":"Matrix-Nat.html#v:kp1"},{"display_html":"<a href=\"#v:kp2\">kp2</a> :: forall e m k. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k))), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k) ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> m) <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> <a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> k))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e (m <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> k) k","name":"kp2","module":"Matrix.Nat","link":"Matrix-Nat.html#v:kp2"},{"display_html":"<a href=\"#v:khatri\">khatri</a> :: forall e cols a b. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> b)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (a <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> b))), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (a <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> b)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> a), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (a <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> b)) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> b), <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (a <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> b) ~ <a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> a) <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> <a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> b))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols a -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols b -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols (a <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:-42-\" title=\"GHC.TypeNats\">*</a> b)","name":"khatri","module":"Matrix.Nat","link":"Matrix-Nat.html#v:khatri"},{"display_html":"<a href=\"#v:identity\">identity</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols))) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols cols","name":"identity","module":"Matrix.Nat","link":"Matrix-Nat.html#v:identity"},{"display_html":"<a href=\"#v:comp\">comp</a> :: <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cr rows -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols cr -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"comp","module":"Matrix.Nat","link":"Matrix-Nat.html#v:comp"},{"display_html":"<a href=\"#v:fromF\">fromF</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> b, <a href=\"/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq\" title=\"Data.Eq\">Eq</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord\" title=\"Data.Ord\">Ord</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> rows) (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)) => (a -> b) -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows","name":"fromF","module":"Matrix.Nat","link":"Matrix-Nat.html#v:fromF"},{"display_html":"<a href=\"#v:fromF-39-\">fromF'</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> b, <a href=\"/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq\" title=\"Data.Eq\">Eq</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Num\" title=\"Prelude\">Num</a> e, <a href=\"/package/base-4.12.0.0/docs/Data-Ord.html#t:Ord\" title=\"Data.Ord\">Ord</a> e, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:Normalize\" title=\"Matrix.Nat\">Normalize</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:Normalize\" title=\"Matrix.Nat\">Normalize</a> b)), <a href=\"Matrix-Nat.html#t:FromLists\" title=\"Matrix.Nat\">FromLists</a> e (<a href=\"Matrix-Nat.html#t:Normalize\" title=\"Matrix.Nat\">Normalize</a> b) (<a href=\"Matrix-Nat.html#t:Normalize\" title=\"Matrix.Nat\">Normalize</a> a)) => (a -> b) -> <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> a) (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> b)","name":"fromF'","module":"Matrix.Nat","link":"Matrix-Nat.html#v:fromF-39-"},{"display_html":"<a href=\"#v:pretty\">pretty</a> :: (<a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"/package/base-4.12.0.0/docs/Text-Show.html#t:Show\" title=\"Text.Show\">Show</a> e) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/Data-String.html#t:String\" title=\"Data.String\">String</a>","name":"pretty","module":"Matrix.Nat","link":"Matrix-Nat.html#v:pretty"},{"display_html":"<a href=\"#v:prettyPrint\">prettyPrint</a> :: (<a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Nat.html#t:Count\" title=\"Matrix.Nat\">Count</a> (<a href=\"Matrix-Nat.html#t:FromNat\" title=\"Matrix.Nat\">FromNat</a> cols)), <a href=\"/package/base-4.12.0.0/docs/Text-Show.html#t:Show\" title=\"Text.Show\">Show</a> e) => <a href=\"Matrix-Nat.html#t:Matrix\" title=\"Matrix.Nat\">Matrix</a> e cols rows -> <a href=\"/package/base-4.12.0.0/docs/System-IO.html#t:IO\" title=\"System.IO\">IO</a> ()","name":"prettyPrint","module":"Matrix.Nat","link":"Matrix-Nat.html#v:prettyPrint"},{"display_html":"newtype <a href=\"#t:Dist\">Dist</a> a = <a href=\"#v:D\">D</a> (<a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () a)","name":"Dist D","module":"Dist","link":"Dist.html#t:Dist"},{"display_html":"type <a href=\"#t:Prob\">Prob</a> = <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Double\" title=\"Prelude\">Double</a>","name":"Prob","module":"Dist","link":"Dist.html#t:Prob"},{"display_html":"<a href=\"#v:fmapD\">fmapD</a> :: (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Bounded\" title=\"Prelude\">Bounded</a> b, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> b, <a href=\"/package/base-4.12.0.0/docs/Data-Eq.html#t:Eq\" title=\"Data.Eq\">Eq</a> b, <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)) => (a -> b) -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> b","name":"fmapD","module":"Dist","link":"Dist.html#v:fmapD"},{"display_html":"<a href=\"#v:unitD\">unitD</a> :: <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> ()","name":"unitD","module":"Dist","link":"Dist.html#v:unitD"},{"display_html":"<a href=\"#v:multD\">multD</a> :: (<a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b))), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b)) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b), <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (a, b) ~ <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a, <a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b)) => <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> b -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> (a, b)","name":"multD","module":"Dist","link":"Dist.html#v:multD"},{"display_html":"<a href=\"#v:selectD\">selectD</a> :: (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) ~ <a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b), <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b) (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> b))) => <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> (<a href=\"/package/base-4.12.0.0/docs/Data-Either.html#t:Either\" title=\"Data.Either\">Either</a> a b) -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> a b -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> b","name":"selectD","module":"Dist","link":"Dist.html#v:selectD"},{"display_html":"<a href=\"#v:returnD\">returnD</a> :: forall a. (<a href=\"/package/base-4.12.0.0/docs/Prelude.html#t:Enum\" title=\"Prelude\">Enum</a> a, <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a), <a href=\"/package/base-4.12.0.0/docs/GHC-TypeNats.html#t:KnownNat\" title=\"GHC.TypeNats\">KnownNat</a> (<a href=\"Matrix-Type.html#t:Count\" title=\"Matrix.Type\">Count</a> a)) => a -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a","name":"returnD","module":"Dist","link":"Dist.html#v:returnD"},{"display_html":"<a href=\"#v:bindD\">bindD</a> :: <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a -> <a href=\"Matrix-Type.html#t:Matrix\" title=\"Matrix.Type\">Matrix</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> a b -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> b","name":"bindD","module":"Dist","link":"Dist.html#v:bindD"},{"display_html":"<a href=\"#v:choose\">choose</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) => <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a","name":"choose","module":"Dist","link":"Dist.html#v:choose"},{"display_html":"<a href=\"#v:shape\">shape</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) => (<a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> -> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a>) -> [a] -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a","name":"shape","module":"Dist","link":"Dist.html#v:shape"},{"display_html":"<a href=\"#v:linear\">linear</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) => [a] -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a","name":"linear","module":"Dist","link":"Dist.html#v:linear"},{"display_html":"<a href=\"#v:uniform\">uniform</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) => [a] -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a","name":"uniform","module":"Dist","link":"Dist.html#v:uniform"},{"display_html":"<a href=\"#v:negExp\">negExp</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) => [a] -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a","name":"negExp","module":"Dist","link":"Dist.html#v:negExp"},{"display_html":"<a href=\"#v:normal\">normal</a> :: <a href=\"Matrix-Type.html#t:FromLists\" title=\"Matrix.Type\">FromLists</a> <a href=\"Dist.html#t:Prob\" title=\"Dist\">Prob</a> () (<a href=\"Matrix-Type.html#t:Normalize\" title=\"Matrix.Type\">Normalize</a> a) => [a] -> <a href=\"Dist.html#t:Dist\" title=\"Dist\">Dist</a> a","name":"normal","module":"Dist","link":"Dist.html#v:normal"},{"display_html":"<a href=\"#v:toValues\">toValues</a> :: forall a. 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1−-- Hoogle documentation, generated by Haddock

2−-- See Hoogle, http://www.haskell.org/hoogle/

3−

4−

5−-- | An inductive matrix definition library à la LAoP

6−--

7−-- Please see the README on GitHub at

8−-- <a>https://github.com/bolt12/laop#readme</a>

9−@package laop

10−@version 0.1.0.0

11−

12−

13−-- | LAoP is a library for algebraic (inductive) construction and

14−-- manipulation of matrices in Haskell. See <a>my Msc Thesis</a> for the

15−-- motivation behind the library, the underlying theory, and

16−-- implementation details.

17−--

18−-- This module provides the <a>Natural</a> data type. The semantic

19−-- associated with this data type is that it's meant to be a restricted

20−-- <a>Int</a> value.

21−module Utils

22−

23−-- | Wrapper around <a>Int</a>s that have a restrictive semantic

24−-- associated. A value of type <tt><a>Natural</a> n m</tt> can only be

25−-- instanciated with some <a>Int</a> <tt>i</tt> that's <tt>n <= i

26−-- <= m</tt>.

27−data Natural (start :: Nat) (end :: Nat)

28−

29−-- | Natural constructor function. Throws a runtime error if the <a>Int</a>

30−-- value is greater than the corresponding <tt>m</tt> or lower than

31−-- <tt>n</tt> in the <tt><a>Natural</a> n m</tt> type.

32−nat :: forall n m. (KnownNat n, KnownNat m) => Int -> Natural n m

33−

34−-- | Auxiliary function that promotes binary <a>Int</a> functions to

35−-- <a>Natural</a> binary functions.

36−coerceNat :: (Int -> Int -> Int) -> Natural a a' -> Natural b b' -> Natural c c'

37−

38−-- | Auxiliary function that promotes ternary (binary) <a>Int</a> functions

39−-- to <a>Natural</a> functions.

40−coerceNat2 :: ((Int, Int) -> Int -> Int) -> (Natural a a', Natural b b') -> Natural c c' -> Natural d d'

41−

42−-- | Auxiliary function that promotes ternary (binary) <a>Int</a> functions

43−-- to <a>Natural</a> functions.

44−coerceNat3 :: (Int -> Int -> a) -> Natural b b' -> Natural c c' -> a

45−

46−

47−-- | The LAoP discipline generalises relations and functions treating them

48−-- as Boolean matrices and in turn consider these as arrows.

49−--

50−-- LAoP is a library for algebraic (inductive) construction and

51−-- manipulation of matrices in Haskell. See <a>my Msc Thesis</a> for the

52−-- motivation behind the library, the underlying theory, and

53−-- implementation details.

54−--

55−-- This module offers a newtype wrapper around <a>Matrix</a> that uses

56−-- arbitrary types instead of canonical data types for the matrices

57−-- dimensions.

58−--

59−-- NOTE: If the types in the dimensions are custom they must need

60−-- to implement a <tt>Generic</tt> instance.

61−module Matrix.Type

62−newtype Matrix e (cols :: Type) (rows :: Type)

63−M :: Matrix e (Normalize cols) (Normalize rows) -> Matrix e

64−

65−-- | Zero type alias

66−type Zero = Void

67−

68−-- | One type alias

69−type One = ()

70−

71−-- | Empty matrix constructor

72−empty :: Matrix e Zero Zero

73−

74−-- | Unit matrix constructor

75−one :: e -> Matrix e One One

76−

77−-- | Matrix <tt>Junc</tt> constructor

78−junc :: Normalize (Either a b) ~ Either (Normalize a) (Normalize b) => Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

79−

80−-- | Matrix <tt>Split</tt> constructor

81−split :: Normalize (Either a b) ~ Either (Normalize a) (Normalize b) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

82−

83−-- | Type family that computes of a given type dimension from a given

84−-- natural

85−--

86−-- Thanks to Li-Yao Xia this type family is super fast.

87−type family FromNat (n :: Nat) :: Type

88−

89−-- | Type family that computes the cardinality of a given type dimension.

90−--

91−-- It can also count the cardinality of custom types that implement the

92−-- <a>Generic</a> instance.

93−type family Count (d :: Type) :: Nat

94−

95−-- | Type family that normalizes the representation of a given data

96−-- structure

97−type family Normalize (d :: Type) :: Type

98−

99−-- | Type class for defining the <tt>fromList</tt> conversion function.

100−--

101−-- Given that it is not possible to branch on types at the term level

102−-- type classes are needed bery much like an inductive definition but on

103−-- types.

104−class FromLists e cols rows

105−

106−-- | Build a matrix out of a list of list of elements. Throws a runtime

107−-- error if the dimensions do not match.

108−fromLists :: FromLists e (Normalize cols) (Normalize rows) => [[e]] -> Matrix e cols rows

109−

110−-- | Converts a matrix to a list of lists of elements.

111−toLists :: Matrix e cols rows -> [[e]]

112−

113−-- | Converts a matrix to a list of elements.

114−toList :: Matrix e cols rows -> [e]

115−

116−-- | Matrix builder function. Constructs a matrix provided with a

117−-- construction function.

118−matrixBuilder :: (FromLists e (Normalize cols) (Normalize rows), KnownNat (Count (Normalize cols)), KnownNat (Count (Normalize rows))) => ((Int, Int) -> e) -> Matrix e cols rows

119−

120−-- | Constructs a row vector matrix

121−row :: FromLists e (Normalize cols) () => [e] -> Matrix e cols One

122−

123−-- | Constructs a column vector matrix

124−col :: FromLists e () (Normalize rows) => [e] -> Matrix e One rows

125−

126−-- | The zero matrix. A matrix wholly filled with zeros.

127−zeros :: (Num e, FromLists e (Normalize cols) (Normalize rows), KnownNat (Count (Normalize cols)), KnownNat (Count (Normalize rows))) => Matrix e cols rows

128−

129−-- | The ones matrix. A matrix wholly filled with ones.

130−--

131−-- Also known as T (Top) matrix.

132−ones :: (Num e, FromLists e (Normalize cols) (Normalize rows), KnownNat (Count (Normalize cols)), KnownNat (Count (Normalize rows))) => Matrix e cols rows

133−

134−-- | The T (Top) row vector matrix.

135−bang :: forall e cols. (Num e, Enum e, FromLists e (Normalize cols) (), KnownNat (Count (Normalize cols))) => Matrix e cols One

136−

137−-- | The constant matrix constructor. A matrix wholly filled with a given

138−-- value.

139−constant :: (Num e, FromLists e (Normalize cols) (Normalize rows), KnownNat (Count (Normalize cols)), KnownNat (Count (Normalize rows))) => e -> Matrix e cols rows

140−

141−-- | Functor instance equivalent function

142−fmapM :: (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e, KnownNat (Count (Normalize a)), KnownNat (Count (Normalize b)), FromLists e (Normalize b) (Normalize a)) => (a -> b) -> Matrix e c a -> Matrix e c b

143−

144−-- | Applicative instance equivalent <tt>unit</tt> function,

145−unitM :: Num e => Matrix e () ()

146−

147−-- | Applicative instance equivalent <tt>unit</tt> function,

148−multM :: (KnownNat (Count (Normalize a)), KnownNat (Count (Normalize b)), KnownNat (Count (Normalize (a, b))), Num e, FromLists e (Normalize (a, b)) (Normalize a), FromLists e (Normalize (a, b)) (Normalize b), Normalize (a, b) ~ Normalize (Normalize a, Normalize b)) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)

149−

150−-- | Selective functors <tt>select</tt> operator equivalent inspired by the

151−-- ArrowMonad solution presented in the paper.

152−selectM :: (Num e, FromLists e (Normalize b) (Normalize b), KnownNat (Count (Normalize b)), Normalize (Either a b) ~ Either (Normalize a) (Normalize b)) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

153−

154−-- | Monad instance equivalent <a>return</a> function,

155−returnM :: forall e a. (Num e, Enum e, Enum a, FromLists e () (Normalize a), KnownNat (Count a)) => a -> Matrix e One a

156−

157−-- | Monad instance equivalent '(>>=)' function,

158−bindM :: Num e => Matrix e a b -> Matrix e b c -> Matrix e a c

159−

160−-- | Obtain the number of columns.

161−--

162−-- NOTE: The <a>KnownNat</a> constaint is needed in order to obtain the

163−-- dimensions in constant time.

164−--

165−-- TODO: A <a>columns</a> function that does not need the <a>KnownNat</a>

166−-- constraint in exchange for performance.

167−columns :: KnownNat (Count (Normalize cols)) => Matrix e cols rows -> Int

168−

169−-- | Obtain the number of rows.

170−--

171−-- NOTE: The <a>KnownNat</a> constaint is needed in order to obtain the

172−-- dimensions in constant time.

173−--

174−-- TODO: A <a>rows</a> function that does not need the <a>KnownNat</a>

175−-- constraint in exchange for performance.

176−rows :: KnownNat (Count (Normalize rows)) => Matrix e cols rows -> Int

177−

178−-- | Matrix transposition.

179−tr :: Matrix e cols rows -> Matrix e rows cols

180−

181−-- | McCarthy's Conditional expresses probabilistic choice.

182−cond :: (Normalize (Normalize a) ~ Normalize a, KnownNat (Count (Normalize a)), FromLists e () (Normalize a), FromLists e (Normalize a) (), FromLists e (Normalize a) (Normalize a), Bounded a, Enum a, Num e, Ord e) => (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b

183−

184−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

185−--

186−-- Law:

187−--

188−-- <pre>

189−-- <tt>Junc</tt> (<tt>Split</tt> a c) (<tt>Split</tt> b d) == <tt>Split</tt> (<tt>Junc</tt> a b) (<tt>Junc</tt> c d)

190−-- </pre>

191−abideJS :: Matrix e cols rows -> Matrix e cols rows

192−

193−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

194−--

195−-- Law:

196−--

197−-- <pre>

198−-- <tt>Split</tt> (<tt>Junc</tt> a b) (<tt>Junc</tt> c d) == <tt>Junc</tt> (<tt>Split</tt> a c) (<tt>Split</tt> b d)

199−-- </pre>

200−abideSJ :: Matrix e cols rows -> Matrix e cols rows

201−

202−-- | Matrix <tt>Split</tt> constructor

203−(===) :: Normalize (Either a b) ~ Either (Normalize a) (Normalize b) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

204−infixl 2 ===

205−

206−-- | Biproduct first component projection

207−p1 :: (Num e, KnownNat (Count (Normalize n)), KnownNat (Count (Normalize m)), FromLists e (Normalize n) (Normalize m), FromLists e (Normalize m) (Normalize m), Normalize (Either m n) ~ Either (Normalize m) (Normalize n)) => Matrix e (Either m n) m

208−

209−-- | Biproduct second component projection

210−p2 :: (Num e, KnownNat (Count (Normalize n)), KnownNat (Count (Normalize m)), FromLists e (Normalize m) (Normalize n), FromLists e (Normalize n) (Normalize n), Normalize (Either m n) ~ Either (Normalize m) (Normalize n)) => Matrix e (Either m n) n

211−

212−-- | Matrix <tt>Junc</tt> constructor

213−(|||) :: Normalize (Either a b) ~ Either (Normalize a) (Normalize b) => Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

214−infixl 3 |||

215−

216−-- | Biproduct first component injection

217−i1 :: (Num e, KnownNat (Count (Normalize n)), KnownNat (Count (Normalize m)), FromLists e (Normalize n) (Normalize m), FromLists e (Normalize m) (Normalize m), Normalize (Either m n) ~ Either (Normalize m) (Normalize n)) => Matrix e m (Either m n)

218−

219−-- | Biproduct second component injection

220−i2 :: (Num e, KnownNat (Count (Normalize n)), KnownNat (Count (Normalize m)), FromLists e (Normalize m) (Normalize n), FromLists e (Normalize n) (Normalize n), Normalize (Either m n) ~ Either (Normalize m) (Normalize n)) => Matrix e n (Either m n)

221−

222−-- | Matrix coproduct functor also known as matrix direct sum.

223−(-|-) :: (Num e, KnownNat (Count (Normalize j)), KnownNat (Count (Normalize k)), FromLists e (Normalize k) (Normalize k), FromLists e (Normalize j) (Normalize k), FromLists e (Normalize k) (Normalize j), FromLists e (Normalize j) (Normalize j), Normalize (Either n m) ~ Either (Normalize n) (Normalize m), Normalize (Either k j) ~ Either (Normalize k) (Normalize j)) => Matrix e n k -> Matrix e m j -> Matrix e (Either n m) (Either k j)

224−infixl 5 -|-

225−

226−-- | Matrix product functor also known as kronecker product

227−(><) :: forall e m p n q. (Num e, KnownNat (Count (Normalize m)), KnownNat (Count (Normalize n)), KnownNat (Count (Normalize p)), KnownNat (Count (Normalize q)), KnownNat (Count (Normalize (m, n))), KnownNat (Count (Normalize (p, q))), FromLists e (Normalize (m, n)) (Normalize m), FromLists e (Normalize (m, n)) (Normalize n), FromLists e (Normalize (p, q)) (Normalize p), FromLists e (Normalize (p, q)) (Normalize q), Normalize (m, n) ~ Normalize (Normalize m, Normalize n), Normalize (p, q) ~ Normalize (Normalize p, Normalize q)) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)

228−infixl 4 ><

229−

230−-- | Khatri Rao product first component projection matrix.

231−kp1 :: forall e m k. (Num e, KnownNat (Count (Normalize m)), KnownNat (Count (Normalize k)), KnownNat (Count (Normalize (m, k))), FromLists e (Normalize (m, k)) (Normalize m), Normalize (m, k) ~ Normalize (Normalize m, Normalize k)) => Matrix e (m, k) m

232−

233−-- | Khatri Rao product second component projection matrix.

234−kp2 :: forall e m k. (Num e, KnownNat (Count (Normalize k)), KnownNat (Count (Normalize m)), KnownNat (Count (Normalize (m, k))), FromLists e (Normalize (m, k)) (Normalize k), Normalize (m, k) ~ Normalize (Normalize m, Normalize k)) => Matrix e (m, k) k

235−

236−-- | Khatri Rao Matrix product also known as matrix pairing.

237−--

238−-- NOTE: That this is not a true categorical product, see for instance:

239−--

240−-- <pre>

241−-- | kp1 <a>comp</a> khatri a b == a

242−-- khatri a b ==> |

243−-- | kp2 <a>comp</a> khatri a b == b

244−-- </pre>

245−--

246−-- Emphasis on the implication symbol.

247−khatri :: forall e cols a b. (Num e, KnownNat (Count (Normalize a)), KnownNat (Count (Normalize b)), KnownNat (Count (Normalize (a, b))), FromLists e (Normalize (a, b)) (Normalize a), FromLists e (Normalize (a, b)) (Normalize b), Normalize (a, b) ~ Normalize (Normalize a, Normalize b)) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)

248−

249−-- | Identity matrix

250−identity :: (Num e, FromLists e (Normalize a) (Normalize a), KnownNat (Count (Normalize a))) => Matrix e a a

251−

252−-- | Matrix composition. Equivalent to matrix-matrix multiplication.

253−--

254−-- This definition takes advantage of divide-and-conquer and fusion laws

255−-- from LAoP.

256−comp :: Num e => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

257−

258−-- | Lifts functions to matrices with arbitrary dimensions.

259−--

260−-- NOTE: Be careful to not ask for a matrix bigger than the cardinality

261−-- of types <tt>a</tt> or <tt>b</tt> allows.

262−fromF :: (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e, KnownNat (Count (Normalize cols)), KnownNat (Count (Normalize rows)), FromLists e (Normalize rows) (Normalize cols)) => (a -> b) -> Matrix e cols rows

263−

264−-- | Lifts functions to matrices with dimensions matching <tt>a</tt> and

265−-- <tt>b</tt> cardinality's.

266−fromF' :: (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e, KnownNat (Count (Normalize a)), KnownNat (Count (Normalize b)), FromLists e (Normalize b) (Normalize a)) => (a -> b) -> Matrix e a b

267−

268−-- | Matrix pretty printer

269−pretty :: (KnownNat (Count (Normalize cols)), Show e) => Matrix e cols rows -> String

270−

271−-- | Matrix pretty printer

272−prettyPrint :: (KnownNat (Count (Normalize cols)), Show e) => Matrix e cols rows -> IO ()

273−instance Control.DeepSeq.NFData e => Control.DeepSeq.NFData (Matrix.Type.Matrix e cols rows)

274−instance GHC.Classes.Ord e => GHC.Classes.Ord (Matrix.Type.Matrix e cols rows)

275−instance GHC.Classes.Eq e => GHC.Classes.Eq (Matrix.Type.Matrix e cols rows)

276−instance GHC.Num.Num e => GHC.Num.Num (Matrix.Type.Matrix e cols rows)

277−instance GHC.Show.Show e => GHC.Show.Show (Matrix.Type.Matrix e cols rows)

278−instance GHC.Num.Num e => Control.Category.Category (Matrix.Type.Matrix e)

279−

280−

281−-- | The LAoP discipline generalises relations and functions treating them

282−-- as Boolean matrices and in turn consider these as arrows.

283−--

284−-- LAoP is a library for algebraic (inductive) construction and

285−-- manipulation of matrices in Haskell. See <a>my Msc Thesis</a> for the

286−-- motivation behind the library, the underlying theory, and

287−-- implementation details.

288−--

289−-- This module offers a newtype wrapper around <a>Matrix</a> that uses

290−-- type level naturals instead of standard data types for the matrices

291−-- dimensions.

292−module Matrix.Nat

293−newtype Matrix e (cols :: Nat) (rows :: Nat)

294−M :: Matrix e (FromNat cols) (FromNat rows) -> Matrix e

295−empty :: Matrix e 0 0

296−one :: e -> Matrix e 1 1

297−junc :: FromNat cols3 ~ Either (FromNat cols1) (FromNat cols2) => Matrix e cols1 rows -> Matrix e cols2 rows -> Matrix e cols3 rows

298−split :: FromNat rows3 ~ Either (FromNat rows1) (FromNat rows2) => Matrix e cols rows1 -> Matrix e cols rows2 -> Matrix e cols rows3

299−

300−-- | Type family that computes of a given type dimension from a given

301−-- natural

302−--

303−-- Thanks to Li-Yao Xia this type family is super fast.

304−type family FromNat (n :: Nat) :: Type

305−

306−-- | Type family that computes the cardinality of a given type dimension.

307−--

308−-- It can also count the cardinality of custom types that implement the

309−-- <a>Generic</a> instance.

310−type family Count (d :: Type) :: Nat

311−

312−-- | Type family that normalizes the representation of a given data

313−-- structure

314−type family Normalize (d :: Type) :: Type

315−

316−-- | Type class for defining the <tt>fromList</tt> conversion function.

317−--

318−-- Given that it is not possible to branch on types at the term level

319−-- type classes are needed bery much like an inductive definition but on

320−-- types.

321−class FromLists e cols rows

322−fromLists :: FromLists e (FromNat cols) (FromNat rows) => [[e]] -> Matrix e cols rows

323−toLists :: Matrix e cols rows -> [[e]]

324−toList :: Matrix e cols rows -> [e]

325−matrixBuilder :: (FromLists e (FromNat cols) (FromNat rows), KnownNat (Count (FromNat cols)), KnownNat (Count (FromNat rows))) => ((Int, Int) -> e) -> Matrix e cols rows

326−row :: FromLists e (FromNat cols) () => [e] -> Matrix e cols 1

327−col :: FromLists e () (FromNat rows) => [e] -> Matrix e 1 rows

328−zeros :: (Num e, FromLists e (FromNat cols) (FromNat rows), KnownNat (Count (FromNat cols)), KnownNat (Count (FromNat rows))) => Matrix e cols rows

329−ones :: (Num e, FromLists e (FromNat cols) (FromNat rows), KnownNat (Count (FromNat cols)), KnownNat (Count (FromNat rows))) => Matrix e cols rows

330−bang :: forall e cols. (Num e, Enum e, FromLists e (FromNat cols) (), KnownNat (Count (FromNat cols))) => Matrix e cols 1

331−constant :: (Num e, FromLists e (FromNat cols) (FromNat rows), KnownNat (Count (FromNat cols)), KnownNat (Count (FromNat rows))) => e -> Matrix e cols rows

332−columns :: KnownNat (Count (FromNat cols)) => Matrix e cols rows -> Int

333−rows :: KnownNat (Count (FromNat rows)) => Matrix e cols rows -> Int

334−tr :: Matrix e cols rows -> Matrix e rows cols

335−select :: (Num e, FromLists e (FromNat rows1) (FromNat rows1), KnownNat (Count (FromNat rows1)), FromNat rows2 ~ FromNat rows1, FromNat cols1 ~ FromNat cols2, FromNat rows3 ~ Either (FromNat cols3) (FromNat rows1)) => Matrix e cols1 rows3 -> Matrix e cols3 rows1 -> Matrix e cols2 rows2

336−cond :: (FromNat (Count (FromNat cols)) ~ FromNat cols, KnownNat (Count (FromNat cols)), FromLists e () (FromNat cols), FromLists e (FromNat cols) (), FromLists e (FromNat cols) (FromNat cols), Bounded a, Enum a, Num e, Ord e) => (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows

337−abideJS :: Matrix e cols rows -> Matrix e cols rows

338−abideSJ :: Matrix e cols rows -> Matrix e cols rows

339−(===) :: FromNat rows3 ~ Either (FromNat rows1) (FromNat rows2) => Matrix e cols rows1 -> Matrix e cols rows2 -> Matrix e cols rows3

340−infixl 2 ===

341−p1 :: (Num e, KnownNat (Count (FromNat n)), KnownNat (Count (FromNat m)), FromLists e (FromNat n) (FromNat m), FromLists e (FromNat m) (FromNat m), FromNat (m + n) ~ Either (FromNat m) (FromNat n)) => Matrix e (m + n) m

342−p2 :: (Num e, KnownNat (Count (FromNat n)), KnownNat (Count (FromNat m)), FromLists e (FromNat m) (FromNat n), FromLists e (FromNat n) (FromNat n), FromNat (m + n) ~ Either (FromNat m) (FromNat n)) => Matrix e (m + n) n

343−(|||) :: FromNat cols3 ~ Either (FromNat cols1) (FromNat cols2) => Matrix e cols1 rows -> Matrix e cols2 rows -> Matrix e cols3 rows

344−infixl 3 |||

345−i1 :: (Num e, KnownNat (Count (FromNat n)), KnownNat (Count (FromNat rows)), FromLists e (FromNat n) (FromNat rows), FromLists e (FromNat rows) (FromNat rows), FromNat (rows + n) ~ Either (FromNat rows) (FromNat n)) => Matrix e rows (rows + n)

346−i2 :: (Num e, KnownNat (Count (FromNat rows)), KnownNat (Count (FromNat m)), FromLists e (FromNat m) (FromNat rows), FromLists e (FromNat rows) (FromNat rows), FromNat (m + rows) ~ Either (FromNat m) (FromNat rows)) => Matrix e rows (m + rows)

347−(-|-) :: (Num e, KnownNat (Count (FromNat j)), KnownNat (Count (FromNat k)), FromLists e (FromNat k) (FromNat k), FromLists e (FromNat j) (FromNat k), FromLists e (FromNat k) (FromNat j), FromLists e (FromNat j) (FromNat j), FromNat (n + m) ~ Either (FromNat n) (FromNat m), FromNat (k + j) ~ Either (FromNat k) (FromNat j)) => Matrix e n k -> Matrix e m j -> Matrix e (n + m) (k + j)

348−infixl 5 -|-

349−(><) :: forall e m p n q. (Num e, KnownNat (Count (FromNat m)), KnownNat (Count (FromNat n)), KnownNat (Count (FromNat p)), KnownNat (Count (FromNat q)), KnownNat (Count (FromNat (m * n))), KnownNat (Count (FromNat (p * q))), FromLists e (FromNat (m * n)) (FromNat m), FromLists e (FromNat (m * n)) (FromNat n), FromLists e (FromNat (p * q)) (FromNat p), FromLists e (FromNat (p * q)) (FromNat q), FromNat (m * n) ~ FromNat (Count (FromNat m) * Count (FromNat n)), FromNat (p * q) ~ FromNat (Count (FromNat p) * Count (FromNat q))) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)

350−infixl 4 ><

351−kp1 :: forall e m k. (Num e, KnownNat (Count (FromNat m)), KnownNat (Count (FromNat k)), KnownNat (Count (FromNat (m * k))), FromLists e (FromNat (m * k)) (FromNat m), FromNat (m * k) ~ FromNat (Count (FromNat m) * Count (FromNat k))) => Matrix e (m * k) m

352−kp2 :: forall e m k. (Num e, KnownNat (Count (FromNat k)), FromLists e (FromNat (m * k)) (FromNat k), KnownNat (Count (FromNat m)), KnownNat (Count (FromNat (m * k))), FromNat (m * k) ~ FromNat (Count (FromNat m) * Count (FromNat k))) => Matrix e (m * k) k

353−khatri :: forall e cols a b. (Num e, KnownNat (Count (FromNat a)), KnownNat (Count (FromNat b)), KnownNat (Count (FromNat (a * b))), FromLists e (FromNat (a * b)) (FromNat a), FromLists e (FromNat (a * b)) (FromNat b), FromNat (a * b) ~ FromNat (Count (FromNat a) * Count (FromNat b))) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)

354−identity :: (Num e, FromLists e (FromNat cols) (FromNat cols), KnownNat (Count (FromNat cols))) => Matrix e cols cols

355−comp :: Num e => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

356−fromF :: (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e, KnownNat (Count (FromNat cols)), KnownNat (Count (FromNat rows)), FromLists e (FromNat rows) (FromNat cols)) => (a -> b) -> Matrix e cols rows

357−fromF' :: (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e, KnownNat (Count (Normalize a)), KnownNat (Count (Normalize b)), FromLists e (Normalize b) (Normalize a)) => (a -> b) -> Matrix e (Count a) (Count b)

358−pretty :: (KnownNat (Count (FromNat cols)), Show e) => Matrix e cols rows -> String

359−prettyPrint :: (KnownNat (Count (FromNat cols)), Show e) => Matrix e cols rows -> IO ()

360−instance Control.DeepSeq.NFData e => Control.DeepSeq.NFData (Matrix.Nat.Matrix e cols rows)

361−instance GHC.Classes.Ord e => GHC.Classes.Ord (Matrix.Nat.Matrix e cols rows)

362−instance GHC.Classes.Eq e => GHC.Classes.Eq (Matrix.Nat.Matrix e cols rows)

363−instance GHC.Num.Num e => GHC.Num.Num (Matrix.Nat.Matrix e cols rows)

364−instance GHC.Show.Show e => GHC.Show.Show (Matrix.Nat.Matrix e cols rows)

365−instance GHC.Num.Num e => Control.Category.Category (Matrix.Nat.Matrix e)

366−

367−

368−-- | LAoP is a library for algebraic (inductive) construction and

369−-- manipulation of matrices in Haskell. See <a>my Msc Thesis</a> for the

370−-- motivation behind the library, the underlying theory, and

371−-- implementation details.

372−--

373−-- This module exports a type synonym <a>Dist</a> that represents a

374−-- stochastic distribution matrix and several distribution construction

375−-- functions.

376−module Dist

377−

378−-- | Type synonym for column vector matrices. This represents a probability

379−-- distribution.

380−newtype Dist a

381−D :: Matrix Prob () a -> Dist a

382−

383−-- | Type synonym for probability value

384−type Prob = Double

385−

386−-- | Functor instance

387−fmapD :: (Bounded a, Bounded b, Enum a, Enum b, Eq b, KnownNat (Count (Normalize a)), KnownNat (Count (Normalize b)), FromLists Prob (Normalize b) (Normalize a)) => (a -> b) -> Dist a -> Dist b

388−

389−-- | Applicative/Monoidal instance <tt>unit</tt> function

390−unitD :: Dist ()

391−

392−-- | Applicative/Monoidal instance <tt>mult</tt> function

393−multD :: (KnownNat (Count (Normalize a)), KnownNat (Count (Normalize b)), KnownNat (Count (Normalize (a, b))), FromLists Prob (Normalize (a, b)) (Normalize a), FromLists Prob (Normalize (a, b)) (Normalize b), Normalize (a, b) ~ Normalize (Normalize a, Normalize b)) => Dist a -> Dist b -> Dist (a, b)

394−

395−-- | Selective instance function

396−selectD :: (Normalize (Either a b) ~ Either (Normalize a) (Normalize b), FromLists Prob (Normalize b) (Normalize b), KnownNat (Count (Normalize b))) => Dist (Either a b) -> Matrix Prob a b -> Dist b

397−

398−-- | Monad instance <a>return</a> function

399−returnD :: forall a. (Enum a, FromLists Prob () (Normalize a), KnownNat (Count a)) => a -> Dist a

400−

401−-- | Monad instance '(>>=)' function

402−bindD :: Dist a -> Matrix Prob a b -> Dist b

403−

404−-- | Constructs a Bernoulli distribution

405−choose :: FromLists Prob () (Normalize a) => Prob -> Dist a

406−

407−-- | Creates a distribution given a shape function

408−shape :: FromLists Prob () (Normalize a) => (Prob -> Prob) -> [a] -> Dist a

409−

410−-- | Constructs a Linear distribution

411−linear :: FromLists Prob () (Normalize a) => [a] -> Dist a

412−

413−-- | Constructs an Uniform distribution

414−uniform :: FromLists Prob () (Normalize a) => [a] -> Dist a

415−

416−-- | Constructs an Negative Exponential distribution

417−negExp :: FromLists Prob () (Normalize a) => [a] -> Dist a

418−

419−-- | Constructs an Normal distribution

420−normal :: FromLists Prob () (Normalize a) => [a] -> Dist a

421−

422−-- | Transforms a <a>Dist</a> into a list of pairs.

423−toValues :: forall a. (Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> [(a, Prob)]

424−

425−-- | Pretty a distribution

426−prettyDist :: forall a. (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> String

427−

428−-- | Pretty Print a distribution

429−prettyPrintDist :: forall a. (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> IO ()

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501− margin: 0;

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513−

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535−

536−.fields .caption { display: none; }

537−

538−.fields p { margin: 0 0; }

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540−/* this seems bulky to me

541−.methods, .constructors {

542− background: #f8f8f8;

543− border: 1px solid #eee;

544−}

545−*/

546−

547−/* @end */

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549−/* @group Auxillary Pages */

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551−

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566−#mini #module-list .caption,

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570−

571−#mini #interface h1,

572−#mini #interface h2,

573−#mini #interface h3,

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579−#mini #interface .top,

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584−#mini #module-list ul {

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586− margin: 0;

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589−#alphabet ul {

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592− margin: 0.5em 0 0;

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596−#alphabet li {

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598− margin: 0 0.25em;

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601−#alphabet a {

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603−}

604−

605−#index .caption,

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608−#index table {

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610−}

611−

612−#index .src {

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615−#index .alt {

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619−}

620−

621−#index td + td {

622− padding-left: 1em;

623−}

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629−

630−#module-list li {

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634−#module-list span.collapser,

635−#module-list span.expander {

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639−#module-list .package {

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642−

643−:target {

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5− top: 3.2em;

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8− width: 44em;

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14−#search.hidden {

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18−#search-form, #search-results {

19− box-shadow: 2px 2px 6px rgb(199, 204, 208);

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26− box-sizing: border-box;

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30− border: 0.05em solid rgb(151, 179, 202);

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33−#search input:focus {

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37−#search p.error {

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39− font-weight: bold;

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41−

42−#search-results {

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44− border: 0.05em solid #b2d5fb;

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48−#search-form input + #search-results {

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58−#search-results > ul {

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62−

63−#search-results > ul > li,

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69−

70−#search-results > ul > li {

71− border-bottom: 1px solid #b2d5fb;

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74−#search-results > ul > li > ul {

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78−.search-module h4 {

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82−.search-module > ul {

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86−.search-module > ul > li > a[href] {

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92−.search-module > ul > li > a[href].active-link {

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96−.search-module a[href]:hover {

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100−.search-result a a {

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104−.search-result ul.subs {

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109−.search-result ul.subs li {

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113−.search-result ul.subs::after {

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115− content: "...";

116− color: rgb(78,98,114);

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120−.more-results {

121− color: rgb(99, 141, 173);

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125−.more-results::before {

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153−.key {

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2−//# sourceMappingURL=quick-jump.min.js.map

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Dist.Internal.htmldeleted−174

@@ -1,174 +0,0 @@

1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>{-# LANGUAGE GADTs #-}

2−<a name="line-2"></a>{-# LANGUAGE AllowAmbiguousTypes #-}

3−<a name="line-3"></a>{-# LANGUAGE TypeApplications #-}

4−<a name="line-4"></a>{-# LANGUAGE KindSignatures #-}

5−<a name="line-5"></a>{-# LANGUAGE ScopedTypeVariables #-}

6−<a name="line-6"></a>{-# LANGUAGE FlexibleContexts #-}

7−<a name="line-7"></a>{-# LANGUAGE DataKinds #-}

8−<a name="line-8"></a>{-# LANGUAGE RankNTypes #-}

9−<a name="line-9"></a>

10−<a name="line-10"></a>module Dist.Internal

11−<a name="line-11"></a> (

12−<a name="line-12"></a> <a href="Dist.Internal.html#Dist">Dist</a>(..),

13−<a name="line-13"></a> <a href="Dist.Internal.html#Prob">Prob</a>,

14−<a name="line-14"></a>

15−<a name="line-15"></a> <a href="Dist.Internal.html#fmapD">fmapD</a>,

16−<a name="line-16"></a> <a href="Dist.Internal.html#unitD">unitD</a>,

17−<a name="line-17"></a> <a href="Dist.Internal.html#multD">multD</a>,

18−<a name="line-18"></a> <a href="Dist.Internal.html#selectD">selectD</a>,

19−<a name="line-19"></a> <a href="Dist.Internal.html#returnD">returnD</a>,

20−<a name="line-20"></a> <a href="Dist.Internal.html#bindD">bindD</a>,

21−<a name="line-21"></a> (<a href="Dist.Internal.html#%3F%3F">??</a>),

22−<a name="line-22"></a>

23−<a name="line-23"></a> <a href="Dist.Internal.html#choose">choose</a>,

24−<a name="line-24"></a> <a href="Dist.Internal.html#shape">shape</a>,

25−<a name="line-25"></a> <a href="Dist.Internal.html#linear">linear</a>,

26−<a name="line-26"></a> <a href="Dist.Internal.html#uniform">uniform</a>,

27−<a name="line-27"></a> <a href="Dist.Internal.html#negExp">negExp</a>,

28−<a name="line-28"></a> <a href="Dist.Internal.html#normal">normal</a>,

29−<a name="line-29"></a> <a href="Dist.Internal.html#toValues">toValues</a>,

30−<a name="line-30"></a> <a href="Dist.Internal.html#prettyDist">prettyDist</a>,

31−<a name="line-31"></a> <a href="Dist.Internal.html#prettyPrintDist">prettyPrintDist</a>

32−<a name="line-32"></a> )

33−<a name="line-33"></a> where

34−<a name="line-34"></a>

35−<a name="line-35"></a>import <a href="Matrix.Type.html">Matrix.Type</a>

36−<a name="line-36"></a>import <a href="Utils.html">Utils</a>

37−<a name="line-37"></a>import GHC.TypeLits

38−<a name="line-38"></a>import Data.Proxy

39−<a name="line-39"></a>import Data.List (sortBy)

40−<a name="line-40"></a>

41−<a name="line-41"></a>-- | Type synonym for probability value

42−<a name="line-42"></a>type <a name="Prob"><a href="Dist.Internal.html#Prob">Prob</a></a> = Double

43−<a name="line-43"></a>

44−<a name="line-44"></a>-- | Type synonym for column vector matrices. This represents a probability

45−<a name="line-45"></a>-- distribution.

46−<a name="line-46"></a>newtype <a name="Dist"><a href="Dist.Internal.html#Dist">Dist</a></a> <a name="local-6989586621679068539"><a href="#local-6989586621679068539">a</a></a> = <a name="D"><a href="Dist.Internal.html#D">D</a></a> (<a href="Matrix.Type.html#Matrix">Matrix</a> <a href="Dist.Internal.html#Prob">Prob</a> () <a href="#local-6989586621679068539">a</a>)

47−<a name="line-47"></a>

48−<a name="line-48"></a>-- | Functor instance

49−<a name="line-49"></a>fmapD ::

50−<a name="line-50"></a> ( Bounded <a href="#local-6989586621679068558">a</a>,

51−<a name="line-51"></a> Bounded <a href="#local-6989586621679068559">b</a>,

52−<a name="line-52"></a> Enum <a href="#local-6989586621679068558">a</a>,

53−<a name="line-53"></a> Enum <a href="#local-6989586621679068559">b</a>,

54−<a name="line-54"></a> Eq <a href="#local-6989586621679068559">b</a>,

55−<a name="line-55"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068558">a</a>)),

56−<a name="line-56"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068559">b</a>)),

57−<a name="line-57"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068559">b</a>) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068558">a</a>)

58−<a name="line-58"></a> )

59−<a name="line-59"></a> =>

60−<a name="line-60"></a> (<a href="#local-6989586621679068558">a</a> -> <a href="#local-6989586621679068559">b</a>) -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068558">a</a> -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068559">b</a>

61−<a name="line-61"></a><a name="fmapD"><a href="Dist.Internal.html#fmapD">fmapD</a></a> <a name="local-6989586621679068560"><a href="#local-6989586621679068560">f</a></a> (<a href="Dist.Internal.html#D">D</a> <a name="local-6989586621679068561"><a href="#local-6989586621679068561">m</a></a>) = <a href="Dist.Internal.html#D">D</a> (<a href="Matrix.Type.html#fromF%27">fromF'</a> <a href="#local-6989586621679068560">f</a> `<a href="Matrix.Type.html#comp">comp</a>` <a href="#local-6989586621679068561">m</a>)

62−<a name="line-62"></a>

63−<a name="line-63"></a>-- | Applicative/Monoidal instance 'unit' function

64−<a name="line-64"></a>unitD :: <a href="Dist.Internal.html#Dist">Dist</a> ()

65−<a name="line-65"></a><a name="unitD"><a href="Dist.Internal.html#unitD">unitD</a></a> = <a href="Dist.Internal.html#D">D</a> (<a href="Matrix.Type.html#one">one</a> 1)

66−<a name="line-66"></a>

67−<a name="line-67"></a>-- | Applicative/Monoidal instance 'mult' function

68−<a name="line-68"></a>multD ::

69−<a name="line-69"></a> ( KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068556">a</a>)),

70−<a name="line-70"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068557">b</a>)),

71−<a name="line-71"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679068556">a</a>, <a href="#local-6989586621679068557">b</a>))),

72−<a name="line-72"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679068556">a</a>, <a href="#local-6989586621679068557">b</a>)) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068556">a</a>),

73−<a name="line-73"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679068556">a</a>, <a href="#local-6989586621679068557">b</a>)) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068557">b</a>),

74−<a name="line-74"></a> Normalize (<a href="#local-6989586621679068556">a</a>, <a href="#local-6989586621679068557">b</a>) ~ <a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068556">a</a>, <a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068557">b</a>)

75−<a name="line-75"></a> ) => <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068556">a</a> -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068557">b</a> -> <a href="Dist.Internal.html#Dist">Dist</a> (<a href="#local-6989586621679068556">a</a>, <a href="#local-6989586621679068557">b</a>)

76−<a name="line-76"></a><a name="multD"><a href="Dist.Internal.html#multD">multD</a></a> (<a href="Dist.Internal.html#D">D</a> <a name="local-6989586621679068562"><a href="#local-6989586621679068562">a</a></a>) (<a href="Dist.Internal.html#D">D</a> <a name="local-6989586621679068563"><a href="#local-6989586621679068563">b</a></a>) = <a href="Dist.Internal.html#D">D</a> (<a href="Matrix.Type.html#khatri">khatri</a> <a href="#local-6989586621679068562">a</a> <a href="#local-6989586621679068563">b</a>)

77−<a name="line-77"></a>

78−<a name="line-78"></a>-- | Selective instance function

79−<a name="line-79"></a>selectD ::

80−<a name="line-80"></a> ( Normalize (Either <a href="#local-6989586621679068554">a</a> <a href="#local-6989586621679068555">b</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068554">a</a>) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068555">b</a>),

81−<a name="line-81"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068555">b</a>) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068555">b</a>),

82−<a name="line-82"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068555">b</a>))

83−<a name="line-83"></a> ) => <a href="Dist.Internal.html#Dist">Dist</a> (Either <a href="#local-6989586621679068554">a</a> <a href="#local-6989586621679068555">b</a>) -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="Dist.Internal.html#Prob">Prob</a> <a href="#local-6989586621679068554">a</a> <a href="#local-6989586621679068555">b</a> -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068555">b</a>

84−<a name="line-84"></a><a name="selectD"><a href="Dist.Internal.html#selectD">selectD</a></a> (<a href="Dist.Internal.html#D">D</a> <a name="local-6989586621679068564"><a href="#local-6989586621679068564">d</a></a>) <a name="local-6989586621679068565"><a href="#local-6989586621679068565">m</a></a> = <a href="Dist.Internal.html#D">D</a> (<a href="Matrix.Type.html#junc">junc</a> <a href="#local-6989586621679068565">m</a> <a href="Matrix.Type.html#identity">identity</a> `<a href="Matrix.Type.html#comp">comp</a>` <a href="#local-6989586621679068564">d</a>)

85−<a name="line-85"></a>

86−<a name="line-86"></a>

87−<a name="line-87"></a>-- | Monad instance 'return' function

88−<a name="line-88"></a>returnD :: forall <a name="local-6989586621679068553"><a href="#local-6989586621679068553">a</a></a> . (Enum <a href="#local-6989586621679068553">a</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068553">a</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679068553">a</a>)) => <a href="#local-6989586621679068553">a</a> -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068553">a</a>

89−<a name="line-89"></a><a name="returnD"><a href="Dist.Internal.html#returnD">returnD</a></a> <a name="local-6989586621679068566"><a href="#local-6989586621679068566">a</a></a> = <a href="Dist.Internal.html#D">D</a> (<a href="Matrix.Type.html#col">col</a> <a href="#local-6989586621679068569">l</a>)

90−<a name="line-90"></a> where

91−<a name="line-91"></a> <a name="local-6989586621679068567"><a href="#local-6989586621679068567">i</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679068553">a</a>))

92−<a name="line-92"></a> <a name="local-6989586621679068568"><a href="#local-6989586621679068568">x</a></a> = fromEnum <a href="#local-6989586621679068566">a</a>

93−<a name="line-93"></a> <a name="local-6989586621679068569"><a href="#local-6989586621679068569">l</a></a> = take <a href="#local-6989586621679068568">x</a> [0,0..] ++ [1] ++ take (<a href="#local-6989586621679068567">i</a> - <a href="#local-6989586621679068568">x</a> - 1) [0,0..]

94−<a name="line-94"></a>

95−<a name="line-95"></a>-- | Monad instance '(>>=)' function

96−<a name="line-96"></a>bindD :: <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068551">a</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="Dist.Internal.html#Prob">Prob</a> <a href="#local-6989586621679068551">a</a> <a href="#local-6989586621679068552">b</a> -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068552">b</a>

97−<a name="line-97"></a><a name="bindD"><a href="Dist.Internal.html#bindD">bindD</a></a> (<a href="Dist.Internal.html#D">D</a> <a name="local-6989586621679068570"><a href="#local-6989586621679068570">d</a></a>) <a name="local-6989586621679068571"><a href="#local-6989586621679068571">m</a></a> = <a href="Dist.Internal.html#D">D</a> (<a href="#local-6989586621679068571">m</a> `<a href="Matrix.Type.html#comp">comp</a>` <a href="#local-6989586621679068570">d</a>)

98−<a name="line-98"></a>

99−<a name="line-99"></a>-- | Extract probabilities given an Event.

100−<a name="line-100"></a>(??) ::

101−<a name="line-101"></a> ( Enum <a href="#local-6989586621679068550">a</a>,

102−<a name="line-102"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679068550">a</a>),

103−<a name="line-103"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068550">a</a>)

104−<a name="line-104"></a> ) => (<a href="#local-6989586621679068550">a</a> -> Bool) -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068550">a</a> -> <a href="Dist.Internal.html#Prob">Prob</a>

105−<a name="line-105"></a>(<a name="%3F%3F"><a href="Dist.Internal.html#%3F%3F">??</a></a>) <a name="local-6989586621679068572"><a href="#local-6989586621679068572">p</a></a> <a name="local-6989586621679068573"><a href="#local-6989586621679068573">d</a></a> =

106−<a name="line-106"></a> let <a name="local-6989586621679068574"><a href="#local-6989586621679068574">l</a></a> = <a href="Dist.Internal.html#toValues">toValues</a> <a href="#local-6989586621679068573">d</a>

107−<a name="line-107"></a> <a name="local-6989586621679068575"><a href="#local-6989586621679068575">x</a></a> = filter (<a href="#local-6989586621679068572">p</a> . fst) <a href="#local-6989586621679068574">l</a>

108−<a name="line-108"></a> in sum . map snd $ <a href="#local-6989586621679068575">x</a>

109−<a name="line-109"></a>

110−<a name="line-110"></a>-- Distribution Construction

111−<a name="line-111"></a>

112−<a name="line-112"></a>-- | Constructs a Bernoulli distribution

113−<a name="line-113"></a>choose :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068549">a</a>)) => <a href="Dist.Internal.html#Prob">Prob</a> -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068549">a</a>

114−<a name="line-114"></a><a name="choose"><a href="Dist.Internal.html#choose">choose</a></a> <a name="local-6989586621679068576"><a href="#local-6989586621679068576">prob</a></a> = <a href="Dist.Internal.html#D">D</a> (<a href="Matrix.Type.html#col">col</a> [<a href="#local-6989586621679068576">prob</a>, 1 - <a href="#local-6989586621679068576">prob</a>])

115−<a name="line-115"></a>

116−<a name="line-116"></a>-- | Creates a distribution given a shape function

117−<a name="line-117"></a>shape :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068548">a</a>)) => (<a href="Dist.Internal.html#Prob">Prob</a> -> <a href="Dist.Internal.html#Prob">Prob</a>) -> [<a href="#local-6989586621679068548">a</a>] -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068548">a</a>

118−<a name="line-118"></a><a name="shape"><a href="Dist.Internal.html#shape">shape</a></a> _ [] = error "Probability.shape: empty list"

119−<a name="line-119"></a>shape <a name="local-6989586621679068577"><a href="#local-6989586621679068577">f</a></a> <a name="local-6989586621679068578"><a href="#local-6989586621679068578">xs</a></a> =

120−<a name="line-120"></a> let <a name="local-6989586621679068579"><a href="#local-6989586621679068579">incr</a></a> = 1 / fromIntegral (length <a href="#local-6989586621679068578">xs</a> - 1)

121−<a name="line-121"></a> <a name="local-6989586621679068580"><a href="#local-6989586621679068580">ps</a></a> = map <a href="#local-6989586621679068577">f</a> (iterate (+<a href="#local-6989586621679068579">incr</a>) 0)

122−<a name="line-122"></a> in <a href="Dist.Internal.html#fromFreqs">fromFreqs</a> (zip <a href="#local-6989586621679068578">xs</a> <a href="#local-6989586621679068580">ps</a>)

123−<a name="line-123"></a>

124−<a name="line-124"></a>-- | Constructs a Linear distribution

125−<a name="line-125"></a>linear :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068547">a</a>)) => [<a href="#local-6989586621679068547">a</a>] -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068547">a</a>

126−<a name="line-126"></a><a name="linear"><a href="Dist.Internal.html#linear">linear</a></a> = <a href="Dist.Internal.html#shape">shape</a> id

127−<a name="line-127"></a>

128−<a name="line-128"></a>-- | Constructs an Uniform distribution

129−<a name="line-129"></a>uniform :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068546">a</a>)) => [<a href="#local-6989586621679068546">a</a>] -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068546">a</a>

130−<a name="line-130"></a><a name="uniform"><a href="Dist.Internal.html#uniform">uniform</a></a> = <a href="Dist.Internal.html#shape">shape</a> (const 1)

131−<a name="line-131"></a>

132−<a name="line-132"></a>-- | Constructs an Negative Exponential distribution

133−<a name="line-133"></a>negExp :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068545">a</a>)) => [<a href="#local-6989586621679068545">a</a>] -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068545">a</a>

134−<a name="line-134"></a><a name="negExp"><a href="Dist.Internal.html#negExp">negExp</a></a> = <a href="Dist.Internal.html#shape">shape</a> (\<a name="local-6989586621679068581"><a href="#local-6989586621679068581">x</a></a> -> exp (-<a href="#local-6989586621679068581">x</a>))

135−<a name="line-135"></a>

136−<a name="line-136"></a>-- | Constructs an Normal distribution

137−<a name="line-137"></a>normal :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068544">a</a>)) => [<a href="#local-6989586621679068544">a</a>] -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068544">a</a>

138−<a name="line-138"></a><a name="normal"><a href="Dist.Internal.html#normal">normal</a></a> = <a href="Dist.Internal.html#shape">shape</a> (<a href="Dist.Internal.html#normalCurve">normalCurve</a> 0.5 0.5)

139−<a name="line-139"></a>

140−<a name="line-140"></a>-- | Transforms a 'Dist' into a list of pairs.

141−<a name="line-141"></a>toValues :: forall <a name="local-6989586621679068543"><a href="#local-6989586621679068543">a</a></a> . (Enum <a href="#local-6989586621679068543">a</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679068543">a</a>), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068543">a</a>)) => <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068543">a</a> -> [(<a href="#local-6989586621679068543">a</a>, <a href="Dist.Internal.html#Prob">Prob</a>)]

142−<a name="line-142"></a><a name="toValues"><a href="Dist.Internal.html#toValues">toValues</a></a> (<a href="Dist.Internal.html#D">D</a> <a name="local-6989586621679068582"><a href="#local-6989586621679068582">d</a></a>) =

143−<a name="line-143"></a> let <a name="local-6989586621679068583"><a href="#local-6989586621679068583">rows</a></a> = fromInteger (natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679068543">a</a>)))

144−<a name="line-144"></a> <a name="local-6989586621679068584"><a href="#local-6989586621679068584">probs</a></a> = <a href="Matrix.Type.html#toList">toList</a> <a href="#local-6989586621679068582">d</a>

145−<a name="line-145"></a> <a name="local-6989586621679068585"><a href="#local-6989586621679068585">res</a></a> = zip (map toEnum [0..<a href="#local-6989586621679068583">rows</a>]) <a href="#local-6989586621679068584">probs</a>

146−<a name="line-146"></a> in <a href="#local-6989586621679068585">res</a>

147−<a name="line-147"></a>

148−<a name="line-148"></a>-- | Pretty a distribution

149−<a name="line-149"></a>prettyDist :: forall <a name="local-6989586621679068542"><a href="#local-6989586621679068542">a</a></a>. (Show <a href="#local-6989586621679068542">a</a>, Enum <a href="#local-6989586621679068542">a</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679068542">a</a>), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068542">a</a>)) => <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068542">a</a> -> String

150−<a name="line-150"></a><a name="prettyDist"><a href="Dist.Internal.html#prettyDist">prettyDist</a></a> <a name="local-6989586621679068586"><a href="#local-6989586621679068586">d</a></a> =

151−<a name="line-151"></a> let <a name="local-6989586621679068592"><a href="#local-6989586621679068592">values</a></a> = sortBy (\(<a name="local-6989586621679068594"><a href="#local-6989586621679068594">a</a></a>, <a name="local-6989586621679068595"><a href="#local-6989586621679068595">p1</a></a>) (<a name="local-6989586621679068596"><a href="#local-6989586621679068596">b</a></a>, <a name="local-6989586621679068597"><a href="#local-6989586621679068597">p2</a></a>) -> compare <a href="#local-6989586621679068597">p2</a> <a href="#local-6989586621679068595">p1</a>) (<a href="Dist.Internal.html#toValues">toValues</a> @<a href="#local-6989586621679068542">a</a> <a href="#local-6989586621679068586">d</a>)

152−<a name="line-152"></a> <a name="local-6989586621679068593"><a href="#local-6989586621679068593">w</a></a> = maximum (map (length . show . fst) <a href="#local-6989586621679068592">values</a>)

153−<a name="line-153"></a> in concatMap

154−<a name="line-154"></a> (\(<a name="local-6989586621679068598"><a href="#local-6989586621679068598">x</a></a>,<a name="local-6989586621679068599"><a href="#local-6989586621679068599">p</a></a>) -> <a href="#local-6989586621679068588">showR</a> <a href="#local-6989586621679068593">w</a> <a href="#local-6989586621679068598">x</a> ++ ' ': <a href="#local-6989586621679068587">showProb</a> <a href="#local-6989586621679068599">p</a> ++ "\n")

155−<a name="line-155"></a> <a href="#local-6989586621679068592">values</a>

156−<a name="line-156"></a> where

157−<a name="line-157"></a> <a name="local-6989586621679068587"><a href="#local-6989586621679068587">showProb</a></a> <a name="local-6989586621679068589"><a href="#local-6989586621679068589">p</a></a> = show (<a href="#local-6989586621679068589">p</a> * 100) ++ "%"

158−<a name="line-158"></a> <a name="local-6989586621679068588"><a href="#local-6989586621679068588">showR</a></a> <a name="local-6989586621679068590"><a href="#local-6989586621679068590">n</a></a> <a name="local-6989586621679068591"><a href="#local-6989586621679068591">x</a></a> = show <a href="#local-6989586621679068591">x</a> ++ " "

159−<a name="line-159"></a>

160−<a name="line-160"></a>-- | Pretty Print a distribution

161−<a name="line-161"></a>prettyPrintDist :: forall <a name="local-6989586621679068541"><a href="#local-6989586621679068541">a</a></a> . (Show <a href="#local-6989586621679068541">a</a>, Enum <a href="#local-6989586621679068541">a</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679068541">a</a>), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068541">a</a>)) => <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068541">a</a> -> IO ()

162−<a name="line-162"></a><a name="prettyPrintDist"><a href="Dist.Internal.html#prettyPrintDist">prettyPrintDist</a></a> = putStrLn . <a href="Dist.Internal.html#prettyDist">prettyDist</a> @<a href="#local-6989586621679068541">a</a>

163−<a name="line-163"></a>

164−<a name="line-164"></a>-- Auxiliary functions

165−<a name="line-165"></a>

166−<a name="line-166"></a>fromFreqs :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="Dist.Internal.html#Prob">Prob</a> () (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679068540">a</a>)) => [(<a href="#local-6989586621679068540">a</a>,<a href="Dist.Internal.html#Prob">Prob</a>)] -> <a href="Dist.Internal.html#Dist">Dist</a> <a href="#local-6989586621679068540">a</a>

167−<a name="line-167"></a><a name="fromFreqs"><a href="Dist.Internal.html#fromFreqs">fromFreqs</a></a> <a name="local-6989586621679068600"><a href="#local-6989586621679068600">xs</a></a> = <a href="Dist.Internal.html#D">D</a> (<a href="Matrix.Type.html#col">col</a> (map (\(<a name="local-6989586621679068602"><a href="#local-6989586621679068602">x</a></a>,<a name="local-6989586621679068603"><a href="#local-6989586621679068603">p</a></a>) -> <a href="#local-6989586621679068603">p</a>/<a href="#local-6989586621679068601">q</a>) <a href="#local-6989586621679068600">xs</a>))

168−<a name="line-168"></a> where <a name="local-6989586621679068601"><a href="#local-6989586621679068601">q</a></a> = sum $ map snd <a href="#local-6989586621679068600">xs</a>

169−<a name="line-169"></a>

170−<a name="line-170"></a>normalCurve :: <a href="Dist.Internal.html#Prob">Prob</a> -> <a href="Dist.Internal.html#Prob">Prob</a> -> <a href="Dist.Internal.html#Prob">Prob</a> -> <a href="Dist.Internal.html#Prob">Prob</a>

171−<a name="line-171"></a><a name="normalCurve"><a href="Dist.Internal.html#normalCurve">normalCurve</a></a> <a name="local-6989586621679068604"><a href="#local-6989586621679068604">mean</a></a> <a name="local-6989586621679068605"><a href="#local-6989586621679068605">dev</a></a> <a name="local-6989586621679068606"><a href="#local-6989586621679068606">x</a></a> =

172−<a name="line-172"></a> let <a name="local-6989586621679068607"><a href="#local-6989586621679068607">u</a></a> = (<a href="#local-6989586621679068606">x</a> - <a href="#local-6989586621679068604">mean</a>) / <a href="#local-6989586621679068605">dev</a>

173−<a name="line-173"></a> in exp (-1/2 * <a href="#local-6989586621679068607">u</a>^(2::Int)) / sqrt (2 * pi)

174−<a name="line-174"></a></pre></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Dist.htmldeleted−70

@@ -1,70 +0,0 @@

1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>-----------------------------------------------------------------------------

2−<a name="line-2"></a>-- |

3−<a name="line-3"></a>-- Module : Dist

4−<a name="line-4"></a>-- Copyright : (c) Armando Santos 2019-2020

5−<a name="line-5"></a>-- Maintainer : armandoifsantos@gmail.com

6−<a name="line-6"></a>-- Stability : experimental

7−<a name="line-7"></a>--

8−<a name="line-8"></a>-- __LAoP__ is a library for algebraic (inductive) construction and manipulation of matrices

9−<a name="line-9"></a>-- in Haskell. See <https://github.com/bolt12/master-thesis my Msc Thesis> for the

10−<a name="line-10"></a>-- motivation behind the library, the underlying theory, and implementation details.

11−<a name="line-11"></a>--

12−<a name="line-12"></a>-- This module exports a type synonym 'Dist'

13−<a name="line-13"></a>-- that represents a stochastic distribution

14−<a name="line-14"></a>-- matrix and several distribution construction

15−<a name="line-15"></a>-- functions.

16−<a name="line-16"></a>--

17−<a name="line-17"></a>-----------------------------------------------------------------------------

18−<a name="line-18"></a>

19−<a name="line-19"></a>module Dist

20−<a name="line-20"></a> ( -- | If the sum of the rows of a column matrix

21−<a name="line-21"></a> -- is equal to 1 then this stochastic matrix

22−<a name="line-22"></a> -- can be seen as a probability distribution.

23−<a name="line-23"></a> --

24−<a name="line-24"></a> --

25−<a name="line-25"></a> -- This module is still experimental but it's

26−<a name="line-26"></a> -- already possible to model probabilistic programming

27−<a name="line-27"></a> -- problems with it. Import 'Matrix.Nat' or 'Matrix.Type'

28−<a name="line-28"></a> -- to access LAoP matrix combinators and then all you have

29−<a name="line-29"></a> -- to do is to define your sample space, either by creating a new data

30−<a name="line-30"></a> -- type or by abstracting it out via 'Natural'.

31−<a name="line-31"></a> --

32−<a name="line-32"></a> -- Write manipulation functions and promote them to matrices via

33−<a name="line-33"></a> -- 'fromF' or 'fromF'' and you're good to go!

34−<a name="line-34"></a>

35−<a name="line-35"></a> -- * 'Dist' and 'Prob' type synonyms

36−<a name="line-36"></a> <a href="Dist.Internal.html#Dist">Dist</a> (..),

37−<a name="line-37"></a> <a href="Dist.Internal.html#Prob">Prob</a>,

38−<a name="line-38"></a>

39−<a name="line-39"></a> -- * Functor instance equivalent functions

40−<a name="line-40"></a> <a href="Dist.Internal.html#fmapD">fmapD</a>,

41−<a name="line-41"></a>

42−<a name="line-42"></a> -- * Applicative equivalent functions

43−<a name="line-43"></a> <a href="Dist.Internal.html#unitD">unitD</a>,

44−<a name="line-44"></a> <a href="Dist.Internal.html#multD">multD</a>,

45−<a name="line-45"></a>

46−<a name="line-46"></a> -- * Selective equivalent functions

47−<a name="line-47"></a> <a href="Dist.Internal.html#selectD">selectD</a>,

48−<a name="line-48"></a>

49−<a name="line-49"></a> -- * Monad equivalent functions

50−<a name="line-50"></a> <a href="Dist.Internal.html#returnD">returnD</a>,

51−<a name="line-51"></a> <a href="Dist.Internal.html#bindD">bindD</a>,

52−<a name="line-52"></a>

53−<a name="line-53"></a> -- * Distribution construction functions

54−<a name="line-54"></a> <a href="Dist.Internal.html#choose">choose</a>,

55−<a name="line-55"></a> <a href="Dist.Internal.html#shape">shape</a>,

56−<a name="line-56"></a> <a href="Dist.Internal.html#linear">linear</a>,

57−<a name="line-57"></a> <a href="Dist.Internal.html#uniform">uniform</a>,

58−<a name="line-58"></a> <a href="Dist.Internal.html#negExp">negExp</a>,

59−<a name="line-59"></a> <a href="Dist.Internal.html#normal">normal</a>,

60−<a name="line-60"></a>

61−<a name="line-61"></a> -- * Converto to list of pairs

62−<a name="line-62"></a> <a href="Dist.Internal.html#toValues">toValues</a>,

63−<a name="line-63"></a> -- * Pretty print distribution

64−<a name="line-64"></a> <a href="Dist.Internal.html#prettyDist">prettyDist</a>,

65−<a name="line-65"></a> <a href="Dist.Internal.html#prettyPrintDist">prettyPrintDist</a>

66−<a name="line-66"></a> )

67−<a name="line-67"></a>where

68−<a name="line-68"></a>

69−<a name="line-69"></a>import <a href="Dist.Internal.html">Dist.Internal</a>

70−<a name="line-70"></a></pre></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Matrix.Internal.htmldeleted−903

@@ -1,903 +0,0 @@

1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>{-# LANGUAGE AllowAmbiguousTypes #-}

2−<a name="line-2"></a>{-# LANGUAGE DataKinds #-}

3−<a name="line-3"></a>{-# LANGUAGE FlexibleContexts #-}

4−<a name="line-4"></a>{-# LANGUAGE FlexibleInstances #-}

5−<a name="line-5"></a>{-# LANGUAGE GADTs #-}

6−<a name="line-6"></a>{-# LANGUAGE InstanceSigs #-}

7−<a name="line-7"></a>{-# LANGUAGE MultiParamTypeClasses #-}

8−<a name="line-8"></a>{-# LANGUAGE NoStarIsType #-}

9−<a name="line-9"></a>{-# LANGUAGE ScopedTypeVariables #-}

10−<a name="line-10"></a>{-# LANGUAGE StandaloneDeriving #-}

11−<a name="line-11"></a>{-# LANGUAGE TypeApplications #-}

12−<a name="line-12"></a>{-# LANGUAGE TypeFamilies #-}

13−<a name="line-13"></a>{-# LANGUAGE TypeOperators #-}

14−<a name="line-14"></a>{-# LANGUAGE UndecidableInstances #-}

15−<a name="line-15"></a>

16−<a name="line-16"></a>-----------------------------------------------------------------------------

17−<a name="line-17"></a>-- |

18−<a name="line-18"></a>-- Module : Matrix.Internal

19−<a name="line-19"></a>-- Copyright : (c) Armando Santos 2019-2020

20−<a name="line-20"></a>-- Maintainer : armandoifsantos@gmail.com

21−<a name="line-21"></a>-- Stability : experimental

22−<a name="line-22"></a>--

23−<a name="line-23"></a>-- The LAoP discipline generalises relations and functions treating them as

24−<a name="line-24"></a>-- Boolean matrices and in turn consider these as arrows.

25−<a name="line-25"></a>--

26−<a name="line-26"></a>-- __LAoP__ is a library for algebraic (inductive) construction and manipulation of matrices

27−<a name="line-27"></a>-- in Haskell. See <https://github.com/bolt12/master-thesis my Msc Thesis> for the

28−<a name="line-28"></a>-- motivation behind the library, the underlying theory, and implementation details.

29−<a name="line-29"></a>--

30−<a name="line-30"></a>-- This module offers many of the combinators mentioned in the work of

31−<a name="line-31"></a>-- Macedo (2012) and Oliveira (2012).

32−<a name="line-32"></a>--

33−<a name="line-33"></a>-- This is an Internal module and it is no supposed to be imported.

34−<a name="line-34"></a>--

35−<a name="line-35"></a>-----------------------------------------------------------------------------

36−<a name="line-36"></a>

37−<a name="line-37"></a>module Matrix.Internal

38−<a name="line-38"></a> ( -- | This definition makes use of the fact that 'Void' is

39−<a name="line-39"></a> -- isomorphic to 0 and '()' to 1 and captures matrix

40−<a name="line-40"></a> -- dimensions as stacks of 'Either's.

41−<a name="line-41"></a> --

42−<a name="line-42"></a> -- There exists two type families that make it easier to write

43−<a name="line-43"></a> -- matrix dimensions: 'FromNat' and 'Count'. This approach

44−<a name="line-44"></a> -- leads to a very straightforward implementation

45−<a name="line-45"></a> -- of LAoP combinators.

46−<a name="line-46"></a>

47−<a name="line-47"></a> -- * Type safe matrix representation

48−<a name="line-48"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> (..),

49−<a name="line-49"></a>

50−<a name="line-50"></a> -- * Primitives

51−<a name="line-51"></a> <a href="Matrix.Internal.html#empty">empty</a>,

52−<a name="line-52"></a> <a href="Matrix.Internal.html#one">one</a>,

53−<a name="line-53"></a> <a href="Matrix.Internal.html#junc">junc</a>,

54−<a name="line-54"></a> <a href="Matrix.Internal.html#split">split</a>,

55−<a name="line-55"></a>

56−<a name="line-56"></a> -- * Auxiliary type families

57−<a name="line-57"></a> <a href="Matrix.Internal.html#FromNat">FromNat</a>,

58−<a name="line-58"></a> <a href="Matrix.Internal.html#Count">Count</a>,

59−<a name="line-59"></a> <a href="Matrix.Internal.html#Normalize">Normalize</a>,

60−<a name="line-60"></a>

61−<a name="line-61"></a> -- * Matrix construction and conversion

62−<a name="line-62"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a>,

63−<a name="line-63"></a> <a href="Matrix.Internal.html#fromLists">fromLists</a>,

64−<a name="line-64"></a> <a href="Matrix.Internal.html#toLists">toLists</a>,

65−<a name="line-65"></a> <a href="Matrix.Internal.html#toList">toList</a>,

66−<a name="line-66"></a> <a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a>,

67−<a name="line-67"></a> <a href="Matrix.Internal.html#row">row</a>,

68−<a name="line-68"></a> <a href="Matrix.Internal.html#col">col</a>,

69−<a name="line-69"></a> <a href="Matrix.Internal.html#zeros">zeros</a>,

70−<a name="line-70"></a> <a href="Matrix.Internal.html#ones">ones</a>,

71−<a name="line-71"></a> <a href="Matrix.Internal.html#bang">bang</a>,

72−<a name="line-72"></a> <a href="Matrix.Internal.html#constant">constant</a>,

73−<a name="line-73"></a>

74−<a name="line-74"></a> -- * Misc

75−<a name="line-75"></a> -- ** Get dimensions

76−<a name="line-76"></a> <a href="Matrix.Internal.html#columns">columns</a>,

77−<a name="line-77"></a> <a href="Matrix.Internal.html#rows">rows</a>,

78−<a name="line-78"></a>

79−<a name="line-79"></a> -- ** Matrix Transposition

80−<a name="line-80"></a> <a href="Matrix.Internal.html#tr">tr</a>,

81−<a name="line-81"></a>

82−<a name="line-82"></a> -- ** Selective operator

83−<a name="line-83"></a> <a href="Matrix.Internal.html#select">select</a>,

84−<a name="line-84"></a>

85−<a name="line-85"></a> -- ** McCarthy's Conditional

86−<a name="line-86"></a> <a href="Matrix.Internal.html#cond">cond</a>,

87−<a name="line-87"></a>

88−<a name="line-88"></a> -- ** Matrix "abiding"

89−<a name="line-89"></a> <a href="Matrix.Internal.html#abideJS">abideJS</a>,

90−<a name="line-90"></a> <a href="Matrix.Internal.html#abideSJ">abideSJ</a>,

91−<a name="line-91"></a>

92−<a name="line-92"></a> -- * Biproduct approach

93−<a name="line-93"></a> -- ** Split

94−<a name="line-94"></a> (<a href="Matrix.Internal.html#%3D%3D%3D">===</a>),

95−<a name="line-95"></a> -- *** Projections

96−<a name="line-96"></a> <a href="Matrix.Internal.html#p1">p1</a>,

97−<a name="line-97"></a> <a href="Matrix.Internal.html#p2">p2</a>,

98−<a name="line-98"></a> -- ** Junc

99−<a name="line-99"></a> (<a href="Matrix.Internal.html#%7C%7C%7C">|||</a>),

100−<a name="line-100"></a> -- *** Injections

101−<a name="line-101"></a> <a href="Matrix.Internal.html#i1">i1</a>,

102−<a name="line-102"></a> <a href="Matrix.Internal.html#i2">i2</a>,

103−<a name="line-103"></a> -- ** Bifunctors

104−<a name="line-104"></a> (<a href="Matrix.Internal.html#-%7C-">-|-</a>),

105−<a name="line-105"></a> (<a href="Matrix.Internal.html#%3E%3C">><</a>),

106−<a name="line-106"></a>

107−<a name="line-107"></a> -- ** Applicative matrix combinators

108−<a name="line-108"></a>

109−<a name="line-109"></a> -- | Note that given the restrictions imposed it is not possible to

110−<a name="line-110"></a> -- implement the standard type classes present in standard Haskell.

111−<a name="line-111"></a> -- *** Matrix pairing projections

112−<a name="line-112"></a> <a href="Matrix.Internal.html#kp1">kp1</a>,

113−<a name="line-113"></a> <a href="Matrix.Internal.html#kp2">kp2</a>,

114−<a name="line-114"></a>

115−<a name="line-115"></a> -- *** Matrix pairing

116−<a name="line-116"></a> <a href="Matrix.Internal.html#khatri">khatri</a>,

117−<a name="line-117"></a>

118−<a name="line-118"></a> -- * Matrix composition and lifting

119−<a name="line-119"></a>

120−<a name="line-120"></a> -- ** Arrow matrix combinators

121−<a name="line-121"></a>

122−<a name="line-122"></a> -- | Note that given the restrictions imposed it is not possible to

123−<a name="line-123"></a> -- implement the standard type classes present in standard Haskell.

124−<a name="line-124"></a> <a href="Matrix.Internal.html#identity">identity</a>,

125−<a name="line-125"></a> <a href="Matrix.Internal.html#comp">comp</a>,

126−<a name="line-126"></a> <a href="Matrix.Internal.html#fromF">fromF</a>,

127−<a name="line-127"></a> <a href="Matrix.Internal.html#fromF%27">fromF'</a>,

128−<a name="line-128"></a>

129−<a name="line-129"></a> -- * Matrix printing

130−<a name="line-130"></a> <a href="Matrix.Internal.html#pretty">pretty</a>,

131−<a name="line-131"></a> <a href="Matrix.Internal.html#prettyPrint">prettyPrint</a>,

132−<a name="line-132"></a>

133−<a name="line-133"></a> -- * Other

134−<a name="line-134"></a> <a href="Matrix.Internal.html#toBool">toBool</a>,

135−<a name="line-135"></a> <a href="Matrix.Internal.html#fromBool">fromBool</a>,

136−<a name="line-136"></a> <a href="Matrix.Internal.html#compRel">compRel</a>,

137−<a name="line-137"></a> <a href="Matrix.Internal.html#fromFRel">fromFRel</a>,

138−<a name="line-138"></a> <a href="Matrix.Internal.html#fromFRel%27">fromFRel'</a>,

139−<a name="line-139"></a> <a href="Matrix.Internal.html#orM">orM</a>,

140−<a name="line-140"></a> <a href="Matrix.Internal.html#andM">andM</a>,

141−<a name="line-141"></a> <a href="Matrix.Internal.html#subM">subM</a>

142−<a name="line-142"></a> )

143−<a name="line-143"></a> where

144−<a name="line-144"></a>

145−<a name="line-145"></a>import <a href="Utils.html">Utils</a>

146−<a name="line-146"></a>import Data.Bool

147−<a name="line-147"></a>import Data.Kind

148−<a name="line-148"></a>import Data.List

149−<a name="line-149"></a>import Data.Proxy

150−<a name="line-150"></a>import Data.Void

151−<a name="line-151"></a>import GHC.TypeLits

152−<a name="line-152"></a>import Data.Type.Equality

153−<a name="line-153"></a>import GHC.Generics

154−<a name="line-154"></a>import Control.DeepSeq

155−<a name="line-155"></a>import Control.Category

156−<a name="line-156"></a>import Prelude hiding ((.))

157−<a name="line-157"></a>

158−<a name="line-158"></a>-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.

159−<a name="line-159"></a>data <a name="Matrix"><a href="Matrix.Internal.html#Matrix">Matrix</a></a> <a name="local-6989586621679038240"><a href="#local-6989586621679038240">e</a></a> <a name="local-6989586621679038241"><a href="#local-6989586621679038241">cols</a></a> <a name="local-6989586621679038242"><a href="#local-6989586621679038242">rows</a></a> where

160−<a name="line-160"></a> <a name="Empty"><a href="Matrix.Internal.html#Empty">Empty</a></a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038243">e</a> Void Void

161−<a name="line-161"></a> <a name="One"><a href="Matrix.Internal.html#One">One</a></a> :: <a href="#local-6989586621679038244">e</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038244">e</a> () ()

162−<a name="line-162"></a> <a name="Junc"><a href="Matrix.Internal.html#Junc">Junc</a></a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038245">e</a> <a href="#local-6989586621679038246">a</a> <a href="#local-6989586621679038247">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038245">e</a> <a href="#local-6989586621679038248">b</a> <a href="#local-6989586621679038247">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038245">e</a> (Either <a href="#local-6989586621679038246">a</a> <a href="#local-6989586621679038248">b</a>) <a href="#local-6989586621679038247">rows</a>

163−<a name="line-163"></a> <a name="Split"><a href="Matrix.Internal.html#Split">Split</a></a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038249">e</a> <a href="#local-6989586621679038250">cols</a> <a href="#local-6989586621679038251">a</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038249">e</a> <a href="#local-6989586621679038250">cols</a> <a href="#local-6989586621679038252">b</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038249">e</a> <a href="#local-6989586621679038250">cols</a> (Either <a href="#local-6989586621679038251">a</a> <a href="#local-6989586621679038252">b</a>)

164−<a name="line-164"></a>

165−<a name="line-165"></a>deriving instance (Show <a href="#local-6989586621679039296">e</a>) => Show (<a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679039296">e</a> <a href="#local-6989586621679039297">cols</a> <a href="#local-6989586621679039298">rows</a>)

166−<a name="line-166"></a>

167−<a name="line-167"></a>-- | Type family that computes the cardinality of a given type dimension.

168−<a name="line-168"></a>--

169−<a name="line-169"></a>-- It can also count the cardinality of custom types that implement the

170−<a name="line-170"></a>-- 'Generic' instance.

171−<a name="line-171"></a>type family <a name="Count"><a href="Matrix.Internal.html#Count">Count</a></a> (<a name="local-6989586621679038215"><a href="#local-6989586621679038215">d</a></a> :: Type) :: Nat where

172−<a name="line-172"></a> Count (<a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679038216">n</a> <a href="#local-6989586621679038217">m</a>) = (m - <a href="#local-6989586621679038216">n</a>) + 1

173−<a name="line-173"></a> Count (Either <a href="#local-6989586621679038218">a</a> <a href="#local-6989586621679038219">b</a>) = (+) (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038218">a</a>) (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038219">b</a>)

174−<a name="line-174"></a> Count (<a href="#local-6989586621679038220">a</a>, <a href="#local-6989586621679038221">b</a>) = (*) (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038220">a</a>) (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038221">b</a>)

175−<a name="line-175"></a> Count (<a href="#local-6989586621679038222">a</a> -> <a href="#local-6989586621679038223">b</a>) = (^) (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038223">b</a>) (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038222">a</a>)

176−<a name="line-176"></a> -- Generics

177−<a name="line-177"></a> Count (M1 _ _ <a href="#local-6989586621679038224">f</a> <a href="#local-6989586621679038225">p</a>) = <a href="Matrix.Internal.html#Count">Count</a> (<a href="#local-6989586621679038224">f</a> <a href="#local-6989586621679038225">p</a>)

178−<a name="line-178"></a> Count (K1 _ _ _) = 1

179−<a name="line-179"></a> Count (V1 _) = 0

180−<a name="line-180"></a> Count (U1 _) = 1

181−<a name="line-181"></a> Count ((:*:) <a href="#local-6989586621679038233">a</a> <a href="#local-6989586621679038234">b</a> <a href="#local-6989586621679038235">p</a>) = Count (<a href="#local-6989586621679038233">a</a> <a href="#local-6989586621679038235">p</a>) * <a href="Matrix.Internal.html#Count">Count</a> (<a href="#local-6989586621679038234">b</a> <a href="#local-6989586621679038235">p</a>)

182−<a name="line-182"></a> Count ((:+:) <a href="#local-6989586621679038236">a</a> <a href="#local-6989586621679038237">b</a> <a href="#local-6989586621679038238">p</a>) = Count (<a href="#local-6989586621679038236">a</a> <a href="#local-6989586621679038238">p</a>) + <a href="Matrix.Internal.html#Count">Count</a> (<a href="#local-6989586621679038237">b</a> <a href="#local-6989586621679038238">p</a>)

183−<a name="line-183"></a> Count <a href="#local-6989586621679038239">d</a> = <a href="Matrix.Internal.html#Count">Count</a> (Rep <a href="#local-6989586621679038239">d</a> R)

184−<a name="line-184"></a>

185−<a name="line-185"></a>-- | Type family that computes of a given type dimension from a given natural

186−<a name="line-186"></a>--

187−<a name="line-187"></a>-- Thanks to Li-Yao Xia this type family is super fast.

188−<a name="line-188"></a>type family <a name="FromNat"><a href="Matrix.Internal.html#FromNat">FromNat</a></a> (<a name="local-6989586621679038213"><a href="#local-6989586621679038213">n</a></a> :: Nat) :: Type where

189−<a name="line-189"></a> FromNat 0 = Void

190−<a name="line-190"></a> FromNat 1 = ()

191−<a name="line-191"></a> FromNat <a href="#local-6989586621679038214">n</a> = <a href="Matrix.Internal.html#FromNat%27">FromNat'</a> (Mod <a href="#local-6989586621679038214">n</a> 2 == 0) (<a href="Matrix.Internal.html#FromNat">FromNat</a> (Div <a href="#local-6989586621679038214">n</a> 2))

192−<a name="line-192"></a>

193−<a name="line-193"></a>type family <a name="FromNat%27"><a href="Matrix.Internal.html#FromNat%27">FromNat'</a></a> (<a name="local-6989586621679038066"><a href="#local-6989586621679038066">b</a></a> :: Bool) (<a name="local-6989586621679038067"><a href="#local-6989586621679038067">m</a></a> :: Type) :: Type where

194−<a name="line-194"></a> FromNat' 'True <a href="#local-6989586621679038068">m</a> = Either <a href="#local-6989586621679038068">m</a> <a href="#local-6989586621679038068">m</a>

195−<a name="line-195"></a> FromNat' 'False <a href="#local-6989586621679038212">m</a> = Either () (Either <a href="#local-6989586621679038212">m</a> <a href="#local-6989586621679038212">m</a>)

196−<a name="line-196"></a>

197−<a name="line-197"></a>-- | Type family that normalizes the representation of a given data

198−<a name="line-198"></a>-- structure

199−<a name="line-199"></a>type family <a name="Normalize"><a href="Matrix.Internal.html#Normalize">Normalize</a></a> (<a name="local-6989586621679038064"><a href="#local-6989586621679038064">d</a></a> :: Type) :: Type where

200−<a name="line-200"></a> Normalize <a href="#local-6989586621679038065">d</a> = <a href="Matrix.Internal.html#FromNat">FromNat</a> (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038065">d</a>)

201−<a name="line-201"></a>

202−<a name="line-202"></a>-- | It isn't possible to implement the 'id' function so it's

203−<a name="line-203"></a>-- implementation is 'undefined'. However 'comp' can be and this partial

204−<a name="line-204"></a>-- class implementation exists just to make the code more readable.

205−<a name="line-205"></a>--

206−<a name="line-206"></a>-- Please use 'identity' instead.

207−<a name="line-207"></a>instance (Num <a href="#local-6989586621679038572">e</a>) => Category (<a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038572">e</a>) where

208−<a name="line-208"></a> <a name="local-8214565720323789508">id</a> = undefined

209−<a name="line-209"></a> (<a name="local-8214565720323789507">.</a>) = <a href="Matrix.Internal.html#comp">comp</a>

210−<a name="line-210"></a>

211−<a name="line-211"></a>instance NFData <a href="#local-6989586621679038564">e</a> => NFData (<a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038564">e</a> <a href="#local-6989586621679038565">cols</a> <a href="#local-6989586621679038566">rows</a>) where

212−<a name="line-212"></a> <a name="local-8214565720323790686">rnf</a> <a href="Matrix.Internal.html#Empty">Empty</a> = ()

213−<a name="line-213"></a> rnf (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038567"><a href="#local-6989586621679038567">e</a></a>) = rnf <a href="#local-6989586621679038567">e</a>

214−<a name="line-214"></a> rnf (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038568"><a href="#local-6989586621679038568">a</a></a> <a name="local-6989586621679038569"><a href="#local-6989586621679038569">b</a></a>) = rnf <a href="#local-6989586621679038568">a</a> `seq` rnf <a href="#local-6989586621679038569">b</a>

215−<a name="line-215"></a> rnf (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038570"><a href="#local-6989586621679038570">a</a></a> <a name="local-6989586621679038571"><a href="#local-6989586621679038571">b</a></a>) = rnf <a href="#local-6989586621679038570">a</a> `seq` rnf <a href="#local-6989586621679038571">b</a>

216−<a name="line-216"></a>

217−<a name="line-217"></a>instance Eq <a href="#local-6989586621679038547">e</a> => Eq (<a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038547">e</a> <a href="#local-6989586621679038548">cols</a> <a href="#local-6989586621679038549">rows</a>) where

218−<a name="line-218"></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a name="local-3458764513820541095">==</a> <a href="Matrix.Internal.html#Empty">Empty</a> = True

219−<a name="line-219"></a> (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038550"><a href="#local-6989586621679038550">a</a></a>) == (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038551"><a href="#local-6989586621679038551">b</a></a>) = <a href="#local-6989586621679038550">a</a> == <a href="#local-6989586621679038551">b</a>

220−<a name="line-220"></a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038552"><a href="#local-6989586621679038552">a</a></a> <a name="local-6989586621679038553"><a href="#local-6989586621679038553">b</a></a>) == (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038554"><a href="#local-6989586621679038554">c</a></a> <a name="local-6989586621679038555"><a href="#local-6989586621679038555">d</a></a>) = <a href="#local-6989586621679038552">a</a> == <a href="#local-6989586621679038554">c</a> && <a href="#local-6989586621679038553">b</a> == <a href="#local-6989586621679038555">d</a>

221−<a name="line-221"></a> (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038556"><a href="#local-6989586621679038556">a</a></a> <a name="local-6989586621679038557"><a href="#local-6989586621679038557">b</a></a>) == (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038558"><a href="#local-6989586621679038558">c</a></a> <a name="local-6989586621679038559"><a href="#local-6989586621679038559">d</a></a>) = <a href="#local-6989586621679038556">a</a> == <a href="#local-6989586621679038558">c</a> && <a href="#local-6989586621679038557">b</a> == <a href="#local-6989586621679038559">d</a>

222−<a name="line-222"></a> <a name="local-6989586621679038560"><a href="#local-6989586621679038560">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) == <a name="local-6989586621679038561"><a href="#local-6989586621679038561">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="#local-6989586621679038560">x</a> == <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038561">y</a>

223−<a name="line-223"></a> <a name="local-6989586621679038562"><a href="#local-6989586621679038562">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) == <a name="local-6989586621679038563"><a href="#local-6989586621679038563">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038562">x</a> == <a href="#local-6989586621679038563">y</a>

224−<a name="line-224"></a>

225−<a name="line-225"></a>instance Num <a href="#local-6989586621679038492">e</a> => Num (<a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038492">e</a> <a href="#local-6989586621679038493">cols</a> <a href="#local-6989586621679038494">rows</a>) where

226−<a name="line-226"></a>

227−<a name="line-227"></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a name="local-8214565720323790403">+</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

228−<a name="line-228"></a> (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038495"><a href="#local-6989586621679038495">a</a></a>) + (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038496"><a href="#local-6989586621679038496">b</a></a>) = <a href="Matrix.Internal.html#One">One</a> (<a href="#local-6989586621679038495">a</a> + <a href="#local-6989586621679038496">b</a>)

229−<a name="line-229"></a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038497"><a href="#local-6989586621679038497">a</a></a> <a name="local-6989586621679038498"><a href="#local-6989586621679038498">b</a></a>) + (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038499"><a href="#local-6989586621679038499">c</a></a> <a name="local-6989586621679038500"><a href="#local-6989586621679038500">d</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="#local-6989586621679038497">a</a> + <a href="#local-6989586621679038499">c</a>) (<a href="#local-6989586621679038498">b</a> + <a href="#local-6989586621679038500">d</a>)

230−<a name="line-230"></a> (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038501"><a href="#local-6989586621679038501">a</a></a> <a name="local-6989586621679038502"><a href="#local-6989586621679038502">b</a></a>) + (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038503"><a href="#local-6989586621679038503">c</a></a> <a name="local-6989586621679038504"><a href="#local-6989586621679038504">d</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="#local-6989586621679038501">a</a> + <a href="#local-6989586621679038503">c</a>) (<a href="#local-6989586621679038502">b</a> + <a href="#local-6989586621679038504">d</a>)

231−<a name="line-231"></a> <a name="local-6989586621679038505"><a href="#local-6989586621679038505">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) + <a name="local-6989586621679038506"><a href="#local-6989586621679038506">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="#local-6989586621679038505">x</a> + <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038506">y</a>

232−<a name="line-232"></a> <a name="local-6989586621679038507"><a href="#local-6989586621679038507">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) + <a name="local-6989586621679038508"><a href="#local-6989586621679038508">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038507">x</a> + <a href="#local-6989586621679038508">y</a>

233−<a name="line-233"></a>

234−<a name="line-234"></a> <a href="Matrix.Internal.html#Empty">Empty</a> - <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

235−<a name="line-235"></a> (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038509"><a href="#local-6989586621679038509">a</a></a>) - (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038510"><a href="#local-6989586621679038510">b</a></a>) = <a href="Matrix.Internal.html#One">One</a> (<a href="#local-6989586621679038509">a</a> - <a href="#local-6989586621679038510">b</a>)

236−<a name="line-236"></a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038511"><a href="#local-6989586621679038511">a</a></a> <a name="local-6989586621679038512"><a href="#local-6989586621679038512">b</a></a>) - (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038513"><a href="#local-6989586621679038513">c</a></a> <a name="local-6989586621679038514"><a href="#local-6989586621679038514">d</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="#local-6989586621679038511">a</a> - <a href="#local-6989586621679038513">c</a>) (<a href="#local-6989586621679038512">b</a> - <a href="#local-6989586621679038514">d</a>)

237−<a name="line-237"></a> (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038515"><a href="#local-6989586621679038515">a</a></a> <a name="local-6989586621679038516"><a href="#local-6989586621679038516">b</a></a>) - (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038517"><a href="#local-6989586621679038517">c</a></a> <a name="local-6989586621679038518"><a href="#local-6989586621679038518">d</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="#local-6989586621679038515">a</a> - <a href="#local-6989586621679038517">c</a>) (<a href="#local-6989586621679038516">b</a> - <a href="#local-6989586621679038518">d</a>)

238−<a name="line-238"></a> <a name="local-6989586621679038519"><a href="#local-6989586621679038519">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) - <a name="local-6989586621679038520"><a href="#local-6989586621679038520">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="#local-6989586621679038519">x</a> - <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038520">y</a>

239−<a name="line-239"></a> <a name="local-6989586621679038521"><a href="#local-6989586621679038521">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) - <a name="local-6989586621679038522"><a href="#local-6989586621679038522">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038521">x</a> - <a href="#local-6989586621679038522">y</a>

240−<a name="line-240"></a>

241−<a name="line-241"></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a name="local-8214565720323790402">*</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

242−<a name="line-242"></a> (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038523"><a href="#local-6989586621679038523">a</a></a>) * (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038524"><a href="#local-6989586621679038524">b</a></a>) = <a href="Matrix.Internal.html#One">One</a> (<a href="#local-6989586621679038523">a</a> * <a href="#local-6989586621679038524">b</a>)

243−<a name="line-243"></a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038525"><a href="#local-6989586621679038525">a</a></a> <a name="local-6989586621679038526"><a href="#local-6989586621679038526">b</a></a>) * (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038527"><a href="#local-6989586621679038527">c</a></a> <a name="local-6989586621679038528"><a href="#local-6989586621679038528">d</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="#local-6989586621679038525">a</a> * <a href="#local-6989586621679038527">c</a>) (<a href="#local-6989586621679038526">b</a> * <a href="#local-6989586621679038528">d</a>)

244−<a name="line-244"></a> (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038529"><a href="#local-6989586621679038529">a</a></a> <a name="local-6989586621679038530"><a href="#local-6989586621679038530">b</a></a>) * (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038531"><a href="#local-6989586621679038531">c</a></a> <a name="local-6989586621679038532"><a href="#local-6989586621679038532">d</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="#local-6989586621679038529">a</a> * <a href="#local-6989586621679038531">c</a>) (<a href="#local-6989586621679038530">b</a> * <a href="#local-6989586621679038532">d</a>)

245−<a name="line-245"></a> <a name="local-6989586621679038533"><a href="#local-6989586621679038533">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) * <a name="local-6989586621679038534"><a href="#local-6989586621679038534">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="#local-6989586621679038533">x</a> * <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038534">y</a>

246−<a name="line-246"></a> <a name="local-6989586621679038535"><a href="#local-6989586621679038535">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) * <a name="local-6989586621679038536"><a href="#local-6989586621679038536">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038535">x</a> * <a href="#local-6989586621679038536">y</a>

247−<a name="line-247"></a>

248−<a name="line-248"></a> <a name="local-8214565720323790401">abs</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

249−<a name="line-249"></a> abs (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038537"><a href="#local-6989586621679038537">a</a></a>) = <a href="Matrix.Internal.html#One">One</a> (abs <a href="#local-6989586621679038537">a</a>)

250−<a name="line-250"></a> abs (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038538"><a href="#local-6989586621679038538">a</a></a> <a name="local-6989586621679038539"><a href="#local-6989586621679038539">b</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (abs <a href="#local-6989586621679038538">a</a>) (abs <a href="#local-6989586621679038539">b</a>)

251−<a name="line-251"></a> abs (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038540"><a href="#local-6989586621679038540">a</a></a> <a name="local-6989586621679038541"><a href="#local-6989586621679038541">b</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (abs <a href="#local-6989586621679038540">a</a>) (abs <a href="#local-6989586621679038541">b</a>)

252−<a name="line-252"></a>

253−<a name="line-253"></a> <a name="local-8214565720323790400">signum</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

254−<a name="line-254"></a> signum (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038542"><a href="#local-6989586621679038542">a</a></a>) = <a href="Matrix.Internal.html#One">One</a> (signum <a href="#local-6989586621679038542">a</a>)

255−<a name="line-255"></a> signum (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038543"><a href="#local-6989586621679038543">a</a></a> <a name="local-6989586621679038544"><a href="#local-6989586621679038544">b</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (signum <a href="#local-6989586621679038543">a</a>) (signum <a href="#local-6989586621679038544">b</a>)

256−<a name="line-256"></a> signum (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038545"><a href="#local-6989586621679038545">a</a></a> <a name="local-6989586621679038546"><a href="#local-6989586621679038546">b</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (signum <a href="#local-6989586621679038545">a</a>) (signum <a href="#local-6989586621679038546">b</a>)

257−<a name="line-257"></a>

258−<a name="line-258"></a>instance Ord <a href="#local-6989586621679038475">e</a> => Ord (<a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038475">e</a> <a href="#local-6989586621679038476">cols</a> <a href="#local-6989586621679038477">rows</a>) where

259−<a name="line-259"></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a name="local-8214565720323790495"><=</a> <a href="Matrix.Internal.html#Empty">Empty</a> = True

260−<a name="line-260"></a> (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038478"><a href="#local-6989586621679038478">a</a></a>) <= (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038479"><a href="#local-6989586621679038479">b</a></a>) = <a href="#local-6989586621679038478">a</a> <= <a href="#local-6989586621679038479">b</a>

261−<a name="line-261"></a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038480"><a href="#local-6989586621679038480">a</a></a> <a name="local-6989586621679038481"><a href="#local-6989586621679038481">b</a></a>) <= (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038482"><a href="#local-6989586621679038482">c</a></a> <a name="local-6989586621679038483"><a href="#local-6989586621679038483">d</a></a>) = (<a href="#local-6989586621679038480">a</a> <= <a href="#local-6989586621679038482">c</a>) && (<a href="#local-6989586621679038481">b</a> <= <a href="#local-6989586621679038483">d</a>)

262−<a name="line-262"></a> (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038484"><a href="#local-6989586621679038484">a</a></a> <a name="local-6989586621679038485"><a href="#local-6989586621679038485">b</a></a>) <= (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038486"><a href="#local-6989586621679038486">c</a></a> <a name="local-6989586621679038487"><a href="#local-6989586621679038487">d</a></a>) = (<a href="#local-6989586621679038484">a</a> <= <a href="#local-6989586621679038486">c</a>) && (<a href="#local-6989586621679038485">b</a> <= <a href="#local-6989586621679038487">d</a>)

263−<a name="line-263"></a> <a name="local-6989586621679038488"><a href="#local-6989586621679038488">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) <= <a name="local-6989586621679038489"><a href="#local-6989586621679038489">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="#local-6989586621679038488">x</a> <= <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038489">y</a>

264−<a name="line-264"></a> <a name="local-6989586621679038490"><a href="#local-6989586621679038490">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) <= <a name="local-6989586621679038491"><a href="#local-6989586621679038491">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038490">x</a> <= <a href="#local-6989586621679038491">y</a>

265−<a name="line-265"></a>

266−<a name="line-266"></a>-- Primitives

267−<a name="line-267"></a>

268−<a name="line-268"></a>-- | Empty matrix constructor

269−<a name="line-269"></a>empty :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038712">e</a> Void Void

270−<a name="line-270"></a><a name="empty"><a href="Matrix.Internal.html#empty">empty</a></a> = <a href="Matrix.Internal.html#Empty">Empty</a>

271−<a name="line-271"></a>

272−<a name="line-272"></a>-- | Unit matrix constructor

273−<a name="line-273"></a>one :: <a href="#local-6989586621679038711">e</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038711">e</a> () ()

274−<a name="line-274"></a><a name="one"><a href="Matrix.Internal.html#one">one</a></a> = <a href="Matrix.Internal.html#One">One</a>

275−<a name="line-275"></a>

276−<a name="line-276"></a>-- | Matrix 'Junc' constructor

277−<a name="line-277"></a>junc :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038707">e</a> <a href="#local-6989586621679038708">a</a> <a href="#local-6989586621679038709">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038707">e</a> <a href="#local-6989586621679038710">b</a> <a href="#local-6989586621679038709">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038707">e</a> (Either <a href="#local-6989586621679038708">a</a> <a href="#local-6989586621679038710">b</a>) <a href="#local-6989586621679038709">rows</a>

278−<a name="line-278"></a><a name="junc"><a href="Matrix.Internal.html#junc">junc</a></a> = <a href="Matrix.Internal.html#Junc">Junc</a>

279−<a name="line-279"></a>

280−<a name="line-280"></a>infixl 3 <a href="Matrix.Internal.html#%7C%7C%7C">|||</a>

281−<a name="line-281"></a>

282−<a name="line-282"></a>-- | Matrix 'Junc' constructor

283−<a name="line-283"></a>(|||) :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038703">e</a> <a href="#local-6989586621679038704">a</a> <a href="#local-6989586621679038705">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038703">e</a> <a href="#local-6989586621679038706">b</a> <a href="#local-6989586621679038705">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038703">e</a> (Either <a href="#local-6989586621679038704">a</a> <a href="#local-6989586621679038706">b</a>) <a href="#local-6989586621679038705">rows</a>

284−<a name="line-284"></a>(<a name="%7C%7C%7C"><a href="Matrix.Internal.html#%7C%7C%7C">|||</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a>

285−<a name="line-285"></a>

286−<a name="line-286"></a>-- | Matrix 'Split' constructor

287−<a name="line-287"></a>split :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038699">e</a> <a href="#local-6989586621679038700">cols</a> <a href="#local-6989586621679038701">a</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038699">e</a> <a href="#local-6989586621679038700">cols</a> <a href="#local-6989586621679038702">b</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038699">e</a> <a href="#local-6989586621679038700">cols</a> (Either <a href="#local-6989586621679038701">a</a> <a href="#local-6989586621679038702">b</a>)

288−<a name="line-288"></a><a name="split"><a href="Matrix.Internal.html#split">split</a></a> = <a href="Matrix.Internal.html#Split">Split</a>

289−<a name="line-289"></a>

290−<a name="line-290"></a>infixl 2 <a href="Matrix.Internal.html#%3D%3D%3D">===</a>

291−<a name="line-291"></a>

292−<a name="line-292"></a>-- | Matrix 'Split' constructor

293−<a name="line-293"></a>(===) :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038695">e</a> <a href="#local-6989586621679038696">cols</a> <a href="#local-6989586621679038697">a</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038695">e</a> <a href="#local-6989586621679038696">cols</a> <a href="#local-6989586621679038698">b</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038695">e</a> <a href="#local-6989586621679038696">cols</a> (Either <a href="#local-6989586621679038697">a</a> <a href="#local-6989586621679038698">b</a>)

294−<a name="line-294"></a>(<a name="%3D%3D%3D"><a href="Matrix.Internal.html#%3D%3D%3D">===</a></a>) = <a href="Matrix.Internal.html#Split">Split</a>

295−<a name="line-295"></a>

296−<a name="line-296"></a>-- Construction

297−<a name="line-297"></a>

298−<a name="line-298"></a>-- | Type class for defining the 'fromList' conversion function.

299−<a name="line-299"></a>--

300−<a name="line-300"></a>-- Given that it is not possible to branch on types at the term level type

301−<a name="line-301"></a>-- classes are needed bery much like an inductive definition but on types.

302−<a name="line-302"></a>class <a name="FromLists"><a href="Matrix.Internal.html#FromLists">FromLists</a></a> <a name="local-6989586621679038061"><a href="#local-6989586621679038061">e</a></a> <a name="local-6989586621679038062"><a href="#local-6989586621679038062">cols</a></a> <a name="local-6989586621679038063"><a href="#local-6989586621679038063">rows</a></a> where

303−<a name="line-303"></a> -- | Build a matrix out of a list of list of elements. Throws a runtime

304−<a name="line-304"></a> -- error if the dimensions do not match.

305−<a name="line-305"></a> <a name="fromLists"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> :: [[<a href="#local-6989586621679038061">e</a>]] -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038061">e</a> <a href="#local-6989586621679038062">cols</a> <a href="#local-6989586621679038063">rows</a>

306−<a name="line-306"></a>

307−<a name="line-307"></a>instance <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038474">e</a> Void Void where

308−<a name="line-308"></a> <a name="local-8214565720323812914"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> [] = <a href="Matrix.Internal.html#Empty">Empty</a>

309−<a name="line-309"></a> fromLists _ = error "Wrong dimensions"

310−<a name="line-310"></a>

311−<a name="line-311"></a>instance {-# OVERLAPPING #-} <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038472">e</a> () () where

312−<a name="line-312"></a> <a name="local-8214565720323812914"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> [[<a name="local-6989586621679038473"><a href="#local-6989586621679038473">e</a></a>]] = <a href="Matrix.Internal.html#One">One</a> <a href="#local-6989586621679038473">e</a>

313−<a name="line-313"></a> fromLists _ = error "Wrong dimensions"

314−<a name="line-314"></a>

315−<a name="line-315"></a>instance {-# OVERLAPPING #-} (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038468">e</a> <a href="#local-6989586621679038469">cols</a> ()) => <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038468">e</a> (Either () <a href="#local-6989586621679038469">cols</a>) () where

316−<a name="line-316"></a> <a name="local-8214565720323812914"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> [<a name="local-6989586621679038470"><a href="#local-6989586621679038470">h</a></a> : <a name="local-6989586621679038471"><a href="#local-6989586621679038471">t</a></a>] = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#One">One</a> <a href="#local-6989586621679038470">h</a>) (<a href="Matrix.Internal.html#fromLists">fromLists</a> [<a href="#local-6989586621679038471">t</a>])

317−<a name="line-317"></a> fromLists _ = error "Wrong dimensions"

318−<a name="line-318"></a>

319−<a name="line-319"></a>instance {-# OVERLAPPABLE #-} (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038463">e</a> <a href="#local-6989586621679038464">a</a> (), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038463">e</a> <a href="#local-6989586621679038465">b</a> (), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038464">a</a>)) => <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038463">e</a> (Either <a href="#local-6989586621679038464">a</a> <a href="#local-6989586621679038465">b</a>) () where

320−<a name="line-320"></a> <a name="local-8214565720323812914"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> [<a name="local-6989586621679038466"><a href="#local-6989586621679038466">l</a></a>] =

321−<a name="line-321"></a> let <a name="local-6989586621679038467"><a href="#local-6989586621679038467">rowsA</a></a> = fromInteger (natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038464">a</a>)))

322−<a name="line-322"></a> in <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#fromLists">fromLists</a> [take <a href="#local-6989586621679038467">rowsA</a> <a href="#local-6989586621679038466">l</a>]) (<a href="Matrix.Internal.html#fromLists">fromLists</a> [drop <a href="#local-6989586621679038467">rowsA</a> <a href="#local-6989586621679038466">l</a>])

323−<a name="line-323"></a> fromLists _ = error "Wrong dimensions"

324−<a name="line-324"></a>

325−<a name="line-325"></a>instance {-# OVERLAPPING #-} (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038459">e</a> () <a href="#local-6989586621679038460">rows</a>) => <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038459">e</a> () (Either () <a href="#local-6989586621679038460">rows</a>) where

326−<a name="line-326"></a> <a name="local-8214565720323812914"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> ([<a name="local-6989586621679038461"><a href="#local-6989586621679038461">h</a></a>] : <a name="local-6989586621679038462"><a href="#local-6989586621679038462">t</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#One">One</a> <a href="#local-6989586621679038461">h</a>) (<a href="Matrix.Internal.html#fromLists">fromLists</a> <a href="#local-6989586621679038462">t</a>)

327−<a name="line-327"></a> fromLists _ = error "Wrong dimensions"

328−<a name="line-328"></a>

329−<a name="line-329"></a>instance {-# OVERLAPPABLE #-} (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038454">e</a> () <a href="#local-6989586621679038455">a</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038454">e</a> () <a href="#local-6989586621679038456">b</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038455">a</a>)) => <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038454">e</a> () (Either <a href="#local-6989586621679038455">a</a> <a href="#local-6989586621679038456">b</a>) where

330−<a name="line-330"></a> <a name="local-8214565720323812914"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> <a name="local-6989586621679038457"><a href="#local-6989586621679038457">l</a></a>@([_] : _) =

331−<a name="line-331"></a> let <a name="local-6989586621679038458"><a href="#local-6989586621679038458">rowsA</a></a> = fromInteger (natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038455">a</a>)))

332−<a name="line-332"></a> in <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#fromLists">fromLists</a> (take <a href="#local-6989586621679038458">rowsA</a> <a href="#local-6989586621679038457">l</a>)) (<a href="Matrix.Internal.html#fromLists">fromLists</a> (drop <a href="#local-6989586621679038458">rowsA</a> <a href="#local-6989586621679038457">l</a>))

333−<a name="line-333"></a> fromLists _ = error "Wrong dimensions"

334−<a name="line-334"></a>

335−<a name="line-335"></a>instance {-# OVERLAPPABLE #-} (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038253">e</a> (Either <a href="#local-6989586621679038254">a</a> <a href="#local-6989586621679038255">b</a>) <a href="#local-6989586621679038256">c</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038253">e</a> (Either <a href="#local-6989586621679038254">a</a> <a href="#local-6989586621679038255">b</a>) <a href="#local-6989586621679038257">d</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038256">c</a>)) => <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038253">e</a> (Either <a href="#local-6989586621679038254">a</a> <a href="#local-6989586621679038255">b</a>) (Either <a href="#local-6989586621679038256">c</a> <a href="#local-6989586621679038257">d</a>) where

336−<a name="line-336"></a> <a name="local-8214565720323812914"><a href="Matrix.Internal.html#fromLists">fromLists</a></a> <a name="local-6989586621679038258"><a href="#local-6989586621679038258">l</a></a>@(<a name="local-6989586621679038259"><a href="#local-6989586621679038259">h</a></a> : <a name="local-6989586621679038260"><a href="#local-6989586621679038260">t</a></a>) =

337−<a name="line-337"></a> let <a name="local-6989586621679038261"><a href="#local-6989586621679038261">lh</a></a> = length <a href="#local-6989586621679038259">h</a>

338−<a name="line-338"></a> <a name="local-6989586621679038262"><a href="#local-6989586621679038262">rowsC</a></a> = fromInteger (natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038256">c</a>)))

339−<a name="line-339"></a> <a name="local-6989586621679038263"><a href="#local-6989586621679038263">condition</a></a> = all (== <a href="#local-6989586621679038261">lh</a>) (map length <a href="#local-6989586621679038260">t</a>)

340−<a name="line-340"></a> in if <a href="#local-6989586621679038261">lh</a> > 0 && <a href="#local-6989586621679038263">condition</a>

341−<a name="line-341"></a> then <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#fromLists">fromLists</a> (take <a href="#local-6989586621679038262">rowsC</a> <a href="#local-6989586621679038258">l</a>)) (<a href="Matrix.Internal.html#fromLists">fromLists</a> (drop <a href="#local-6989586621679038262">rowsC</a> <a href="#local-6989586621679038258">l</a>))

342−<a name="line-342"></a> else error "Not all rows have the same length"

343−<a name="line-343"></a>

344−<a name="line-344"></a>-- | Matrix builder function. Constructs a matrix provided with

345−<a name="line-345"></a>-- a construction function.

346−<a name="line-346"></a>matrixBuilder ::

347−<a name="line-347"></a> forall <a name="local-6989586621679038692"><a href="#local-6989586621679038692">e</a></a> <a name="local-6989586621679038693"><a href="#local-6989586621679038693">cols</a></a> <a name="local-6989586621679038694"><a href="#local-6989586621679038694">rows</a></a>.

348−<a name="line-348"></a> ( <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038692">e</a> <a href="#local-6989586621679038693">cols</a> <a href="#local-6989586621679038694">rows</a>,

349−<a name="line-349"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038693">cols</a>),

350−<a name="line-350"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038694">rows</a>)

351−<a name="line-351"></a> ) =>

352−<a name="line-352"></a> ((Int, Int) -> <a href="#local-6989586621679038692">e</a>) ->

353−<a name="line-353"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038692">e</a> <a href="#local-6989586621679038693">cols</a> <a href="#local-6989586621679038694">rows</a>

354−<a name="line-354"></a><a name="matrixBuilder"><a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a></a> <a name="local-6989586621679038713"><a href="#local-6989586621679038713">f</a></a> =

355−<a name="line-355"></a> let <a name="local-6989586621679038714"><a href="#local-6989586621679038714">c</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038693">cols</a>))

356−<a name="line-356"></a> <a name="local-6989586621679038715"><a href="#local-6989586621679038715">r</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038694">rows</a>))

357−<a name="line-357"></a> <a name="local-6989586621679038716"><a href="#local-6989586621679038716">positions</a></a> = [(<a href="#local-6989586621679038717">a</a>, <a href="#local-6989586621679038718">b</a>) | <a name="local-6989586621679038717"><a href="#local-6989586621679038717">a</a></a> <- [0 .. (<a href="#local-6989586621679038715">r</a> - 1)], <a name="local-6989586621679038718"><a href="#local-6989586621679038718">b</a></a> <- [0 .. (<a href="#local-6989586621679038714">c</a> - 1)]]

358−<a name="line-358"></a> in <a href="Matrix.Internal.html#fromLists">fromLists</a> . map (map <a href="#local-6989586621679038713">f</a>) . groupBy (\(<a name="local-6989586621679038874"><a href="#local-6989586621679038874">x</a></a>, _) (<a name="local-6989586621679038875"><a href="#local-6989586621679038875">w</a></a>, _) -> <a href="#local-6989586621679038874">x</a> == <a href="#local-6989586621679038875">w</a>) $ <a href="#local-6989586621679038716">positions</a>

359−<a name="line-359"></a>

360−<a name="line-360"></a>-- | Constructs a column vector matrix

361−<a name="line-361"></a>col :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038690">e</a> () <a href="#local-6989586621679038691">rows</a>) => [<a href="#local-6989586621679038690">e</a>] -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038690">e</a> () <a href="#local-6989586621679038691">rows</a>

362−<a name="line-362"></a><a name="col"><a href="Matrix.Internal.html#col">col</a></a> = <a href="Matrix.Internal.html#fromLists">fromLists</a> . map (: [])

363−<a name="line-363"></a>

364−<a name="line-364"></a>-- | Constructs a row vector matrix

365−<a name="line-365"></a>row :: (<a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038688">e</a> <a href="#local-6989586621679038689">cols</a> ()) => [<a href="#local-6989586621679038688">e</a>] -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038688">e</a> <a href="#local-6989586621679038689">cols</a> ()

366−<a name="line-366"></a><a name="row"><a href="Matrix.Internal.html#row">row</a></a> = <a href="Matrix.Internal.html#fromLists">fromLists</a> . (: [])

367−<a name="line-367"></a>

368−<a name="line-368"></a>-- | Lifts functions to matrices with arbitrary dimensions.

369−<a name="line-369"></a>--

370−<a name="line-370"></a>-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

371−<a name="line-371"></a>-- types @a@ or @b@ allows.

372−<a name="line-372"></a>fromF ::

373−<a name="line-373"></a> forall <a name="local-6989586621679038683"><a href="#local-6989586621679038683">a</a></a> <a name="local-6989586621679038684"><a href="#local-6989586621679038684">b</a></a> <a name="local-6989586621679038685"><a href="#local-6989586621679038685">cols</a></a> <a name="local-6989586621679038686"><a href="#local-6989586621679038686">rows</a></a> <a name="local-6989586621679038687"><a href="#local-6989586621679038687">e</a></a>.

374−<a name="line-374"></a> ( Bounded <a href="#local-6989586621679038683">a</a>,

375−<a name="line-375"></a> Bounded <a href="#local-6989586621679038684">b</a>,

376−<a name="line-376"></a> Enum <a href="#local-6989586621679038683">a</a>,

377−<a name="line-377"></a> Enum <a href="#local-6989586621679038684">b</a>,

378−<a name="line-378"></a> Eq <a href="#local-6989586621679038684">b</a>,

379−<a name="line-379"></a> Num <a href="#local-6989586621679038687">e</a>,

380−<a name="line-380"></a> Ord <a href="#local-6989586621679038687">e</a>,

381−<a name="line-381"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038685">cols</a>),

382−<a name="line-382"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038686">rows</a>),

383−<a name="line-383"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038687">e</a> <a href="#local-6989586621679038686">rows</a> <a href="#local-6989586621679038685">cols</a>

384−<a name="line-384"></a> ) =>

385−<a name="line-385"></a> (<a href="#local-6989586621679038683">a</a> -> <a href="#local-6989586621679038684">b</a>) ->

386−<a name="line-386"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038687">e</a> <a href="#local-6989586621679038685">cols</a> <a href="#local-6989586621679038686">rows</a>

387−<a name="line-387"></a><a name="fromF"><a href="Matrix.Internal.html#fromF">fromF</a></a> <a name="local-6989586621679038876"><a href="#local-6989586621679038876">f</a></a> =

388−<a name="line-388"></a> let <a name="local-6989586621679038880"><a href="#local-6989586621679038880">minA</a></a> = minBound @<a href="#local-6989586621679038683">a</a>

389−<a name="line-389"></a> <a name="local-6989586621679038881"><a href="#local-6989586621679038881">maxA</a></a> = maxBound @<a href="#local-6989586621679038683">a</a>

390−<a name="line-390"></a> <a name="local-6989586621679038882"><a href="#local-6989586621679038882">minB</a></a> = minBound @<a href="#local-6989586621679038684">b</a>

391−<a name="line-391"></a> <a name="local-6989586621679038883"><a href="#local-6989586621679038883">maxB</a></a> = maxBound @<a href="#local-6989586621679038684">b</a>

392−<a name="line-392"></a> <a name="local-6989586621679038884"><a href="#local-6989586621679038884">ccols</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038685">cols</a>))

393−<a name="line-393"></a> <a name="local-6989586621679038885"><a href="#local-6989586621679038885">rrows</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038686">rows</a>))

394−<a name="line-394"></a> <a name="local-6989586621679038886"><a href="#local-6989586621679038886">elementsA</a></a> = take <a href="#local-6989586621679038884">ccols</a> [<a href="#local-6989586621679038880">minA</a> .. <a href="#local-6989586621679038881">maxA</a>]

395−<a name="line-395"></a> <a name="local-6989586621679038887"><a href="#local-6989586621679038887">elementsB</a></a> = take <a href="#local-6989586621679038885">rrows</a> [<a href="#local-6989586621679038882">minB</a> .. <a href="#local-6989586621679038883">maxB</a>]

396−<a name="line-396"></a> <a name="local-6989586621679038888"><a href="#local-6989586621679038888">combinations</a></a> = (,) <$> <a href="#local-6989586621679038886">elementsA</a> <*> <a href="#local-6989586621679038887">elementsB</a>

397−<a name="line-397"></a> <a name="local-6989586621679038889"><a href="#local-6989586621679038889">combAp</a></a> = map snd . sort . map (\(<a name="local-6989586621679038920"><a href="#local-6989586621679038920">a</a></a>, <a name="local-6989586621679038921"><a href="#local-6989586621679038921">b</a></a>) -> if <a href="#local-6989586621679038876">f</a> <a href="#local-6989586621679038920">a</a> == <a href="#local-6989586621679038921">b</a>

398−<a name="line-398"></a> then ((fromEnum <a href="#local-6989586621679038920">a</a>, fromEnum <a href="#local-6989586621679038921">b</a>), 1)

399−<a name="line-399"></a> else ((fromEnum <a href="#local-6989586621679038920">a</a>, fromEnum <a href="#local-6989586621679038921">b</a>), 0)) $ <a href="#local-6989586621679038888">combinations</a>

400−<a name="line-400"></a> <a name="local-6989586621679038890"><a href="#local-6989586621679038890">mList</a></a> = <a href="#local-6989586621679038877">buildList</a> <a href="#local-6989586621679038889">combAp</a> <a href="#local-6989586621679038885">rrows</a>

401−<a name="line-401"></a> in <a href="Matrix.Internal.html#tr">tr</a> $ <a href="Matrix.Internal.html#fromLists">fromLists</a> <a href="#local-6989586621679038890">mList</a>

402−<a name="line-402"></a> where

403−<a name="line-403"></a> <a name="local-6989586621679038877"><a href="#local-6989586621679038877">buildList</a></a> [] _ = []

404−<a name="line-404"></a> buildList <a name="local-6989586621679038878"><a href="#local-6989586621679038878">l</a></a> <a name="local-6989586621679038879"><a href="#local-6989586621679038879">r</a></a> = take <a href="#local-6989586621679038879">r</a> <a href="#local-6989586621679038878">l</a> : <a href="#local-6989586621679038877">buildList</a> (drop <a href="#local-6989586621679038879">r</a> <a href="#local-6989586621679038878">l</a>) <a href="#local-6989586621679038879">r</a>

405−<a name="line-405"></a>

406−<a name="line-406"></a>-- | Lifts functions to matrices with dimensions matching @a@ and @b@

407−<a name="line-407"></a>-- cardinality's.

408−<a name="line-408"></a>fromF' ::

409−<a name="line-409"></a> forall <a name="local-6989586621679038680"><a href="#local-6989586621679038680">a</a></a> <a name="local-6989586621679038681"><a href="#local-6989586621679038681">b</a></a> <a name="local-6989586621679038682"><a href="#local-6989586621679038682">e</a></a>.

410−<a name="line-410"></a> ( Bounded <a href="#local-6989586621679038680">a</a>,

411−<a name="line-411"></a> Bounded <a href="#local-6989586621679038681">b</a>,

412−<a name="line-412"></a> Enum <a href="#local-6989586621679038680">a</a>,

413−<a name="line-413"></a> Enum <a href="#local-6989586621679038681">b</a>,

414−<a name="line-414"></a> Eq <a href="#local-6989586621679038681">b</a>,

415−<a name="line-415"></a> Num <a href="#local-6989586621679038682">e</a>,

416−<a name="line-416"></a> Ord <a href="#local-6989586621679038682">e</a>,

417−<a name="line-417"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038680">a</a>)),

418−<a name="line-418"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038681">b</a>)),

419−<a name="line-419"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038682">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038681">b</a>) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038680">a</a>)

420−<a name="line-420"></a> ) =>

421−<a name="line-421"></a> (<a href="#local-6989586621679038680">a</a> -> <a href="#local-6989586621679038681">b</a>) ->

422−<a name="line-422"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038682">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038680">a</a>) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038681">b</a>)

423−<a name="line-423"></a><a name="fromF%27"><a href="Matrix.Internal.html#fromF%27">fromF'</a></a> <a name="local-6989586621679038922"><a href="#local-6989586621679038922">f</a></a> =

424−<a name="line-424"></a> let <a name="local-6989586621679038926"><a href="#local-6989586621679038926">minA</a></a> = minBound @<a href="#local-6989586621679038680">a</a>

425−<a name="line-425"></a> <a name="local-6989586621679038927"><a href="#local-6989586621679038927">maxA</a></a> = maxBound @<a href="#local-6989586621679038680">a</a>

426−<a name="line-426"></a> <a name="local-6989586621679038928"><a href="#local-6989586621679038928">minB</a></a> = minBound @<a href="#local-6989586621679038681">b</a>

427−<a name="line-427"></a> <a name="local-6989586621679038929"><a href="#local-6989586621679038929">maxB</a></a> = maxBound @<a href="#local-6989586621679038681">b</a>

428−<a name="line-428"></a> <a name="local-6989586621679038930"><a href="#local-6989586621679038930">ccols</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038680">a</a>)))

429−<a name="line-429"></a> <a name="local-6989586621679038931"><a href="#local-6989586621679038931">rrows</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038681">b</a>)))

430−<a name="line-430"></a> <a name="local-6989586621679038932"><a href="#local-6989586621679038932">elementsA</a></a> = take <a href="#local-6989586621679038930">ccols</a> [<a href="#local-6989586621679038926">minA</a> .. <a href="#local-6989586621679038927">maxA</a>]

431−<a name="line-431"></a> <a name="local-6989586621679038933"><a href="#local-6989586621679038933">elementsB</a></a> = take <a href="#local-6989586621679038931">rrows</a> [<a href="#local-6989586621679038928">minB</a> .. <a href="#local-6989586621679038929">maxB</a>]

432−<a name="line-432"></a> <a name="local-6989586621679038934"><a href="#local-6989586621679038934">combinations</a></a> = (,) <$> <a href="#local-6989586621679038932">elementsA</a> <*> <a href="#local-6989586621679038933">elementsB</a>

433−<a name="line-433"></a> <a name="local-6989586621679038935"><a href="#local-6989586621679038935">combAp</a></a> = map snd . sort . map (\(<a name="local-6989586621679038937"><a href="#local-6989586621679038937">a</a></a>, <a name="local-6989586621679038938"><a href="#local-6989586621679038938">b</a></a>) -> if <a href="#local-6989586621679038922">f</a> <a href="#local-6989586621679038937">a</a> == <a href="#local-6989586621679038938">b</a>

434−<a name="line-434"></a> then ((fromEnum <a href="#local-6989586621679038937">a</a>, fromEnum <a href="#local-6989586621679038938">b</a>), 1)

435−<a name="line-435"></a> else ((fromEnum <a href="#local-6989586621679038937">a</a>, fromEnum <a href="#local-6989586621679038938">b</a>), 0)) $ <a href="#local-6989586621679038934">combinations</a>

436−<a name="line-436"></a> <a name="local-6989586621679038936"><a href="#local-6989586621679038936">mList</a></a> = <a href="#local-6989586621679038923">buildList</a> <a href="#local-6989586621679038935">combAp</a> <a href="#local-6989586621679038931">rrows</a>

437−<a name="line-437"></a> in <a href="Matrix.Internal.html#tr">tr</a> $ <a href="Matrix.Internal.html#fromLists">fromLists</a> <a href="#local-6989586621679038936">mList</a>

438−<a name="line-438"></a> where

439−<a name="line-439"></a> <a name="local-6989586621679038923"><a href="#local-6989586621679038923">buildList</a></a> [] _ = []

440−<a name="line-440"></a> buildList <a name="local-6989586621679038924"><a href="#local-6989586621679038924">l</a></a> <a name="local-6989586621679038925"><a href="#local-6989586621679038925">r</a></a> = take <a href="#local-6989586621679038925">r</a> <a href="#local-6989586621679038924">l</a> : <a href="#local-6989586621679038923">buildList</a> (drop <a href="#local-6989586621679038925">r</a> <a href="#local-6989586621679038924">l</a>) <a href="#local-6989586621679038925">r</a>

441−<a name="line-441"></a>

442−<a name="line-442"></a>-- Conversion

443−<a name="line-443"></a>

444−<a name="line-444"></a>-- | Converts a matrix to a list of lists of elements.

445−<a name="line-445"></a>toLists :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038677">e</a> <a href="#local-6989586621679038678">cols</a> <a href="#local-6989586621679038679">rows</a> -> [[<a href="#local-6989586621679038677">e</a>]]

446−<a name="line-446"></a><a name="toLists"><a href="Matrix.Internal.html#toLists">toLists</a></a> <a href="Matrix.Internal.html#Empty">Empty</a> = []

447−<a name="line-447"></a>toLists (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038939"><a href="#local-6989586621679038939">e</a></a>) = [[<a href="#local-6989586621679038939">e</a>]]

448−<a name="line-448"></a>toLists (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038940"><a href="#local-6989586621679038940">l</a></a> <a name="local-6989586621679038941"><a href="#local-6989586621679038941">r</a></a>) = <a href="Matrix.Internal.html#toLists">toLists</a> <a href="#local-6989586621679038940">l</a> ++ <a href="Matrix.Internal.html#toLists">toLists</a> <a href="#local-6989586621679038941">r</a>

449−<a name="line-449"></a>toLists (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038942"><a href="#local-6989586621679038942">l</a></a> <a name="local-6989586621679038943"><a href="#local-6989586621679038943">r</a></a>) = zipWith (++) (<a href="Matrix.Internal.html#toLists">toLists</a> <a href="#local-6989586621679038942">l</a>) (<a href="Matrix.Internal.html#toLists">toLists</a> <a href="#local-6989586621679038943">r</a>)

450−<a name="line-450"></a>

451−<a name="line-451"></a>-- | Converts a matrix to a list of elements.

452−<a name="line-452"></a>toList :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038674">e</a> <a href="#local-6989586621679038675">cols</a> <a href="#local-6989586621679038676">rows</a> -> [<a href="#local-6989586621679038674">e</a>]

453−<a name="line-453"></a><a name="toList"><a href="Matrix.Internal.html#toList">toList</a></a> = concat . <a href="Matrix.Internal.html#toLists">toLists</a>

454−<a name="line-454"></a>

455−<a name="line-455"></a>-- Zeros Matrix

456−<a name="line-456"></a>

457−<a name="line-457"></a>-- | The zero matrix. A matrix wholly filled with zeros.

458−<a name="line-458"></a>zeros :: (Num <a href="#local-6989586621679038671">e</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038671">e</a> <a href="#local-6989586621679038672">cols</a> <a href="#local-6989586621679038673">rows</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038672">cols</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038673">rows</a>)) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038671">e</a> <a href="#local-6989586621679038672">cols</a> <a href="#local-6989586621679038673">rows</a>

459−<a name="line-459"></a><a name="zeros"><a href="Matrix.Internal.html#zeros">zeros</a></a> = <a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a> (const 0)

460−<a name="line-460"></a>

461−<a name="line-461"></a>-- Ones Matrix

462−<a name="line-462"></a>

463−<a name="line-463"></a>-- | The ones matrix. A matrix wholly filled with ones.

464−<a name="line-464"></a>--

465−<a name="line-465"></a>-- Also known as T (Top) matrix.

466−<a name="line-466"></a>ones :: (Num <a href="#local-6989586621679038668">e</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038668">e</a> <a href="#local-6989586621679038669">cols</a> <a href="#local-6989586621679038670">rows</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038669">cols</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038670">rows</a>)) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038668">e</a> <a href="#local-6989586621679038669">cols</a> <a href="#local-6989586621679038670">rows</a>

467−<a name="line-467"></a><a name="ones"><a href="Matrix.Internal.html#ones">ones</a></a> = <a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a> (const 1)

468−<a name="line-468"></a>

469−<a name="line-469"></a>-- Const Matrix

470−<a name="line-470"></a>

471−<a name="line-471"></a>-- | The constant matrix constructor. A matrix wholly filled with a given

472−<a name="line-472"></a>-- value.

473−<a name="line-473"></a>constant :: (Num <a href="#local-6989586621679038665">e</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038665">e</a> <a href="#local-6989586621679038666">cols</a> <a href="#local-6989586621679038667">rows</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038666">cols</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038667">rows</a>)) => <a href="#local-6989586621679038665">e</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038665">e</a> <a href="#local-6989586621679038666">cols</a> <a href="#local-6989586621679038667">rows</a>

474−<a name="line-474"></a><a name="constant"><a href="Matrix.Internal.html#constant">constant</a></a> <a name="local-6989586621679038944"><a href="#local-6989586621679038944">e</a></a> = <a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a> (const <a href="#local-6989586621679038944">e</a>)

475−<a name="line-475"></a>

476−<a name="line-476"></a>-- Bang Matrix

477−<a name="line-477"></a>

478−<a name="line-478"></a>-- | The T (Top) row vector matrix.

479−<a name="line-479"></a>bang :: forall <a name="local-6989586621679038663"><a href="#local-6989586621679038663">e</a></a> <a name="local-6989586621679038664"><a href="#local-6989586621679038664">cols</a></a>. (Num <a href="#local-6989586621679038663">e</a>, Enum <a href="#local-6989586621679038663">e</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038663">e</a> <a href="#local-6989586621679038664">cols</a> (), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038664">cols</a>)) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038663">e</a> <a href="#local-6989586621679038664">cols</a> ()

480−<a name="line-480"></a><a name="bang"><a href="Matrix.Internal.html#bang">bang</a></a> =

481−<a name="line-481"></a> let <a name="local-6989586621679038945"><a href="#local-6989586621679038945">c</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038664">cols</a>))

482−<a name="line-482"></a> in <a href="Matrix.Internal.html#fromLists">fromLists</a> [take <a href="#local-6989586621679038945">c</a> [1, 1 ..]]

483−<a name="line-483"></a>

484−<a name="line-484"></a>-- Identity Matrix

485−<a name="line-485"></a>

486−<a name="line-486"></a>-- | Identity matrix.

487−<a name="line-487"></a>identity :: (Num <a href="#local-6989586621679038661">e</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038661">e</a> <a href="#local-6989586621679038662">cols</a> <a href="#local-6989586621679038662">cols</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038662">cols</a>)) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038661">e</a> <a href="#local-6989586621679038662">cols</a> <a href="#local-6989586621679038662">cols</a>

488−<a name="line-488"></a><a name="identity"><a href="Matrix.Internal.html#identity">identity</a></a> = <a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a> (bool 0 1 . uncurry (==))

489−<a name="line-489"></a>

490−<a name="line-490"></a>-- Matrix composition (MMM)

491−<a name="line-491"></a>

492−<a name="line-492"></a>-- | Matrix composition. Equivalent to matrix-matrix multiplication.

493−<a name="line-493"></a>--

494−<a name="line-494"></a>-- This definition takes advantage of divide-and-conquer and fusion laws

495−<a name="line-495"></a>-- from LAoP.

496−<a name="line-496"></a>comp :: (Num <a href="#local-6989586621679038657">e</a>) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038657">e</a> <a href="#local-6989586621679038658">cr</a> <a href="#local-6989586621679038659">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038657">e</a> <a href="#local-6989586621679038660">cols</a> <a href="#local-6989586621679038658">cr</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038657">e</a> <a href="#local-6989586621679038660">cols</a> <a href="#local-6989586621679038659">rows</a>

497−<a name="line-497"></a><a name="comp"><a href="Matrix.Internal.html#comp">comp</a></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

498−<a name="line-498"></a>comp (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038946"><a href="#local-6989586621679038946">a</a></a>) (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038947"><a href="#local-6989586621679038947">b</a></a>) = <a href="Matrix.Internal.html#One">One</a> (<a href="#local-6989586621679038946">a</a> * <a href="#local-6989586621679038947">b</a>)

499−<a name="line-499"></a>comp (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038948"><a href="#local-6989586621679038948">a</a></a> <a name="local-6989586621679038949"><a href="#local-6989586621679038949">b</a></a>) (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038950"><a href="#local-6989586621679038950">c</a></a> <a name="local-6989586621679038951"><a href="#local-6989586621679038951">d</a></a>) = <a href="Matrix.Internal.html#comp">comp</a> <a href="#local-6989586621679038948">a</a> <a href="#local-6989586621679038950">c</a> + <a href="Matrix.Internal.html#comp">comp</a> <a href="#local-6989586621679038949">b</a> <a href="#local-6989586621679038951">d</a> -- Divide-and-conquer law

500−<a name="line-500"></a>comp (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038952"><a href="#local-6989586621679038952">a</a></a> <a name="local-6989586621679038953"><a href="#local-6989586621679038953">b</a></a>) <a name="local-6989586621679038954"><a href="#local-6989586621679038954">c</a></a> = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#comp">comp</a> <a href="#local-6989586621679038952">a</a> <a href="#local-6989586621679038954">c</a>) (<a href="Matrix.Internal.html#comp">comp</a> <a href="#local-6989586621679038953">b</a> <a href="#local-6989586621679038954">c</a>) -- Split fusion law

501−<a name="line-501"></a>comp <a name="local-6989586621679038955"><a href="#local-6989586621679038955">c</a></a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038956"><a href="#local-6989586621679038956">a</a></a> <a name="local-6989586621679038957"><a href="#local-6989586621679038957">b</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#comp">comp</a> <a href="#local-6989586621679038955">c</a> <a href="#local-6989586621679038956">a</a>) (<a href="Matrix.Internal.html#comp">comp</a> <a href="#local-6989586621679038955">c</a> <a href="#local-6989586621679038957">b</a>) -- Junc fusion law

502−<a name="line-502"></a>

503−<a name="line-503"></a>-- Projections

504−<a name="line-504"></a>

505−<a name="line-505"></a>-- | Biproduct first component projection

506−<a name="line-506"></a>p1 :: forall <a name="local-6989586621679038654"><a href="#local-6989586621679038654">e</a></a> <a name="local-6989586621679038655"><a href="#local-6989586621679038655">m</a></a> <a name="local-6989586621679038656"><a href="#local-6989586621679038656">n</a></a>. (Num <a href="#local-6989586621679038654">e</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038656">n</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038655">m</a>), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038654">e</a> <a href="#local-6989586621679038656">n</a> <a href="#local-6989586621679038655">m</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038654">e</a> <a href="#local-6989586621679038655">m</a> <a href="#local-6989586621679038655">m</a>) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038654">e</a> (Either <a href="#local-6989586621679038655">m</a> <a href="#local-6989586621679038656">n</a>) <a href="#local-6989586621679038655">m</a>

507−<a name="line-507"></a><a name="p1"><a href="Matrix.Internal.html#p1">p1</a></a> =

508−<a name="line-508"></a> let <a name="local-6989586621679038958"><a href="#local-6989586621679038958">iden</a></a> = <a href="Matrix.Internal.html#identity">identity</a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038654">e</a> <a href="#local-6989586621679038655">m</a> <a href="#local-6989586621679038655">m</a>

509−<a name="line-509"></a> <a name="local-6989586621679038959"><a href="#local-6989586621679038959">zero</a></a> = <a href="Matrix.Internal.html#zeros">zeros</a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038654">e</a> <a href="#local-6989586621679038656">n</a> <a href="#local-6989586621679038655">m</a>

510−<a name="line-510"></a> in <a href="Matrix.Internal.html#junc">junc</a> <a href="#local-6989586621679038958">iden</a> <a href="#local-6989586621679038959">zero</a>

511−<a name="line-511"></a>

512−<a name="line-512"></a>-- | Biproduct second component projection

513−<a name="line-513"></a>p2 :: forall <a name="local-6989586621679038651"><a href="#local-6989586621679038651">e</a></a> <a name="local-6989586621679038652"><a href="#local-6989586621679038652">m</a></a> <a name="local-6989586621679038653"><a href="#local-6989586621679038653">n</a></a>. (Num <a href="#local-6989586621679038651">e</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038653">n</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038652">m</a>), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038651">e</a> <a href="#local-6989586621679038652">m</a> <a href="#local-6989586621679038653">n</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038651">e</a> <a href="#local-6989586621679038653">n</a> <a href="#local-6989586621679038653">n</a>) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038651">e</a> (Either <a href="#local-6989586621679038652">m</a> <a href="#local-6989586621679038653">n</a>) <a href="#local-6989586621679038653">n</a>

514−<a name="line-514"></a><a name="p2"><a href="Matrix.Internal.html#p2">p2</a></a> =

515−<a name="line-515"></a> let <a name="local-6989586621679038960"><a href="#local-6989586621679038960">iden</a></a> = <a href="Matrix.Internal.html#identity">identity</a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038651">e</a> <a href="#local-6989586621679038653">n</a> <a href="#local-6989586621679038653">n</a>

516−<a name="line-516"></a> <a name="local-6989586621679038961"><a href="#local-6989586621679038961">zero</a></a> = <a href="Matrix.Internal.html#zeros">zeros</a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038651">e</a> <a href="#local-6989586621679038652">m</a> <a href="#local-6989586621679038653">n</a>

517−<a name="line-517"></a> in <a href="Matrix.Internal.html#junc">junc</a> <a href="#local-6989586621679038961">zero</a> <a href="#local-6989586621679038960">iden</a>

518−<a name="line-518"></a>

519−<a name="line-519"></a>-- Injections

520−<a name="line-520"></a>

521−<a name="line-521"></a>-- | Biproduct first component injection

522−<a name="line-522"></a>i1 :: (Num <a href="#local-6989586621679038648">e</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038649">n</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038650">m</a>), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038648">e</a> <a href="#local-6989586621679038649">n</a> <a href="#local-6989586621679038650">m</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038648">e</a> <a href="#local-6989586621679038650">m</a> <a href="#local-6989586621679038650">m</a>) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038648">e</a> <a href="#local-6989586621679038650">m</a> (Either <a href="#local-6989586621679038650">m</a> <a href="#local-6989586621679038649">n</a>)

523−<a name="line-523"></a><a name="i1"><a href="Matrix.Internal.html#i1">i1</a></a> = <a href="Matrix.Internal.html#tr">tr</a> <a href="Matrix.Internal.html#p1">p1</a>

524−<a name="line-524"></a>

525−<a name="line-525"></a>-- | Biproduct second component injection

526−<a name="line-526"></a>i2 :: (Num <a href="#local-6989586621679038645">e</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038646">n</a>), KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038647">m</a>), <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038645">e</a> <a href="#local-6989586621679038647">m</a> <a href="#local-6989586621679038646">n</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038645">e</a> <a href="#local-6989586621679038646">n</a> <a href="#local-6989586621679038646">n</a>) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038645">e</a> <a href="#local-6989586621679038646">n</a> (Either <a href="#local-6989586621679038647">m</a> <a href="#local-6989586621679038646">n</a>)

527−<a name="line-527"></a><a name="i2"><a href="Matrix.Internal.html#i2">i2</a></a> = <a href="Matrix.Internal.html#tr">tr</a> <a href="Matrix.Internal.html#p2">p2</a>

528−<a name="line-528"></a>

529−<a name="line-529"></a>-- Dimensions

530−<a name="line-530"></a>

531−<a name="line-531"></a>-- | Obtain the number of rows.

532−<a name="line-532"></a>--

533−<a name="line-533"></a>-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

534−<a name="line-534"></a>-- dimensions in constant time.

535−<a name="line-535"></a>--

536−<a name="line-536"></a>-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

537−<a name="line-537"></a>-- exchange for performance.

538−<a name="line-538"></a>rows :: forall <a name="local-6989586621679038642"><a href="#local-6989586621679038642">e</a></a> <a name="local-6989586621679038643"><a href="#local-6989586621679038643">cols</a></a> <a name="local-6989586621679038644"><a href="#local-6989586621679038644">rows</a></a>. (KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038644">rows</a>)) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038642">e</a> <a href="#local-6989586621679038643">cols</a> <a href="#local-6989586621679038644">rows</a> -> Int

539−<a name="line-539"></a><a name="rows"><a href="Matrix.Internal.html#rows">rows</a></a> _ = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038644">rows</a>))

540−<a name="line-540"></a>

541−<a name="line-541"></a>-- | Obtain the number of columns.

542−<a name="line-542"></a>--

543−<a name="line-543"></a>-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

544−<a name="line-544"></a>-- dimensions in constant time.

545−<a name="line-545"></a>--

546−<a name="line-546"></a>-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

547−<a name="line-547"></a>-- exchange for performance.

548−<a name="line-548"></a>columns :: forall <a name="local-6989586621679038639"><a href="#local-6989586621679038639">e</a></a> <a name="local-6989586621679038640"><a href="#local-6989586621679038640">cols</a></a> <a name="local-6989586621679038641"><a href="#local-6989586621679038641">rows</a></a>. (KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038640">cols</a>)) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038639">e</a> <a href="#local-6989586621679038640">cols</a> <a href="#local-6989586621679038641">rows</a> -> Int

549−<a name="line-549"></a><a name="columns"><a href="Matrix.Internal.html#columns">columns</a></a> _ = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038640">cols</a>))

550−<a name="line-550"></a>

551−<a name="line-551"></a>-- Coproduct Bifunctor

552−<a name="line-552"></a>

553−<a name="line-553"></a>infixl 5 <a href="Matrix.Internal.html#-%7C-">-|-</a>

554−<a name="line-554"></a>

555−<a name="line-555"></a>-- | Matrix coproduct functor also known as matrix direct sum.

556−<a name="line-556"></a>(-|-) ::

557−<a name="line-557"></a> forall <a name="local-6989586621679038634"><a href="#local-6989586621679038634">e</a></a> <a name="local-6989586621679038635"><a href="#local-6989586621679038635">n</a></a> <a name="local-6989586621679038636"><a href="#local-6989586621679038636">k</a></a> <a name="local-6989586621679038637"><a href="#local-6989586621679038637">m</a></a> <a name="local-6989586621679038638"><a href="#local-6989586621679038638">j</a></a>.

558−<a name="line-558"></a> ( Num <a href="#local-6989586621679038634">e</a>,

559−<a name="line-559"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038638">j</a>),

560−<a name="line-560"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038636">k</a>),

561−<a name="line-561"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038634">e</a> <a href="#local-6989586621679038636">k</a> <a href="#local-6989586621679038636">k</a>,

562−<a name="line-562"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038634">e</a> <a href="#local-6989586621679038638">j</a> <a href="#local-6989586621679038636">k</a>,

563−<a name="line-563"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038634">e</a> <a href="#local-6989586621679038636">k</a> <a href="#local-6989586621679038638">j</a>,

564−<a name="line-564"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038634">e</a> <a href="#local-6989586621679038638">j</a> <a href="#local-6989586621679038638">j</a>

565−<a name="line-565"></a> ) =>

566−<a name="line-566"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038634">e</a> <a href="#local-6989586621679038635">n</a> <a href="#local-6989586621679038636">k</a> ->

567−<a name="line-567"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038634">e</a> <a href="#local-6989586621679038637">m</a> <a href="#local-6989586621679038638">j</a> ->

568−<a name="line-568"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038634">e</a> (Either <a href="#local-6989586621679038635">n</a> <a href="#local-6989586621679038637">m</a>) (Either <a href="#local-6989586621679038636">k</a> <a href="#local-6989586621679038638">j</a>)

569−<a name="line-569"></a>(<a name="-%7C-"><a href="Matrix.Internal.html#-%7C-">-|-</a></a>) <a name="local-6989586621679038962"><a href="#local-6989586621679038962">a</a></a> <a name="local-6989586621679038963"><a href="#local-6989586621679038963">b</a></a> = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#i1">i1</a> . <a href="#local-6989586621679038962">a</a>) (<a href="Matrix.Internal.html#i2">i2</a> . <a href="#local-6989586621679038963">b</a>)

570−<a name="line-570"></a>

571−<a name="line-571"></a>-- Khatri Rao Product and projections

572−<a name="line-572"></a>

573−<a name="line-573"></a>-- | Khatri Rao product first component projection matrix.

574−<a name="line-574"></a>kp1 ::

575−<a name="line-575"></a> forall <a name="local-6989586621679038631"><a href="#local-6989586621679038631">e</a></a> <a name="local-6989586621679038632"><a href="#local-6989586621679038632">m</a></a> <a name="local-6989586621679038633"><a href="#local-6989586621679038633">k</a></a> .

576−<a name="line-576"></a> ( Num <a href="#local-6989586621679038631">e</a>,

577−<a name="line-577"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038633">k</a>),

578−<a name="line-578"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038631">e</a> (<a href="Matrix.Internal.html#FromNat">FromNat</a> (Count <a href="#local-6989586621679038632">m</a> * <a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038633">k</a>)) <a href="#local-6989586621679038632">m</a>,

579−<a name="line-579"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038632">m</a>),

580−<a name="line-580"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038632">m</a>, <a href="#local-6989586621679038633">k</a>)))

581−<a name="line-581"></a> ) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038631">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038632">m</a>, <a href="#local-6989586621679038633">k</a>)) <a href="#local-6989586621679038632">m</a>

582−<a name="line-582"></a><a name="kp1"><a href="Matrix.Internal.html#kp1">kp1</a></a> = <a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a> <a href="#local-6989586621679038965">f</a>

583−<a name="line-583"></a> where

584−<a name="line-584"></a> <a name="local-6989586621679038964"><a href="#local-6989586621679038964">offset</a></a> = fromInteger (natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038633">k</a>)))

585−<a name="line-585"></a> <a name="local-6989586621679038965"><a href="#local-6989586621679038965">f</a></a> (<a name="local-6989586621679038966"><a href="#local-6989586621679038966">x</a></a>, <a name="local-6989586621679038967"><a href="#local-6989586621679038967">y</a></a>)

586−<a name="line-586"></a> | <a href="#local-6989586621679038967">y</a> >= (<a href="#local-6989586621679038966">x</a> * <a href="#local-6989586621679038964">offset</a>) && <a href="#local-6989586621679038967">y</a> <= (<a href="#local-6989586621679038966">x</a> * <a href="#local-6989586621679038964">offset</a> + <a href="#local-6989586621679038964">offset</a> - 1) = 1

587−<a name="line-587"></a> | otherwise = 0

588−<a name="line-588"></a>

589−<a name="line-589"></a>-- | Khatri Rao product second component projection matrix.

590−<a name="line-590"></a>kp2 ::

591−<a name="line-591"></a> forall <a name="local-6989586621679038628"><a href="#local-6989586621679038628">e</a></a> <a name="local-6989586621679038629"><a href="#local-6989586621679038629">m</a></a> <a name="local-6989586621679038630"><a href="#local-6989586621679038630">k</a></a> .

592−<a name="line-592"></a> ( Num <a href="#local-6989586621679038628">e</a>,

593−<a name="line-593"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038630">k</a>),

594−<a name="line-594"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038628">e</a> (<a href="Matrix.Internal.html#FromNat">FromNat</a> (Count <a href="#local-6989586621679038629">m</a> * <a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038630">k</a>)) <a href="#local-6989586621679038630">k</a>,

595−<a name="line-595"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038629">m</a>),

596−<a name="line-596"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038629">m</a>, <a href="#local-6989586621679038630">k</a>)))

597−<a name="line-597"></a> ) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038628">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038629">m</a>, <a href="#local-6989586621679038630">k</a>)) <a href="#local-6989586621679038630">k</a>

598−<a name="line-598"></a><a name="kp2"><a href="Matrix.Internal.html#kp2">kp2</a></a> = <a href="Matrix.Internal.html#matrixBuilder">matrixBuilder</a> <a href="#local-6989586621679038969">f</a>

599−<a name="line-599"></a> where

600−<a name="line-600"></a> <a name="local-6989586621679038968"><a href="#local-6989586621679038968">offset</a></a> = fromInteger (natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038630">k</a>)))

601−<a name="line-601"></a> <a name="local-6989586621679038969"><a href="#local-6989586621679038969">f</a></a> (<a name="local-6989586621679038970"><a href="#local-6989586621679038970">x</a></a>, <a name="local-6989586621679038971"><a href="#local-6989586621679038971">y</a></a>)

602−<a name="line-602"></a> | <a href="#local-6989586621679038970">x</a> == <a href="#local-6989586621679038971">y</a> || mod (<a href="#local-6989586621679038971">y</a> - <a href="#local-6989586621679038970">x</a>) <a href="#local-6989586621679038968">offset</a> == 0 = 1

603−<a name="line-603"></a> | otherwise = 0

604−<a name="line-604"></a>

605−<a name="line-605"></a>-- | Khatri Rao Matrix product also known as matrix pairing.

606−<a name="line-606"></a>--

607−<a name="line-607"></a>-- NOTE: That this is not a true categorical product, see for instance:

608−<a name="line-608"></a>--

609−<a name="line-609"></a>-- @

610−<a name="line-610"></a>-- | kp1 . khatri a b == a

611−<a name="line-611"></a>-- khatri a b ==> |

612−<a name="line-612"></a>-- | kp2 . khatri a b == b

613−<a name="line-613"></a>-- @

614−<a name="line-614"></a>--

615−<a name="line-615"></a>-- __Emphasis__ on the implication symbol.

616−<a name="line-616"></a>khatri ::

617−<a name="line-617"></a> forall <a name="local-6989586621679038624"><a href="#local-6989586621679038624">e</a></a> <a name="local-6989586621679038625"><a href="#local-6989586621679038625">cols</a></a> <a name="local-6989586621679038626"><a href="#local-6989586621679038626">a</a></a> <a name="local-6989586621679038627"><a href="#local-6989586621679038627">b</a></a>.

618−<a name="line-618"></a> ( Num <a href="#local-6989586621679038624">e</a>,

619−<a name="line-619"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038626">a</a>),

620−<a name="line-620"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038627">b</a>),

621−<a name="line-621"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038626">a</a>, <a href="#local-6989586621679038627">b</a>))),

622−<a name="line-622"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038624">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038626">a</a>, <a href="#local-6989586621679038627">b</a>)) <a href="#local-6989586621679038626">a</a>,

623−<a name="line-623"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038624">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038626">a</a>, <a href="#local-6989586621679038627">b</a>)) <a href="#local-6989586621679038627">b</a>

624−<a name="line-624"></a> ) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038624">e</a> <a href="#local-6989586621679038625">cols</a> <a href="#local-6989586621679038626">a</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038624">e</a> <a href="#local-6989586621679038625">cols</a> <a href="#local-6989586621679038627">b</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038624">e</a> <a href="#local-6989586621679038625">cols</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038626">a</a>, <a href="#local-6989586621679038627">b</a>))

625−<a name="line-625"></a><a name="khatri"><a href="Matrix.Internal.html#khatri">khatri</a></a> <a name="local-6989586621679038972"><a href="#local-6989586621679038972">a</a></a> <a name="local-6989586621679038973"><a href="#local-6989586621679038973">b</a></a> =

626−<a name="line-626"></a> let <a name="local-6989586621679038974"><a href="#local-6989586621679038974">kp1'</a></a> = <a href="Matrix.Internal.html#kp1">kp1</a> @<a href="#local-6989586621679038624">e</a> @<a href="#local-6989586621679038626">a</a> @<a href="#local-6989586621679038627">b</a>

627−<a name="line-627"></a> <a name="local-6989586621679038975"><a href="#local-6989586621679038975">kp2'</a></a> = <a href="Matrix.Internal.html#kp2">kp2</a> @<a href="#local-6989586621679038624">e</a> @<a href="#local-6989586621679038626">a</a> @<a href="#local-6989586621679038627">b</a>

628−<a name="line-628"></a> in (<a href="Matrix.Internal.html#tr">tr</a> <a href="#local-6989586621679038974">kp1'</a>) . <a href="#local-6989586621679038972">a</a> * (<a href="Matrix.Internal.html#tr">tr</a> <a href="#local-6989586621679038975">kp2'</a>) . <a href="#local-6989586621679038973">b</a>

629−<a name="line-629"></a>

630−<a name="line-630"></a>-- Product Bifunctor (Kronecker)

631−<a name="line-631"></a>

632−<a name="line-632"></a>infixl 4 <a href="Matrix.Internal.html#%3E%3C">><</a>

633−<a name="line-633"></a>

634−<a name="line-634"></a>-- | Matrix product functor also known as kronecker product

635−<a name="line-635"></a>(><) ::

636−<a name="line-636"></a> forall <a name="local-6989586621679038619"><a href="#local-6989586621679038619">e</a></a> <a name="local-6989586621679038620"><a href="#local-6989586621679038620">m</a></a> <a name="local-6989586621679038621"><a href="#local-6989586621679038621">p</a></a> <a name="local-6989586621679038622"><a href="#local-6989586621679038622">n</a></a> <a name="local-6989586621679038623"><a href="#local-6989586621679038623">q</a></a>.

637−<a name="line-637"></a> ( Num <a href="#local-6989586621679038619">e</a>,

638−<a name="line-638"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038620">m</a>),

639−<a name="line-639"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038622">n</a>),

640−<a name="line-640"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038621">p</a>),

641−<a name="line-641"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038623">q</a>),

642−<a name="line-642"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038620">m</a>, <a href="#local-6989586621679038622">n</a>))),

643−<a name="line-643"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038619">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038620">m</a>, <a href="#local-6989586621679038622">n</a>)) <a href="#local-6989586621679038620">m</a>,

644−<a name="line-644"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038619">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038620">m</a>, <a href="#local-6989586621679038622">n</a>)) <a href="#local-6989586621679038622">n</a>,

645−<a name="line-645"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038621">p</a>, <a href="#local-6989586621679038623">q</a>))),

646−<a name="line-646"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038619">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038621">p</a>, <a href="#local-6989586621679038623">q</a>)) <a href="#local-6989586621679038621">p</a>,

647−<a name="line-647"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038619">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038621">p</a>, <a href="#local-6989586621679038623">q</a>)) <a href="#local-6989586621679038623">q</a>

648−<a name="line-648"></a> )

649−<a name="line-649"></a> => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038619">e</a> <a href="#local-6989586621679038620">m</a> <a href="#local-6989586621679038621">p</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038619">e</a> <a href="#local-6989586621679038622">n</a> <a href="#local-6989586621679038623">q</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038619">e</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038620">m</a>, <a href="#local-6989586621679038622">n</a>)) (<a href="Matrix.Internal.html#Normalize">Normalize</a> (<a href="#local-6989586621679038621">p</a>, <a href="#local-6989586621679038623">q</a>))

650−<a name="line-650"></a>(<a name="%3E%3C"><a href="Matrix.Internal.html#%3E%3C">><</a></a>) <a name="local-6989586621679038976"><a href="#local-6989586621679038976">a</a></a> <a name="local-6989586621679038977"><a href="#local-6989586621679038977">b</a></a> =

651−<a name="line-651"></a> let <a name="local-6989586621679038978"><a href="#local-6989586621679038978">kp1'</a></a> = <a href="Matrix.Internal.html#kp1">kp1</a> @<a href="#local-6989586621679038619">e</a> @<a href="#local-6989586621679038620">m</a> @<a href="#local-6989586621679038622">n</a>

652−<a name="line-652"></a> <a name="local-6989586621679038979"><a href="#local-6989586621679038979">kp2'</a></a> = <a href="Matrix.Internal.html#kp2">kp2</a> @<a href="#local-6989586621679038619">e</a> @<a href="#local-6989586621679038620">m</a> @<a href="#local-6989586621679038622">n</a>

653−<a name="line-653"></a> in <a href="Matrix.Internal.html#khatri">khatri</a> (<a href="#local-6989586621679038976">a</a> . <a href="#local-6989586621679038978">kp1'</a>) (<a href="#local-6989586621679038977">b</a> . <a href="#local-6989586621679038979">kp2'</a>)

654−<a name="line-654"></a>

655−<a name="line-655"></a>-- Matrix abide Junc Split

656−<a name="line-656"></a>

657−<a name="line-657"></a>-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

658−<a name="line-658"></a>--

659−<a name="line-659"></a>-- Law:

660−<a name="line-660"></a>--

661−<a name="line-661"></a>-- @

662−<a name="line-662"></a>-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

663−<a name="line-663"></a>-- @

664−<a name="line-664"></a>abideJS :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038616">e</a> <a href="#local-6989586621679038617">cols</a> <a href="#local-6989586621679038618">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038616">e</a> <a href="#local-6989586621679038617">cols</a> <a href="#local-6989586621679038618">rows</a>

665−<a name="line-665"></a><a name="abideJS"><a href="Matrix.Internal.html#abideJS">abideJS</a></a> (<a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038980"><a href="#local-6989586621679038980">a</a></a> <a name="local-6989586621679038981"><a href="#local-6989586621679038981">c</a></a>) (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038982"><a href="#local-6989586621679038982">b</a></a> <a name="local-6989586621679038983"><a href="#local-6989586621679038983">d</a></a>)) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038980">a</a>) (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038982">b</a>)) (<a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038981">c</a>) (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038983">d</a>)) -- Junc-Split abide law

666−<a name="line-666"></a>abideJS <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

667−<a name="line-667"></a>abideJS (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038984"><a href="#local-6989586621679038984">e</a></a>) = <a href="Matrix.Internal.html#One">One</a> <a href="#local-6989586621679038984">e</a>

668−<a name="line-668"></a>abideJS (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038985"><a href="#local-6989586621679038985">a</a></a> <a name="local-6989586621679038986"><a href="#local-6989586621679038986">b</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038985">a</a>) (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038986">b</a>)

669−<a name="line-669"></a>abideJS (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038987"><a href="#local-6989586621679038987">a</a></a> <a name="local-6989586621679038988"><a href="#local-6989586621679038988">b</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038987">a</a>) (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679038988">b</a>)

670−<a name="line-670"></a>

671−<a name="line-671"></a>-- Matrix abide Split Junc

672−<a name="line-672"></a>

673−<a name="line-673"></a>-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

674−<a name="line-674"></a>--

675−<a name="line-675"></a>-- @

676−<a name="line-676"></a>-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

677−<a name="line-677"></a>-- @

678−<a name="line-678"></a>abideSJ :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038613">e</a> <a href="#local-6989586621679038614">cols</a> <a href="#local-6989586621679038615">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038613">e</a> <a href="#local-6989586621679038614">cols</a> <a href="#local-6989586621679038615">rows</a>

679−<a name="line-679"></a><a name="abideSJ"><a href="Matrix.Internal.html#abideSJ">abideSJ</a></a> (<a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038989"><a href="#local-6989586621679038989">a</a></a> <a name="local-6989586621679038990"><a href="#local-6989586621679038990">b</a></a>) (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038991"><a href="#local-6989586621679038991">c</a></a> <a name="local-6989586621679038992"><a href="#local-6989586621679038992">d</a></a>)) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038989">a</a>) (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038991">c</a>)) (<a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038990">b</a>) (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038992">d</a>)) -- Split-Junc abide law

680−<a name="line-680"></a>abideSJ <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

681−<a name="line-681"></a>abideSJ (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038993"><a href="#local-6989586621679038993">e</a></a>) = <a href="Matrix.Internal.html#One">One</a> <a href="#local-6989586621679038993">e</a>

682−<a name="line-682"></a>abideSJ (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038994"><a href="#local-6989586621679038994">a</a></a> <a name="local-6989586621679038995"><a href="#local-6989586621679038995">b</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038994">a</a>) (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038995">b</a>)

683−<a name="line-683"></a>abideSJ (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679038996"><a href="#local-6989586621679038996">a</a></a> <a name="local-6989586621679038997"><a href="#local-6989586621679038997">b</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038996">a</a>) (<a href="Matrix.Internal.html#abideSJ">abideSJ</a> <a href="#local-6989586621679038997">b</a>)

684−<a name="line-684"></a>

685−<a name="line-685"></a>-- Matrix transposition

686−<a name="line-686"></a>

687−<a name="line-687"></a>-- | Matrix transposition.

688−<a name="line-688"></a>tr :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038610">e</a> <a href="#local-6989586621679038611">cols</a> <a href="#local-6989586621679038612">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038610">e</a> <a href="#local-6989586621679038612">rows</a> <a href="#local-6989586621679038611">cols</a>

689−<a name="line-689"></a><a name="tr"><a href="Matrix.Internal.html#tr">tr</a></a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

690−<a name="line-690"></a>tr (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679038998"><a href="#local-6989586621679038998">e</a></a>) = <a href="Matrix.Internal.html#One">One</a> <a href="#local-6989586621679038998">e</a>

691−<a name="line-691"></a>tr (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679038999"><a href="#local-6989586621679038999">a</a></a> <a name="local-6989586621679039000"><a href="#local-6989586621679039000">b</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#tr">tr</a> <a href="#local-6989586621679038999">a</a>) (<a href="Matrix.Internal.html#tr">tr</a> <a href="#local-6989586621679039000">b</a>)

692−<a name="line-692"></a>tr (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039001"><a href="#local-6989586621679039001">a</a></a> <a name="local-6989586621679039002"><a href="#local-6989586621679039002">b</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#tr">tr</a> <a href="#local-6989586621679039001">a</a>) (<a href="Matrix.Internal.html#tr">tr</a> <a href="#local-6989586621679039002">b</a>)

693−<a name="line-693"></a>

694−<a name="line-694"></a>-- Selective 'select' operator

695−<a name="line-695"></a>

696−<a name="line-696"></a>-- | Selective functors 'select' operator equivalent inspired by the

697−<a name="line-697"></a>-- ArrowMonad solution presented in the paper.

698−<a name="line-698"></a>select :: (Num <a href="#local-6989586621679038606">e</a>, <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038606">e</a> <a href="#local-6989586621679038607">b</a> <a href="#local-6989586621679038607">b</a>, KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038607">b</a>)) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038606">e</a> <a href="#local-6989586621679038608">cols</a> (Either <a href="#local-6989586621679038609">a</a> <a href="#local-6989586621679038607">b</a>) -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038606">e</a> <a href="#local-6989586621679038609">a</a> <a href="#local-6989586621679038607">b</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038606">e</a> <a href="#local-6989586621679038608">cols</a> <a href="#local-6989586621679038607">b</a>

699−<a name="line-699"></a><a name="select"><a href="Matrix.Internal.html#select">select</a></a> <a name="local-6989586621679039003"><a href="#local-6989586621679039003">m</a></a> <a name="local-6989586621679039004"><a href="#local-6989586621679039004">y</a></a> = <a href="Matrix.Internal.html#junc">junc</a> <a href="#local-6989586621679039004">y</a> <a href="Matrix.Internal.html#identity">identity</a> . <a href="#local-6989586621679039003">m</a>

700−<a name="line-700"></a>

701−<a name="line-701"></a>-- McCarthy's Conditional

702−<a name="line-702"></a>

703−<a name="line-703"></a>-- | McCarthy's Conditional expresses probabilistic choice.

704−<a name="line-704"></a>cond ::

705−<a name="line-705"></a> ( cols ~ <a href="Matrix.Internal.html#FromNat">FromNat</a> (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038602">cols</a>),

706−<a name="line-706"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038602">cols</a>),

707−<a name="line-707"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038603">e</a> () <a href="#local-6989586621679038602">cols</a>,

708−<a name="line-708"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038603">e</a> <a href="#local-6989586621679038602">cols</a> (),

709−<a name="line-709"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038603">e</a> <a href="#local-6989586621679038602">cols</a> <a href="#local-6989586621679038602">cols</a>,

710−<a name="line-710"></a> Bounded <a href="#local-6989586621679038604">a</a>,

711−<a name="line-711"></a> Enum <a href="#local-6989586621679038604">a</a>,

712−<a name="line-712"></a> Num <a href="#local-6989586621679038603">e</a>,

713−<a name="line-713"></a> Ord <a href="#local-6989586621679038603">e</a>

714−<a name="line-714"></a> )

715−<a name="line-715"></a> =>

716−<a name="line-716"></a> (<a href="#local-6989586621679038604">a</a> -> Bool) -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038603">e</a> <a href="#local-6989586621679038602">cols</a> <a href="#local-6989586621679038605">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038603">e</a> <a href="#local-6989586621679038602">cols</a> <a href="#local-6989586621679038605">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038603">e</a> <a href="#local-6989586621679038602">cols</a> <a href="#local-6989586621679038605">rows</a>

717−<a name="line-717"></a><a name="cond"><a href="Matrix.Internal.html#cond">cond</a></a> <a name="local-6989586621679039005"><a href="#local-6989586621679039005">p</a></a> <a name="local-6989586621679039006"><a href="#local-6989586621679039006">f</a></a> <a name="local-6989586621679039007"><a href="#local-6989586621679039007">g</a></a> = <a href="Matrix.Internal.html#junc">junc</a> <a href="#local-6989586621679039006">f</a> <a href="#local-6989586621679039007">g</a> . <a href="Matrix.Internal.html#grd">grd</a> <a href="#local-6989586621679039005">p</a>

718−<a name="line-718"></a>

719−<a name="line-719"></a>grd ::

720−<a name="line-720"></a> ( q ~ <a href="Matrix.Internal.html#FromNat">FromNat</a> (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038599">q</a>),

721−<a name="line-721"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038599">q</a>),

722−<a name="line-722"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038600">e</a> () <a href="#local-6989586621679038599">q</a>,

723−<a name="line-723"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038600">e</a> <a href="#local-6989586621679038599">q</a> (),

724−<a name="line-724"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038600">e</a> <a href="#local-6989586621679038599">q</a> <a href="#local-6989586621679038599">q</a>,

725−<a name="line-725"></a> Bounded <a href="#local-6989586621679038601">a</a>,

726−<a name="line-726"></a> Enum <a href="#local-6989586621679038601">a</a>,

727−<a name="line-727"></a> Num <a href="#local-6989586621679038600">e</a>,

728−<a name="line-728"></a> Ord <a href="#local-6989586621679038600">e</a>

729−<a name="line-729"></a> )

730−<a name="line-730"></a> =>

731−<a name="line-731"></a> (<a href="#local-6989586621679038601">a</a> -> Bool) -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038600">e</a> <a href="#local-6989586621679038599">q</a> (Either <a href="#local-6989586621679038599">q</a> <a href="#local-6989586621679038599">q</a>)

732−<a name="line-732"></a><a name="grd"><a href="Matrix.Internal.html#grd">grd</a></a> <a name="local-6989586621679039008"><a href="#local-6989586621679039008">f</a></a> = <a href="Matrix.Internal.html#split">split</a> (<a href="Matrix.Internal.html#corr">corr</a> <a href="#local-6989586621679039008">f</a>) (<a href="Matrix.Internal.html#corr">corr</a> (not . <a href="#local-6989586621679039008">f</a>))

733−<a name="line-733"></a>

734−<a name="line-734"></a>corr ::

735−<a name="line-735"></a> forall <a name="local-6989586621679038596"><a href="#local-6989586621679038596">e</a></a> <a name="local-6989586621679038597"><a href="#local-6989586621679038597">a</a></a> <a name="local-6989586621679038598"><a href="#local-6989586621679038598">q</a></a> .

736−<a name="line-736"></a> ( q ~ <a href="Matrix.Internal.html#FromNat">FromNat</a> (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038598">q</a>),

737−<a name="line-737"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038598">q</a>),

738−<a name="line-738"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038596">e</a> () <a href="#local-6989586621679038598">q</a>,

739−<a name="line-739"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038596">e</a> <a href="#local-6989586621679038598">q</a> (),

740−<a name="line-740"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> <a href="#local-6989586621679038596">e</a> <a href="#local-6989586621679038598">q</a> <a href="#local-6989586621679038598">q</a>,

741−<a name="line-741"></a> Bounded <a href="#local-6989586621679038597">a</a>,

742−<a name="line-742"></a> Enum <a href="#local-6989586621679038597">a</a>,

743−<a name="line-743"></a> Num <a href="#local-6989586621679038596">e</a>,

744−<a name="line-744"></a> Ord <a href="#local-6989586621679038596">e</a>

745−<a name="line-745"></a> )

746−<a name="line-746"></a> => (<a href="#local-6989586621679038597">a</a> -> Bool) -> <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038596">e</a> <a href="#local-6989586621679038598">q</a> <a href="#local-6989586621679038598">q</a>

747−<a name="line-747"></a><a name="corr"><a href="Matrix.Internal.html#corr">corr</a></a> <a name="local-6989586621679039009"><a href="#local-6989586621679039009">p</a></a> = let <a name="local-6989586621679039010"><a href="#local-6989586621679039010">f</a></a> = <a href="Matrix.Internal.html#fromF">fromF</a> <a href="#local-6989586621679039009">p</a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038596">e</a> <a href="#local-6989586621679038598">q</a> ()

748−<a name="line-748"></a> in <a href="Matrix.Internal.html#khatri">khatri</a> <a href="#local-6989586621679039010">f</a> (<a href="Matrix.Internal.html#identity">identity</a> :: <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038596">e</a> <a href="#local-6989586621679038598">q</a> <a href="#local-6989586621679038598">q</a>)

749−<a name="line-749"></a>

750−<a name="line-750"></a>-- Pretty print

751−<a name="line-751"></a>

752−<a name="line-752"></a>prettyAux :: Show <a href="#local-6989586621679038595">e</a> => [[<a href="#local-6989586621679038595">e</a>]] -> [[<a href="#local-6989586621679038595">e</a>]] -> String

753−<a name="line-753"></a><a name="prettyAux"><a href="Matrix.Internal.html#prettyAux">prettyAux</a></a> [] _ = ""

754−<a name="line-754"></a>prettyAux [[<a name="local-6989586621679039011"><a href="#local-6989586621679039011">e</a></a>]] <a name="local-6989586621679039012"><a href="#local-6989586621679039012">m</a></a> = "│ " ++ <a href="#local-6989586621679039015">fill</a> (show <a href="#local-6989586621679039011">e</a>) ++ " │\n"

755−<a name="line-755"></a> where

756−<a name="line-756"></a> <a name="local-6989586621679039013"><a href="#local-6989586621679039013">v</a></a> = fmap show <a href="#local-6989586621679039012">m</a>

757−<a name="line-757"></a> <a name="local-6989586621679039014"><a href="#local-6989586621679039014">widest</a></a> = maximum $ fmap length <a href="#local-6989586621679039013">v</a>

758−<a name="line-758"></a> <a name="local-6989586621679039015"><a href="#local-6989586621679039015">fill</a></a> <a name="local-6989586621679039016"><a href="#local-6989586621679039016">str</a></a> = replicate (<a href="#local-6989586621679039014">widest</a> - length <a href="#local-6989586621679039016">str</a> - 2) ' ' ++ <a href="#local-6989586621679039016">str</a>

759−<a name="line-759"></a>prettyAux [<a name="local-6989586621679039017"><a href="#local-6989586621679039017">h</a></a>] <a name="local-6989586621679039018"><a href="#local-6989586621679039018">m</a></a> = "│ " ++ <a href="#local-6989586621679039021">fill</a> (unwords $ map show <a href="#local-6989586621679039017">h</a>) ++ " │\n"

760−<a name="line-760"></a> where

761−<a name="line-761"></a> <a name="local-6989586621679039019"><a href="#local-6989586621679039019">v</a></a> = fmap show <a href="#local-6989586621679039018">m</a>

762−<a name="line-762"></a> <a name="local-6989586621679039020"><a href="#local-6989586621679039020">widest</a></a> = maximum $ fmap length <a href="#local-6989586621679039019">v</a>

763−<a name="line-763"></a> <a name="local-6989586621679039021"><a href="#local-6989586621679039021">fill</a></a> <a name="local-6989586621679039022"><a href="#local-6989586621679039022">str</a></a> = replicate (<a href="#local-6989586621679039020">widest</a> - length <a href="#local-6989586621679039022">str</a> - 2) ' ' ++ <a href="#local-6989586621679039022">str</a>

764−<a name="line-764"></a>prettyAux (<a name="local-6989586621679039023"><a href="#local-6989586621679039023">h</a></a> : <a name="local-6989586621679039024"><a href="#local-6989586621679039024">t</a></a>) <a name="local-6989586621679039025"><a href="#local-6989586621679039025">l</a></a> = "│ " ++ <a href="#local-6989586621679039028">fill</a> (unwords $ map show <a href="#local-6989586621679039023">h</a>) ++ " │\n" ++

765−<a name="line-765"></a> <a href="Matrix.Internal.html#prettyAux">prettyAux</a> <a href="#local-6989586621679039024">t</a> <a href="#local-6989586621679039025">l</a>

766−<a name="line-766"></a> where

767−<a name="line-767"></a> <a name="local-6989586621679039026"><a href="#local-6989586621679039026">v</a></a> = fmap show <a href="#local-6989586621679039025">l</a>

768−<a name="line-768"></a> <a name="local-6989586621679039027"><a href="#local-6989586621679039027">widest</a></a> = maximum $ fmap length <a href="#local-6989586621679039026">v</a>

769−<a name="line-769"></a> <a name="local-6989586621679039028"><a href="#local-6989586621679039028">fill</a></a> <a name="local-6989586621679039029"><a href="#local-6989586621679039029">str</a></a> = replicate (<a href="#local-6989586621679039027">widest</a> - length <a href="#local-6989586621679039029">str</a> - 2) ' ' ++ <a href="#local-6989586621679039029">str</a>

770−<a name="line-770"></a>

771−<a name="line-771"></a>-- | Matrix pretty printer

772−<a name="line-772"></a>pretty :: (KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038592">cols</a>), Show <a href="#local-6989586621679038593">e</a>) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038593">e</a> <a href="#local-6989586621679038592">cols</a> <a href="#local-6989586621679038594">rows</a> -> String

773−<a name="line-773"></a><a name="pretty"><a href="Matrix.Internal.html#pretty">pretty</a></a> <a name="local-6989586621679039030"><a href="#local-6989586621679039030">m</a></a> = "┌ " ++ unwords (replicate (<a href="Matrix.Internal.html#columns">columns</a> <a href="#local-6989586621679039030">m</a>) <a href="#local-6989586621679039034">blank</a>) ++ " ┐\n" ++

774−<a name="line-774"></a> <a href="Matrix.Internal.html#prettyAux">prettyAux</a> (<a href="Matrix.Internal.html#toLists">toLists</a> <a href="#local-6989586621679039030">m</a>) (<a href="Matrix.Internal.html#toLists">toLists</a> <a href="#local-6989586621679039030">m</a>) ++

775−<a name="line-775"></a> "└ " ++ unwords (replicate (<a href="Matrix.Internal.html#columns">columns</a> <a href="#local-6989586621679039030">m</a>) <a href="#local-6989586621679039034">blank</a>) ++ " ┘"

776−<a name="line-776"></a> where

777−<a name="line-777"></a> <a name="local-6989586621679039031"><a href="#local-6989586621679039031">v</a></a> = fmap show (<a href="Matrix.Internal.html#toList">toList</a> <a href="#local-6989586621679039030">m</a>)

778−<a name="line-778"></a> <a name="local-6989586621679039032"><a href="#local-6989586621679039032">widest</a></a> = maximum $ fmap length <a href="#local-6989586621679039031">v</a>

779−<a name="line-779"></a> <a name="local-6989586621679039033"><a href="#local-6989586621679039033">fill</a></a> <a name="local-6989586621679039035"><a href="#local-6989586621679039035">str</a></a> = replicate (<a href="#local-6989586621679039032">widest</a> - length <a href="#local-6989586621679039035">str</a>) ' ' ++ <a href="#local-6989586621679039035">str</a>

780−<a name="line-780"></a> <a name="local-6989586621679039034"><a href="#local-6989586621679039034">blank</a></a> = <a href="#local-6989586621679039033">fill</a> ""

781−<a name="line-781"></a>

782−<a name="line-782"></a>-- | Matrix pretty printer

783−<a name="line-783"></a>prettyPrint :: (KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038589">cols</a>), Show <a href="#local-6989586621679038590">e</a>) => <a href="Matrix.Internal.html#Matrix">Matrix</a> <a href="#local-6989586621679038590">e</a> <a href="#local-6989586621679038589">cols</a> <a href="#local-6989586621679038591">rows</a> -> IO ()

784−<a name="line-784"></a><a name="prettyPrint"><a href="Matrix.Internal.html#prettyPrint">prettyPrint</a></a> = putStrLn . <a href="Matrix.Internal.html#pretty">pretty</a>

785−<a name="line-785"></a>

786−<a name="line-786"></a>-- Relational operators functions

787−<a name="line-787"></a>

788−<a name="line-788"></a>-- | Helper conversion function

789−<a name="line-789"></a>toBool :: (Num <a href="#local-6989586621679038588">e</a>, Eq <a href="#local-6989586621679038588">e</a>) => <a href="#local-6989586621679038588">e</a> -> Bool

790−<a name="line-790"></a><a name="toBool"><a href="Matrix.Internal.html#toBool">toBool</a></a> <a name="local-6989586621679039207"><a href="#local-6989586621679039207">n</a></a>

791−<a name="line-791"></a> | <a href="#local-6989586621679039207">n</a> == 0 = False

792−<a name="line-792"></a> | <a href="#local-6989586621679039207">n</a> == 1 = True

793−<a name="line-793"></a>

794−<a name="line-794"></a>-- | Helper conversion function

795−<a name="line-795"></a>fromBool :: Bool -> <a href="Utils.Internal.html#Natural">Natural</a> 0 1

796−<a name="line-796"></a><a name="fromBool"><a href="Matrix.Internal.html#fromBool">fromBool</a></a> True = <a href="Utils.Internal.html#nat">nat</a> 1

797−<a name="line-797"></a>fromBool False = <a href="Utils.Internal.html#nat">nat</a> 0

798−<a name="line-798"></a>

799−<a name="line-799"></a>-- | Relational addition

800−<a name="line-800"></a>orM :: <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038586">cols</a> <a href="#local-6989586621679038587">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038586">cols</a> <a href="#local-6989586621679038587">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038586">cols</a> <a href="#local-6989586621679038587">rows</a>

801−<a name="line-801"></a><a name="orM"><a href="Matrix.Internal.html#orM">orM</a></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

802−<a name="line-802"></a>orM (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039208"><a href="#local-6989586621679039208">a</a></a>) (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039209"><a href="#local-6989586621679039209">b</a></a>) = <a href="Matrix.Internal.html#One">One</a> (<a href="Matrix.Internal.html#fromBool">fromBool</a> (<a href="Matrix.Internal.html#toBool">toBool</a> <a href="#local-6989586621679039208">a</a> || <a href="Matrix.Internal.html#toBool">toBool</a> <a href="#local-6989586621679039209">b</a>))

803−<a name="line-803"></a>orM (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039210"><a href="#local-6989586621679039210">a</a></a> <a name="local-6989586621679039211"><a href="#local-6989586621679039211">b</a></a>) (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039212"><a href="#local-6989586621679039212">c</a></a> <a name="local-6989586621679039213"><a href="#local-6989586621679039213">d</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#orM">orM</a> <a href="#local-6989586621679039210">a</a> <a href="#local-6989586621679039212">c</a>) (<a href="Matrix.Internal.html#orM">orM</a> <a href="#local-6989586621679039211">b</a> <a href="#local-6989586621679039213">d</a>)

804−<a name="line-804"></a>orM (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039214"><a href="#local-6989586621679039214">a</a></a> <a name="local-6989586621679039215"><a href="#local-6989586621679039215">b</a></a>) (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039216"><a href="#local-6989586621679039216">c</a></a> <a name="local-6989586621679039217"><a href="#local-6989586621679039217">d</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#orM">orM</a> <a href="#local-6989586621679039214">a</a> <a href="#local-6989586621679039216">c</a>) (<a href="Matrix.Internal.html#orM">orM</a> <a href="#local-6989586621679039215">b</a> <a href="#local-6989586621679039217">d</a>)

805−<a name="line-805"></a>orM <a name="local-6989586621679039218"><a href="#local-6989586621679039218">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) <a name="local-6989586621679039219"><a href="#local-6989586621679039219">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="Matrix.Internal.html#orM">orM</a> <a href="#local-6989586621679039218">x</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679039219">y</a>)

806−<a name="line-806"></a>orM <a name="local-6989586621679039220"><a href="#local-6989586621679039220">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) <a name="local-6989586621679039221"><a href="#local-6989586621679039221">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#orM">orM</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679039220">x</a>) <a href="#local-6989586621679039221">y</a>

807−<a name="line-807"></a>

808−<a name="line-808"></a>-- | Relational multiplication

809−<a name="line-809"></a>andM :: <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038584">cols</a> <a href="#local-6989586621679038585">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038584">cols</a> <a href="#local-6989586621679038585">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038584">cols</a> <a href="#local-6989586621679038585">rows</a>

810−<a name="line-810"></a><a name="andM"><a href="Matrix.Internal.html#andM">andM</a></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

811−<a name="line-811"></a>andM (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039222"><a href="#local-6989586621679039222">a</a></a>) (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039223"><a href="#local-6989586621679039223">b</a></a>) = <a href="Matrix.Internal.html#One">One</a> (<a href="Matrix.Internal.html#fromBool">fromBool</a> (<a href="Matrix.Internal.html#toBool">toBool</a> <a href="#local-6989586621679039222">a</a> && <a href="Matrix.Internal.html#toBool">toBool</a> <a href="#local-6989586621679039223">b</a>))

812−<a name="line-812"></a>andM (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039224"><a href="#local-6989586621679039224">a</a></a> <a name="local-6989586621679039225"><a href="#local-6989586621679039225">b</a></a>) (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039226"><a href="#local-6989586621679039226">c</a></a> <a name="local-6989586621679039227"><a href="#local-6989586621679039227">d</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#andM">andM</a> <a href="#local-6989586621679039224">a</a> <a href="#local-6989586621679039226">c</a>) (<a href="Matrix.Internal.html#andM">andM</a> <a href="#local-6989586621679039225">b</a> <a href="#local-6989586621679039227">d</a>)

813−<a name="line-813"></a>andM (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039228"><a href="#local-6989586621679039228">a</a></a> <a name="local-6989586621679039229"><a href="#local-6989586621679039229">b</a></a>) (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039230"><a href="#local-6989586621679039230">c</a></a> <a name="local-6989586621679039231"><a href="#local-6989586621679039231">d</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#andM">andM</a> <a href="#local-6989586621679039228">a</a> <a href="#local-6989586621679039230">c</a>) (<a href="Matrix.Internal.html#andM">andM</a> <a href="#local-6989586621679039229">b</a> <a href="#local-6989586621679039231">d</a>)

814−<a name="line-814"></a>andM <a name="local-6989586621679039232"><a href="#local-6989586621679039232">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) <a name="local-6989586621679039233"><a href="#local-6989586621679039233">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="Matrix.Internal.html#andM">andM</a> <a href="#local-6989586621679039232">x</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679039233">y</a>)

815−<a name="line-815"></a>andM <a name="local-6989586621679039234"><a href="#local-6989586621679039234">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) <a name="local-6989586621679039235"><a href="#local-6989586621679039235">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#andM">andM</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679039234">x</a>) <a href="#local-6989586621679039235">y</a>

816−<a name="line-816"></a>

817−<a name="line-817"></a>-- | Relational subtraction

818−<a name="line-818"></a>subM :: <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038582">cols</a> <a href="#local-6989586621679038583">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038582">cols</a> <a href="#local-6989586621679038583">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038582">cols</a> <a href="#local-6989586621679038583">rows</a>

819−<a name="line-819"></a><a name="subM"><a href="Matrix.Internal.html#subM">subM</a></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

820−<a name="line-820"></a>subM (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039236"><a href="#local-6989586621679039236">a</a></a>) (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039237"><a href="#local-6989586621679039237">b</a></a>) = if <a href="#local-6989586621679039236">a</a> - <a href="#local-6989586621679039237">b</a> < <a href="Utils.Internal.html#nat">nat</a> 0 then <a href="Matrix.Internal.html#One">One</a> (<a href="Utils.Internal.html#nat">nat</a> 0) else <a href="Matrix.Internal.html#One">One</a> (<a href="#local-6989586621679039236">a</a> - <a href="#local-6989586621679039237">b</a>)

821−<a name="line-821"></a>subM (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039238"><a href="#local-6989586621679039238">a</a></a> <a name="local-6989586621679039239"><a href="#local-6989586621679039239">b</a></a>) (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039240"><a href="#local-6989586621679039240">c</a></a> <a name="local-6989586621679039241"><a href="#local-6989586621679039241">d</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#subM">subM</a> <a href="#local-6989586621679039238">a</a> <a href="#local-6989586621679039240">c</a>) (<a href="Matrix.Internal.html#subM">subM</a> <a href="#local-6989586621679039239">b</a> <a href="#local-6989586621679039241">d</a>)

822−<a name="line-822"></a>subM (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039242"><a href="#local-6989586621679039242">a</a></a> <a name="local-6989586621679039243"><a href="#local-6989586621679039243">b</a></a>) (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039244"><a href="#local-6989586621679039244">c</a></a> <a name="local-6989586621679039245"><a href="#local-6989586621679039245">d</a></a>) = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#subM">subM</a> <a href="#local-6989586621679039242">a</a> <a href="#local-6989586621679039244">c</a>) (<a href="Matrix.Internal.html#subM">subM</a> <a href="#local-6989586621679039243">b</a> <a href="#local-6989586621679039245">d</a>)

823−<a name="line-823"></a>subM <a name="local-6989586621679039246"><a href="#local-6989586621679039246">x</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) <a name="local-6989586621679039247"><a href="#local-6989586621679039247">y</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) = <a href="Matrix.Internal.html#subM">subM</a> <a href="#local-6989586621679039246">x</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679039247">y</a>)

824−<a name="line-824"></a>subM <a name="local-6989586621679039248"><a href="#local-6989586621679039248">x</a></a>@(<a href="Matrix.Internal.html#Junc">Junc</a> _ _) <a name="local-6989586621679039249"><a href="#local-6989586621679039249">y</a></a>@(<a href="Matrix.Internal.html#Split">Split</a> _ _) = <a href="Matrix.Internal.html#subM">subM</a> (<a href="Matrix.Internal.html#abideJS">abideJS</a> <a href="#local-6989586621679039248">x</a>) <a href="#local-6989586621679039249">y</a>

825−<a name="line-825"></a>

826−<a name="line-826"></a>-- | Matrix relational composition.

827−<a name="line-827"></a>compRel :: <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038579">cr</a> <a href="#local-6989586621679038580">rows</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038581">cols</a> <a href="#local-6989586621679038579">cr</a> -> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038581">cols</a> <a href="#local-6989586621679038580">rows</a>

828−<a name="line-828"></a><a name="compRel"><a href="Matrix.Internal.html#compRel">compRel</a></a> <a href="Matrix.Internal.html#Empty">Empty</a> <a href="Matrix.Internal.html#Empty">Empty</a> = <a href="Matrix.Internal.html#Empty">Empty</a>

829−<a name="line-829"></a>compRel (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039250"><a href="#local-6989586621679039250">a</a></a>) (<a href="Matrix.Internal.html#One">One</a> <a name="local-6989586621679039251"><a href="#local-6989586621679039251">b</a></a>) = <a href="Matrix.Internal.html#One">One</a> (<a href="Matrix.Internal.html#fromBool">fromBool</a> (<a href="Matrix.Internal.html#toBool">toBool</a> <a href="#local-6989586621679039250">a</a> && <a href="Matrix.Internal.html#toBool">toBool</a> <a href="#local-6989586621679039251">b</a>))

830−<a name="line-830"></a>compRel (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039252"><a href="#local-6989586621679039252">a</a></a> <a name="local-6989586621679039253"><a href="#local-6989586621679039253">b</a></a>) (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039254"><a href="#local-6989586621679039254">c</a></a> <a name="local-6989586621679039255"><a href="#local-6989586621679039255">d</a></a>) = <a href="Matrix.Internal.html#orM">orM</a> (<a href="Matrix.Internal.html#compRel">compRel</a> <a href="#local-6989586621679039252">a</a> <a href="#local-6989586621679039254">c</a>) (<a href="Matrix.Internal.html#compRel">compRel</a> <a href="#local-6989586621679039253">b</a> <a href="#local-6989586621679039255">d</a>) -- Divide-and-conquer law

831−<a name="line-831"></a>compRel (<a href="Matrix.Internal.html#Split">Split</a> <a name="local-6989586621679039256"><a href="#local-6989586621679039256">a</a></a> <a name="local-6989586621679039257"><a href="#local-6989586621679039257">b</a></a>) <a name="local-6989586621679039258"><a href="#local-6989586621679039258">c</a></a> = <a href="Matrix.Internal.html#Split">Split</a> (<a href="Matrix.Internal.html#compRel">compRel</a> <a href="#local-6989586621679039256">a</a> <a href="#local-6989586621679039258">c</a>) (<a href="Matrix.Internal.html#compRel">compRel</a> <a href="#local-6989586621679039257">b</a> <a href="#local-6989586621679039258">c</a>) -- Split fusion law

832−<a name="line-832"></a>compRel <a name="local-6989586621679039259"><a href="#local-6989586621679039259">c</a></a> (<a href="Matrix.Internal.html#Junc">Junc</a> <a name="local-6989586621679039260"><a href="#local-6989586621679039260">a</a></a> <a name="local-6989586621679039261"><a href="#local-6989586621679039261">b</a></a>) = <a href="Matrix.Internal.html#Junc">Junc</a> (<a href="Matrix.Internal.html#compRel">compRel</a> <a href="#local-6989586621679039259">c</a> <a href="#local-6989586621679039260">a</a>) (<a href="Matrix.Internal.html#compRel">compRel</a> <a href="#local-6989586621679039259">c</a> <a href="#local-6989586621679039261">b</a>) -- Junc fusion law

833−<a name="line-833"></a>

834−<a name="line-834"></a>-- | Lifts functions to relations with arbitrary dimensions.

835−<a name="line-835"></a>--

836−<a name="line-836"></a>-- NOTE: Be careful to not ask for a relation bigger than the cardinality of

837−<a name="line-837"></a>-- types @a@ or @b@ allows.

838−<a name="line-838"></a>fromFRel ::

839−<a name="line-839"></a> forall <a name="local-6989586621679038575"><a href="#local-6989586621679038575">a</a></a> <a name="local-6989586621679038576"><a href="#local-6989586621679038576">b</a></a> <a name="local-6989586621679038577"><a href="#local-6989586621679038577">cols</a></a> <a name="local-6989586621679038578"><a href="#local-6989586621679038578">rows</a></a>.

840−<a name="line-840"></a> ( Bounded <a href="#local-6989586621679038575">a</a>,

841−<a name="line-841"></a> Bounded <a href="#local-6989586621679038576">b</a>,

842−<a name="line-842"></a> Enum <a href="#local-6989586621679038575">a</a>,

843−<a name="line-843"></a> Enum <a href="#local-6989586621679038576">b</a>,

844−<a name="line-844"></a> Eq <a href="#local-6989586621679038576">b</a>,

845−<a name="line-845"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038577">cols</a>),

846−<a name="line-846"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038578">rows</a>),

847−<a name="line-847"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038578">rows</a> <a href="#local-6989586621679038577">cols</a>

848−<a name="line-848"></a> ) =>

849−<a name="line-849"></a> (<a href="#local-6989586621679038575">a</a> -> <a href="#local-6989586621679038576">b</a>) ->

850−<a name="line-850"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) <a href="#local-6989586621679038577">cols</a> <a href="#local-6989586621679038578">rows</a>

851−<a name="line-851"></a><a name="fromFRel"><a href="Matrix.Internal.html#fromFRel">fromFRel</a></a> <a name="local-6989586621679039262"><a href="#local-6989586621679039262">f</a></a> =

852−<a name="line-852"></a> let <a name="local-6989586621679039266"><a href="#local-6989586621679039266">minA</a></a> = minBound @<a href="#local-6989586621679038575">a</a>

853−<a name="line-853"></a> <a name="local-6989586621679039267"><a href="#local-6989586621679039267">maxA</a></a> = maxBound @<a href="#local-6989586621679038575">a</a>

854−<a name="line-854"></a> <a name="local-6989586621679039268"><a href="#local-6989586621679039268">minB</a></a> = minBound @<a href="#local-6989586621679038576">b</a>

855−<a name="line-855"></a> <a name="local-6989586621679039269"><a href="#local-6989586621679039269">maxB</a></a> = maxBound @<a href="#local-6989586621679038576">b</a>

856−<a name="line-856"></a> <a name="local-6989586621679039270"><a href="#local-6989586621679039270">ccols</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038577">cols</a>))

857−<a name="line-857"></a> <a name="local-6989586621679039271"><a href="#local-6989586621679039271">rrows</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> <a href="#local-6989586621679038578">rows</a>))

858−<a name="line-858"></a> <a name="local-6989586621679039272"><a href="#local-6989586621679039272">elementsA</a></a> = take <a href="#local-6989586621679039270">ccols</a> [<a href="#local-6989586621679039266">minA</a> .. <a href="#local-6989586621679039267">maxA</a>]

859−<a name="line-859"></a> <a name="local-6989586621679039273"><a href="#local-6989586621679039273">elementsB</a></a> = take <a href="#local-6989586621679039271">rrows</a> [<a href="#local-6989586621679039268">minB</a> .. <a href="#local-6989586621679039269">maxB</a>]

860−<a name="line-860"></a> <a name="local-6989586621679039274"><a href="#local-6989586621679039274">combinations</a></a> = (,) <$> <a href="#local-6989586621679039272">elementsA</a> <*> <a href="#local-6989586621679039273">elementsB</a>

861−<a name="line-861"></a> <a name="local-6989586621679039275"><a href="#local-6989586621679039275">combAp</a></a> = map snd . sort . map (\(<a name="local-6989586621679039277"><a href="#local-6989586621679039277">a</a></a>, <a name="local-6989586621679039278"><a href="#local-6989586621679039278">b</a></a>) -> if <a href="#local-6989586621679039262">f</a> <a href="#local-6989586621679039277">a</a> == <a href="#local-6989586621679039278">b</a>

862−<a name="line-862"></a> then ((fromEnum <a href="#local-6989586621679039277">a</a>, fromEnum <a href="#local-6989586621679039278">b</a>), <a href="Utils.Internal.html#nat">nat</a> 1)

863−<a name="line-863"></a> else ((fromEnum <a href="#local-6989586621679039277">a</a>, fromEnum <a href="#local-6989586621679039278">b</a>), <a href="Utils.Internal.html#nat">nat</a> 0)) $ <a href="#local-6989586621679039274">combinations</a>

864−<a name="line-864"></a> <a name="local-6989586621679039276"><a href="#local-6989586621679039276">mList</a></a> = <a href="#local-6989586621679039263">buildList</a> <a href="#local-6989586621679039275">combAp</a> <a href="#local-6989586621679039271">rrows</a>

865−<a name="line-865"></a> in <a href="Matrix.Internal.html#tr">tr</a> $ <a href="Matrix.Internal.html#fromLists">fromLists</a> <a href="#local-6989586621679039276">mList</a>

866−<a name="line-866"></a> where

867−<a name="line-867"></a> <a name="local-6989586621679039263"><a href="#local-6989586621679039263">buildList</a></a> [] _ = []

868−<a name="line-868"></a> buildList <a name="local-6989586621679039264"><a href="#local-6989586621679039264">l</a></a> <a name="local-6989586621679039265"><a href="#local-6989586621679039265">r</a></a> = take <a href="#local-6989586621679039265">r</a> <a href="#local-6989586621679039264">l</a> : <a href="#local-6989586621679039263">buildList</a> (drop <a href="#local-6989586621679039265">r</a> <a href="#local-6989586621679039264">l</a>) <a href="#local-6989586621679039265">r</a>

869−<a name="line-869"></a>

870−<a name="line-870"></a>-- | Lifts functions to relations with dimensions matching @a@ and @b@

871−<a name="line-871"></a>-- cardinality's.

872−<a name="line-872"></a>fromFRel' ::

873−<a name="line-873"></a> forall <a name="local-6989586621679038573"><a href="#local-6989586621679038573">a</a></a> <a name="local-6989586621679038574"><a href="#local-6989586621679038574">b</a></a>.

874−<a name="line-874"></a> ( Bounded <a href="#local-6989586621679038573">a</a>,

875−<a name="line-875"></a> Bounded <a href="#local-6989586621679038574">b</a>,

876−<a name="line-876"></a> Enum <a href="#local-6989586621679038573">a</a>,

877−<a name="line-877"></a> Enum <a href="#local-6989586621679038574">b</a>,

878−<a name="line-878"></a> Eq <a href="#local-6989586621679038574">b</a>,

879−<a name="line-879"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038573">a</a>)),

880−<a name="line-880"></a> KnownNat (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038574">b</a>)),

881−<a name="line-881"></a> <a href="Matrix.Internal.html#FromLists">FromLists</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038574">b</a>) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038573">a</a>)

882−<a name="line-882"></a> ) =>

883−<a name="line-883"></a> (<a href="#local-6989586621679038573">a</a> -> <a href="#local-6989586621679038574">b</a>) ->

884−<a name="line-884"></a> <a href="Matrix.Internal.html#Matrix">Matrix</a> (<a href="Utils.Internal.html#Natural">Natural</a> 0 1) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038573">a</a>) (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038574">b</a>)

885−<a name="line-885"></a><a name="fromFRel%27"><a href="Matrix.Internal.html#fromFRel%27">fromFRel'</a></a> <a name="local-6989586621679039279"><a href="#local-6989586621679039279">f</a></a> =

886−<a name="line-886"></a> let <a name="local-6989586621679039283"><a href="#local-6989586621679039283">minA</a></a> = minBound @<a href="#local-6989586621679038573">a</a>

887−<a name="line-887"></a> <a name="local-6989586621679039284"><a href="#local-6989586621679039284">maxA</a></a> = maxBound @<a href="#local-6989586621679038573">a</a>

888−<a name="line-888"></a> <a name="local-6989586621679039285"><a href="#local-6989586621679039285">minB</a></a> = minBound @<a href="#local-6989586621679038574">b</a>

889−<a name="line-889"></a> <a name="local-6989586621679039286"><a href="#local-6989586621679039286">maxB</a></a> = maxBound @<a href="#local-6989586621679038574">b</a>

890−<a name="line-890"></a> <a name="local-6989586621679039287"><a href="#local-6989586621679039287">ccols</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038573">a</a>)))

891−<a name="line-891"></a> <a name="local-6989586621679039288"><a href="#local-6989586621679039288">rrows</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">Count</a> (<a href="Matrix.Internal.html#Normalize">Normalize</a> <a href="#local-6989586621679038574">b</a>)))

892−<a name="line-892"></a> <a name="local-6989586621679039289"><a href="#local-6989586621679039289">elementsA</a></a> = take <a href="#local-6989586621679039287">ccols</a> [<a href="#local-6989586621679039283">minA</a> .. <a href="#local-6989586621679039284">maxA</a>]

893−<a name="line-893"></a> <a name="local-6989586621679039290"><a href="#local-6989586621679039290">elementsB</a></a> = take <a href="#local-6989586621679039288">rrows</a> [<a href="#local-6989586621679039285">minB</a> .. <a href="#local-6989586621679039286">maxB</a>]

894−<a name="line-894"></a> <a name="local-6989586621679039291"><a href="#local-6989586621679039291">combinations</a></a> = (,) <$> <a href="#local-6989586621679039289">elementsA</a> <*> <a href="#local-6989586621679039290">elementsB</a>

895−<a name="line-895"></a> <a name="local-6989586621679039292"><a href="#local-6989586621679039292">combAp</a></a> = map snd . sort . map (\(<a name="local-6989586621679039294"><a href="#local-6989586621679039294">a</a></a>, <a name="local-6989586621679039295"><a href="#local-6989586621679039295">b</a></a>) -> if <a href="#local-6989586621679039279">f</a> <a href="#local-6989586621679039294">a</a> == <a href="#local-6989586621679039295">b</a>

896−<a name="line-896"></a> then ((fromEnum <a href="#local-6989586621679039294">a</a>, fromEnum <a href="#local-6989586621679039295">b</a>), <a href="Utils.Internal.html#nat">nat</a> 1)

897−<a name="line-897"></a> else ((fromEnum <a href="#local-6989586621679039294">a</a>, fromEnum <a href="#local-6989586621679039295">b</a>), <a href="Utils.Internal.html#nat">nat</a> 0)) $ <a href="#local-6989586621679039291">combinations</a>

898−<a name="line-898"></a> <a name="local-6989586621679039293"><a href="#local-6989586621679039293">mList</a></a> = <a href="#local-6989586621679039280">buildList</a> <a href="#local-6989586621679039292">combAp</a> <a href="#local-6989586621679039288">rrows</a>

899−<a name="line-899"></a> in <a href="Matrix.Internal.html#tr">tr</a> $ <a href="Matrix.Internal.html#fromLists">fromLists</a> <a href="#local-6989586621679039293">mList</a>

900−<a name="line-900"></a> where

901−<a name="line-901"></a> <a name="local-6989586621679039280"><a href="#local-6989586621679039280">buildList</a></a> [] _ = []

902−<a name="line-902"></a> buildList <a name="local-6989586621679039281"><a href="#local-6989586621679039281">l</a></a> <a name="local-6989586621679039282"><a href="#local-6989586621679039282">r</a></a> = take <a href="#local-6989586621679039282">r</a> <a href="#local-6989586621679039281">l</a> : <a href="#local-6989586621679039280">buildList</a> (drop <a href="#local-6989586621679039282">r</a> <a href="#local-6989586621679039281">l</a>) <a href="#local-6989586621679039282">r</a>

903−<a name="line-903"></a></pre></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Matrix.Nat.htmldeleted−484

@@ -1,484 +0,0 @@

1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>{-# LANGUAGE AllowAmbiguousTypes #-}

2−<a name="line-2"></a>{-# LANGUAGE DataKinds #-}

3−<a name="line-3"></a>{-# LANGUAGE DerivingVia #-}

4−<a name="line-4"></a>{-# LANGUAGE FlexibleContexts #-}

5−<a name="line-5"></a>{-# LANGUAGE FlexibleInstances #-}

6−<a name="line-6"></a>{-# LANGUAGE GADTs #-}

7−<a name="line-7"></a>{-# LANGUAGE KindSignatures #-}

8−<a name="line-8"></a>{-# LANGUAGE MultiParamTypeClasses #-}

9−<a name="line-9"></a>{-# LANGUAGE NoStarIsType #-}

10−<a name="line-10"></a>{-# LANGUAGE RankNTypes #-}

11−<a name="line-11"></a>{-# LANGUAGE ScopedTypeVariables #-}

12−<a name="line-12"></a>{-# LANGUAGE TypeApplications #-}

13−<a name="line-13"></a>{-# LANGUAGE TypeOperators #-}

14−<a name="line-14"></a>{-# LANGUAGE UndecidableInstances #-}

15−<a name="line-15"></a>

16−<a name="line-16"></a>-----------------------------------------------------------------------------

17−<a name="line-17"></a>-- |

18−<a name="line-18"></a>-- Module : Matrix.Nat

19−<a name="line-19"></a>-- Copyright : (c) Armando Santos 2019-2020

20−<a name="line-20"></a>-- Maintainer : armandoifsantos@gmail.com

21−<a name="line-21"></a>-- Stability : experimental

22−<a name="line-22"></a>--

23−<a name="line-23"></a>-- The LAoP discipline generalises relations and functions treating them as

24−<a name="line-24"></a>-- Boolean matrices and in turn consider these as arrows.

25−<a name="line-25"></a>--

26−<a name="line-26"></a>-- __LAoP__ is a library for algebraic (inductive) construction and manipulation of matrices

27−<a name="line-27"></a>-- in Haskell. See <https://github.com/bolt12/master-thesis my Msc Thesis> for the

28−<a name="line-28"></a>-- motivation behind the library, the underlying theory, and implementation details.

29−<a name="line-29"></a>--

30−<a name="line-30"></a>-- This module offers a newtype wrapper around 'Matrix.Type.Matrix' that

31−<a name="line-31"></a>-- uses type level naturals instead of standard data types for the matrices

32−<a name="line-32"></a>-- dimensions.

33−<a name="line-33"></a>--

34−<a name="line-34"></a>-----------------------------------------------------------------------------

35−<a name="line-35"></a>

36−<a name="line-36"></a>module Matrix.Nat

37−<a name="line-37"></a> ( -- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.

38−<a name="line-38"></a> --

39−<a name="line-39"></a> -- LAoP generalises relations and functions treating them as

40−<a name="line-40"></a> -- Boolean matrices and in turn consider these as arrows.

41−<a name="line-41"></a> -- This library offers many of the combinators mentioned in the work of

42−<a name="line-42"></a> -- Macedo (2012) and Oliveira (2012).

43−<a name="line-43"></a> --

44−<a name="line-44"></a> -- This definition is a wrapper around 'Matrix.Type' but

45−<a name="line-45"></a> -- dimensions are type level Naturals. Type inference might not

46−<a name="line-46"></a> -- be as desired.

47−<a name="line-47"></a> --

48−<a name="line-48"></a> -- There exists two type families that make it easier to write

49−<a name="line-49"></a> -- matrix dimensions: 'FromNat' and 'Count'. This approach

50−<a name="line-50"></a> -- leads to a very straightforward implementation

51−<a name="line-51"></a> -- of LAoP combinators.

52−<a name="line-52"></a>

53−<a name="line-53"></a> -- * Type safe matrix representation

54−<a name="line-54"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> (..),

55−<a name="line-55"></a>

56−<a name="line-56"></a> -- * Primitives

57−<a name="line-57"></a> <a href="Matrix.Nat.html#empty">empty</a>,

58−<a name="line-58"></a> <a href="Matrix.Nat.html#one">one</a>,

59−<a name="line-59"></a> <a href="Matrix.Nat.html#junc">junc</a>,

60−<a name="line-60"></a> <a href="Matrix.Nat.html#split">split</a>,

61−<a name="line-61"></a>

62−<a name="line-62"></a> -- * Auxiliary type families

63−<a name="line-63"></a> <a href="Matrix.Internal.html#FromNat">I.FromNat</a>,

64−<a name="line-64"></a> <a href="Matrix.Internal.html#Count">I.Count</a>,

65−<a name="line-65"></a> <a href="Matrix.Internal.html#Normalize">I.Normalize</a>,

66−<a name="line-66"></a>

67−<a name="line-67"></a> -- * Matrix construction and conversion

68−<a name="line-68"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a>,

69−<a name="line-69"></a> <a href="Matrix.Nat.html#fromLists">fromLists</a>,

70−<a name="line-70"></a> <a href="Matrix.Nat.html#toLists">toLists</a>,

71−<a name="line-71"></a> <a href="Matrix.Nat.html#toList">toList</a>,

72−<a name="line-72"></a> <a href="Matrix.Nat.html#matrixBuilder">matrixBuilder</a>,

73−<a name="line-73"></a> <a href="Matrix.Nat.html#row">row</a>,

74−<a name="line-74"></a> <a href="Matrix.Nat.html#col">col</a>,

75−<a name="line-75"></a> <a href="Matrix.Nat.html#zeros">zeros</a>,

76−<a name="line-76"></a> <a href="Matrix.Nat.html#ones">ones</a>,

77−<a name="line-77"></a> <a href="Matrix.Nat.html#bang">bang</a>,

78−<a name="line-78"></a> <a href="Matrix.Nat.html#constant">constant</a>,

79−<a name="line-79"></a>

80−<a name="line-80"></a> -- * Misc

81−<a name="line-81"></a> -- ** Get dimensions

82−<a name="line-82"></a> <a href="Matrix.Nat.html#columns">columns</a>,

83−<a name="line-83"></a> <a href="Matrix.Nat.html#rows">rows</a>,

84−<a name="line-84"></a>

85−<a name="line-85"></a> -- ** Matrix Transposition

86−<a name="line-86"></a> <a href="Matrix.Nat.html#tr">tr</a>,

87−<a name="line-87"></a>

88−<a name="line-88"></a> -- ** Selective operator

89−<a name="line-89"></a> <a href="Matrix.Nat.html#select">select</a>,

90−<a name="line-90"></a>

91−<a name="line-91"></a> -- ** McCarthy's Conditional

92−<a name="line-92"></a> <a href="Matrix.Nat.html#cond">cond</a>,

93−<a name="line-93"></a>

94−<a name="line-94"></a> -- ** Matrix "abiding"

95−<a name="line-95"></a> <a href="Matrix.Nat.html#abideJS">abideJS</a>,

96−<a name="line-96"></a> <a href="Matrix.Nat.html#abideSJ">abideSJ</a>,

97−<a name="line-97"></a>

98−<a name="line-98"></a> -- * Biproduct approach

99−<a name="line-99"></a> -- ** Split

100−<a name="line-100"></a> (<a href="Matrix.Nat.html#%3D%3D%3D">===</a>),

101−<a name="line-101"></a> -- *** Projections

102−<a name="line-102"></a> <a href="Matrix.Nat.html#p1">p1</a>,

103−<a name="line-103"></a> <a href="Matrix.Nat.html#p2">p2</a>,

104−<a name="line-104"></a> -- ** Junc

105−<a name="line-105"></a> (<a href="Matrix.Nat.html#%7C%7C%7C">|||</a>),

106−<a name="line-106"></a> -- *** Injections

107−<a name="line-107"></a> <a href="Matrix.Nat.html#i1">i1</a>,

108−<a name="line-108"></a> <a href="Matrix.Nat.html#i2">i2</a>,

109−<a name="line-109"></a> -- ** Bifunctors

110−<a name="line-110"></a> (<a href="Matrix.Nat.html#-%7C-">-|-</a>),

111−<a name="line-111"></a> (<a href="Matrix.Nat.html#%3E%3C">><</a>),

112−<a name="line-112"></a>

113−<a name="line-113"></a> -- ** Applicative matrix combinators

114−<a name="line-114"></a>

115−<a name="line-115"></a> -- | Note that given the restrictions imposed it is not possible to

116−<a name="line-116"></a> -- implement the standard type classes present in standard Haskell.

117−<a name="line-117"></a> -- *** Matrix pairing projections

118−<a name="line-118"></a> <a href="Matrix.Nat.html#kp1">kp1</a>,

119−<a name="line-119"></a> <a href="Matrix.Nat.html#kp2">kp2</a>,

120−<a name="line-120"></a>

121−<a name="line-121"></a> -- *** Matrix pairing

122−<a name="line-122"></a> <a href="Matrix.Nat.html#khatri">khatri</a>,

123−<a name="line-123"></a>

124−<a name="line-124"></a> -- * Matrix composition and lifting

125−<a name="line-125"></a>

126−<a name="line-126"></a> -- ** Arrow matrix combinators

127−<a name="line-127"></a>

128−<a name="line-128"></a> -- | Note that given the restrictions imposed it is not possible to

129−<a name="line-129"></a> -- implement the standard type classes present in standard Haskell.

130−<a name="line-130"></a> <a href="Matrix.Nat.html#identity">identity</a>,

131−<a name="line-131"></a> <a href="Matrix.Nat.html#comp">comp</a>,

132−<a name="line-132"></a> <a href="Matrix.Nat.html#fromF">fromF</a>,

133−<a name="line-133"></a> <a href="Matrix.Nat.html#fromF%27">fromF'</a>,

134−<a name="line-134"></a>

135−<a name="line-135"></a> -- * Matrix printing

136−<a name="line-136"></a> <a href="Matrix.Nat.html#pretty">pretty</a>,

137−<a name="line-137"></a> <a href="Matrix.Nat.html#prettyPrint">prettyPrint</a>

138−<a name="line-138"></a> )

139−<a name="line-139"></a>where

140−<a name="line-140"></a>

141−<a name="line-141"></a>import Data.Proxy

142−<a name="line-142"></a>import GHC.TypeLits

143−<a name="line-143"></a>import Control.DeepSeq

144−<a name="line-144"></a>import qualified Control.Category as C

145−<a name="line-145"></a>import qualified <a href="Matrix.Internal.html">Matrix.Internal</a> as I

146−<a name="line-146"></a>

147−<a name="line-147"></a>newtype <a name="Matrix"><a href="Matrix.Nat.html#Matrix">Matrix</a></a> <a name="local-6989586621679063540"><a href="#local-6989586621679063540">e</a></a> (<a name="local-6989586621679063541"><a href="#local-6989586621679063541">cols</a></a> :: Nat) (<a name="local-6989586621679063542"><a href="#local-6989586621679063542">rows</a></a> :: Nat) = <a name="M"><a href="Matrix.Nat.html#M">M</a></a> (<a href="Matrix.Internal.html#Matrix">I.Matrix</a> <a href="#local-6989586621679063540">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063541">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063542">rows</a>))

148−<a name="line-148"></a> deriving (Show, Num, Eq, Ord, NFData) via (<a href="Matrix.Internal.html#Matrix">I.Matrix</a> <a href="#local-6989586621679063540">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063541">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063542">rows</a>))

149−<a name="line-149"></a>

150−<a name="line-150"></a>-- | It isn't possible to implement the 'id' function so it's

151−<a name="line-151"></a>-- implementation is 'undefined'. However 'comp' can be and this partial

152−<a name="line-152"></a>-- class implementation exists just to make the code more readable.

153−<a name="line-153"></a>--

154−<a name="line-154"></a>-- Please use 'identity' instead.

155−<a name="line-155"></a>instance (Num <a href="#local-6989586621679063543">e</a>) => C.Category (<a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063543">e</a>) where

156−<a name="line-156"></a> <a name="local-8214565720323789508">id</a> = undefined

157−<a name="line-157"></a> (<a name="local-8214565720323789507">.</a>) = <a href="Matrix.Nat.html#comp">comp</a>

158−<a name="line-158"></a>

159−<a name="line-159"></a>-- Primitives

160−<a name="line-160"></a>

161−<a name="line-161"></a>empty :: <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063670">e</a> 0 0

162−<a name="line-162"></a><a name="empty"><a href="Matrix.Nat.html#empty">empty</a></a> = <a href="Matrix.Nat.html#M">M</a> <a href="Matrix.Internal.html#Empty">I.Empty</a>

163−<a name="line-163"></a>

164−<a name="line-164"></a>one :: <a href="#local-6989586621679063669">e</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063669">e</a> 1 1

165−<a name="line-165"></a><a name="one"><a href="Matrix.Nat.html#one">one</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#One">I.One</a>

166−<a name="line-166"></a>

167−<a name="line-167"></a>junc ::

168−<a name="line-168"></a> (I.FromNat <a href="#local-6989586621679063664">cols3</a> ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063665">cols1</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063666">cols2</a>)) =>

169−<a name="line-169"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063667">e</a> <a href="#local-6989586621679063665">cols1</a> <a href="#local-6989586621679063668">rows</a> ->

170−<a name="line-170"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063667">e</a> <a href="#local-6989586621679063666">cols2</a> <a href="#local-6989586621679063668">rows</a> ->

171−<a name="line-171"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063667">e</a> <a href="#local-6989586621679063664">cols3</a> <a href="#local-6989586621679063668">rows</a>

172−<a name="line-172"></a><a name="junc"><a href="Matrix.Nat.html#junc">junc</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063671"><a href="#local-6989586621679063671">a</a></a>) (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063672"><a href="#local-6989586621679063672">b</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#Junc">I.Junc</a> <a href="#local-6989586621679063671">a</a> <a href="#local-6989586621679063672">b</a>)

173−<a name="line-173"></a>

174−<a name="line-174"></a>infixl 3 <a href="Matrix.Nat.html#%7C%7C%7C">|||</a>

175−<a name="line-175"></a>(|||) ::

176−<a name="line-176"></a> (I.FromNat <a href="#local-6989586621679063659">cols3</a> ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063660">cols1</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063661">cols2</a>)) =>

177−<a name="line-177"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063662">e</a> <a href="#local-6989586621679063660">cols1</a> <a href="#local-6989586621679063663">rows</a> ->

178−<a name="line-178"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063662">e</a> <a href="#local-6989586621679063661">cols2</a> <a href="#local-6989586621679063663">rows</a> ->

179−<a name="line-179"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063662">e</a> <a href="#local-6989586621679063659">cols3</a> <a href="#local-6989586621679063663">rows</a>

180−<a name="line-180"></a>(<a name="%7C%7C%7C"><a href="Matrix.Nat.html#%7C%7C%7C">|||</a></a>) = <a href="Matrix.Nat.html#junc">junc</a>

181−<a name="line-181"></a>

182−<a name="line-182"></a>split ::

183−<a name="line-183"></a> (I.FromNat <a href="#local-6989586621679063654">rows3</a> ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063655">rows1</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063656">rows2</a>)) =>

184−<a name="line-184"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063657">e</a> <a href="#local-6989586621679063658">cols</a> <a href="#local-6989586621679063655">rows1</a> ->

185−<a name="line-185"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063657">e</a> <a href="#local-6989586621679063658">cols</a> <a href="#local-6989586621679063656">rows2</a> ->

186−<a name="line-186"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063657">e</a> <a href="#local-6989586621679063658">cols</a> <a href="#local-6989586621679063654">rows3</a>

187−<a name="line-187"></a><a name="split"><a href="Matrix.Nat.html#split">split</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063673"><a href="#local-6989586621679063673">a</a></a>) (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063674"><a href="#local-6989586621679063674">b</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#Split">I.Split</a> <a href="#local-6989586621679063673">a</a> <a href="#local-6989586621679063674">b</a>)

188−<a name="line-188"></a>

189−<a name="line-189"></a>infixl 2 <a href="Matrix.Nat.html#%3D%3D%3D">===</a>

190−<a name="line-190"></a>(===) ::

191−<a name="line-191"></a> (I.FromNat <a href="#local-6989586621679063649">rows3</a> ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063650">rows1</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063651">rows2</a>)) =>

192−<a name="line-192"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063652">e</a> <a href="#local-6989586621679063653">cols</a> <a href="#local-6989586621679063650">rows1</a> ->

193−<a name="line-193"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063652">e</a> <a href="#local-6989586621679063653">cols</a> <a href="#local-6989586621679063651">rows2</a> ->

194−<a name="line-194"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063652">e</a> <a href="#local-6989586621679063653">cols</a> <a href="#local-6989586621679063649">rows3</a>

195−<a name="line-195"></a>(<a name="%3D%3D%3D"><a href="Matrix.Nat.html#%3D%3D%3D">===</a></a>) = <a href="Matrix.Nat.html#split">split</a>

196−<a name="line-196"></a>

197−<a name="line-197"></a>-- Construction

198−<a name="line-198"></a>

199−<a name="line-199"></a>fromLists :: (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063646">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063647">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063648">rows</a>)) => [[<a href="#local-6989586621679063646">e</a>]] -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063646">e</a> <a href="#local-6989586621679063647">cols</a> <a href="#local-6989586621679063648">rows</a>

200−<a name="line-200"></a><a name="fromLists"><a href="Matrix.Nat.html#fromLists">fromLists</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#fromLists">I.fromLists</a>

201−<a name="line-201"></a>

202−<a name="line-202"></a>matrixBuilder ::

203−<a name="line-203"></a> (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063643">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063644">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063645">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063644">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063645">rows</a>))) =>

204−<a name="line-204"></a> ((Int, Int) -> <a href="#local-6989586621679063643">e</a>) ->

205−<a name="line-205"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063643">e</a> <a href="#local-6989586621679063644">cols</a> <a href="#local-6989586621679063645">rows</a>

206−<a name="line-206"></a><a name="matrixBuilder"><a href="Matrix.Nat.html#matrixBuilder">matrixBuilder</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#matrixBuilder">I.matrixBuilder</a>

207−<a name="line-207"></a>

208−<a name="line-208"></a>col :: (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063641">e</a> () (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063642">rows</a>)) => [<a href="#local-6989586621679063641">e</a>] -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063641">e</a> 1 <a href="#local-6989586621679063642">rows</a>

209−<a name="line-209"></a><a name="col"><a href="Matrix.Nat.html#col">col</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#col">I.col</a>

210−<a name="line-210"></a>

211−<a name="line-211"></a>row :: (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063639">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063640">cols</a>) ()) => [<a href="#local-6989586621679063639">e</a>] -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063639">e</a> <a href="#local-6989586621679063640">cols</a> 1

212−<a name="line-212"></a><a name="row"><a href="Matrix.Nat.html#row">row</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#row">I.row</a>

213−<a name="line-213"></a>

214−<a name="line-214"></a>fromF ::

215−<a name="line-215"></a> ( Bounded <a href="#local-6989586621679063634">a</a>,

216−<a name="line-216"></a> Bounded <a href="#local-6989586621679063635">b</a>,

217−<a name="line-217"></a> Enum <a href="#local-6989586621679063634">a</a>,

218−<a name="line-218"></a> Enum <a href="#local-6989586621679063635">b</a>,

219−<a name="line-219"></a> Eq <a href="#local-6989586621679063635">b</a>,

220−<a name="line-220"></a> Num <a href="#local-6989586621679063636">e</a>,

221−<a name="line-221"></a> Ord <a href="#local-6989586621679063636">e</a>,

222−<a name="line-222"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063637">cols</a>)),

223−<a name="line-223"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063638">rows</a>)),

224−<a name="line-224"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063636">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063638">rows</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063637">cols</a>)

225−<a name="line-225"></a> ) =>

226−<a name="line-226"></a> (<a href="#local-6989586621679063634">a</a> -> <a href="#local-6989586621679063635">b</a>) ->

227−<a name="line-227"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063636">e</a> <a href="#local-6989586621679063637">cols</a> <a href="#local-6989586621679063638">rows</a>

228−<a name="line-228"></a><a name="fromF"><a href="Matrix.Nat.html#fromF">fromF</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#fromF">I.fromF</a>

229−<a name="line-229"></a>

230−<a name="line-230"></a>fromF' ::

231−<a name="line-231"></a> ( Bounded <a href="#local-6989586621679063631">a</a>,

232−<a name="line-232"></a> Bounded <a href="#local-6989586621679063632">b</a>,

233−<a name="line-233"></a> Enum <a href="#local-6989586621679063631">a</a>,

234−<a name="line-234"></a> Enum <a href="#local-6989586621679063632">b</a>,

235−<a name="line-235"></a> Eq <a href="#local-6989586621679063632">b</a>,

236−<a name="line-236"></a> Num <a href="#local-6989586621679063633">e</a>,

237−<a name="line-237"></a> Ord <a href="#local-6989586621679063633">e</a>,

238−<a name="line-238"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679063631">a</a>)),

239−<a name="line-239"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679063632">b</a>)),

240−<a name="line-240"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063633">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679063632">b</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679063631">a</a>)

241−<a name="line-241"></a> ) =>

242−<a name="line-242"></a> (<a href="#local-6989586621679063631">a</a> -> <a href="#local-6989586621679063632">b</a>) ->

243−<a name="line-243"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063633">e</a> (<a href="Matrix.Internal.html#Count">I.Count</a> <a href="#local-6989586621679063631">a</a>) (<a href="Matrix.Internal.html#Count">I.Count</a> <a href="#local-6989586621679063632">b</a>)

244−<a name="line-244"></a><a name="fromF%27"><a href="Matrix.Nat.html#fromF%27">fromF'</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#fromF%27">I.fromF'</a>

245−<a name="line-245"></a>

246−<a name="line-246"></a>-- Conversion

247−<a name="line-247"></a>

248−<a name="line-248"></a>toLists :: <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063628">e</a> <a href="#local-6989586621679063629">cols</a> <a href="#local-6989586621679063630">rows</a> -> [[<a href="#local-6989586621679063628">e</a>]]

249−<a name="line-249"></a><a name="toLists"><a href="Matrix.Nat.html#toLists">toLists</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063675"><a href="#local-6989586621679063675">m</a></a>) = <a href="Matrix.Internal.html#toLists">I.toLists</a> <a href="#local-6989586621679063675">m</a>

250−<a name="line-250"></a>

251−<a name="line-251"></a>toList :: <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063625">e</a> <a href="#local-6989586621679063626">cols</a> <a href="#local-6989586621679063627">rows</a> -> [<a href="#local-6989586621679063625">e</a>]

252−<a name="line-252"></a><a name="toList"><a href="Matrix.Nat.html#toList">toList</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063676"><a href="#local-6989586621679063676">m</a></a>) = <a href="Matrix.Internal.html#toList">I.toList</a> <a href="#local-6989586621679063676">m</a>

253−<a name="line-253"></a>

254−<a name="line-254"></a>-- Zeros Matrix

255−<a name="line-255"></a>

256−<a name="line-256"></a>zeros ::

257−<a name="line-257"></a> (Num <a href="#local-6989586621679063622">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063622">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063623">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063624">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063623">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063624">rows</a>))) =>

258−<a name="line-258"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063622">e</a> <a href="#local-6989586621679063623">cols</a> <a href="#local-6989586621679063624">rows</a>

259−<a name="line-259"></a><a name="zeros"><a href="Matrix.Nat.html#zeros">zeros</a></a> = <a href="Matrix.Nat.html#M">M</a> <a href="Matrix.Internal.html#zeros">I.zeros</a>

260−<a name="line-260"></a>

261−<a name="line-261"></a>-- Ones Matrix

262−<a name="line-262"></a>

263−<a name="line-263"></a>ones ::

264−<a name="line-264"></a> (Num <a href="#local-6989586621679063619">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063619">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063620">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063621">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063620">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063621">rows</a>))) =>

265−<a name="line-265"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063619">e</a> <a href="#local-6989586621679063620">cols</a> <a href="#local-6989586621679063621">rows</a>

266−<a name="line-266"></a><a name="ones"><a href="Matrix.Nat.html#ones">ones</a></a> = <a href="Matrix.Nat.html#M">M</a> <a href="Matrix.Internal.html#ones">I.ones</a>

267−<a name="line-267"></a>

268−<a name="line-268"></a>-- Const Matrix

269−<a name="line-269"></a>

270−<a name="line-270"></a>constant ::

271−<a name="line-271"></a> (Num <a href="#local-6989586621679063616">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063616">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063617">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063618">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063617">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063618">rows</a>))) =>

272−<a name="line-272"></a> <a href="#local-6989586621679063616">e</a> ->

273−<a name="line-273"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063616">e</a> <a href="#local-6989586621679063617">cols</a> <a href="#local-6989586621679063618">rows</a>

274−<a name="line-274"></a><a name="constant"><a href="Matrix.Nat.html#constant">constant</a></a> = <a href="Matrix.Nat.html#M">M</a> . <a href="Matrix.Internal.html#constant">I.constant</a>

275−<a name="line-275"></a>

276−<a name="line-276"></a>-- Bang Matrix

277−<a name="line-277"></a>

278−<a name="line-278"></a>bang ::

279−<a name="line-279"></a> forall <a name="local-6989586621679063614"><a href="#local-6989586621679063614">e</a></a> <a name="local-6989586621679063615"><a href="#local-6989586621679063615">cols</a></a>.

280−<a name="line-280"></a> (Num <a href="#local-6989586621679063614">e</a>, Enum <a href="#local-6989586621679063614">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063614">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063615">cols</a>) (), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063615">cols</a>))) =>

281−<a name="line-281"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063614">e</a> <a href="#local-6989586621679063615">cols</a> 1

282−<a name="line-282"></a><a name="bang"><a href="Matrix.Nat.html#bang">bang</a></a> = <a href="Matrix.Nat.html#M">M</a> <a href="Matrix.Internal.html#bang">I.bang</a>

283−<a name="line-283"></a>

284−<a name="line-284"></a>-- Identity Matrix

285−<a name="line-285"></a>

286−<a name="line-286"></a>identity ::

287−<a name="line-287"></a> (Num <a href="#local-6989586621679063612">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063612">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063613">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063613">cols</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063613">cols</a>))) =>

288−<a name="line-288"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063612">e</a> <a href="#local-6989586621679063613">cols</a> <a href="#local-6989586621679063613">cols</a>

289−<a name="line-289"></a><a name="identity"><a href="Matrix.Nat.html#identity">identity</a></a> = <a href="Matrix.Nat.html#M">M</a> <a href="Matrix.Internal.html#identity">I.identity</a>

290−<a name="line-290"></a>

291−<a name="line-291"></a>-- Matrix composition (MMM)

292−<a name="line-292"></a>

293−<a name="line-293"></a>comp :: (Num <a href="#local-6989586621679063608">e</a>) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063608">e</a> <a href="#local-6989586621679063609">cr</a> <a href="#local-6989586621679063610">rows</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063608">e</a> <a href="#local-6989586621679063611">cols</a> <a href="#local-6989586621679063609">cr</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063608">e</a> <a href="#local-6989586621679063611">cols</a> <a href="#local-6989586621679063610">rows</a>

294−<a name="line-294"></a><a name="comp"><a href="Matrix.Nat.html#comp">comp</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063677"><a href="#local-6989586621679063677">a</a></a>) (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063678"><a href="#local-6989586621679063678">b</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#comp">I.comp</a> <a href="#local-6989586621679063677">a</a> <a href="#local-6989586621679063678">b</a>)

295−<a name="line-295"></a>

296−<a name="line-296"></a>p1 ::

297−<a name="line-297"></a> ( Num <a href="#local-6989586621679063605">e</a>,

298−<a name="line-298"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063606">n</a>)),

299−<a name="line-299"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063607">m</a>)),

300−<a name="line-300"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063605">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063606">n</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063607">m</a>),

301−<a name="line-301"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063605">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063607">m</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063607">m</a>),

302−<a name="line-302"></a> I.FromNat (m + <a href="#local-6989586621679063606">n</a>) ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063607">m</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063606">n</a>)

303−<a name="line-303"></a> ) =>

304−<a name="line-304"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063605">e</a> (m + <a href="#local-6989586621679063606">n</a>) <a href="#local-6989586621679063607">m</a>

305−<a name="line-305"></a><a name="p1"><a href="Matrix.Nat.html#p1">p1</a></a> = <a href="Matrix.Nat.html#M">M</a> <a href="Matrix.Internal.html#p1">I.p1</a>

306−<a name="line-306"></a>

307−<a name="line-307"></a>p2 ::

308−<a name="line-308"></a> ( Num <a href="#local-6989586621679063602">e</a>,

309−<a name="line-309"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063603">n</a>)),

310−<a name="line-310"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063604">m</a>)),

311−<a name="line-311"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063602">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063604">m</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063603">n</a>),

312−<a name="line-312"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063602">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063603">n</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063603">n</a>),

313−<a name="line-313"></a> I.FromNat (m + <a href="#local-6989586621679063603">n</a>) ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063604">m</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063603">n</a>)

314−<a name="line-314"></a> ) =>

315−<a name="line-315"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063602">e</a> (m + <a href="#local-6989586621679063603">n</a>) <a href="#local-6989586621679063603">n</a>

316−<a name="line-316"></a><a name="p2"><a href="Matrix.Nat.html#p2">p2</a></a> = <a href="Matrix.Nat.html#M">M</a> <a href="Matrix.Internal.html#p2">I.p2</a>

317−<a name="line-317"></a>

318−<a name="line-318"></a>-- Injections

319−<a name="line-319"></a>

320−<a name="line-320"></a>i1 ::

321−<a name="line-321"></a> ( Num <a href="#local-6989586621679063599">e</a>,

322−<a name="line-322"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063600">n</a>)),

323−<a name="line-323"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063601">rows</a>)),

324−<a name="line-324"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063599">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063600">n</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063601">rows</a>),

325−<a name="line-325"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063599">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063601">rows</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063601">rows</a>),

326−<a name="line-326"></a> I.FromNat (rows + <a href="#local-6989586621679063600">n</a>) ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063601">rows</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063600">n</a>)

327−<a name="line-327"></a> ) =>

328−<a name="line-328"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063599">e</a> <a href="#local-6989586621679063601">rows</a> (rows + <a href="#local-6989586621679063600">n</a>)

329−<a name="line-329"></a><a name="i1"><a href="Matrix.Nat.html#i1">i1</a></a> = <a href="Matrix.Nat.html#tr">tr</a> <a href="Matrix.Nat.html#p1">p1</a>

330−<a name="line-330"></a>

331−<a name="line-331"></a>i2 ::

332−<a name="line-332"></a> ( Num <a href="#local-6989586621679063596">e</a>,

333−<a name="line-333"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063597">rows</a>)),

334−<a name="line-334"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063598">m</a>)),

335−<a name="line-335"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063596">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063598">m</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063597">rows</a>),

336−<a name="line-336"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063596">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063597">rows</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063597">rows</a>),

337−<a name="line-337"></a> I.FromNat (m + <a href="#local-6989586621679063597">rows</a>) ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063598">m</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063597">rows</a>)

338−<a name="line-338"></a> ) =>

339−<a name="line-339"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063596">e</a> <a href="#local-6989586621679063597">rows</a> (m + <a href="#local-6989586621679063597">rows</a>)

340−<a name="line-340"></a><a name="i2"><a href="Matrix.Nat.html#i2">i2</a></a> = <a href="Matrix.Nat.html#tr">tr</a> <a href="Matrix.Nat.html#p2">p2</a>

341−<a name="line-341"></a>

342−<a name="line-342"></a>-- Dimensions

343−<a name="line-343"></a>

344−<a name="line-344"></a>rows :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063593">rows</a>))) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063594">e</a> <a href="#local-6989586621679063595">cols</a> <a href="#local-6989586621679063593">rows</a> -> Int

345−<a name="line-345"></a><a name="rows"><a href="Matrix.Nat.html#rows">rows</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063679"><a href="#local-6989586621679063679">m</a></a>) = <a href="Matrix.Internal.html#rows">I.rows</a> <a href="#local-6989586621679063679">m</a>

346−<a name="line-346"></a>

347−<a name="line-347"></a>columns :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063590">cols</a>))) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063591">e</a> <a href="#local-6989586621679063590">cols</a> <a href="#local-6989586621679063592">rows</a> -> Int

348−<a name="line-348"></a><a name="columns"><a href="Matrix.Nat.html#columns">columns</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063680"><a href="#local-6989586621679063680">m</a></a>) = <a href="Matrix.Internal.html#columns">I.columns</a> <a href="#local-6989586621679063680">m</a>

349−<a name="line-349"></a>

350−<a name="line-350"></a>-- Coproduct Bifunctor

351−<a name="line-351"></a>

352−<a name="line-352"></a>infixl 5 <a href="Matrix.Nat.html#-%7C-">-|-</a>

353−<a name="line-353"></a>

354−<a name="line-354"></a>(-|-) ::

355−<a name="line-355"></a> ( Num <a href="#local-6989586621679063585">e</a>,

356−<a name="line-356"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063586">j</a>)),

357−<a name="line-357"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063587">k</a>)),

358−<a name="line-358"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063585">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063587">k</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063587">k</a>),

359−<a name="line-359"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063585">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063586">j</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063587">k</a>),

360−<a name="line-360"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063585">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063587">k</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063586">j</a>),

361−<a name="line-361"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063585">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063586">j</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063586">j</a>),

362−<a name="line-362"></a> I.FromNat (n + <a href="#local-6989586621679063589">m</a>) ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063588">n</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063589">m</a>),

363−<a name="line-363"></a> I.FromNat (k + <a href="#local-6989586621679063586">j</a>) ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063587">k</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063586">j</a>)

364−<a name="line-364"></a> ) =>

365−<a name="line-365"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063585">e</a> <a href="#local-6989586621679063588">n</a> <a href="#local-6989586621679063587">k</a> ->

366−<a name="line-366"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063585">e</a> <a href="#local-6989586621679063589">m</a> <a href="#local-6989586621679063586">j</a> ->

367−<a name="line-367"></a> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063585">e</a> (n + <a href="#local-6989586621679063589">m</a>) (k + <a href="#local-6989586621679063586">j</a>)

368−<a name="line-368"></a>(<a name="-%7C-"><a href="Matrix.Nat.html#-%7C-">-|-</a></a>) (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063681"><a href="#local-6989586621679063681">a</a></a>) (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063682"><a href="#local-6989586621679063682">b</a></a>) = <a href="Matrix.Nat.html#M">M</a> ((<a href="Matrix.Internal.html#-%7C-">I.-|-</a>) <a href="#local-6989586621679063681">a</a> <a href="#local-6989586621679063682">b</a>)

369−<a name="line-369"></a>

370−<a name="line-370"></a>-- Khatri Rao Product and projections

371−<a name="line-371"></a>

372−<a name="line-372"></a>kp1 ::

373−<a name="line-373"></a> forall <a name="local-6989586621679063582"><a href="#local-6989586621679063582">e</a></a> <a name="local-6989586621679063583"><a href="#local-6989586621679063583">m</a></a> <a name="local-6989586621679063584"><a href="#local-6989586621679063584">k</a></a> .

374−<a name="line-374"></a> ( Num <a href="#local-6989586621679063582">e</a>,

375−<a name="line-375"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063583">m</a>)),

376−<a name="line-376"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063584">k</a>)),

377−<a name="line-377"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (m * <a href="#local-6989586621679063584">k</a>))),

378−<a name="line-378"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063582">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (m * <a href="#local-6989586621679063584">k</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063583">m</a>),

379−<a name="line-379"></a> I.FromNat (m * <a href="#local-6989586621679063584">k</a>) ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> (I.Count (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063583">m</a>) * <a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063584">k</a>))

380−<a name="line-380"></a> ) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063582">e</a> (m * <a href="#local-6989586621679063584">k</a>) <a href="#local-6989586621679063583">m</a>

381−<a name="line-381"></a><a name="kp1"><a href="Matrix.Nat.html#kp1">kp1</a></a> = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#kp1">I.kp1</a> @<a href="#local-6989586621679063582">e</a> @(<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063583">m</a>) @(<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063584">k</a>))

382−<a name="line-382"></a>

383−<a name="line-383"></a>kp2 ::

384−<a name="line-384"></a> forall <a name="local-6989586621679063579"><a href="#local-6989586621679063579">e</a></a> <a name="local-6989586621679063580"><a href="#local-6989586621679063580">m</a></a> <a name="local-6989586621679063581"><a href="#local-6989586621679063581">k</a></a>.

385−<a name="line-385"></a> ( Num <a href="#local-6989586621679063579">e</a>,

386−<a name="line-386"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063581">k</a>)),

387−<a name="line-387"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063579">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (m * <a href="#local-6989586621679063581">k</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063581">k</a>),

388−<a name="line-388"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063580">m</a>)),

389−<a name="line-389"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (m * <a href="#local-6989586621679063581">k</a>))),

390−<a name="line-390"></a> I.FromNat (m * <a href="#local-6989586621679063581">k</a>) ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> (I.Count (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063580">m</a>) * <a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063581">k</a>))

391−<a name="line-391"></a> ) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063579">e</a> (m * <a href="#local-6989586621679063581">k</a>) <a href="#local-6989586621679063581">k</a>

392−<a name="line-392"></a><a name="kp2"><a href="Matrix.Nat.html#kp2">kp2</a></a> = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#kp2">I.kp2</a> @<a href="#local-6989586621679063579">e</a> @(<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063580">m</a>) @(<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063581">k</a>))

393−<a name="line-393"></a>

394−<a name="line-394"></a>khatri ::

395−<a name="line-395"></a> forall <a name="local-6989586621679063575"><a href="#local-6989586621679063575">e</a></a> <a name="local-6989586621679063576"><a href="#local-6989586621679063576">cols</a></a> <a name="local-6989586621679063577"><a href="#local-6989586621679063577">a</a></a> <a name="local-6989586621679063578"><a href="#local-6989586621679063578">b</a></a>.

396−<a name="line-396"></a> ( Num <a href="#local-6989586621679063575">e</a>,

397−<a name="line-397"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063577">a</a>)),

398−<a name="line-398"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063578">b</a>)),

399−<a name="line-399"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (a * <a href="#local-6989586621679063578">b</a>))),

400−<a name="line-400"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063575">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (a * <a href="#local-6989586621679063578">b</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063577">a</a>),

401−<a name="line-401"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063575">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (a * <a href="#local-6989586621679063578">b</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063578">b</a>),

402−<a name="line-402"></a> I.FromNat (a * <a href="#local-6989586621679063578">b</a>) ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> (I.Count (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063577">a</a>) * <a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063578">b</a>))

403−<a name="line-403"></a> ) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063575">e</a> <a href="#local-6989586621679063576">cols</a> <a href="#local-6989586621679063577">a</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063575">e</a> <a href="#local-6989586621679063576">cols</a> <a href="#local-6989586621679063578">b</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063575">e</a> <a href="#local-6989586621679063576">cols</a> (a * <a href="#local-6989586621679063578">b</a>)

404−<a name="line-404"></a><a name="khatri"><a href="Matrix.Nat.html#khatri">khatri</a></a> <a name="local-6989586621679063683"><a href="#local-6989586621679063683">a</a></a> <a name="local-6989586621679063684"><a href="#local-6989586621679063684">b</a></a> =

405−<a name="line-405"></a> let <a name="local-6989586621679063685"><a href="#local-6989586621679063685">kp1'</a></a> = <a href="Matrix.Nat.html#kp1">kp1</a> @<a href="#local-6989586621679063575">e</a> @<a href="#local-6989586621679063577">a</a> @<a href="#local-6989586621679063578">b</a>

406−<a name="line-406"></a> <a name="local-6989586621679063686"><a href="#local-6989586621679063686">kp2'</a></a> = <a href="Matrix.Nat.html#kp2">kp2</a> @<a href="#local-6989586621679063575">e</a> @<a href="#local-6989586621679063577">a</a> @<a href="#local-6989586621679063578">b</a>

407−<a name="line-407"></a> in <a href="Matrix.Nat.html#comp">comp</a> (<a href="Matrix.Nat.html#tr">tr</a> <a href="#local-6989586621679063685">kp1'</a>) <a href="#local-6989586621679063683">a</a> * <a href="Matrix.Nat.html#comp">comp</a> (<a href="Matrix.Nat.html#tr">tr</a> <a href="#local-6989586621679063686">kp2'</a>) <a href="#local-6989586621679063684">b</a>

408−<a name="line-408"></a>

409−<a name="line-409"></a>-- Product Bifunctor (Kronecker)

410−<a name="line-410"></a>

411−<a name="line-411"></a>infixl 4 <a href="Matrix.Nat.html#%3E%3C">><</a>

412−<a name="line-412"></a>

413−<a name="line-413"></a>(><) ::

414−<a name="line-414"></a> forall <a name="local-6989586621679063570"><a href="#local-6989586621679063570">e</a></a> <a name="local-6989586621679063571"><a href="#local-6989586621679063571">m</a></a> <a name="local-6989586621679063572"><a href="#local-6989586621679063572">p</a></a> <a name="local-6989586621679063573"><a href="#local-6989586621679063573">n</a></a> <a name="local-6989586621679063574"><a href="#local-6989586621679063574">q</a></a>.

415−<a name="line-415"></a> ( Num <a href="#local-6989586621679063570">e</a>,

416−<a name="line-416"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063571">m</a>)),

417−<a name="line-417"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063573">n</a>)),

418−<a name="line-418"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063572">p</a>)),

419−<a name="line-419"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063574">q</a>)),

420−<a name="line-420"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (m * <a href="#local-6989586621679063573">n</a>))),

421−<a name="line-421"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (p * <a href="#local-6989586621679063574">q</a>))),

422−<a name="line-422"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063570">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (m * <a href="#local-6989586621679063573">n</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063571">m</a>),

423−<a name="line-423"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063570">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (m * <a href="#local-6989586621679063573">n</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063573">n</a>),

424−<a name="line-424"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063570">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (p * <a href="#local-6989586621679063574">q</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063572">p</a>),

425−<a name="line-425"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063570">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> (p * <a href="#local-6989586621679063574">q</a>)) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063574">q</a>),

426−<a name="line-426"></a> I.FromNat (m * <a href="#local-6989586621679063573">n</a>) ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> (I.Count (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063571">m</a>) * <a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063573">n</a>)),

427−<a name="line-427"></a> I.FromNat (p * <a href="#local-6989586621679063574">q</a>) ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> (I.Count (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063572">p</a>) * <a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063574">q</a>))

428−<a name="line-428"></a> ) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063570">e</a> <a href="#local-6989586621679063571">m</a> <a href="#local-6989586621679063572">p</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063570">e</a> <a href="#local-6989586621679063573">n</a> <a href="#local-6989586621679063574">q</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063570">e</a> (m * <a href="#local-6989586621679063573">n</a>) (p * <a href="#local-6989586621679063574">q</a>)

429−<a name="line-429"></a>(<a name="%3E%3C"><a href="Matrix.Nat.html#%3E%3C">><</a></a>) <a name="local-6989586621679063687"><a href="#local-6989586621679063687">a</a></a> <a name="local-6989586621679063688"><a href="#local-6989586621679063688">b</a></a> =

430−<a name="line-430"></a> let <a name="local-6989586621679063689"><a href="#local-6989586621679063689">kp1'</a></a> = <a href="Matrix.Nat.html#kp1">kp1</a> @<a href="#local-6989586621679063570">e</a> @<a href="#local-6989586621679063571">m</a> @<a href="#local-6989586621679063573">n</a>

431−<a name="line-431"></a> <a name="local-6989586621679063690"><a href="#local-6989586621679063690">kp2'</a></a> = <a href="Matrix.Nat.html#kp2">kp2</a> @<a href="#local-6989586621679063570">e</a> @<a href="#local-6989586621679063571">m</a> @<a href="#local-6989586621679063573">n</a>

432−<a name="line-432"></a> in <a href="Matrix.Nat.html#khatri">khatri</a> (<a href="Matrix.Nat.html#comp">comp</a> <a href="#local-6989586621679063687">a</a> <a href="#local-6989586621679063689">kp1'</a>) (<a href="Matrix.Nat.html#comp">comp</a> <a href="#local-6989586621679063688">b</a> <a href="#local-6989586621679063690">kp2'</a>)

433−<a name="line-433"></a>

434−<a name="line-434"></a>-- Matrix abide Junc Split

435−<a name="line-435"></a>

436−<a name="line-436"></a>abideJS :: <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063567">e</a> <a href="#local-6989586621679063568">cols</a> <a href="#local-6989586621679063569">rows</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063567">e</a> <a href="#local-6989586621679063568">cols</a> <a href="#local-6989586621679063569">rows</a>

437−<a name="line-437"></a><a name="abideJS"><a href="Matrix.Nat.html#abideJS">abideJS</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063691"><a href="#local-6989586621679063691">m</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#abideJS">I.abideJS</a> <a href="#local-6989586621679063691">m</a>)

438−<a name="line-438"></a>

439−<a name="line-439"></a>-- Matrix abide Split Junc

440−<a name="line-440"></a>

441−<a name="line-441"></a>abideSJ :: <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063564">e</a> <a href="#local-6989586621679063565">cols</a> <a href="#local-6989586621679063566">rows</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063564">e</a> <a href="#local-6989586621679063565">cols</a> <a href="#local-6989586621679063566">rows</a>

442−<a name="line-442"></a><a name="abideSJ"><a href="Matrix.Nat.html#abideSJ">abideSJ</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063692"><a href="#local-6989586621679063692">m</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#abideSJ">I.abideSJ</a> <a href="#local-6989586621679063692">m</a>)

443−<a name="line-443"></a>

444−<a name="line-444"></a>-- Matrix transposition

445−<a name="line-445"></a>

446−<a name="line-446"></a>tr :: <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063561">e</a> <a href="#local-6989586621679063562">cols</a> <a href="#local-6989586621679063563">rows</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063561">e</a> <a href="#local-6989586621679063563">rows</a> <a href="#local-6989586621679063562">cols</a>

447−<a name="line-447"></a><a name="tr"><a href="Matrix.Nat.html#tr">tr</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063693"><a href="#local-6989586621679063693">m</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#tr">I.tr</a> <a href="#local-6989586621679063693">m</a>)

448−<a name="line-448"></a>

449−<a name="line-449"></a>-- Selective 'select' operator

450−<a name="line-450"></a>select ::

451−<a name="line-451"></a> ( Num <a href="#local-6989586621679063554">e</a>,

452−<a name="line-452"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063554">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063555">rows1</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063555">rows1</a>),

453−<a name="line-453"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063555">rows1</a>)),

454−<a name="line-454"></a> I.FromNat <a href="#local-6989586621679063556">rows2</a> ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063555">rows1</a>,

455−<a name="line-455"></a> I.FromNat <a href="#local-6989586621679063557">cols1</a> ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063558">cols2</a>,

456−<a name="line-456"></a> I.FromNat <a href="#local-6989586621679063559">rows3</a> ~ Either (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063560">cols3</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063555">rows1</a>)

457−<a name="line-457"></a> ) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063554">e</a> <a href="#local-6989586621679063557">cols1</a> <a href="#local-6989586621679063559">rows3</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063554">e</a> <a href="#local-6989586621679063560">cols3</a> <a href="#local-6989586621679063555">rows1</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063554">e</a> <a href="#local-6989586621679063558">cols2</a> <a href="#local-6989586621679063556">rows2</a>

458−<a name="line-458"></a><a name="select"><a href="Matrix.Nat.html#select">select</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063694"><a href="#local-6989586621679063694">m</a></a>) (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063695"><a href="#local-6989586621679063695">y</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#select">I.select</a> <a href="#local-6989586621679063694">m</a> <a href="#local-6989586621679063695">y</a>)

459−<a name="line-459"></a>

460−<a name="line-460"></a>-- McCarthy's Conditional

461−<a name="line-461"></a>

462−<a name="line-462"></a>cond ::

463−<a name="line-463"></a> ( I.FromNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063550">cols</a>)) ~ <a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063550">cols</a>,

464−<a name="line-464"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063550">cols</a>)),

465−<a name="line-465"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063551">e</a> () (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063550">cols</a>),

466−<a name="line-466"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063551">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063550">cols</a>) (),

467−<a name="line-467"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679063551">e</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063550">cols</a>) (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063550">cols</a>),

468−<a name="line-468"></a> Bounded <a href="#local-6989586621679063552">a</a>,

469−<a name="line-469"></a> Enum <a href="#local-6989586621679063552">a</a>,

470−<a name="line-470"></a> Num <a href="#local-6989586621679063551">e</a>,

471−<a name="line-471"></a> Ord <a href="#local-6989586621679063551">e</a>

472−<a name="line-472"></a> )

473−<a name="line-473"></a> =>

474−<a name="line-474"></a> (<a href="#local-6989586621679063552">a</a> -> Bool) -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063551">e</a> <a href="#local-6989586621679063550">cols</a> <a href="#local-6989586621679063553">rows</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063551">e</a> <a href="#local-6989586621679063550">cols</a> <a href="#local-6989586621679063553">rows</a> -> <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063551">e</a> <a href="#local-6989586621679063550">cols</a> <a href="#local-6989586621679063553">rows</a>

475−<a name="line-475"></a><a name="cond"><a href="Matrix.Nat.html#cond">cond</a></a> <a name="local-6989586621679063696"><a href="#local-6989586621679063696">p</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063697"><a href="#local-6989586621679063697">a</a></a>) (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063698"><a href="#local-6989586621679063698">b</a></a>) = <a href="Matrix.Nat.html#M">M</a> (<a href="Matrix.Internal.html#cond">I.cond</a> <a href="#local-6989586621679063696">p</a> <a href="#local-6989586621679063697">a</a> <a href="#local-6989586621679063698">b</a>)

476−<a name="line-476"></a>

477−<a name="line-477"></a>-- Pretty print

478−<a name="line-478"></a>

479−<a name="line-479"></a>pretty :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063547">cols</a>)), Show <a href="#local-6989586621679063548">e</a>) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063548">e</a> <a href="#local-6989586621679063547">cols</a> <a href="#local-6989586621679063549">rows</a> -> String

480−<a name="line-480"></a><a name="pretty"><a href="Matrix.Nat.html#pretty">pretty</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063699"><a href="#local-6989586621679063699">m</a></a>) = <a href="Matrix.Internal.html#pretty">I.pretty</a> <a href="#local-6989586621679063699">m</a>

481−<a name="line-481"></a>

482−<a name="line-482"></a>prettyPrint :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#FromNat">I.FromNat</a> <a href="#local-6989586621679063544">cols</a>)), Show <a href="#local-6989586621679063545">e</a>) => <a href="Matrix.Nat.html#Matrix">Matrix</a> <a href="#local-6989586621679063545">e</a> <a href="#local-6989586621679063544">cols</a> <a href="#local-6989586621679063546">rows</a> -> IO ()

483−<a name="line-483"></a><a name="prettyPrint"><a href="Matrix.Nat.html#prettyPrint">prettyPrint</a></a> (<a href="Matrix.Nat.html#M">M</a> <a name="local-6989586621679063700"><a href="#local-6989586621679063700">m</a></a>) = <a href="Matrix.Internal.html#prettyPrint">I.prettyPrint</a> <a href="#local-6989586621679063700">m</a>

484−<a name="line-484"></a></pre></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Matrix.Type.htmldeleted−648

@@ -1,648 +0,0 @@

1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>{-# LANGUAGE AllowAmbiguousTypes #-}

2−<a name="line-2"></a>{-# LANGUAGE DataKinds #-}

3−<a name="line-3"></a>{-# LANGUAGE DerivingVia #-}

4−<a name="line-4"></a>{-# LANGUAGE FlexibleContexts #-}

5−<a name="line-5"></a>{-# LANGUAGE FlexibleInstances #-}

6−<a name="line-6"></a>{-# LANGUAGE GADTs #-}

7−<a name="line-7"></a>{-# LANGUAGE KindSignatures #-}

8−<a name="line-8"></a>{-# LANGUAGE MultiParamTypeClasses #-}

9−<a name="line-9"></a>{-# LANGUAGE NoStarIsType #-}

10−<a name="line-10"></a>{-# LANGUAGE RankNTypes #-}

11−<a name="line-11"></a>{-# LANGUAGE ScopedTypeVariables #-}

12−<a name="line-12"></a>{-# LANGUAGE TypeApplications #-}

13−<a name="line-13"></a>{-# LANGUAGE TypeOperators #-}

14−<a name="line-14"></a>{-# LANGUAGE UndecidableInstances #-}

15−<a name="line-15"></a>

16−<a name="line-16"></a>-----------------------------------------------------------------------------

17−<a name="line-17"></a>-- |

18−<a name="line-18"></a>-- Module : Matrix.Type

19−<a name="line-19"></a>-- Copyright : (c) Armando Santos 2019-2020

20−<a name="line-20"></a>-- Maintainer : armandoifsantos@gmail.com

21−<a name="line-21"></a>-- Stability : experimental

22−<a name="line-22"></a>--

23−<a name="line-23"></a>-- The LAoP discipline generalises relations and functions treating them as

24−<a name="line-24"></a>-- Boolean matrices and in turn consider these as arrows.

25−<a name="line-25"></a>--

26−<a name="line-26"></a>-- __LAoP__ is a library for algebraic (inductive) construction and manipulation of matrices

27−<a name="line-27"></a>-- in Haskell. See <https://github.com/bolt12/master-thesis my Msc Thesis> for the

28−<a name="line-28"></a>-- motivation behind the library, the underlying theory, and implementation details.

29−<a name="line-29"></a>--

30−<a name="line-30"></a>-- This module offers a newtype wrapper around 'Matrix.Type.Matrix' that

31−<a name="line-31"></a>-- uses arbitrary types instead of canonical data types for the matrices

32−<a name="line-32"></a>-- dimensions.

33−<a name="line-33"></a>--

34−<a name="line-34"></a>-- __NOTE__: If the types in the dimensions are custom they must need to

35−<a name="line-35"></a>-- implement a 'Generic' instance.

36−<a name="line-36"></a>--

37−<a name="line-37"></a>-----------------------------------------------------------------------------

38−<a name="line-38"></a>

39−<a name="line-39"></a>module Matrix.Type

40−<a name="line-40"></a> ( -- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.

41−<a name="line-41"></a> --

42−<a name="line-42"></a> -- LAoP generalises relations and functions treating them as

43−<a name="line-43"></a> -- Boolean matrices and in turn consider these as arrows.

44−<a name="line-44"></a> -- This library offers many of the combinators mentioned in the work of

45−<a name="line-45"></a> -- Macedo (2012) and Oliveira (2012).

46−<a name="line-46"></a> --

47−<a name="line-47"></a> -- This definition is a wrapper around 'Matrix.Type' but

48−<a name="line-48"></a> -- dimensions are arbitrary data types. Type inference might not

49−<a name="line-49"></a> -- be as desired.

50−<a name="line-50"></a>

51−<a name="line-51"></a> -- __NOTE__: If the types in the dimensions are custom they must need to

52−<a name="line-52"></a> -- implement a 'Generic' instance.

53−<a name="line-53"></a> --

54−<a name="line-54"></a>

55−<a name="line-55"></a> -- * Type safe matrix representation

56−<a name="line-56"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> (..),

57−<a name="line-57"></a>

58−<a name="line-58"></a> -- * Type aliases

59−<a name="line-59"></a> <a href="Matrix.Type.html#Zero">Zero</a>,

60−<a name="line-60"></a> <a href="Matrix.Type.html#One">One</a>,

61−<a name="line-61"></a>

62−<a name="line-62"></a> -- * Primitives

63−<a name="line-63"></a> <a href="Matrix.Type.html#empty">empty</a>,

64−<a name="line-64"></a> <a href="Matrix.Type.html#one">one</a>,

65−<a name="line-65"></a> <a href="Matrix.Type.html#junc">junc</a>,

66−<a name="line-66"></a> <a href="Matrix.Type.html#split">split</a>,

67−<a name="line-67"></a>

68−<a name="line-68"></a> -- * Auxiliary type families

69−<a name="line-69"></a> <a href="Matrix.Internal.html#FromNat">I.FromNat</a>,

70−<a name="line-70"></a> <a href="Matrix.Internal.html#Count">I.Count</a>,

71−<a name="line-71"></a> <a href="Matrix.Internal.html#Normalize">I.Normalize</a>,

72−<a name="line-72"></a>

73−<a name="line-73"></a> -- * Matrix construction and conversion

74−<a name="line-74"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a>,

75−<a name="line-75"></a> <a href="Matrix.Type.html#fromLists">fromLists</a>,

76−<a name="line-76"></a> <a href="Matrix.Type.html#toLists">toLists</a>,

77−<a name="line-77"></a> <a href="Matrix.Type.html#toList">toList</a>,

78−<a name="line-78"></a> <a href="Matrix.Type.html#matrixBuilder">matrixBuilder</a>,

79−<a name="line-79"></a> <a href="Matrix.Type.html#row">row</a>,

80−<a name="line-80"></a> <a href="Matrix.Type.html#col">col</a>,

81−<a name="line-81"></a> <a href="Matrix.Type.html#zeros">zeros</a>,

82−<a name="line-82"></a> <a href="Matrix.Type.html#ones">ones</a>,

83−<a name="line-83"></a> <a href="Matrix.Type.html#bang">bang</a>,

84−<a name="line-84"></a> <a href="Matrix.Type.html#constant">constant</a>,

85−<a name="line-85"></a>

86−<a name="line-86"></a> -- * Functor instance equivalent function

87−<a name="line-87"></a> <a href="Matrix.Type.html#fmapM">fmapM</a>,

88−<a name="line-88"></a>

89−<a name="line-89"></a> -- * Applicative/Monoidal instance equivalent functions

90−<a name="line-90"></a> <a href="Matrix.Type.html#unitM">unitM</a>,

91−<a name="line-91"></a> <a href="Matrix.Type.html#multM">multM</a>,

92−<a name="line-92"></a>

93−<a name="line-93"></a> -- * Selective equivalent instance function

94−<a name="line-94"></a> <a href="Matrix.Type.html#selectM">selectM</a>,

95−<a name="line-95"></a>

96−<a name="line-96"></a> -- * Monad equivalent instance function

97−<a name="line-97"></a> <a href="Matrix.Type.html#returnM">returnM</a>,

98−<a name="line-98"></a> <a href="Matrix.Type.html#bindM">bindM</a>,

99−<a name="line-99"></a>

100−<a name="line-100"></a> -- * Misc

101−<a name="line-101"></a> -- ** Get dimensions

102−<a name="line-102"></a> <a href="Matrix.Type.html#columns">columns</a>,

103−<a name="line-103"></a> <a href="Matrix.Type.html#rows">rows</a>,

104−<a name="line-104"></a>

105−<a name="line-105"></a> -- ** Matrix Transposition

106−<a name="line-106"></a> <a href="Matrix.Type.html#tr">tr</a>,

107−<a name="line-107"></a>

108−<a name="line-108"></a> -- ** McCarthy's Conditional

109−<a name="line-109"></a> <a href="Matrix.Type.html#cond">cond</a>,

110−<a name="line-110"></a>

111−<a name="line-111"></a> -- ** Matrix "abiding"

112−<a name="line-112"></a> <a href="Matrix.Type.html#abideJS">abideJS</a>,

113−<a name="line-113"></a> <a href="Matrix.Type.html#abideSJ">abideSJ</a>,

114−<a name="line-114"></a>

115−<a name="line-115"></a> -- * Biproduct approach

116−<a name="line-116"></a> -- ** Split

117−<a name="line-117"></a> (<a href="Matrix.Type.html#%3D%3D%3D">===</a>),

118−<a name="line-118"></a> -- *** Projections

119−<a name="line-119"></a> <a href="Matrix.Type.html#p1">p1</a>,

120−<a name="line-120"></a> <a href="Matrix.Type.html#p2">p2</a>,

121−<a name="line-121"></a> -- ** Junc

122−<a name="line-122"></a> (<a href="Matrix.Type.html#%7C%7C%7C">|||</a>),

123−<a name="line-123"></a> -- *** Injections

124−<a name="line-124"></a> <a href="Matrix.Type.html#i1">i1</a>,

125−<a name="line-125"></a> <a href="Matrix.Type.html#i2">i2</a>,

126−<a name="line-126"></a> -- ** Bifunctors

127−<a name="line-127"></a> (<a href="Matrix.Type.html#-%7C-">-|-</a>),

128−<a name="line-128"></a> (<a href="Matrix.Type.html#%3E%3C">><</a>),

129−<a name="line-129"></a>

130−<a name="line-130"></a> -- ** Applicative matrix combinators

131−<a name="line-131"></a>

132−<a name="line-132"></a> -- | Note that given the restrictions imposed it is not possible to

133−<a name="line-133"></a> -- implement the standard type classes present in standard Haskell.

134−<a name="line-134"></a> -- *** Matrix pairing projections

135−<a name="line-135"></a> <a href="Matrix.Type.html#kp1">kp1</a>,

136−<a name="line-136"></a> <a href="Matrix.Type.html#kp2">kp2</a>,

137−<a name="line-137"></a>

138−<a name="line-138"></a> -- *** Matrix pairing

139−<a name="line-139"></a> <a href="Matrix.Type.html#khatri">khatri</a>,

140−<a name="line-140"></a>

141−<a name="line-141"></a> -- * Matrix composition and lifting

142−<a name="line-142"></a>

143−<a name="line-143"></a> -- ** Arrow matrix combinators

144−<a name="line-144"></a>

145−<a name="line-145"></a> -- | Note that given the restrictions imposed it is not possible to

146−<a name="line-146"></a> -- implement the standard type classes present in standard Haskell.

147−<a name="line-147"></a> <a href="Matrix.Type.html#identity">identity</a>,

148−<a name="line-148"></a> <a href="Matrix.Type.html#comp">comp</a>,

149−<a name="line-149"></a> <a href="Matrix.Type.html#fromF">fromF</a>,

150−<a name="line-150"></a> <a href="Matrix.Type.html#fromF%27">fromF'</a>,

151−<a name="line-151"></a>

152−<a name="line-152"></a> -- * Matrix printing

153−<a name="line-153"></a> <a href="Matrix.Type.html#pretty">pretty</a>,

154−<a name="line-154"></a> <a href="Matrix.Type.html#prettyPrint">prettyPrint</a>

155−<a name="line-155"></a> )

156−<a name="line-156"></a>where

157−<a name="line-157"></a>

158−<a name="line-158"></a>import Data.Void

159−<a name="line-159"></a>import Data.Proxy

160−<a name="line-160"></a>import Data.Kind

161−<a name="line-161"></a>import GHC.TypeLits

162−<a name="line-162"></a>import Control.DeepSeq

163−<a name="line-163"></a>import qualified Control.Category as C

164−<a name="line-164"></a>import qualified <a href="Matrix.Internal.html">Matrix.Internal</a> as I

165−<a name="line-165"></a>

166−<a name="line-166"></a>newtype <a name="Matrix"><a href="Matrix.Type.html#Matrix">Matrix</a></a> <a name="local-6989586621679046459"><a href="#local-6989586621679046459">e</a></a> (<a name="local-6989586621679046460"><a href="#local-6989586621679046460">cols</a></a> :: Type) (<a name="local-6989586621679046461"><a href="#local-6989586621679046461">rows</a></a> :: Type) = <a name="M"><a href="Matrix.Type.html#M">M</a></a> (<a href="Matrix.Internal.html#Matrix">I.Matrix</a> <a href="#local-6989586621679046459">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046460">cols</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046461">rows</a>))

167−<a name="line-167"></a> deriving (Show, Num, Eq, Ord, NFData) via (<a href="Matrix.Internal.html#Matrix">I.Matrix</a> <a href="#local-6989586621679046459">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046460">cols</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046461">rows</a>))

168−<a name="line-168"></a>

169−<a name="line-169"></a>-- | It isn't possible to implement the 'id' function so it's

170−<a name="line-170"></a>-- implementation is 'undefined'. However 'comp' can be and this partial

171−<a name="line-171"></a>-- class implementation exists just to make the code more readable.

172−<a name="line-172"></a>--

173−<a name="line-173"></a>-- Please use 'identity' instead.

174−<a name="line-174"></a>instance (Num <a href="#local-6989586621679046462">e</a>) => C.Category (<a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046462">e</a>) where

175−<a name="line-175"></a> <a name="local-8214565720323789508">id</a> = undefined

176−<a name="line-176"></a> (<a name="local-8214565720323789507">.</a>) = <a href="Matrix.Type.html#comp">comp</a>

177−<a name="line-177"></a>

178−<a name="line-178"></a>-- | Zero type alias

179−<a name="line-179"></a>type <a name="Zero"><a href="Matrix.Type.html#Zero">Zero</a></a> = Void

180−<a name="line-180"></a>

181−<a name="line-181"></a>-- | One type alias

182−<a name="line-182"></a>type <a name="One"><a href="Matrix.Type.html#One">One</a></a> = ()

183−<a name="line-183"></a>

184−<a name="line-184"></a>-- Primitives

185−<a name="line-185"></a>

186−<a name="line-186"></a>-- | Empty matrix constructor

187−<a name="line-187"></a>empty :: <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046596">e</a> <a href="Matrix.Type.html#Zero">Zero</a> <a href="Matrix.Type.html#Zero">Zero</a>

188−<a name="line-188"></a><a name="empty"><a href="Matrix.Type.html#empty">empty</a></a> = <a href="Matrix.Type.html#M">M</a> <a href="Matrix.Internal.html#Empty">I.Empty</a>

189−<a name="line-189"></a>

190−<a name="line-190"></a>-- | Unit matrix constructor

191−<a name="line-191"></a>one :: <a href="#local-6989586621679046595">e</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046595">e</a> <a href="Matrix.Type.html#One">One</a> <a href="Matrix.Type.html#One">One</a>

192−<a name="line-192"></a><a name="one"><a href="Matrix.Type.html#one">one</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#One">I.One</a>

193−<a name="line-193"></a>

194−<a name="line-194"></a>-- | Matrix 'Junc' constructor

195−<a name="line-195"></a>junc ::

196−<a name="line-196"></a> (I.Normalize (Either <a href="#local-6989586621679046591">a</a> <a href="#local-6989586621679046592">b</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046591">a</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046592">b</a>)) =>

197−<a name="line-197"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046593">e</a> <a href="#local-6989586621679046591">a</a> <a href="#local-6989586621679046594">rows</a> ->

198−<a name="line-198"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046593">e</a> <a href="#local-6989586621679046592">b</a> <a href="#local-6989586621679046594">rows</a> ->

199−<a name="line-199"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046593">e</a> (Either <a href="#local-6989586621679046591">a</a> <a href="#local-6989586621679046592">b</a>) <a href="#local-6989586621679046594">rows</a>

200−<a name="line-200"></a><a name="junc"><a href="Matrix.Type.html#junc">junc</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046597"><a href="#local-6989586621679046597">a</a></a>) (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046598"><a href="#local-6989586621679046598">b</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#Junc">I.Junc</a> <a href="#local-6989586621679046597">a</a> <a href="#local-6989586621679046598">b</a>)

201−<a name="line-201"></a>

202−<a name="line-202"></a>infixl 3 <a href="Matrix.Type.html#%7C%7C%7C">|||</a>

203−<a name="line-203"></a>-- | Matrix 'Junc' constructor

204−<a name="line-204"></a>(|||) ::

205−<a name="line-205"></a> (I.Normalize (Either <a href="#local-6989586621679046587">a</a> <a href="#local-6989586621679046588">b</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046587">a</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046588">b</a>)) =>

206−<a name="line-206"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046589">e</a> <a href="#local-6989586621679046587">a</a> <a href="#local-6989586621679046590">rows</a> ->

207−<a name="line-207"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046589">e</a> <a href="#local-6989586621679046588">b</a> <a href="#local-6989586621679046590">rows</a> ->

208−<a name="line-208"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046589">e</a> (Either <a href="#local-6989586621679046587">a</a> <a href="#local-6989586621679046588">b</a>) <a href="#local-6989586621679046590">rows</a>

209−<a name="line-209"></a>(<a name="%7C%7C%7C"><a href="Matrix.Type.html#%7C%7C%7C">|||</a></a>) = <a href="Matrix.Type.html#junc">junc</a>

210−<a name="line-210"></a>

211−<a name="line-211"></a>-- | Matrix 'Split' constructor

212−<a name="line-212"></a>split ::

213−<a name="line-213"></a> (I.Normalize (Either <a href="#local-6989586621679046583">a</a> <a href="#local-6989586621679046584">b</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046583">a</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046584">b</a>)) =>

214−<a name="line-214"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046585">e</a> <a href="#local-6989586621679046586">cols</a> <a href="#local-6989586621679046583">a</a> ->

215−<a name="line-215"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046585">e</a> <a href="#local-6989586621679046586">cols</a> <a href="#local-6989586621679046584">b</a> ->

216−<a name="line-216"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046585">e</a> <a href="#local-6989586621679046586">cols</a> (Either <a href="#local-6989586621679046583">a</a> <a href="#local-6989586621679046584">b</a>)

217−<a name="line-217"></a><a name="split"><a href="Matrix.Type.html#split">split</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046599"><a href="#local-6989586621679046599">a</a></a>) (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046600"><a href="#local-6989586621679046600">b</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#Split">I.Split</a> <a href="#local-6989586621679046599">a</a> <a href="#local-6989586621679046600">b</a>)

218−<a name="line-218"></a>

219−<a name="line-219"></a>infixl 2 <a href="Matrix.Type.html#%3D%3D%3D">===</a>

220−<a name="line-220"></a>-- | Matrix 'Split' constructor

221−<a name="line-221"></a>(===) ::

222−<a name="line-222"></a> (I.Normalize (Either <a href="#local-6989586621679046579">a</a> <a href="#local-6989586621679046580">b</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046579">a</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046580">b</a>)) =>

223−<a name="line-223"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046581">e</a> <a href="#local-6989586621679046582">cols</a> <a href="#local-6989586621679046579">a</a> ->

224−<a name="line-224"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046581">e</a> <a href="#local-6989586621679046582">cols</a> <a href="#local-6989586621679046580">b</a> ->

225−<a name="line-225"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046581">e</a> <a href="#local-6989586621679046582">cols</a> (Either <a href="#local-6989586621679046579">a</a> <a href="#local-6989586621679046580">b</a>)

226−<a name="line-226"></a>(<a name="%3D%3D%3D"><a href="Matrix.Type.html#%3D%3D%3D">===</a></a>) = <a href="Matrix.Type.html#split">split</a>

227−<a name="line-227"></a>

228−<a name="line-228"></a>-- Functor hierarchy

229−<a name="line-229"></a>

230−<a name="line-230"></a>-- | Functor instance equivalent function

231−<a name="line-231"></a>fmapM ::

232−<a name="line-232"></a> ( Bounded <a href="#local-6989586621679046575">a</a>,

233−<a name="line-233"></a> Bounded <a href="#local-6989586621679046576">b</a>,

234−<a name="line-234"></a> Enum <a href="#local-6989586621679046575">a</a>,

235−<a name="line-235"></a> Enum <a href="#local-6989586621679046576">b</a>,

236−<a name="line-236"></a> Eq <a href="#local-6989586621679046576">b</a>,

237−<a name="line-237"></a> Num <a href="#local-6989586621679046577">e</a>,

238−<a name="line-238"></a> Ord <a href="#local-6989586621679046577">e</a>,

239−<a name="line-239"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046575">a</a>)),

240−<a name="line-240"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046576">b</a>)),

241−<a name="line-241"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046577">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046576">b</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046575">a</a>)

242−<a name="line-242"></a> )

243−<a name="line-243"></a> =>

244−<a name="line-244"></a> (<a href="#local-6989586621679046575">a</a> -> <a href="#local-6989586621679046576">b</a>) -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046577">e</a> <a href="#local-6989586621679046578">c</a> <a href="#local-6989586621679046575">a</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046577">e</a> <a href="#local-6989586621679046578">c</a> <a href="#local-6989586621679046576">b</a>

245−<a name="line-245"></a><a name="fmapM"><a href="Matrix.Type.html#fmapM">fmapM</a></a> <a name="local-6989586621679046601"><a href="#local-6989586621679046601">f</a></a> <a name="local-6989586621679046602"><a href="#local-6989586621679046602">m</a></a> = <a href="Matrix.Type.html#fromF%27">fromF'</a> <a href="#local-6989586621679046601">f</a> `<a href="Matrix.Type.html#comp">comp</a>` <a href="#local-6989586621679046602">m</a>

246−<a name="line-246"></a>

247−<a name="line-247"></a>-- | Applicative instance equivalent 'unit' function,

248−<a name="line-248"></a>unitM :: (Num <a href="#local-6989586621679046574">e</a>) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046574">e</a> () ()

249−<a name="line-249"></a><a name="unitM"><a href="Matrix.Type.html#unitM">unitM</a></a> = <a href="Matrix.Type.html#one">one</a> 1

250−<a name="line-250"></a>

251−<a name="line-251"></a>-- | Applicative instance equivalent 'unit' function,

252−<a name="line-252"></a>multM ::

253−<a name="line-253"></a> ( KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046570">a</a>)),

254−<a name="line-254"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046571">b</a>)),

255−<a name="line-255"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046570">a</a>, <a href="#local-6989586621679046571">b</a>))),

256−<a name="line-256"></a> Num <a href="#local-6989586621679046572">e</a>,

257−<a name="line-257"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046572">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046570">a</a>, <a href="#local-6989586621679046571">b</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046570">a</a>),

258−<a name="line-258"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046572">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046570">a</a>, <a href="#local-6989586621679046571">b</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046571">b</a>),

259−<a name="line-259"></a> I.Normalize (<a href="#local-6989586621679046570">a</a>, <a href="#local-6989586621679046571">b</a>) ~ <a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046570">a</a>, <a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046571">b</a>)

260−<a name="line-260"></a> ) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046572">e</a> <a href="#local-6989586621679046573">c</a> <a href="#local-6989586621679046570">a</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046572">e</a> <a href="#local-6989586621679046573">c</a> <a href="#local-6989586621679046571">b</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046572">e</a> <a href="#local-6989586621679046573">c</a> (<a href="#local-6989586621679046570">a</a>, <a href="#local-6989586621679046571">b</a>)

261−<a name="line-261"></a><a name="multM"><a href="Matrix.Type.html#multM">multM</a></a> <a name="local-6989586621679046603"><a href="#local-6989586621679046603">a</a></a> <a name="local-6989586621679046604"><a href="#local-6989586621679046604">b</a></a> = <a href="Matrix.Type.html#khatri">khatri</a> <a href="#local-6989586621679046603">a</a> <a href="#local-6989586621679046604">b</a>

262−<a name="line-262"></a>

263−<a name="line-263"></a>

264−<a name="line-264"></a>-- | Monad instance equivalent 'return' function,

265−<a name="line-265"></a>returnM ::

266−<a name="line-266"></a> forall <a name="local-6989586621679046568"><a href="#local-6989586621679046568">e</a></a> <a name="local-6989586621679046569"><a href="#local-6989586621679046569">a</a></a> .

267−<a name="line-267"></a> ( Num <a href="#local-6989586621679046568">e</a>,

268−<a name="line-268"></a> Enum <a href="#local-6989586621679046568">e</a>,

269−<a name="line-269"></a> Enum <a href="#local-6989586621679046569">a</a>,

270−<a name="line-270"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046568">e</a> () (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046569">a</a>),

271−<a name="line-271"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> <a href="#local-6989586621679046569">a</a>)

272−<a name="line-272"></a> ) => <a href="#local-6989586621679046569">a</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046568">e</a> <a href="Matrix.Type.html#One">One</a> <a href="#local-6989586621679046569">a</a>

273−<a name="line-273"></a><a name="returnM"><a href="Matrix.Type.html#returnM">returnM</a></a> <a name="local-6989586621679046605"><a href="#local-6989586621679046605">a</a></a> = <a href="Matrix.Type.html#col">col</a> <a href="#local-6989586621679046608">l</a>

274−<a name="line-274"></a> where

275−<a name="line-275"></a> <a name="local-6989586621679046606"><a href="#local-6989586621679046606">i</a></a> = fromInteger $ natVal (Proxy :: Proxy (<a href="Matrix.Internal.html#Count">I.Count</a> <a href="#local-6989586621679046569">a</a>))

276−<a name="line-276"></a> <a name="local-6989586621679046607"><a href="#local-6989586621679046607">x</a></a> = fromEnum <a href="#local-6989586621679046605">a</a>

277−<a name="line-277"></a> <a name="local-6989586621679046608"><a href="#local-6989586621679046608">l</a></a> = take <a href="#local-6989586621679046607">x</a> [0,0..] ++ [1] ++ take (<a href="#local-6989586621679046606">i</a> - <a href="#local-6989586621679046607">x</a> - 1) [0,0..]

278−<a name="line-278"></a>

279−<a name="line-279"></a>-- | Monad instance equivalent '(>>=)' function,

280−<a name="line-280"></a>bindM :: (Num <a href="#local-6989586621679046564">e</a>) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046564">e</a> <a href="#local-6989586621679046565">a</a> <a href="#local-6989586621679046566">b</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046564">e</a> <a href="#local-6989586621679046566">b</a> <a href="#local-6989586621679046567">c</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046564">e</a> <a href="#local-6989586621679046565">a</a> <a href="#local-6989586621679046567">c</a>

281−<a name="line-281"></a><a name="bindM"><a href="Matrix.Type.html#bindM">bindM</a></a> = flip <a href="Matrix.Type.html#comp">comp</a>

282−<a name="line-282"></a>

283−<a name="line-283"></a>-- Construction

284−<a name="line-284"></a>

285−<a name="line-285"></a>-- | Build a matrix out of a list of list of elements. Throws a runtime

286−<a name="line-286"></a>-- error if the dimensions do not match.

287−<a name="line-287"></a>fromLists :: (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046561">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046562">cols</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046563">rows</a>)) => [[<a href="#local-6989586621679046561">e</a>]] -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046561">e</a> <a href="#local-6989586621679046562">cols</a> <a href="#local-6989586621679046563">rows</a>

288−<a name="line-288"></a><a name="fromLists"><a href="Matrix.Type.html#fromLists">fromLists</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#fromLists">I.fromLists</a>

289−<a name="line-289"></a>

290−<a name="line-290"></a>-- | Matrix builder function. Constructs a matrix provided with

291−<a name="line-291"></a>-- a construction function.

292−<a name="line-292"></a>matrixBuilder ::

293−<a name="line-293"></a> (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046558">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046559">cols</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046560">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046559">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046560">rows</a>))) =>

294−<a name="line-294"></a> ((Int, Int) -> <a href="#local-6989586621679046558">e</a>) ->

295−<a name="line-295"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046558">e</a> <a href="#local-6989586621679046559">cols</a> <a href="#local-6989586621679046560">rows</a>

296−<a name="line-296"></a><a name="matrixBuilder"><a href="Matrix.Type.html#matrixBuilder">matrixBuilder</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#matrixBuilder">I.matrixBuilder</a>

297−<a name="line-297"></a>

298−<a name="line-298"></a>-- | Constructs a column vector matrix

299−<a name="line-299"></a>col :: (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046556">e</a> () (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046557">rows</a>)) => [<a href="#local-6989586621679046556">e</a>] -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046556">e</a> <a href="Matrix.Type.html#One">One</a> <a href="#local-6989586621679046557">rows</a>

300−<a name="line-300"></a><a name="col"><a href="Matrix.Type.html#col">col</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#col">I.col</a>

301−<a name="line-301"></a>

302−<a name="line-302"></a>-- | Constructs a row vector matrix

303−<a name="line-303"></a>row :: (<a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046554">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046555">cols</a>) ()) => [<a href="#local-6989586621679046554">e</a>] -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046554">e</a> <a href="#local-6989586621679046555">cols</a> <a href="Matrix.Type.html#One">One</a>

304−<a name="line-304"></a><a name="row"><a href="Matrix.Type.html#row">row</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#row">I.row</a>

305−<a name="line-305"></a>

306−<a name="line-306"></a>-- | Lifts functions to matrices with arbitrary dimensions.

307−<a name="line-307"></a>--

308−<a name="line-308"></a>-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

309−<a name="line-309"></a>-- types @a@ or @b@ allows.

310−<a name="line-310"></a>fromF ::

311−<a name="line-311"></a> ( Bounded <a href="#local-6989586621679046549">a</a>,

312−<a name="line-312"></a> Bounded <a href="#local-6989586621679046550">b</a>,

313−<a name="line-313"></a> Enum <a href="#local-6989586621679046549">a</a>,

314−<a name="line-314"></a> Enum <a href="#local-6989586621679046550">b</a>,

315−<a name="line-315"></a> Eq <a href="#local-6989586621679046550">b</a>,

316−<a name="line-316"></a> Num <a href="#local-6989586621679046551">e</a>,

317−<a name="line-317"></a> Ord <a href="#local-6989586621679046551">e</a>,

318−<a name="line-318"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046552">cols</a>)),

319−<a name="line-319"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046553">rows</a>)),

320−<a name="line-320"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046551">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046553">rows</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046552">cols</a>)

321−<a name="line-321"></a> ) =>

322−<a name="line-322"></a> (<a href="#local-6989586621679046549">a</a> -> <a href="#local-6989586621679046550">b</a>) ->

323−<a name="line-323"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046551">e</a> <a href="#local-6989586621679046552">cols</a> <a href="#local-6989586621679046553">rows</a>

324−<a name="line-324"></a><a name="fromF"><a href="Matrix.Type.html#fromF">fromF</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#fromF">I.fromF</a>

325−<a name="line-325"></a>

326−<a name="line-326"></a>-- | Lifts functions to matrices with dimensions matching @a@ and @b@

327−<a name="line-327"></a>-- cardinality's.

328−<a name="line-328"></a>fromF' ::

329−<a name="line-329"></a> ( Bounded <a href="#local-6989586621679046546">a</a>,

330−<a name="line-330"></a> Bounded <a href="#local-6989586621679046547">b</a>,

331−<a name="line-331"></a> Enum <a href="#local-6989586621679046546">a</a>,

332−<a name="line-332"></a> Enum <a href="#local-6989586621679046547">b</a>,

333−<a name="line-333"></a> Eq <a href="#local-6989586621679046547">b</a>,

334−<a name="line-334"></a> Num <a href="#local-6989586621679046548">e</a>,

335−<a name="line-335"></a> Ord <a href="#local-6989586621679046548">e</a>,

336−<a name="line-336"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046546">a</a>)),

337−<a name="line-337"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046547">b</a>)),

338−<a name="line-338"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046548">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046547">b</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046546">a</a>)

339−<a name="line-339"></a> ) =>

340−<a name="line-340"></a> (<a href="#local-6989586621679046546">a</a> -> <a href="#local-6989586621679046547">b</a>) ->

341−<a name="line-341"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046548">e</a> <a href="#local-6989586621679046546">a</a> <a href="#local-6989586621679046547">b</a>

342−<a name="line-342"></a><a name="fromF%27"><a href="Matrix.Type.html#fromF%27">fromF'</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#fromF%27">I.fromF'</a>

343−<a name="line-343"></a>

344−<a name="line-344"></a>-- Conversion

345−<a name="line-345"></a>

346−<a name="line-346"></a>-- | Converts a matrix to a list of lists of elements.

347−<a name="line-347"></a>toLists :: <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046543">e</a> <a href="#local-6989586621679046544">cols</a> <a href="#local-6989586621679046545">rows</a> -> [[<a href="#local-6989586621679046543">e</a>]]

348−<a name="line-348"></a><a name="toLists"><a href="Matrix.Type.html#toLists">toLists</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046609"><a href="#local-6989586621679046609">m</a></a>) = <a href="Matrix.Internal.html#toLists">I.toLists</a> <a href="#local-6989586621679046609">m</a>

349−<a name="line-349"></a>

350−<a name="line-350"></a>-- | Converts a matrix to a list of elements.

351−<a name="line-351"></a>toList :: <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046540">e</a> <a href="#local-6989586621679046541">cols</a> <a href="#local-6989586621679046542">rows</a> -> [<a href="#local-6989586621679046540">e</a>]

352−<a name="line-352"></a><a name="toList"><a href="Matrix.Type.html#toList">toList</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046610"><a href="#local-6989586621679046610">m</a></a>) = <a href="Matrix.Internal.html#toList">I.toList</a> <a href="#local-6989586621679046610">m</a>

353−<a name="line-353"></a>

354−<a name="line-354"></a>-- Zeros Matrix

355−<a name="line-355"></a>

356−<a name="line-356"></a>-- | The zero matrix. A matrix wholly filled with zeros.

357−<a name="line-357"></a>zeros ::

358−<a name="line-358"></a> (Num <a href="#local-6989586621679046537">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046537">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046538">cols</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046539">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046538">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046539">rows</a>))) =>

359−<a name="line-359"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046537">e</a> <a href="#local-6989586621679046538">cols</a> <a href="#local-6989586621679046539">rows</a>

360−<a name="line-360"></a><a name="zeros"><a href="Matrix.Type.html#zeros">zeros</a></a> = <a href="Matrix.Type.html#M">M</a> <a href="Matrix.Internal.html#zeros">I.zeros</a>

361−<a name="line-361"></a>

362−<a name="line-362"></a>-- Ones Matrix

363−<a name="line-363"></a>

364−<a name="line-364"></a>-- | The ones matrix. A matrix wholly filled with ones.

365−<a name="line-365"></a>--

366−<a name="line-366"></a>-- Also known as T (Top) matrix.

367−<a name="line-367"></a>ones ::

368−<a name="line-368"></a> (Num <a href="#local-6989586621679046534">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046534">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046535">cols</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046536">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046535">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046536">rows</a>))) =>

369−<a name="line-369"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046534">e</a> <a href="#local-6989586621679046535">cols</a> <a href="#local-6989586621679046536">rows</a>

370−<a name="line-370"></a><a name="ones"><a href="Matrix.Type.html#ones">ones</a></a> = <a href="Matrix.Type.html#M">M</a> <a href="Matrix.Internal.html#ones">I.ones</a>

371−<a name="line-371"></a>

372−<a name="line-372"></a>-- Const Matrix

373−<a name="line-373"></a>

374−<a name="line-374"></a>-- | The constant matrix constructor. A matrix wholly filled with a given

375−<a name="line-375"></a>-- value.

376−<a name="line-376"></a>constant ::

377−<a name="line-377"></a> (Num <a href="#local-6989586621679046531">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046531">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046532">cols</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046533">rows</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046532">cols</a>)), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046533">rows</a>))) =>

378−<a name="line-378"></a> <a href="#local-6989586621679046531">e</a> ->

379−<a name="line-379"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046531">e</a> <a href="#local-6989586621679046532">cols</a> <a href="#local-6989586621679046533">rows</a>

380−<a name="line-380"></a><a name="constant"><a href="Matrix.Type.html#constant">constant</a></a> = <a href="Matrix.Type.html#M">M</a> . <a href="Matrix.Internal.html#constant">I.constant</a>

381−<a name="line-381"></a>

382−<a name="line-382"></a>-- Bang Matrix

383−<a name="line-383"></a>

384−<a name="line-384"></a>-- | The T (Top) row vector matrix.

385−<a name="line-385"></a>bang ::

386−<a name="line-386"></a> forall <a name="local-6989586621679046529"><a href="#local-6989586621679046529">e</a></a> <a name="local-6989586621679046530"><a href="#local-6989586621679046530">cols</a></a>.

387−<a name="line-387"></a> (Num <a href="#local-6989586621679046529">e</a>, Enum <a href="#local-6989586621679046529">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046529">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046530">cols</a>) (), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046530">cols</a>))) =>

388−<a name="line-388"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046529">e</a> <a href="#local-6989586621679046530">cols</a> <a href="Matrix.Type.html#One">One</a>

389−<a name="line-389"></a><a name="bang"><a href="Matrix.Type.html#bang">bang</a></a> = <a href="Matrix.Type.html#M">M</a> <a href="Matrix.Internal.html#bang">I.bang</a>

390−<a name="line-390"></a>

391−<a name="line-391"></a>-- Identity Matrix

392−<a name="line-392"></a>

393−<a name="line-393"></a>-- | Identity matrix

394−<a name="line-394"></a>identity ::

395−<a name="line-395"></a> (Num <a href="#local-6989586621679046527">e</a>, <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046527">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046528">a</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046528">a</a>), KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046528">a</a>))) =>

396−<a name="line-396"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046527">e</a> <a href="#local-6989586621679046528">a</a> <a href="#local-6989586621679046528">a</a>

397−<a name="line-397"></a><a name="identity"><a href="Matrix.Type.html#identity">identity</a></a> = <a href="Matrix.Type.html#M">M</a> <a href="Matrix.Internal.html#identity">I.identity</a>

398−<a name="line-398"></a>

399−<a name="line-399"></a>-- Matrix composition (MMM)

400−<a name="line-400"></a>

401−<a name="line-401"></a>-- | Matrix composition. Equivalent to matrix-matrix multiplication.

402−<a name="line-402"></a>--

403−<a name="line-403"></a>-- This definition takes advantage of divide-and-conquer and fusion laws

404−<a name="line-404"></a>-- from LAoP.

405−<a name="line-405"></a>comp :: (Num <a href="#local-6989586621679046523">e</a>) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046523">e</a> <a href="#local-6989586621679046524">cr</a> <a href="#local-6989586621679046525">rows</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046523">e</a> <a href="#local-6989586621679046526">cols</a> <a href="#local-6989586621679046524">cr</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046523">e</a> <a href="#local-6989586621679046526">cols</a> <a href="#local-6989586621679046525">rows</a>

406−<a name="line-406"></a><a name="comp"><a href="Matrix.Type.html#comp">comp</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046611"><a href="#local-6989586621679046611">a</a></a>) (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046612"><a href="#local-6989586621679046612">b</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#comp">I.comp</a> <a href="#local-6989586621679046611">a</a> <a href="#local-6989586621679046612">b</a>)

407−<a name="line-407"></a>

408−<a name="line-408"></a>-- | Biproduct first component projection

409−<a name="line-409"></a>p1 ::

410−<a name="line-410"></a> ( Num <a href="#local-6989586621679046520">e</a>,

411−<a name="line-411"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046521">n</a>)),

412−<a name="line-412"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046522">m</a>)),

413−<a name="line-413"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046520">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046521">n</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046522">m</a>),

414−<a name="line-414"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046520">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046522">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046522">m</a>),

415−<a name="line-415"></a> I.Normalize (Either <a href="#local-6989586621679046522">m</a> <a href="#local-6989586621679046521">n</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046522">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046521">n</a>)

416−<a name="line-416"></a> ) =>

417−<a name="line-417"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046520">e</a> (Either <a href="#local-6989586621679046522">m</a> <a href="#local-6989586621679046521">n</a>) <a href="#local-6989586621679046522">m</a>

418−<a name="line-418"></a><a name="p1"><a href="Matrix.Type.html#p1">p1</a></a> = <a href="Matrix.Type.html#M">M</a> <a href="Matrix.Internal.html#p1">I.p1</a>

419−<a name="line-419"></a>

420−<a name="line-420"></a>-- | Biproduct second component projection

421−<a name="line-421"></a>p2 ::

422−<a name="line-422"></a> ( Num <a href="#local-6989586621679046517">e</a>,

423−<a name="line-423"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046518">n</a>)),

424−<a name="line-424"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046519">m</a>)),

425−<a name="line-425"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046517">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046519">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046518">n</a>),

426−<a name="line-426"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046517">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046518">n</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046518">n</a>),

427−<a name="line-427"></a> I.Normalize (Either <a href="#local-6989586621679046519">m</a> <a href="#local-6989586621679046518">n</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046519">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046518">n</a>)

428−<a name="line-428"></a> ) =>

429−<a name="line-429"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046517">e</a> (Either <a href="#local-6989586621679046519">m</a> <a href="#local-6989586621679046518">n</a>) <a href="#local-6989586621679046518">n</a>

430−<a name="line-430"></a><a name="p2"><a href="Matrix.Type.html#p2">p2</a></a> = <a href="Matrix.Type.html#M">M</a> <a href="Matrix.Internal.html#p2">I.p2</a>

431−<a name="line-431"></a>

432−<a name="line-432"></a>-- Injections

433−<a name="line-433"></a>

434−<a name="line-434"></a>-- | Biproduct first component injection

435−<a name="line-435"></a>i1 ::

436−<a name="line-436"></a> ( Num <a href="#local-6989586621679046514">e</a>,

437−<a name="line-437"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046515">n</a>)),

438−<a name="line-438"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046516">m</a>)),

439−<a name="line-439"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046514">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046515">n</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046516">m</a>),

440−<a name="line-440"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046514">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046516">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046516">m</a>),

441−<a name="line-441"></a> I.Normalize (Either <a href="#local-6989586621679046516">m</a> <a href="#local-6989586621679046515">n</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046516">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046515">n</a>)

442−<a name="line-442"></a> ) =>

443−<a name="line-443"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046514">e</a> <a href="#local-6989586621679046516">m</a> (Either <a href="#local-6989586621679046516">m</a> <a href="#local-6989586621679046515">n</a>)

444−<a name="line-444"></a><a name="i1"><a href="Matrix.Type.html#i1">i1</a></a> = <a href="Matrix.Type.html#tr">tr</a> <a href="Matrix.Type.html#p1">p1</a>

445−<a name="line-445"></a>

446−<a name="line-446"></a>-- | Biproduct second component injection

447−<a name="line-447"></a>i2 ::

448−<a name="line-448"></a> ( Num <a href="#local-6989586621679046511">e</a>,

449−<a name="line-449"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046512">n</a>)),

450−<a name="line-450"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046513">m</a>)),

451−<a name="line-451"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046511">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046513">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046512">n</a>),

452−<a name="line-452"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046511">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046512">n</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046512">n</a>),

453−<a name="line-453"></a> I.Normalize (Either <a href="#local-6989586621679046513">m</a> <a href="#local-6989586621679046512">n</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046513">m</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046512">n</a>)

454−<a name="line-454"></a> ) =>

455−<a name="line-455"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046511">e</a> <a href="#local-6989586621679046512">n</a> (Either <a href="#local-6989586621679046513">m</a> <a href="#local-6989586621679046512">n</a>)

456−<a name="line-456"></a><a name="i2"><a href="Matrix.Type.html#i2">i2</a></a> = <a href="Matrix.Type.html#tr">tr</a> <a href="Matrix.Type.html#p2">p2</a>

457−<a name="line-457"></a>

458−<a name="line-458"></a>-- Dimensions

459−<a name="line-459"></a>

460−<a name="line-460"></a>-- | Obtain the number of rows.

461−<a name="line-461"></a>--

462−<a name="line-462"></a>-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

463−<a name="line-463"></a>-- dimensions in constant time.

464−<a name="line-464"></a>--

465−<a name="line-465"></a>-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

466−<a name="line-466"></a>-- exchange for performance.

467−<a name="line-467"></a>rows :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046508">rows</a>))) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046509">e</a> <a href="#local-6989586621679046510">cols</a> <a href="#local-6989586621679046508">rows</a> -> Int

468−<a name="line-468"></a><a name="rows"><a href="Matrix.Type.html#rows">rows</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046613"><a href="#local-6989586621679046613">m</a></a>) = <a href="Matrix.Internal.html#rows">I.rows</a> <a href="#local-6989586621679046613">m</a>

469−<a name="line-469"></a>

470−<a name="line-470"></a>-- | Obtain the number of columns.

471−<a name="line-471"></a>--

472−<a name="line-472"></a>-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

473−<a name="line-473"></a>-- dimensions in constant time.

474−<a name="line-474"></a>--

475−<a name="line-475"></a>-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

476−<a name="line-476"></a>-- exchange for performance.

477−<a name="line-477"></a>columns :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046505">cols</a>))) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046506">e</a> <a href="#local-6989586621679046505">cols</a> <a href="#local-6989586621679046507">rows</a> -> Int

478−<a name="line-478"></a><a name="columns"><a href="Matrix.Type.html#columns">columns</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046614"><a href="#local-6989586621679046614">m</a></a>) = <a href="Matrix.Internal.html#columns">I.columns</a> <a href="#local-6989586621679046614">m</a>

479−<a name="line-479"></a>

480−<a name="line-480"></a>-- Coproduct Bifunctor

481−<a name="line-481"></a>

482−<a name="line-482"></a>infixl 5 <a href="Matrix.Type.html#-%7C-">-|-</a>

483−<a name="line-483"></a>

484−<a name="line-484"></a>-- | Matrix coproduct functor also known as matrix direct sum.

485−<a name="line-485"></a>(-|-) ::

486−<a name="line-486"></a> ( Num <a href="#local-6989586621679046500">e</a>,

487−<a name="line-487"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046501">j</a>)),

488−<a name="line-488"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046502">k</a>)),

489−<a name="line-489"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046500">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046502">k</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046502">k</a>),

490−<a name="line-490"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046500">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046501">j</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046502">k</a>),

491−<a name="line-491"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046500">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046502">k</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046501">j</a>),

492−<a name="line-492"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046500">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046501">j</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046501">j</a>),

493−<a name="line-493"></a> I.Normalize (Either <a href="#local-6989586621679046503">n</a> <a href="#local-6989586621679046504">m</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046503">n</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046504">m</a>),

494−<a name="line-494"></a> I.Normalize (Either <a href="#local-6989586621679046502">k</a> <a href="#local-6989586621679046501">j</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046502">k</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046501">j</a>)

495−<a name="line-495"></a> ) =>

496−<a name="line-496"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046500">e</a> <a href="#local-6989586621679046503">n</a> <a href="#local-6989586621679046502">k</a> ->

497−<a name="line-497"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046500">e</a> <a href="#local-6989586621679046504">m</a> <a href="#local-6989586621679046501">j</a> ->

498−<a name="line-498"></a> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046500">e</a> (Either <a href="#local-6989586621679046503">n</a> <a href="#local-6989586621679046504">m</a>) (Either <a href="#local-6989586621679046502">k</a> <a href="#local-6989586621679046501">j</a>)

499−<a name="line-499"></a>(<a name="-%7C-"><a href="Matrix.Type.html#-%7C-">-|-</a></a>) (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046615"><a href="#local-6989586621679046615">a</a></a>) (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046616"><a href="#local-6989586621679046616">b</a></a>) = <a href="Matrix.Type.html#M">M</a> ((<a href="Matrix.Internal.html#-%7C-">I.-|-</a>) <a href="#local-6989586621679046615">a</a> <a href="#local-6989586621679046616">b</a>)

500−<a name="line-500"></a>

501−<a name="line-501"></a>-- Khatri Rao Product and projections

502−<a name="line-502"></a>

503−<a name="line-503"></a>-- | Khatri Rao product first component projection matrix.

504−<a name="line-504"></a>kp1 ::

505−<a name="line-505"></a> forall <a name="local-6989586621679046497"><a href="#local-6989586621679046497">e</a></a> <a name="local-6989586621679046498"><a href="#local-6989586621679046498">m</a></a> <a name="local-6989586621679046499"><a href="#local-6989586621679046499">k</a></a> .

506−<a name="line-506"></a> ( Num <a href="#local-6989586621679046497">e</a>,

507−<a name="line-507"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046498">m</a>)),

508−<a name="line-508"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046499">k</a>)),

509−<a name="line-509"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046498">m</a>, <a href="#local-6989586621679046499">k</a>))),

510−<a name="line-510"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046497">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046498">m</a>, <a href="#local-6989586621679046499">k</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046498">m</a>),

511−<a name="line-511"></a> I.Normalize (<a href="#local-6989586621679046498">m</a>, <a href="#local-6989586621679046499">k</a>) ~ <a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046498">m</a>, <a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046499">k</a>)

512−<a name="line-512"></a> ) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046497">e</a> (<a href="#local-6989586621679046498">m</a>, <a href="#local-6989586621679046499">k</a>) <a href="#local-6989586621679046498">m</a>

513−<a name="line-513"></a><a name="kp1"><a href="Matrix.Type.html#kp1">kp1</a></a> = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#kp1">I.kp1</a> @<a href="#local-6989586621679046497">e</a> @(<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046498">m</a>) @(<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046499">k</a>))

514−<a name="line-514"></a>

515−<a name="line-515"></a>-- | Khatri Rao product second component projection matrix.

516−<a name="line-516"></a>kp2 ::

517−<a name="line-517"></a> forall <a name="local-6989586621679046494"><a href="#local-6989586621679046494">e</a></a> <a name="local-6989586621679046495"><a href="#local-6989586621679046495">m</a></a> <a name="local-6989586621679046496"><a href="#local-6989586621679046496">k</a></a>.

518−<a name="line-518"></a> ( Num <a href="#local-6989586621679046494">e</a>,

519−<a name="line-519"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046496">k</a>)),

520−<a name="line-520"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046495">m</a>)),

521−<a name="line-521"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046495">m</a>, <a href="#local-6989586621679046496">k</a>))),

522−<a name="line-522"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046494">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046495">m</a>, <a href="#local-6989586621679046496">k</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046496">k</a>),

523−<a name="line-523"></a> I.Normalize (<a href="#local-6989586621679046495">m</a>, <a href="#local-6989586621679046496">k</a>) ~ <a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046495">m</a>, <a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046496">k</a>)

524−<a name="line-524"></a> ) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046494">e</a> (<a href="#local-6989586621679046495">m</a>, <a href="#local-6989586621679046496">k</a>) <a href="#local-6989586621679046496">k</a>

525−<a name="line-525"></a><a name="kp2"><a href="Matrix.Type.html#kp2">kp2</a></a> = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#kp2">I.kp2</a> @<a href="#local-6989586621679046494">e</a> @(<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046495">m</a>) @(<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046496">k</a>))

526−<a name="line-526"></a>

527−<a name="line-527"></a>-- | Khatri Rao Matrix product also known as matrix pairing.

528−<a name="line-528"></a>--

529−<a name="line-529"></a>-- NOTE: That this is not a true categorical product, see for instance:

530−<a name="line-530"></a>--

531−<a name="line-531"></a>-- @

532−<a name="line-532"></a>-- | kp1 `comp` khatri a b == a

533−<a name="line-533"></a>-- khatri a b ==> |

534−<a name="line-534"></a>-- | kp2 `comp` khatri a b == b

535−<a name="line-535"></a>-- @

536−<a name="line-536"></a>--

537−<a name="line-537"></a>-- __Emphasis__ on the implication symbol.

538−<a name="line-538"></a>khatri ::

539−<a name="line-539"></a> forall <a name="local-6989586621679046490"><a href="#local-6989586621679046490">e</a></a> <a name="local-6989586621679046491"><a href="#local-6989586621679046491">cols</a></a> <a name="local-6989586621679046492"><a href="#local-6989586621679046492">a</a></a> <a name="local-6989586621679046493"><a href="#local-6989586621679046493">b</a></a>.

540−<a name="line-540"></a> ( Num <a href="#local-6989586621679046490">e</a>,

541−<a name="line-541"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046492">a</a>)),

542−<a name="line-542"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046493">b</a>)),

543−<a name="line-543"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046492">a</a>, <a href="#local-6989586621679046493">b</a>))),

544−<a name="line-544"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046490">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046492">a</a>, <a href="#local-6989586621679046493">b</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046492">a</a>),

545−<a name="line-545"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046490">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046492">a</a>, <a href="#local-6989586621679046493">b</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046493">b</a>),

546−<a name="line-546"></a> I.Normalize (<a href="#local-6989586621679046492">a</a>, <a href="#local-6989586621679046493">b</a>) ~ <a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046492">a</a>, <a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046493">b</a>)

547−<a name="line-547"></a> ) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046490">e</a> <a href="#local-6989586621679046491">cols</a> <a href="#local-6989586621679046492">a</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046490">e</a> <a href="#local-6989586621679046491">cols</a> <a href="#local-6989586621679046493">b</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046490">e</a> <a href="#local-6989586621679046491">cols</a> (<a href="#local-6989586621679046492">a</a>, <a href="#local-6989586621679046493">b</a>)

548−<a name="line-548"></a><a name="khatri"><a href="Matrix.Type.html#khatri">khatri</a></a> <a name="local-6989586621679046617"><a href="#local-6989586621679046617">a</a></a> <a name="local-6989586621679046618"><a href="#local-6989586621679046618">b</a></a> =

549−<a name="line-549"></a> let <a name="local-6989586621679046619"><a href="#local-6989586621679046619">kp1'</a></a> = <a href="Matrix.Type.html#kp1">kp1</a> @<a href="#local-6989586621679046490">e</a> @<a href="#local-6989586621679046492">a</a> @<a href="#local-6989586621679046493">b</a>

550−<a name="line-550"></a> <a name="local-6989586621679046620"><a href="#local-6989586621679046620">kp2'</a></a> = <a href="Matrix.Type.html#kp2">kp2</a> @<a href="#local-6989586621679046490">e</a> @<a href="#local-6989586621679046492">a</a> @<a href="#local-6989586621679046493">b</a>

551−<a name="line-551"></a> in <a href="Matrix.Type.html#comp">comp</a> (<a href="Matrix.Type.html#tr">tr</a> <a href="#local-6989586621679046619">kp1'</a>) <a href="#local-6989586621679046617">a</a> * <a href="Matrix.Type.html#comp">comp</a> (<a href="Matrix.Type.html#tr">tr</a> <a href="#local-6989586621679046620">kp2'</a>) <a href="#local-6989586621679046618">b</a>

552−<a name="line-552"></a>

553−<a name="line-553"></a>-- Product Bifunctor (Kronecker)

554−<a name="line-554"></a>

555−<a name="line-555"></a>infixl 4 <a href="Matrix.Type.html#%3E%3C">><</a>

556−<a name="line-556"></a>

557−<a name="line-557"></a>-- | Matrix product functor also known as kronecker product

558−<a name="line-558"></a>(><) ::

559−<a name="line-559"></a> forall <a name="local-6989586621679046485"><a href="#local-6989586621679046485">e</a></a> <a name="local-6989586621679046486"><a href="#local-6989586621679046486">m</a></a> <a name="local-6989586621679046487"><a href="#local-6989586621679046487">p</a></a> <a name="local-6989586621679046488"><a href="#local-6989586621679046488">n</a></a> <a name="local-6989586621679046489"><a href="#local-6989586621679046489">q</a></a>.

560−<a name="line-560"></a> ( Num <a href="#local-6989586621679046485">e</a>,

561−<a name="line-561"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046486">m</a>)),

562−<a name="line-562"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046488">n</a>)),

563−<a name="line-563"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046487">p</a>)),

564−<a name="line-564"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046489">q</a>)),

565−<a name="line-565"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046486">m</a>, <a href="#local-6989586621679046488">n</a>))),

566−<a name="line-566"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046487">p</a>, <a href="#local-6989586621679046489">q</a>))),

567−<a name="line-567"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046485">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046486">m</a>, <a href="#local-6989586621679046488">n</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046486">m</a>),

568−<a name="line-568"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046485">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046486">m</a>, <a href="#local-6989586621679046488">n</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046488">n</a>),

569−<a name="line-569"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046485">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046487">p</a>, <a href="#local-6989586621679046489">q</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046487">p</a>),

570−<a name="line-570"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046485">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="#local-6989586621679046487">p</a>, <a href="#local-6989586621679046489">q</a>)) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046489">q</a>),

571−<a name="line-571"></a> I.Normalize (<a href="#local-6989586621679046486">m</a>, <a href="#local-6989586621679046488">n</a>) ~ <a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046486">m</a>, <a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046488">n</a>),

572−<a name="line-572"></a> I.Normalize (<a href="#local-6989586621679046487">p</a>, <a href="#local-6989586621679046489">q</a>) ~ <a href="Matrix.Internal.html#Normalize">I.Normalize</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046487">p</a>, <a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046489">q</a>)

573−<a name="line-573"></a> ) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046485">e</a> <a href="#local-6989586621679046486">m</a> <a href="#local-6989586621679046487">p</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046485">e</a> <a href="#local-6989586621679046488">n</a> <a href="#local-6989586621679046489">q</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046485">e</a> (<a href="#local-6989586621679046486">m</a>, <a href="#local-6989586621679046488">n</a>) (<a href="#local-6989586621679046487">p</a>, <a href="#local-6989586621679046489">q</a>)

574−<a name="line-574"></a>(<a name="%3E%3C"><a href="Matrix.Type.html#%3E%3C">><</a></a>) <a name="local-6989586621679046621"><a href="#local-6989586621679046621">a</a></a> <a name="local-6989586621679046622"><a href="#local-6989586621679046622">b</a></a> =

575−<a name="line-575"></a> let <a name="local-6989586621679046623"><a href="#local-6989586621679046623">kp1'</a></a> = <a href="Matrix.Type.html#kp1">kp1</a> @<a href="#local-6989586621679046485">e</a> @<a href="#local-6989586621679046486">m</a> @<a href="#local-6989586621679046488">n</a>

576−<a name="line-576"></a> <a name="local-6989586621679046624"><a href="#local-6989586621679046624">kp2'</a></a> = <a href="Matrix.Type.html#kp2">kp2</a> @<a href="#local-6989586621679046485">e</a> @<a href="#local-6989586621679046486">m</a> @<a href="#local-6989586621679046488">n</a>

577−<a name="line-577"></a> in <a href="Matrix.Type.html#khatri">khatri</a> (<a href="Matrix.Type.html#comp">comp</a> <a href="#local-6989586621679046621">a</a> <a href="#local-6989586621679046623">kp1'</a>) (<a href="Matrix.Type.html#comp">comp</a> <a href="#local-6989586621679046622">b</a> <a href="#local-6989586621679046624">kp2'</a>)

578−<a name="line-578"></a>

579−<a name="line-579"></a>-- Matrix abide Junc Split

580−<a name="line-580"></a>

581−<a name="line-581"></a>-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

582−<a name="line-582"></a>--

583−<a name="line-583"></a>-- Law:

584−<a name="line-584"></a>--

585−<a name="line-585"></a>-- @

586−<a name="line-586"></a>-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

587−<a name="line-587"></a>-- @

588−<a name="line-588"></a>abideJS :: <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046482">e</a> <a href="#local-6989586621679046483">cols</a> <a href="#local-6989586621679046484">rows</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046482">e</a> <a href="#local-6989586621679046483">cols</a> <a href="#local-6989586621679046484">rows</a>

589−<a name="line-589"></a><a name="abideJS"><a href="Matrix.Type.html#abideJS">abideJS</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046625"><a href="#local-6989586621679046625">m</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#abideJS">I.abideJS</a> <a href="#local-6989586621679046625">m</a>)

590−<a name="line-590"></a>

591−<a name="line-591"></a>-- Matrix abide Split Junc

592−<a name="line-592"></a>

593−<a name="line-593"></a>-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

594−<a name="line-594"></a>--

595−<a name="line-595"></a>-- Law:

596−<a name="line-596"></a>--

597−<a name="line-597"></a>-- @

598−<a name="line-598"></a>-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

599−<a name="line-599"></a>-- @

600−<a name="line-600"></a>abideSJ :: <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046479">e</a> <a href="#local-6989586621679046480">cols</a> <a href="#local-6989586621679046481">rows</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046479">e</a> <a href="#local-6989586621679046480">cols</a> <a href="#local-6989586621679046481">rows</a>

601−<a name="line-601"></a><a name="abideSJ"><a href="Matrix.Type.html#abideSJ">abideSJ</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046626"><a href="#local-6989586621679046626">m</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#abideSJ">I.abideSJ</a> <a href="#local-6989586621679046626">m</a>)

602−<a name="line-602"></a>

603−<a name="line-603"></a>-- Matrix transposition

604−<a name="line-604"></a>

605−<a name="line-605"></a>-- | Matrix transposition.

606−<a name="line-606"></a>tr :: <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046476">e</a> <a href="#local-6989586621679046477">cols</a> <a href="#local-6989586621679046478">rows</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046476">e</a> <a href="#local-6989586621679046478">rows</a> <a href="#local-6989586621679046477">cols</a>

607−<a name="line-607"></a><a name="tr"><a href="Matrix.Type.html#tr">tr</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046627"><a href="#local-6989586621679046627">m</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#tr">I.tr</a> <a href="#local-6989586621679046627">m</a>)

608−<a name="line-608"></a>

609−<a name="line-609"></a>-- Selective 'select' operator

610−<a name="line-610"></a>

611−<a name="line-611"></a>-- | Selective functors 'select' operator equivalent inspired by the

612−<a name="line-612"></a>-- ArrowMonad solution presented in the paper.

613−<a name="line-613"></a>selectM ::

614−<a name="line-614"></a> ( Num <a href="#local-6989586621679046472">e</a>,

615−<a name="line-615"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046472">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046473">b</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046473">b</a>),

616−<a name="line-616"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046473">b</a>)),

617−<a name="line-617"></a> I.Normalize (Either <a href="#local-6989586621679046474">a</a> <a href="#local-6989586621679046473">b</a>) ~ Either (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046474">a</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046473">b</a>)

618−<a name="line-618"></a> ) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046472">e</a> <a href="#local-6989586621679046475">cols</a> (Either <a href="#local-6989586621679046474">a</a> <a href="#local-6989586621679046473">b</a>) -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046472">e</a> <a href="#local-6989586621679046474">a</a> <a href="#local-6989586621679046473">b</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046472">e</a> <a href="#local-6989586621679046475">cols</a> <a href="#local-6989586621679046473">b</a>

619−<a name="line-619"></a><a name="selectM"><a href="Matrix.Type.html#selectM">selectM</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046628"><a href="#local-6989586621679046628">m</a></a>) (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046629"><a href="#local-6989586621679046629">y</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#select">I.select</a> <a href="#local-6989586621679046628">m</a> <a href="#local-6989586621679046629">y</a>)

620−<a name="line-620"></a>

621−<a name="line-621"></a>-- McCarthy's Conditional

622−<a name="line-622"></a>

623−<a name="line-623"></a>-- | McCarthy's Conditional expresses probabilistic choice.

624−<a name="line-624"></a>cond ::

625−<a name="line-625"></a> ( I.Normalize (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046469">a</a>) ~ <a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046469">a</a>,

626−<a name="line-626"></a> KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046469">a</a>)),

627−<a name="line-627"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046470">e</a> () (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046469">a</a>),

628−<a name="line-628"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046470">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046469">a</a>) (),

629−<a name="line-629"></a> <a href="Matrix.Internal.html#FromLists">I.FromLists</a> <a href="#local-6989586621679046470">e</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046469">a</a>) (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046469">a</a>),

630−<a name="line-630"></a> Bounded <a href="#local-6989586621679046469">a</a>,

631−<a name="line-631"></a> Enum <a href="#local-6989586621679046469">a</a>,

632−<a name="line-632"></a> Num <a href="#local-6989586621679046470">e</a>,

633−<a name="line-633"></a> Ord <a href="#local-6989586621679046470">e</a>

634−<a name="line-634"></a> )

635−<a name="line-635"></a> =>

636−<a name="line-636"></a> (<a href="#local-6989586621679046469">a</a> -> Bool) -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046470">e</a> <a href="#local-6989586621679046469">a</a> <a href="#local-6989586621679046471">b</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046470">e</a> <a href="#local-6989586621679046469">a</a> <a href="#local-6989586621679046471">b</a> -> <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046470">e</a> <a href="#local-6989586621679046469">a</a> <a href="#local-6989586621679046471">b</a>

637−<a name="line-637"></a><a name="cond"><a href="Matrix.Type.html#cond">cond</a></a> <a name="local-6989586621679046630"><a href="#local-6989586621679046630">p</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046631"><a href="#local-6989586621679046631">a</a></a>) (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046632"><a href="#local-6989586621679046632">b</a></a>) = <a href="Matrix.Type.html#M">M</a> (<a href="Matrix.Internal.html#cond">I.cond</a> <a href="#local-6989586621679046630">p</a> <a href="#local-6989586621679046631">a</a> <a href="#local-6989586621679046632">b</a>)

638−<a name="line-638"></a>

639−<a name="line-639"></a>-- Pretty print

640−<a name="line-640"></a>

641−<a name="line-641"></a>-- | Matrix pretty printer

642−<a name="line-642"></a>pretty :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046466">cols</a>)), Show <a href="#local-6989586621679046467">e</a>) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046467">e</a> <a href="#local-6989586621679046466">cols</a> <a href="#local-6989586621679046468">rows</a> -> String

643−<a name="line-643"></a><a name="pretty"><a href="Matrix.Type.html#pretty">pretty</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046633"><a href="#local-6989586621679046633">m</a></a>) = <a href="Matrix.Internal.html#pretty">I.pretty</a> <a href="#local-6989586621679046633">m</a>

644−<a name="line-644"></a>

645−<a name="line-645"></a>-- | Matrix pretty printer

646−<a name="line-646"></a>prettyPrint :: (KnownNat (<a href="Matrix.Internal.html#Count">I.Count</a> (<a href="Matrix.Internal.html#Normalize">I.Normalize</a> <a href="#local-6989586621679046463">cols</a>)), Show <a href="#local-6989586621679046464">e</a>) => <a href="Matrix.Type.html#Matrix">Matrix</a> <a href="#local-6989586621679046464">e</a> <a href="#local-6989586621679046463">cols</a> <a href="#local-6989586621679046465">rows</a> -> IO ()

647−<a name="line-647"></a><a name="prettyPrint"><a href="Matrix.Type.html#prettyPrint">prettyPrint</a></a> (<a href="Matrix.Type.html#M">M</a> <a name="local-6989586621679046634"><a href="#local-6989586621679046634">m</a></a>) = <a href="Matrix.Internal.html#prettyPrint">I.prettyPrint</a> <a href="#local-6989586621679046634">m</a>

648−<a name="line-648"></a></pre></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Paths_laop.htmldeleted−51

@@ -1,51 +0,0 @@

1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>{-# LANGUAGE CPP #-}

2−<a name="line-2"></a>{-# LANGUAGE NoRebindableSyntax #-}

3−<a name="line-3"></a>{-# OPTIONS_GHC -fno-warn-missing-import-lists #-}

4−<a name="line-4"></a>module Paths_laop (

5−<a name="line-5"></a> <a href="Paths_laop.html#version">version</a>,

6−<a name="line-6"></a> <a href="Paths_laop.html#getBinDir">getBinDir</a>, <a href="Paths_laop.html#getLibDir">getLibDir</a>, <a href="Paths_laop.html#getDynLibDir">getDynLibDir</a>, <a href="Paths_laop.html#getDataDir">getDataDir</a>, <a href="Paths_laop.html#getLibexecDir">getLibexecDir</a>,

7−<a name="line-7"></a> <a href="Paths_laop.html#getDataFileName">getDataFileName</a>, <a href="Paths_laop.html#getSysconfDir">getSysconfDir</a>

8−<a name="line-8"></a> ) where

9−<a name="line-9"></a>

10−<a name="line-10"></a>import qualified Control.Exception as Exception

11−<a name="line-11"></a>import Data.Version (Version(..))

12−<a name="line-12"></a>import System.Environment (getEnv)

13−<a name="line-13"></a>import Prelude

14−<a name="line-14"></a>

15−<a name="line-15"></a>#if defined(VERSION_base)

16−

17−<a name="line-17"></a>#if MIN_VERSION_base(4,0,0)

18−catchIO :: IO <a href="#local-6989586621679020005">a</a> -> (Exception.IOException -> IO <a href="#local-6989586621679020005">a</a>) -> IO <a href="#local-6989586621679020005">a</a>

19−<a name="line-19"></a>#else

20−catchIO :: IO a -> (Exception.Exception -> IO a) -> IO a

21−<a name="line-21"></a>#endif

22−

23−<a name="line-23"></a>#else

24−catchIO :: IO a -> (Exception.IOException -> IO a) -> IO a

25−<a name="line-25"></a>#endif

26−<a name="catchIO"><a href="Paths_laop.html#catchIO">catchIO</a></a> = Exception.catch

27−<a name="line-27"></a>

28−<a name="line-28"></a>version :: Version

29−<a name="line-29"></a><a name="version"><a href="Paths_laop.html#version">version</a></a> = Version [0,1,0,0] []

30−<a name="line-30"></a>bindir, libdir, dynlibdir, datadir, libexecdir, sysconfdir :: FilePath

31−<a name="line-31"></a>

32−<a name="line-32"></a><a name="bindir"><a href="Paths_laop.html#bindir">bindir</a></a> = "/home/bolt/.cabal/bin"

33−<a name="line-33"></a><a name="libdir"><a href="Paths_laop.html#libdir">libdir</a></a> = "/home/bolt/.cabal/lib/x86_64-linux-ghc-8.6.5/laop-0.1.0.0-inplace"

34−<a name="line-34"></a><a name="dynlibdir"><a href="Paths_laop.html#dynlibdir">dynlibdir</a></a> = "/home/bolt/.cabal/lib/x86_64-linux-ghc-8.6.5"

35−<a name="line-35"></a><a name="datadir"><a href="Paths_laop.html#datadir">datadir</a></a> = "/home/bolt/.cabal/share/x86_64-linux-ghc-8.6.5/laop-0.1.0.0"

36−<a name="line-36"></a><a name="libexecdir"><a href="Paths_laop.html#libexecdir">libexecdir</a></a> = "/home/bolt/.cabal/libexec/x86_64-linux-ghc-8.6.5/laop-0.1.0.0"

37−<a name="line-37"></a><a name="sysconfdir"><a href="Paths_laop.html#sysconfdir">sysconfdir</a></a> = "/home/bolt/.cabal/etc"

38−<a name="line-38"></a>

39−<a name="line-39"></a>getBinDir, getLibDir, getDynLibDir, getDataDir, getLibexecDir, getSysconfDir :: IO FilePath

40−<a name="line-40"></a><a name="getBinDir"><a href="Paths_laop.html#getBinDir">getBinDir</a></a> = <a href="Paths_laop.html#catchIO">catchIO</a> (getEnv "laop_bindir") (\_ -> return <a href="Paths_laop.html#bindir">bindir</a>)

41−<a name="line-41"></a><a name="getLibDir"><a href="Paths_laop.html#getLibDir">getLibDir</a></a> = <a href="Paths_laop.html#catchIO">catchIO</a> (getEnv "laop_libdir") (\_ -> return <a href="Paths_laop.html#libdir">libdir</a>)

42−<a name="line-42"></a><a name="getDynLibDir"><a href="Paths_laop.html#getDynLibDir">getDynLibDir</a></a> = <a href="Paths_laop.html#catchIO">catchIO</a> (getEnv "laop_dynlibdir") (\_ -> return <a href="Paths_laop.html#dynlibdir">dynlibdir</a>)

43−<a name="line-43"></a><a name="getDataDir"><a href="Paths_laop.html#getDataDir">getDataDir</a></a> = <a href="Paths_laop.html#catchIO">catchIO</a> (getEnv "laop_datadir") (\_ -> return <a href="Paths_laop.html#datadir">datadir</a>)

44−<a name="line-44"></a><a name="getLibexecDir"><a href="Paths_laop.html#getLibexecDir">getLibexecDir</a></a> = <a href="Paths_laop.html#catchIO">catchIO</a> (getEnv "laop_libexecdir") (\_ -> return <a href="Paths_laop.html#libexecdir">libexecdir</a>)

45−<a name="line-45"></a><a name="getSysconfDir"><a href="Paths_laop.html#getSysconfDir">getSysconfDir</a></a> = <a href="Paths_laop.html#catchIO">catchIO</a> (getEnv "laop_sysconfdir") (\_ -> return <a href="Paths_laop.html#sysconfdir">sysconfdir</a>)

46−<a name="line-46"></a>

47−<a name="line-47"></a>getDataFileName :: FilePath -> IO FilePath

48−<a name="line-48"></a><a name="getDataFileName"><a href="Paths_laop.html#getDataFileName">getDataFileName</a></a> <a name="local-6989586621679020854"><a href="#local-6989586621679020854">name</a></a> = do

49−<a name="line-49"></a> <a name="local-6989586621679020855"><a href="#local-6989586621679020855">dir</a></a> <- <a href="Paths_laop.html#getDataDir">getDataDir</a>

50−<a name="line-50"></a> return (<a href="#local-6989586621679020855">dir</a> ++ "/" ++ <a href="#local-6989586621679020854">name</a>)

51−<a name="line-51"></a></pre></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Utils.Internal.htmldeleted−177

@@ -1,177 +0,0 @@

1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>{-# LANGUAGE TypeApplications #-}

2−<a name="line-2"></a>{-# LANGUAGE DataKinds #-}

3−<a name="line-3"></a>{-# LANGUAGE FlexibleInstances #-}

4−<a name="line-4"></a>{-# LANGUAGE GeneralizedNewtypeDeriving #-}

5−<a name="line-5"></a>{-# LANGUAGE KindSignatures #-}

6−<a name="line-6"></a>{-# LANGUAGE ScopedTypeVariables #-}

7−<a name="line-7"></a>

8−<a name="line-8"></a>module Utils.Internal

9−<a name="line-9"></a> ( -- | This is an internal module and it is not meant to be imported.

10−<a name="line-10"></a> --

11−<a name="line-11"></a> -- Utility module that provides the 'Natural' data type.

12−<a name="line-12"></a> -- The semantic associated with this data type is that

13−<a name="line-13"></a> -- it's meant to be a restricted 'Int' value. For example

14−<a name="line-14"></a> -- the type @Natural 6@ can only be instanciated with @Nat n@

15−<a name="line-15"></a> -- where @0 <= n <= 6@. Why, You might ask, because with normal

16−<a name="line-16"></a> -- 'Int's it is not possible to have a decent @Enum (Int, Int)@

17−<a name="line-17"></a> -- instance. See the following probabilistic programming model as and

18−<a name="line-18"></a> -- example:

19−<a name="line-19"></a> --

20−<a name="line-20"></a> -- We want to calculate the probability of the sum of two dice throws.

21−<a name="line-21"></a> -- To do this we start by defining the sample space:

22−<a name="line-22"></a> --

23−<a name="line-23"></a> -- @

24−<a name="line-24"></a> -- type SampleSpace = Int -- We think 'Int' are enough

25−<a name="line-25"></a> --

26−<a name="line-26"></a> -- die :: Dist Int 6

27−<a name="line-27"></a> -- die = unifrom [1..6]

28−<a name="line-28"></a> --

29−<a name="line-29"></a> -- -- Promote 'Int' addition to a matrix

30−<a name="line-30"></a> -- addM = fromF (uncurry (+)) -- Impossible

31−<a name="line-31"></a> -- @

32−<a name="line-32"></a> --

33−<a name="line-33"></a> -- The last line is impossible because @(Int, Int)@ does not have

34−<a name="line-34"></a> -- a good 'Enum' instance: @[(0, 1), (0, 2), .. (0, maxBound), (1, 0),

35−<a name="line-35"></a> -- ..]@. And we'd like the addition matrix to be of 36 columns by 12

36−<a name="line-36"></a> -- rows but limited to integers up to @6@!

37−<a name="line-37"></a> --

38−<a name="line-38"></a> -- One way to solve this issue is by defining and auxilary data type to

39−<a name="line-39"></a> -- represent the sample space:

40−<a name="line-40"></a> --

41−<a name="line-41"></a> -- @

42−<a name="line-42"></a> -- data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6

43−<a name="line-43"></a> -- deriving (Show, Eq, Enum, Bounded) -- Enum and Bounded are

44−<a name="line-44"></a> -- important

45−<a name="line-45"></a> -- @

46−<a name="line-46"></a> --

47−<a name="line-47"></a> -- And write the sample space addition function:

48−<a name="line-48"></a> --

49−<a name="line-49"></a> -- @

50−<a name="line-50"></a> -- ssAdd :: SampleSpace -> SampleSpace -> Int

51−<a name="line-51"></a> -- ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)

52−<a name="line-52"></a> -- @

53−<a name="line-53"></a> --

54−<a name="line-54"></a> -- And then promote that function to matrix and everything is alright:

55−<a name="line-55"></a> --

56−<a name="line-56"></a> -- @

57−<a name="line-57"></a> -- ssAddM = fromF' (uncurry ssAdd)

58−<a name="line-58"></a> --

59−<a name="line-59"></a> -- dieSumProb = ssAddM `comp` (khatri die die)

60−<a name="line-60"></a> -- @

61−<a name="line-61"></a> --

62−<a name="line-62"></a> -- This is a nice solution for small sample spaces. But for larger ones

63−<a name="line-63"></a> -- it is not feasible to write a data type with hundreds of constructors

64−<a name="line-64"></a> -- and then write manipulation functions that need to deal with them.

65−<a name="line-65"></a> -- To mitigate this limitation the 'Natural' type comes a long way and

66−<a name="line-66"></a> -- allows one to model the sample in an easier way. See for instance:

67−<a name="line-67"></a> --

68−<a name="line-68"></a> -- @

69−<a name="line-69"></a> -- ssAdd :: Natural 6 -> Natural 6 -> Natural 12

70−<a name="line-70"></a> -- ssAdd = coerceNat (+)

71−<a name="line-71"></a> --

72−<a name="line-72"></a> -- ssAddM = fromF' (uncurry sumSS)

73−<a name="line-73"></a> --

74−<a name="line-74"></a> -- die :: Dist (Natural 6) 6

75−<a name="line-75"></a> -- die = uniform [nat @6 1 .. nat 6]

76−<a name="line-76"></a> --

77−<a name="line-77"></a> -- dieSumProb = ssAddM `comp` (khatri die die)

78−<a name="line-78"></a> -- @

79−<a name="line-79"></a> --

80−<a name="line-80"></a> -- * 'Natural' data type

81−<a name="line-81"></a> <a href="Utils.Internal.html#Natural">Natural</a>,

82−<a name="line-82"></a> <a href="Utils.Internal.html#nat">nat</a>,

83−<a name="line-83"></a>

84−<a name="line-84"></a> -- * Coerce auxiliar functions to help promote 'Int' typed functions to

85−<a name="line-85"></a> -- 'Natural' typed functions.

86−<a name="line-86"></a> <a href="Utils.Internal.html#coerceNat">coerceNat</a>,

87−<a name="line-87"></a> <a href="Utils.Internal.html#coerceNat2">coerceNat2</a>,

88−<a name="line-88"></a> <a href="Utils.Internal.html#coerceNat3">coerceNat3</a>,

89−<a name="line-89"></a> )

90−<a name="line-90"></a>where

91−<a name="line-91"></a>

92−<a name="line-92"></a>import Data.Coerce

93−<a name="line-93"></a>import Data.Proxy

94−<a name="line-94"></a>import GHC.TypeLits

95−<a name="line-95"></a>import Control.DeepSeq

96−<a name="line-96"></a>

97−<a name="line-97"></a>-- | Wrapper around 'Int's that have a restrictive semantic associated.

98−<a name="line-98"></a>-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'

99−<a name="line-99"></a>-- @i@ that's @n <= i <= m@.

100−<a name="line-100"></a>newtype <a name="Natural"><a href="Utils.Internal.html#Natural">Natural</a></a> (<a name="local-6989586621679023463"><a href="#local-6989586621679023463">start</a></a> :: Nat) (<a name="local-6989586621679023464"><a href="#local-6989586621679023464">end</a></a> :: Nat) = <a name="Nat"><a href="Utils.Internal.html#Nat">Nat</a></a> Int

101−<a name="line-101"></a> deriving (Show, Read, Eq, Ord, NFData)

102−<a name="line-102"></a>

103−<a name="line-103"></a>-- | Throws a runtime error if any of the operations overflows or

104−<a name="line-104"></a>-- underflows.

105−<a name="line-105"></a>instance (KnownNat <a href="#local-6989586621679031022">n</a>, KnownNat <a href="#local-6989586621679031023">m</a>) => Num (<a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031022">n</a> <a href="#local-6989586621679031023">m</a>) where

106−<a name="line-106"></a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031024"><a href="#local-6989586621679031024">a</a></a>) <a name="local-8214565720323790403">+</a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031025"><a href="#local-6989586621679031025">b</a></a>) = <a href="Utils.Internal.html#nat">nat</a> @<a href="#local-6989586621679031022">n</a> @<a href="#local-6989586621679031023">m</a> (<a href="#local-6989586621679031024">a</a> + <a href="#local-6989586621679031025">b</a>)

107−<a name="line-107"></a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031026"><a href="#local-6989586621679031026">a</a></a>) - (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031027"><a href="#local-6989586621679031027">b</a></a>) = <a href="Utils.Internal.html#nat">nat</a> @<a href="#local-6989586621679031022">n</a> @<a href="#local-6989586621679031023">m</a> (<a href="#local-6989586621679031026">a</a> - <a href="#local-6989586621679031027">b</a>)

108−<a name="line-108"></a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031028"><a href="#local-6989586621679031028">a</a></a>) <a name="local-8214565720323790402">*</a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031029"><a href="#local-6989586621679031029">b</a></a>) = <a href="Utils.Internal.html#nat">nat</a> @<a href="#local-6989586621679031022">n</a> @<a href="#local-6989586621679031023">m</a> (<a href="#local-6989586621679031028">a</a> * <a href="#local-6989586621679031029">b</a>)

109−<a name="line-109"></a> <a name="local-8214565720323790401">abs</a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031030"><a href="#local-6989586621679031030">a</a></a>) = <a href="Utils.Internal.html#nat">nat</a> @<a href="#local-6989586621679031022">n</a> @<a href="#local-6989586621679031023">m</a> (abs <a href="#local-6989586621679031030">a</a>)

110−<a name="line-110"></a> <a name="local-8214565720323790400">signum</a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031031"><a href="#local-6989586621679031031">a</a></a>) = <a href="Utils.Internal.html#nat">nat</a> @<a href="#local-6989586621679031022">n</a> @<a href="#local-6989586621679031023">m</a> (signum <a href="#local-6989586621679031031">a</a>)

111−<a name="line-111"></a> <a name="local-3458764513820541088">fromInteger</a> <a name="local-6989586621679031032"><a href="#local-6989586621679031032">i</a></a> = <a href="Utils.Internal.html#nat">nat</a> @<a href="#local-6989586621679031022">n</a> @<a href="#local-6989586621679031023">m</a> (fromInteger <a href="#local-6989586621679031032">i</a>)

112−<a name="line-112"></a>

113−<a name="line-113"></a>-- | Natural constructor function. Throws a runtime error if the 'Int'

114−<a name="line-114"></a>-- value is greater than the corresponding @m@ or lower than @n@ in the @'Natural' n m@ type.

115−<a name="line-115"></a>nat :: forall <a name="local-6989586621679031052"><a href="#local-6989586621679031052">n</a></a> <a name="local-6989586621679031053"><a href="#local-6989586621679031053">m</a></a> . (KnownNat <a href="#local-6989586621679031052">n</a>, KnownNat <a href="#local-6989586621679031053">m</a>) => Int -> <a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031052">n</a> <a href="#local-6989586621679031053">m</a>

116−<a name="line-116"></a><a name="nat"><a href="Utils.Internal.html#nat">nat</a></a> <a name="local-6989586621679031054"><a href="#local-6989586621679031054">i</a></a> =

117−<a name="line-117"></a> let <a name="local-6989586621679031055"><a href="#local-6989586621679031055">start</a></a> = fromInteger (natVal (Proxy :: Proxy <a href="#local-6989586621679031052">n</a>))

118−<a name="line-118"></a> <a name="local-6989586621679031056"><a href="#local-6989586621679031056">end</a></a> = fromInteger (natVal (Proxy :: Proxy <a href="#local-6989586621679031053">m</a>))

119−<a name="line-119"></a> in if <a href="#local-6989586621679031055">start</a> <= <a href="#local-6989586621679031054">i</a> && <a href="#local-6989586621679031054">i</a> <= <a href="#local-6989586621679031056">end</a>

120−<a name="line-120"></a> then <a href="Utils.Internal.html#Nat">Nat</a> <a href="#local-6989586621679031054">i</a>

121−<a name="line-121"></a> else error "Off limits"

122−<a name="line-122"></a>

123−<a name="line-123"></a>-- | Auxiliary function that promotes binary 'Int' functions to 'Natural'

124−<a name="line-124"></a>-- binary functions.

125−<a name="line-125"></a>coerceNat :: (Int -> Int -> Int) -> (<a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031046">a</a> <a href="#local-6989586621679031047">a'</a> -> <a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031048">b</a> <a href="#local-6989586621679031049">b'</a> -> <a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031050">c</a> <a href="#local-6989586621679031051">c'</a>)

126−<a name="line-126"></a><a name="coerceNat"><a href="Utils.Internal.html#coerceNat">coerceNat</a></a> = coerce

127−<a name="line-127"></a>

128−<a name="line-128"></a>-- | Auxiliary function that promotes ternary (binary) 'Int' functions to 'Natural'

129−<a name="line-129"></a>-- functions.

130−<a name="line-130"></a>coerceNat2 :: ((Int, Int) -> Int -> Int) -> ((<a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031038">a</a> <a href="#local-6989586621679031039">a'</a>, <a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031040">b</a> <a href="#local-6989586621679031041">b'</a>) -> <a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031042">c</a> <a href="#local-6989586621679031043">c'</a> -> <a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031044">d</a> <a href="#local-6989586621679031045">d'</a>)

131−<a name="line-131"></a><a name="coerceNat2"><a href="Utils.Internal.html#coerceNat2">coerceNat2</a></a> = coerce

132−<a name="line-132"></a>

133−<a name="line-133"></a>-- | Auxiliary function that promotes ternary (binary) 'Int' functions to 'Natural'

134−<a name="line-134"></a>-- functions.

135−<a name="line-135"></a>coerceNat3 :: (Int -> Int -> <a href="#local-6989586621679031033">a</a>) -> (<a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031034">b</a> <a href="#local-6989586621679031035">b'</a> -> <a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031036">c</a> <a href="#local-6989586621679031037">c'</a> -> <a href="#local-6989586621679031033">a</a>)

136−<a name="line-136"></a><a name="coerceNat3"><a href="Utils.Internal.html#coerceNat3">coerceNat3</a></a> = coerce

137−<a name="line-137"></a>

138−<a name="line-138"></a>instance (KnownNat <a href="#local-6989586621679031020">n</a>, KnownNat <a href="#local-6989586621679031021">m</a>) => Bounded (<a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679031020">n</a> <a href="#local-6989586621679031021">m</a>) where

139−<a name="line-139"></a> <a name="local-8214565720323790309">minBound</a> = <a href="Utils.Internal.html#Nat">Nat</a> $ fromInteger (natVal (Proxy :: Proxy <a href="#local-6989586621679031020">n</a>))

140−<a name="line-140"></a> <a name="local-8214565720323790308">maxBound</a> = <a href="Utils.Internal.html#Nat">Nat</a> $ fromInteger (natVal (Proxy :: Proxy <a href="#local-6989586621679031021">m</a>))

141−<a name="line-141"></a>

142−<a name="line-142"></a>instance (KnownNat <a href="#local-6989586621679030849">n</a>, KnownNat <a href="#local-6989586621679030850">m</a>) => Enum (<a href="Utils.Internal.html#Natural">Natural</a> <a href="#local-6989586621679030849">n</a> <a href="#local-6989586621679030850">m</a>) where

143−<a name="line-143"></a> <a name="local-8214565720323790311">toEnum</a> <a name="local-6989586621679031001"><a href="#local-6989586621679031001">i</a></a> =

144−<a name="line-144"></a> let <a name="local-6989586621679031002"><a href="#local-6989586621679031002">start</a></a> = fromInteger (natVal (Proxy :: Proxy <a href="#local-6989586621679030849">n</a>))

145−<a name="line-145"></a> in <a href="Utils.Internal.html#nat">nat</a> (<a href="#local-6989586621679031002">start</a> + <a href="#local-6989586621679031001">i</a>)

146−<a name="line-146"></a> -- | Throws a runtime error if the value is off limits

147−<a name="line-147"></a> <a name="local-8214565720323790310">fromEnum</a> (<a href="Utils.Internal.html#Nat">Nat</a> <a name="local-6989586621679031003"><a href="#local-6989586621679031003">nat</a></a>) =

148−<a name="line-148"></a> let <a name="local-6989586621679031004"><a href="#local-6989586621679031004">start</a></a> = fromInteger (natVal (Proxy :: Proxy <a href="#local-6989586621679030849">n</a>))

149−<a name="line-149"></a> <a name="local-6989586621679031005"><a href="#local-6989586621679031005">end</a></a> = fromInteger (natVal (Proxy :: Proxy <a href="#local-6989586621679030850">m</a>))

150−<a name="line-150"></a> in if <a href="#local-6989586621679031004">start</a> <= <a href="#local-6989586621679031003">nat</a> && <a href="#local-6989586621679031003">nat</a> <= <a href="#local-6989586621679031005">end</a>

151−<a name="line-151"></a> then <a href="#local-6989586621679031003">nat</a> - <a href="#local-6989586621679031004">start</a>

152−<a name="line-152"></a> else error "Off limits"

153−<a name="line-153"></a>

154−<a name="line-154"></a>-- | Optimized 'Enum' instance for tuples that comply with the given

155−<a name="line-155"></a>-- constraints.

156−<a name="line-156"></a>instance

157−<a name="line-157"></a> ( Enum <a href="#local-6989586621679029190">a</a>,

158−<a name="line-158"></a> Enum <a href="#local-6989586621679029191">b</a>,

159−<a name="line-159"></a> Bounded <a href="#local-6989586621679029191">b</a>

160−<a name="line-160"></a> ) =>

161−<a name="line-161"></a> Enum (<a href="#local-6989586621679029190">a</a>, <a href="#local-6989586621679029191">b</a>)

162−<a name="line-162"></a> where

163−<a name="line-163"></a>

164−<a name="line-164"></a> <a name="local-8214565720323790311">toEnum</a> <a name="local-6989586621679029192"><a href="#local-6989586621679029192">i</a></a> =

165−<a name="line-165"></a> let (<a name="local-6989586621679029193"><a href="#local-6989586621679029193">listB</a></a> :: [<a href="#local-6989586621679029191">b</a>]) = [minBound .. maxBound]

166−<a name="line-166"></a> <a name="local-6989586621679029194"><a href="#local-6989586621679029194">lengthB</a></a> = length <a href="#local-6989586621679029193">listB</a>

167−<a name="line-167"></a> <a name="local-6989586621679029195"><a href="#local-6989586621679029195">fstI</a></a> = div <a href="#local-6989586621679029192">i</a> <a href="#local-6989586621679029194">lengthB</a>

168−<a name="line-168"></a> <a name="local-6989586621679029196"><a href="#local-6989586621679029196">sndI</a></a> = mod <a href="#local-6989586621679029192">i</a> <a href="#local-6989586621679029194">lengthB</a>

169−<a name="line-169"></a> in (toEnum <a href="#local-6989586621679029195">fstI</a>, toEnum <a href="#local-6989586621679029196">sndI</a>)

170−<a name="line-170"></a>

171−<a name="line-171"></a> <a name="local-8214565720323790310">fromEnum</a> (<a name="local-6989586621679030843"><a href="#local-6989586621679030843">a</a></a>, <a name="local-6989586621679030844"><a href="#local-6989586621679030844">b</a></a>) =

172−<a name="line-172"></a> let (<a name="local-6989586621679030845"><a href="#local-6989586621679030845">listB</a></a> :: [<a href="#local-6989586621679029191">b</a>]) = [minBound .. maxBound]

173−<a name="line-173"></a> <a name="local-6989586621679030846"><a href="#local-6989586621679030846">lengthB</a></a> = length <a href="#local-6989586621679030845">listB</a>

174−<a name="line-174"></a> <a name="local-6989586621679030847"><a href="#local-6989586621679030847">fstI</a></a> = fromEnum <a href="#local-6989586621679030843">a</a>

175−<a name="line-175"></a> <a name="local-6989586621679030848"><a href="#local-6989586621679030848">sndI</a></a> = fromEnum <a href="#local-6989586621679030844">b</a>

176−<a name="line-176"></a> in <a href="#local-6989586621679030847">fstI</a> * <a href="#local-6989586621679030846">lengthB</a> + <a href="#local-6989586621679030848">sndI</a>

177−<a name="line-177"></a></pre></body></html>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/Utils.htmldeleted−101

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1−<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><link rel="stylesheet" type="text/css" href="style.css" /><script type="text/javascript" src="highlight.js"></script></head><body><pre>-----------------------------------------------------------------------------

2−<a name="line-2"></a>-- |

3−<a name="line-3"></a>-- Module : Utils

4−<a name="line-4"></a>-- Copyright : (c) Armando Santos 2019-2020

5−<a name="line-5"></a>-- Maintainer : armandoifsantos@gmail.com

6−<a name="line-6"></a>-- Stability : experimental

7−<a name="line-7"></a>--

8−<a name="line-8"></a>-- __LAoP__ is a library for algebraic (inductive) construction and manipulation of matrices

9−<a name="line-9"></a>-- in Haskell. See <https://github.com/bolt12/master-thesis my Msc Thesis> for the

10−<a name="line-10"></a>-- motivation behind the library, the underlying theory, and implementation details.

11−<a name="line-11"></a>--

12−<a name="line-12"></a>-- This module provides the 'Natural' data type.

13−<a name="line-13"></a>-- The semantic associated with this data type is that

14−<a name="line-14"></a>-- it's meant to be a restricted 'Int' value.

15−<a name="line-15"></a>--

16−<a name="line-16"></a>-----------------------------------------------------------------------------

17−<a name="line-17"></a>

18−<a name="line-18"></a>module Utils

19−<a name="line-19"></a> ( -- | Utility module that provides the 'Natural' data type.

20−<a name="line-20"></a> -- The semantic associated with this data type is that

21−<a name="line-21"></a> -- it's meant to be a restricted 'Int' value. For example

22−<a name="line-22"></a> -- the type @Natural 1 6@ can only be instanciated with @nat n@

23−<a name="line-23"></a> -- where @1 <= n <= 6@. Why, You might ask, because with normal

24−<a name="line-24"></a> -- 'Int's it is not possible to have a decent @Enum (Int, Int)@

25−<a name="line-25"></a> -- instance. See the following probabilistic programming model as and

26−<a name="line-26"></a> -- example:

27−<a name="line-27"></a> --

28−<a name="line-28"></a> -- We want to calculate the probability of the sum of two dice throws.

29−<a name="line-29"></a> -- To do this we start by defining the sample space:

30−<a name="line-30"></a> --

31−<a name="line-31"></a> -- @

32−<a name="line-32"></a> -- type SampleSpace = Int -- We think 'Int' are enough

33−<a name="line-33"></a> --

34−<a name="line-34"></a> -- die :: Dist Int 6

35−<a name="line-35"></a> -- die = unifrom [1..6]

36−<a name="line-36"></a> --

37−<a name="line-37"></a> -- -- Promote 'Int' addition to a matrix

38−<a name="line-38"></a> -- addM = fromF (uncurry (+)) -- Impossible

39−<a name="line-39"></a> -- @

40−<a name="line-40"></a> --

41−<a name="line-41"></a> -- The last line is impossible because @(Int, Int)@ does not have

42−<a name="line-42"></a> -- a good 'Enum' instance: @[(0, 1), (0, 2), .. (0, maxBound), (1, 0),

43−<a name="line-43"></a> -- ..]@. And we'd like the addition matrix to be of 36 columns by 12

44−<a name="line-44"></a> -- rows but limited to integers up to @6@!

45−<a name="line-45"></a> --

46−<a name="line-46"></a> -- One way to solve this issue is by defining and auxilary data type to

47−<a name="line-47"></a> -- represent the sample space:

48−<a name="line-48"></a> --

49−<a name="line-49"></a> -- @

50−<a name="line-50"></a> -- data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6

51−<a name="line-51"></a> -- deriving (Show, Eq, Enum, Bounded) -- Enum and Bounded are

52−<a name="line-52"></a> -- important

53−<a name="line-53"></a> -- @

54−<a name="line-54"></a> --

55−<a name="line-55"></a> -- And write the sample space addition function:

56−<a name="line-56"></a> --

57−<a name="line-57"></a> -- @

58−<a name="line-58"></a> -- ssAdd :: SampleSpace -> SampleSpace -> Int

59−<a name="line-59"></a> -- ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)

60−<a name="line-60"></a> -- @

61−<a name="line-61"></a> --

62−<a name="line-62"></a> -- And then promote that function to matrix and everything is alright:

63−<a name="line-63"></a> --

64−<a name="line-64"></a> -- @

65−<a name="line-65"></a> -- ssAddM = fromF' (uncurry ssAdd)

66−<a name="line-66"></a> --

67−<a name="line-67"></a> -- dieSumProb = ssAddM `comp` (khatri die die)

68−<a name="line-68"></a> -- @

69−<a name="line-69"></a> --

70−<a name="line-70"></a> -- This is a nice solution for small sample spaces. But for larger ones

71−<a name="line-71"></a> -- it is not feasible to write a data type with hundreds of constructors

72−<a name="line-72"></a> -- and then write manipulation functions that need to deal with them.

73−<a name="line-73"></a> -- To mitigate this limitation the 'Natural' type comes a long way and

74−<a name="line-74"></a> -- allows one to model the sample in an easier way. See for instance:

75−<a name="line-75"></a> --

76−<a name="line-76"></a> -- @

77−<a name="line-77"></a> -- ssAdd :: Natural 1 6 -> Natural 1 6 -> Natural 1 12

78−<a name="line-78"></a> -- ssAdd = coerceNat (+)

79−<a name="line-79"></a> --

80−<a name="line-80"></a> -- ssAddM = fromF' (uncurry sumSS)

81−<a name="line-81"></a> --

82−<a name="line-82"></a> -- die :: Dist (Natural 1 6) 6

83−<a name="line-83"></a> -- die = uniform [nat @1 @6 1 .. nat 6]

84−<a name="line-84"></a> --

85−<a name="line-85"></a> -- dieSumProb = ssAddM `comp` (khatri die die)

86−<a name="line-86"></a> -- @

87−<a name="line-87"></a> --

88−<a name="line-88"></a> -- * 'Natural' data type

89−<a name="line-89"></a> <a href="Utils.Internal.html#Natural">Natural</a>,

90−<a name="line-90"></a> <a href="Utils.Internal.html#nat">nat</a>,

91−<a name="line-91"></a>

92−<a name="line-92"></a> -- * Coerce auxiliar functions to help promote 'Int' typed functions to

93−<a name="line-93"></a> -- 'Natural' typed functions.

94−<a name="line-94"></a> <a href="Utils.Internal.html#coerceNat">coerceNat</a>,

95−<a name="line-95"></a> <a href="Utils.Internal.html#coerceNat2">coerceNat2</a>,

96−<a name="line-96"></a> <a href="Utils.Internal.html#coerceNat3">coerceNat3</a>,

97−<a name="line-97"></a> )

98−<a name="line-98"></a>where

99−<a name="line-99"></a>

100−<a name="line-100"></a>import <a href="Utils.Internal.html">Utils.Internal</a>

101−<a name="line-101"></a></pre></body></html>

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1−{-# LANGUAGE CPP #-}

2−{-# LANGUAGE NoRebindableSyntax #-}

3−{-# OPTIONS_GHC -fno-warn-missing-import-lists #-}

4−module Paths_fake_package (

5− version,

6− getBinDir, getLibDir, getDynLibDir, getDataDir, getLibexecDir,

7− getDataFileName, getSysconfDir

8− ) where

9−

10−import qualified Control.Exception as Exception

11−import Data.Version (Version(..))

12−import System.Environment (getEnv)

13−import Prelude

14−

15−#if defined(VERSION_base)

16−

17−#if MIN_VERSION_base(4,0,0)

18−catchIO :: IO a -> (Exception.IOException -> IO a) -> IO a

19−#else

20−catchIO :: IO a -> (Exception.Exception -> IO a) -> IO a

21−#endif

22−

23−#else

24−catchIO :: IO a -> (Exception.IOException -> IO a) -> IO a

25−#endif

26−catchIO = Exception.catch

27−

28−version :: Version

29−version = Version [0] []

30−bindir, libdir, dynlibdir, datadir, libexecdir, sysconfdir :: FilePath

31−

32−bindir = "/home/bolt/.cabal/bin"

33−libdir = "/home/bolt/.cabal/lib/x86_64-linux-ghc-8.6.5/fake-package-0-inplace-script"

34−dynlibdir = "/home/bolt/.cabal/lib/x86_64-linux-ghc-8.6.5"

35−datadir = "/home/bolt/.cabal/share/x86_64-linux-ghc-8.6.5/fake-package-0"

36−libexecdir = "/home/bolt/.cabal/libexec/x86_64-linux-ghc-8.6.5/fake-package-0"

37−sysconfdir = "/home/bolt/.cabal/etc"

38−

39−getBinDir, getLibDir, getDynLibDir, getDataDir, getLibexecDir, getSysconfDir :: IO FilePath

40−getBinDir = catchIO (getEnv "fake_package_bindir") (\_ -> return bindir)

41−getLibDir = catchIO (getEnv "fake_package_libdir") (\_ -> return libdir)

42−getDynLibDir = catchIO (getEnv "fake_package_dynlibdir") (\_ -> return dynlibdir)

43−getDataDir = catchIO (getEnv "fake_package_datadir") (\_ -> return datadir)

44−getLibexecDir = catchIO (getEnv "fake_package_libexecdir") (\_ -> return libexecdir)

45−getSysconfDir = catchIO (getEnv "fake_package_sysconfdir") (\_ -> return sysconfdir)

46−

47−getDataFileName :: FilePath -> IO FilePath

48−getDataFileName name = do

49− dir <- getDataDir

50− return (dir ++ "/" ++ name)

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@@ -1,118 +0,0 @@

1−/* DO NOT EDIT: This file is automatically generated by Cabal */

2−

3−/* package fake-package-0 */

4−#ifndef VERSION_fake_package

5−#define VERSION_fake_package "0"

6−#endif /* VERSION_fake_package */

7−#ifndef MIN_VERSION_fake_package

8−#define MIN_VERSION_fake_package(major1,major2,minor) (\

9− (major1) < 0 || \

10− (major1) == 0 && (major2) < 0 || \

11− (major1) == 0 && (major2) == 0 && (minor) <= 0)

12−#endif /* MIN_VERSION_fake_package */

13−

14−/* tool gcc-9.2.0 */

15−#ifndef TOOL_VERSION_gcc

16−#define TOOL_VERSION_gcc "9.2.0"

17−#endif /* TOOL_VERSION_gcc */

18−#ifndef MIN_TOOL_VERSION_gcc

19−#define MIN_TOOL_VERSION_gcc(major1,major2,minor) (\

20− (major1) < 9 || \

21− (major1) == 9 && (major2) < 2 || \

22− (major1) == 9 && (major2) == 2 && (minor) <= 0)

23−#endif /* MIN_TOOL_VERSION_gcc */

24−

25−/* tool ghc-8.6.5 */

26−#ifndef TOOL_VERSION_ghc

27−#define TOOL_VERSION_ghc "8.6.5"

28−#endif /* TOOL_VERSION_ghc */

29−#ifndef MIN_TOOL_VERSION_ghc

30−#define MIN_TOOL_VERSION_ghc(major1,major2,minor) (\

31− (major1) < 8 || \

32− (major1) == 8 && (major2) < 6 || \

33− (major1) == 8 && (major2) == 6 && (minor) <= 5)

34−#endif /* MIN_TOOL_VERSION_ghc */

35−

36−/* tool ghc-pkg-8.6.5 */

37−#ifndef TOOL_VERSION_ghc_pkg

38−#define TOOL_VERSION_ghc_pkg "8.6.5"

39−#endif /* TOOL_VERSION_ghc_pkg */

40−#ifndef MIN_TOOL_VERSION_ghc_pkg

41−#define MIN_TOOL_VERSION_ghc_pkg(major1,major2,minor) (\

42− (major1) < 8 || \

43− (major1) == 8 && (major2) < 6 || \

44− (major1) == 8 && (major2) == 6 && (minor) <= 5)

45−#endif /* MIN_TOOL_VERSION_ghc_pkg */

46−

47−/* tool haddock-2.22.0 */

48−#ifndef TOOL_VERSION_haddock

49−#define TOOL_VERSION_haddock "2.22.0"

50−#endif /* TOOL_VERSION_haddock */

51−#ifndef MIN_TOOL_VERSION_haddock

52−#define MIN_TOOL_VERSION_haddock(major1,major2,minor) (\

53− (major1) < 2 || \

54− (major1) == 2 && (major2) < 22 || \

55− (major1) == 2 && (major2) == 22 && (minor) <= 0)

56−#endif /* MIN_TOOL_VERSION_haddock */

57−

58−/* tool hpc-0.67 */

59−#ifndef TOOL_VERSION_hpc

60−#define TOOL_VERSION_hpc "0.67"

61−#endif /* TOOL_VERSION_hpc */

62−#ifndef MIN_TOOL_VERSION_hpc

63−#define MIN_TOOL_VERSION_hpc(major1,major2,minor) (\

64− (major1) < 0 || \

65− (major1) == 0 && (major2) < 67 || \

66− (major1) == 0 && (major2) == 67 && (minor) <= 0)

67−#endif /* MIN_TOOL_VERSION_hpc */

68−

69−/* tool hsc2hs-0.68.5 */

70−#ifndef TOOL_VERSION_hsc2hs

71−#define TOOL_VERSION_hsc2hs "0.68.5"

72−#endif /* TOOL_VERSION_hsc2hs */

73−#ifndef MIN_TOOL_VERSION_hsc2hs

74−#define MIN_TOOL_VERSION_hsc2hs(major1,major2,minor) (\

75− (major1) < 0 || \

76− (major1) == 0 && (major2) < 68 || \

77− (major1) == 0 && (major2) == 68 && (minor) <= 5)

78−#endif /* MIN_TOOL_VERSION_hsc2hs */

79−

80−/* tool pkg-config-1.6.3 */

81−#ifndef TOOL_VERSION_pkg_config

82−#define TOOL_VERSION_pkg_config "1.6.3"

83−#endif /* TOOL_VERSION_pkg_config */

84−#ifndef MIN_TOOL_VERSION_pkg_config

85−#define MIN_TOOL_VERSION_pkg_config(major1,major2,minor) (\

86− (major1) < 1 || \

87− (major1) == 1 && (major2) < 6 || \

88− (major1) == 1 && (major2) == 6 && (minor) <= 3)

89−#endif /* MIN_TOOL_VERSION_pkg_config */

90−

91−/* tool runghc-8.6.5 */

92−#ifndef TOOL_VERSION_runghc

93−#define TOOL_VERSION_runghc "8.6.5"

94−#endif /* TOOL_VERSION_runghc */

95−#ifndef MIN_TOOL_VERSION_runghc

96−#define MIN_TOOL_VERSION_runghc(major1,major2,minor) (\

97− (major1) < 8 || \

98− (major1) == 8 && (major2) < 6 || \

99− (major1) == 8 && (major2) == 6 && (minor) <= 5)

100−#endif /* MIN_TOOL_VERSION_runghc */

101−

102−/* tool strip-2.33 */

103−#ifndef TOOL_VERSION_strip

104−#define TOOL_VERSION_strip "2.33"

105−#endif /* TOOL_VERSION_strip */

106−#ifndef MIN_TOOL_VERSION_strip

107−#define MIN_TOOL_VERSION_strip(major1,major2,minor) (\

108− (major1) < 2 || \

109− (major1) == 2 && (major2) < 33 || \

110− (major1) == 2 && (major2) == 33 && (minor) <= 0)

111−#endif /* MIN_TOOL_VERSION_strip */

112−

113−#ifndef CURRENT_COMPONENT_ID

114−#define CURRENT_COMPONENT_ID "fake-package-0-inplace-script"

115−#endif /* CURRENT_COMPONENT_ID */

116−#ifndef CURRENT_PACKAGE_VERSION

117−#define CURRENT_PACKAGE_VERSION "0"

118−#endif /* CURRENT_PACKAGE_VERSION */

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1−{"cabal-version":"2.4.1.0","cabal-lib-version":"2.4.1.0","compiler-id":"ghc-8.6.5","os":"linux","arch":"x86_64","install-plan":[{"type":"pre-existing","id":"array-0.5.3.0","pkg-name":"array","pkg-version":"0.5.3.0","depends":["base-4.12.0.0"]},{"type":"pre-existing","id":"base-4.12.0.0","pkg-name":"base","pkg-version":"4.12.0.0","depends":["ghc-prim-0.5.3","integer-gmp-1.0.2.0","rts"]},{"type":"pre-existing","id":"deepseq-1.4.4.0","pkg-name":"deepseq","pkg-version":"1.4.4.0","depends":["array-0.5.3.0","base-4.12.0.0"]},{"type":"configured","id":"fake-package-0-inplace-script","pkg-name":"fake-package","pkg-version":"0","flags":{},"style":"local","pkg-src":{"type":"local","path":"/tmp/cabal-repl.-15730"},"dist-dir":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/dist-newstyle/build/x86_64-linux/ghc-8.6.5/fake-package-0/x/script","depends":[],"exe-depends":[],"component-name":"exe:script","bin-file":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/dist-newstyle/build/x86_64-linux/ghc-8.6.5/fake-package-0/x/script/build/script/script"},{"type":"pre-existing","id":"ghc-prim-0.5.3","pkg-name":"ghc-prim","pkg-version":"0.5.3","depends":["rts"]},{"type":"pre-existing","id":"integer-gmp-1.0.2.0","pkg-name":"integer-gmp","pkg-version":"1.0.2.0","depends":["ghc-prim-0.5.3"]},{"type":"configured","id":"laop-0.1.0.0-inplace","pkg-name":"laop","pkg-version":"0.1.0.0","flags":{},"style":"local","pkg-src":{"type":"local","path":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/."},"dist-dir":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/dist-newstyle/build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0","depends":["base-4.12.0.0","deepseq-1.4.4.0"],"exe-depends":[],"component-name":"lib"},{"type":"configured","id":"laop-0.1.0.0-inplace-laop-exe","pkg-name":"laop","pkg-version":"0.1.0.0","flags":{},"style":"local","pkg-src":{"type":"local","path":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/."},"dist-dir":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/dist-newstyle/build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/x/laop-exe","depends":["base-4.12.0.0","deepseq-1.4.4.0","laop-0.1.0.0-inplace"],"exe-depends":[],"component-name":"exe:laop-exe","bin-file":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/dist-newstyle/build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/x/laop-exe/build/laop-exe/laop-exe"},{"type":"configured","id":"laop-0.1.0.0-inplace-laop-test","pkg-name":"laop","pkg-version":"0.1.0.0","flags":{},"style":"local","pkg-src":{"type":"local","path":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/."},"dist-dir":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/dist-newstyle/build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/t/laop-test","depends":["base-4.12.0.0","laop-0.1.0.0-inplace"],"exe-depends":[],"component-name":"test:laop-test","bin-file":"/home/bolt/Desktop/Bolt/Playground/Haskell/laop/dist-newstyle/build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/t/laop-test/build/laop-test/laop-test"},{"type":"pre-existing","id":"rts","pkg-name":"rts","pkg-version":"1.0","depends":[]}]}

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hackage.shdeleted−10

@@ -1,10 +0,0 @@

1−#!/bin/sh

2−set -e

3−

4−dir=$(mktemp -d dist-docs.XXXXXX)

5−trap 'rm -r "$dir"' EXIT

6−

7−# assumes cabal 2.4 or later

8−cabal v2-haddock --builddir="$dir" --haddock-for-hackage --enable-doc

9−

10−cabal upload -d $dir/*-docs.tar.gz

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laop.cabalnew file+89

@@ -0,0 +1,89 @@

1+cabal-version: 2.4

2+name: laop

3+version: 0.1.1.1

4+synopsis: Matrix programming library

5+description: Matrix programming library that uses a type-safe inductive definition

6+ inspired by the Linear Algebra of Programming

7+homepage: https://github.com/bolt12/laop

8+bug-reports: https://github.com/bolt12/laop/issues

9+license: MIT

10+license-file: LICENSE

11+author: Armando Santos

12+maintainer: Armando Santos <armandoifsantos@gmail.com>

13+copyright: 2020 Armando Santos

14+category: Data

15+build-type: Simple

16+extra-doc-files: README.md

17+ CHANGELOG.md

18+tested-with: GHC == 8.6.5

19+ GHC == 8.8.3

20+

21+source-repository head

22+ type: git

23+ location: https://github.com/bolt12/laop.git

24+

25+common common-options

26+ build-depends: base >= 4.12.0.0 && < 5,

27+ deepseq

28+

29+ ghc-options: -Wall

30+ -Wcompat

31+ -Widentities

32+ -Wincomplete-uni-patterns

33+ -Wincomplete-record-updates

34+ if impl(ghc >= 8.0)

35+ ghc-options: -Wredundant-constraints

36+ if impl(ghc >= 8.2)

37+ ghc-options: -fhide-source-paths

38+ if impl(ghc >= 8.4)

39+ ghc-options: -Wmissing-export-lists

40+ -Wpartial-fields

41+ if impl(ghc >= 8.8)

42+ ghc-options: -Wmissing-deriving-strategies

43+

44+ default-language: Haskell2010

45+

46+library

47+ import: common-options

48+ hs-source-dirs: src

49+ exposed-modules: LAoP.Dist

50+ LAoP.Dist.Internal

51+ LAoP.Matrix.Internal

52+ LAoP.Matrix.Nat

53+ LAoP.Matrix.Type

54+ LAoP.Relation

55+ LAoP.Relation.Internal

56+ LAoP.Utils

57+ LAoP.Utils.Internal

58+

59+test-suite laop-test

60+ import: common-options

61+ type: exitcode-stdio-1.0

62+ hs-source-dirs: test

63+ other-modules: Examples.Readme

64+ main-is: Spec.hs

65+ build-depends: laop

66+ ghc-options: -threaded

67+ -rtsopts

68+ -with-rtsopts=-N

69+ -funfolding-use-threshold=16

70+ -fexcess-precision

71+ -optc-O3

72+ -optc-ffast-math

73+

74+benchmark laop-benchmark

75+ import: common-options

76+ type: exitcode-stdio-1.0

77+ hs-source-dirs: benchmark

78+ main-is: Main.hs

79+ build-depends: criterion

80+ , QuickCheck

81+ , deepseq

82+ , laop

83+ ghc-options: -threaded

84+ -rtsopts

85+ -with-rtsopts=-N

86+ -funfolding-use-threshold=16

87+ -fexcess-precision

88+ -optc-O3

89+ -optc-ffast-math

package.yamldeleted−48

@@ -1,48 +0,0 @@

1−name: laop

2−version: 0.1.0.0

3−github: "bolt12/laop"

4−license: BSD3

5−author: "Armando Santos"

6−maintainer: "armandoifsantos@gmail.com"

7−copyright: "MIT"

8−

9−extra-source-files:

10−- README.md

11−- ChangeLog.md

12−

13−# Metadata used when publishing your package

14−# synopsis: Short description of your package

15−# category: Web

16−

17−# To avoid duplicated efforts in documentation and dealing with the

18−# complications of embedding Haddock markup inside cabal files, it is

19−# common to point users to the README.md file.

20−description: Please see the README on GitHub at <https://github.com/bolt12/laop#readme>

21−

22−dependencies:

23−- base >= 4.7 && < 5

24−

25−library:

26− source-dirs: src

27−

28−executables:

29− laop-exe:

30− main: Main.hs

31− source-dirs: app

32− ghc-options:

33− - -threaded

34− - -rtsopts

35− - -with-rtsopts=-N

36− dependencies:

37− - laop

38−

39−tests:

40− laop-test:

41− main: Spec.hs

42− source-dirs: test

43− ghc-options:

44− - -threaded

45− - -rtsopts

46− - -with-rtsopts=-N

47− dependencies:

48− - laop

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src/LAoP/Dist.hs+10

@@ -36,6 +36,14 @@ module LAoP.Dist

3636 Dist (..),

3737 Prob,

3838

39+ -- * Constraint type synonyms

40+ Countable,

41+ CountableN,

42+ CountableDimsN,

43+ FLN,

44+ Liftable,

45+ TrivialP,

46+

3947 -- * Functor instance equivalent functions

4048 fmapD,

4149

@@ -45,6 +53,8 @@ module LAoP.Dist

4553

4654 -- * Selective equivalent functions

4755 selectD,

56+ branchD,

57+ ifD,

4858

4959 -- * Monad equivalent functions

5060 returnD,

src/LAoP/Dist/Internal.hs+87−36

@@ -1,3 +1,5 @@

1+{-# LANGUAGE DerivingVia #-}

2+{-# LANGUAGE ConstraintKinds #-}

13{-# LANGUAGE GADTs #-}

24{-# LANGUAGE AllowAmbiguousTypes #-}

35{-# LANGUAGE TypeApplications #-}

@@ -8,18 +10,27 @@

810{-# LANGUAGE RankNTypes #-}

911

1012module LAoP.Dist.Internal

11− (

13+ (

1214 Dist(..),

1315 Prob,

1416

17+ Countable,

18+ CountableN,

19+ CountableDimsN,

20+ FLN,

21+ Liftable,

22+ TrivialP,

23+

1524 fmapD,

1625 unitD,

1726 multD,

1827 selectD,

28+ branchD,

29+ ifD,

1930 returnD,

2031 bindD,

2132 (??),

22−

33+

2334 choose,

2435 shape,

2536 linear,

@@ -32,11 +43,15 @@ module LAoP.Dist.Internal

3243 )

3344 where

3445

35−import LAoP.Matrix.Type

46+import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDims, CountableN, CountableDimsN, Liftable, FLN)

47+import Prelude hiding (id, (.))

48+import qualified LAoP.Matrix.Internal as I

3649import LAoP.Utils

3750import GHC.TypeLits

3851import Data.Proxy

3952import Data.List (sortBy)

53+import Control.DeepSeq

54+import Data.Bool

4055

4156-- | Type synonym for probability value

4257type Prob = Double

@@ -44,17 +59,21 @@ type Prob = Double

4459-- | Type synonym for column vector matrices. This represents a probability

4560-- distribution.

4661newtype Dist a = D (Matrix Prob () a)

62+ deriving (Show, Num, Eq, Ord, NFData) via (Matrix Prob () a)

63+

64+-- | Constraint type synonyms to keep the type signatures less convoluted

65+type Countable a = KnownNat (I.Count a)

66+type CountableN a = KnownNat (I.Count (I.Normalize a))

67+type CountableDimsN a b = (CountableN a, CountableN b)

68+type FLN a b = I.FL (I.Normalize a) (I.Normalize b)

69+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Prob, Ord Prob)

70+type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)

4771

4872-- | Functor instance

4973fmapD ::

50− ( Bounded a,

51− Bounded b,

52− Enum a,

53− Enum b,

54− Eq b,

55− KnownNat (Count (Normalize a)),

56− KnownNat (Count (Normalize b)),

57− FromLists Prob (Normalize b) (Normalize a)

74+ ( Liftable a b,

75+ CountableDimsN a b,

76+ FLN b a

5877 )

5978 =>

6079 (a -> b) -> Dist a -> Dist b

@@ -66,26 +85,58 @@ unitD = D (one 1)

6685

6786-- | Applicative/Monoidal instance 'mult' function

6887multD ::

69− ( KnownNat (Count (Normalize a)),

70− KnownNat (Count (Normalize b)),

71− KnownNat (Count (Normalize (a, b))),

72− FromLists Prob (Normalize (a, b)) (Normalize a),

73− FromLists Prob (Normalize (a, b)) (Normalize b),

74− Normalize (a, b) ~ Normalize (Normalize a, Normalize b)

88+ ( CountableDimsN a b,

89+ CountableN (a, b),

90+ FLN (a, b) a,

91+ FLN (a, b) b,

92+ TrivialP a b

7593 ) => Dist a -> Dist b -> Dist (a, b)

76−multD (D a) (D b) = D (khatri a b)

94+multD (D a) (D b) = D (kr a b)

7795

7896-- | Selective instance function

7997selectD ::

80− ( Normalize (Either a b) ~ Either (Normalize a) (Normalize b),

81− FromLists Prob (Normalize b) (Normalize b),

82− KnownNat (Count (Normalize b))

98+ ( FLN b b,

99+ CountableN b

83100 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b

84−selectD (D d) m = D (junc m identity `comp` d)

85−

101+selectD (D d) m = D (selectM d m)

102+

103+-- | Chooses which of the two given effectful

104+-- functions to apply to a given argument;

105+branchD ::

106+ ( Num e,

107+ CountableDimsN a b,

108+ CountableDimsN c (Either b c),

109+ FLN c b,

110+ FLN a b,

111+ FLN a a,

112+ FLN b b,

113+ FLN c c,

114+ FLN b a,

115+ FLN b c,

116+ FLN (Either b c) b,

117+ FLN (Either b c) c

118+ )

119+ => Dist (Either a b) -> Matrix Prob a c -> Matrix Prob b c -> Dist c

120+branchD x l r = f x `selectD` g l `selectD` r

121+ where

122+ f (D m) = D (fork (tr i1) (i1 `comp` tr i2) `comp` m)

123+ g m = i2 `comp` m

124+

125+-- | Branch on a Boolean value, skipping unnecessary computations.

126+ifD ::

127+ ( CountableDimsN a (Either () a),

128+ FLN a a,

129+ FLN a (),

130+ FLN () a,

131+ FLN (Either () a) a

132+ )

133+ => Dist Bool -> Dist a -> Dist a -> Dist a

134+ifD x (D t) (D e) = branchD x' t e

135+ where

136+ x' = bool (Right ()) (Left ()) `fmapD` x

86137

87138-- | Monad instance 'return' function

88−returnD :: forall a . (Enum a, FromLists Prob () (Normalize a), KnownNat (Count a)) => a -> Dist a

139+returnD :: forall a . (Enum a, FLN () a, Countable a) => a -> Dist a

89140returnD a = D (col l)

90141 where

91142 i = fromInteger $ natVal (Proxy :: Proxy (Count a))

@@ -99,8 +150,8 @@ bindD (D d) m = D (m `comp` d)

99150-- | Extract probabilities given an Event.

100151(??) ::

101152 ( Enum a,

102− KnownNat (Count a),

103− FromLists Prob () (Normalize a)

153+ Countable a,

154+ FLN () a

104155 ) => (a -> Bool) -> Dist a -> Prob

105156(??) p d =

106157 let l = toValues d

@@ -110,11 +161,11 @@ bindD (D d) m = D (m `comp` d)

110161-- Distribution Construction

111162

112163-- | Constructs a Bernoulli distribution

113−choose :: (FromLists Prob () (Normalize a)) => Prob -> Dist a

164+choose :: (FLN () a) => Prob -> Dist a

114165choose prob = D (col [prob, 1 - prob])

115166

116167-- | Creates a distribution given a shape function

117−shape :: (FromLists Prob () (Normalize a)) => (Prob -> Prob) -> [a] -> Dist a

168+shape :: (FLN () a) => (Prob -> Prob) -> [a] -> Dist a

118169shape _ [] = error "Probability.shape: empty list"

119170shape f xs =

120171 let incr = 1 / fromIntegral (length xs - 1)

@@ -122,23 +173,23 @@ shape f xs =

122173 in fromFreqs (zip xs ps)

123174

124175-- | Constructs a Linear distribution

125−linear :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

176+linear :: (FLN () a) => [a] -> Dist a

126177linear = shape id

127178

128179-- | Constructs an Uniform distribution

129−uniform :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

180+uniform :: (FLN () a) => [a] -> Dist a

130181uniform = shape (const 1)

131182

132183-- | Constructs an Negative Exponential distribution

133−negExp :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

184+negExp :: (FLN () a) => [a] -> Dist a

134185negExp = shape (\x -> exp (-x))

135186

136187-- | Constructs an Normal distribution

137−normal :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

188+normal :: (FLN () a) => [a] -> Dist a

138189normal = shape (normalCurve 0.5 0.5)

139190

140191-- | Transforms a 'Dist' into a list of pairs.

141−toValues :: forall a . (Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> [(a, Prob)]

192+toValues :: forall a . (Enum a, Countable a, FLN () a) => Dist a -> [(a, Prob)]

142193toValues (D d) =

143194 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))

144195 probs = toList d

@@ -146,7 +197,7 @@ toValues (D d) =

146197 in res

147198

148199-- | Pretty a distribution

149−prettyDist :: forall a. (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> String

200+prettyDist :: forall a. (Show a, Enum a, Countable a, FLN () a) => Dist a -> String

150201prettyDist d =

151202 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)

152203 w = maximum (map (length . show . fst) values)

@@ -158,12 +209,12 @@ prettyDist d =

158209 showR n x = show x ++ " "

159210

160211-- | Pretty Print a distribution

161−prettyPrintDist :: forall a . (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> IO ()

212+prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FLN () a) => Dist a -> IO ()

162213prettyPrintDist = putStrLn . prettyDist @a

163214

164215-- Auxiliary functions

165216

166−fromFreqs :: (FromLists Prob () (Normalize a)) => [(a,Prob)] -> Dist a

217+fromFreqs :: (FLN () a) => [(a,Prob)] -> Dist a

167218fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))

168219 where q = sum $ map snd xs

169220

src/LAoP/Matrix/Internal.hs+445−436

@@ -1,4 +1,6 @@

1+{-# LANGUAGE StrictData #-}

12{-# LANGUAGE TypeFamilyDependencies #-}

3+{-# LANGUAGE ConstraintKinds #-}

24{-# LANGUAGE AllowAmbiguousTypes #-}

35{-# LANGUAGE DataKinds #-}

46{-# LANGUAGE FlexibleContexts #-}

@@ -28,7 +30,7 @@

2830-- motivation behind the library, the underlying theory, and implementation details.

2931--

3032-- This module offers many of the combinators mentioned in the work of

31−-- Macedo (2012) and Oliveira (2012).

33+-- Macedo (2012) and Oliveira (2012).

3234--

3335-- This is an Internal module and it is no supposed to be imported.

3436--

@@ -41,17 +43,26 @@ module LAoP.Matrix.Internal

4143 --

4244 -- There exists two type families that make it easier to write

4345 -- matrix dimensions: 'FromNat' and 'Count'. This approach

44− -- leads to a very straightforward implementation

45− -- of LAoP combinators.

46+ -- leads to a very straightforward implementation

47+ -- of LAoP combinators.

4648

4749 -- * Type safe matrix representation

4850 Matrix (..),

4951

52+ -- * Constraint type aliases

53+ Countable,

54+ CountableDims,

55+ CountableN,

56+ CountableDimsN,

57+ FL,

58+ FLN,

59+ Liftable,

60+ Trivial,

61+

5062 -- * Primitives

51− empty,

5263 one,

53− junc,

54− split,

64+ join,

65+ fork,

5566

5667 -- * Auxiliary type families

5768 FromNat,

@@ -64,6 +75,7 @@ module LAoP.Matrix.Internal

6475 toLists,

6576 toList,

6677 matrixBuilder,

78+ matrixBuilder',

6779 row,

6880 col,

6981 zeros,

@@ -74,28 +86,38 @@ module LAoP.Matrix.Internal

7486 -- * Misc

7587 -- ** Get dimensions

7688 columns,

89+ columns',

7790 rows,

91+ rows',

7892

7993 -- ** Matrix Transposition

8094 tr,

8195

96+ -- ** Scalar multiplication/division of matrices

97+ (.|),

98+ (./),

99+

82100 -- ** Selective operator

83− select,

101+ select,

102+ branch,

84103

85104 -- ** McCarthy's Conditional

86105 cond,

87106

88107 -- ** Matrix "abiding"

89− abideJS,

90− abideSJ,

108+ abideJF,

109+ abideFJ,

110+

111+ -- ** Zip matrices

112+ zipWithM,

91113

92114 -- * Biproduct approach

93− -- ** Split

115+ -- ** Fork

94116 (===),

95117 -- *** Projections

96118 p1,

97119 p2,

98− -- ** Junc

120+ -- ** Join

99121 (|||),

100122 -- *** Injections

101123 i1,

@@ -108,12 +130,13 @@ module LAoP.Matrix.Internal

108130

109131 -- | Note that given the restrictions imposed it is not possible to

110132 -- implement the standard type classes present in standard Haskell.

133+

111134 -- *** Matrix pairing projections

112− kp1,

113− kp2,

135+ fstM,

136+ sndM,

114137

115138 -- *** Matrix pairing

116− khatri,

139+ kr,

117140

118141 -- * Matrix composition and lifting

119142

@@ -121,10 +144,10 @@ module LAoP.Matrix.Internal

121144

122145 -- | Note that given the restrictions imposed it is not possible to

123146 -- implement the standard type classes present in standard Haskell.

124− identity,

147+ iden,

125148 comp,

126− fromF,

127149 fromF',

150+ fromF,

128151

129152 -- * Matrix printing

130153 pretty,

@@ -151,21 +174,22 @@ import LAoP.Utils.Internal

151174import Data.Bool

152175import Data.Kind

153176import Data.List

177+import Data.Maybe

154178import Data.Proxy

179+import Data.Typeable

155180import Data.Void

156181import GHC.TypeLits

157182import Data.Type.Equality

158183import GHC.Generics

159184import Control.DeepSeq

160−import Control.Category

161−import Prelude hiding ((.))

185+import Prelude hiding (id, (.))

186+import qualified Prelude (id, (.))

162187

163188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.

164189data Matrix e cols rows where

165− Empty :: Matrix e Void Void

166190 One :: e -> Matrix e () ()

167− Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

168− Split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

191+ Join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

192+ Fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

169193

170194deriving instance (Show e) => Show (Matrix e cols rows)

171195

@@ -175,221 +199,214 @@ deriving instance (Show e) => Show (Matrix e cols rows)

175199-- 'Generic' instance.

176200type family Count (d :: Type) :: Nat where

177201 Count (Natural n m) = (m - n) + 1

178− Count (Either a b) = (+) (Count a) (Count b)

179− Count (a, b) = (*) (Count a) (Count b)

180− Count (a -> b) = (^) (Count b) (Count a)

202+ Count (List a) = (^) 2 (Count a)

203+ Count (Either a b) = (+) (Count a) (Count b)

204+ Count (a, b) = (*) (Count a) (Count b)

205+ Count (a -> b) = (^) (Count b) (Count a)

181206 -- Generics

182− Count (M1 _ _ f p) = Count (f p)

183− Count (K1 _ _ _) = 1

184− Count (V1 _) = 0

185− Count (U1 _) = 1

207+ Count (M1 _ _ f p) = Count (f p)

208+ Count (K1 _ _ _) = 1

209+ Count (V1 _) = 0

210+ Count (U1 _) = 1

186211 Count ((:*:) a b p) = Count (a p) * Count (b p)

187212 Count ((:+:) a b p) = Count (a p) + Count (b p)

188− Count d = Count (Rep d R)

213+ Count d = Count (Rep d R)

189214

190215-- | Type family that computes of a given type dimension from a given natural

191216--

192217-- Thanks to Li-Yao Xia this type family is super fast.

193218type family FromNat (n :: Nat) :: Type where

194− FromNat 0 = Void

195219 FromNat 1 = ()

196220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))

197221

198222type family FromNat' (b :: Bool) (m :: Type) :: Type where

199− FromNat' 'True m = Either m m

223+ FromNat' 'True m = Either m m

200224 FromNat' 'False m = Either () (Either m m)

201225

202226-- | Type family that normalizes the representation of a given data

203227-- structure

204228type family Normalize (d :: Type) :: Type where

205− Normalize d = FromNat (Count d)

206−

207−-- | It isn't possible to implement the 'id' function so it's

208−-- implementation is 'undefined'. However 'comp' can be and this partial

209−-- class implementation exists just to make the code more readable.

210−--

211−-- Please use 'identity' instead.

229+ Normalize (Either a b) = Either (Normalize a) (Normalize b)

230+ Normalize d = FromNat (Count d)

231+

232+-- | Constraint type synonyms to keep the type signatures less convoluted

233+type Countable a = KnownNat (Count a)

234+type CountableN a = KnownNat (Count (Normalize a))

235+type CountableDims a b = (Countable a, Countable b)

236+type CountableDimsN a b = (CountableN a, CountableN b)

237+type FL a b = FromLists a b

238+type FLN a b = FromLists (Normalize a) (Normalize b)

239+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

240+type Trivial a = FromNat (Count a) ~ a

241+

242+-- | It is possible to implement a constrained version of the category type

243+-- class.

212244instance (Num e) => Category (Matrix e) where

213− id = undefined

214− (.) = comp

245+ type Object (Matrix e) a = (FL a a, Countable a)

246+ id = iden

247+ (.) = comp

215248

216249instance NFData e => NFData (Matrix e cols rows) where

217− rnf Empty = ()

218− rnf (One e) = rnf e

219− rnf (Junc a b) = rnf a `seq` rnf b

220− rnf (Split a b) = rnf a `seq` rnf b

250+ rnf (One e) = rnf e

251+ rnf (Join a b) = rnf a `seq` rnf b

252+ rnf (Fork a b) = rnf a `seq` rnf b

221253

222254instance Eq e => Eq (Matrix e cols rows) where

223− Empty == Empty = True

224− (One a) == (One b) = a == b

225− (Junc a b) == (Junc c d) = a == c && b == d

226− (Split a b) == (Split c d) = a == c && b == d

227− x@(Split _ _) == y@(Junc _ _) = x == abideJS y

228− x@(Junc _ _) == y@(Split _ _) = abideJS x == y

255+ (One a) == (One b) = a == b

256+ (Join a b) == (Join c d) = a == c && b == d

257+ (Fork a b) == (Fork c d) = a == c && b == d

258+ x@(Fork _ _) == y@(Join _ _) = x == abideJF y

259+ x@(Join _ _) == y@(Fork _ _) = abideJF x == y

229260

230261instance Num e => Num (Matrix e cols rows) where

262+ a + b = zipWithM (+) a b

263+

264+ a - b = zipWithM (-) a b

265+

266+ a * b = zipWithM (*) a b

267+

268+ abs (One a) = One (abs a)

269+ abs (Join a b) = Join (abs a) (abs b)

270+ abs (Fork a b) = Fork (abs a) (abs b)

231271

232− Empty + Empty = Empty

233− (One a) + (One b) = One (a + b)

234− (Junc a b) + (Junc c d) = Junc (a + c) (b + d)

235− (Split a b) + (Split c d) = Split (a + c) (b + d)

236− x@(Split _ _) + y@(Junc _ _) = x + abideJS y

237− x@(Junc _ _) + y@(Split _ _) = abideJS x + y

238−

239− Empty - Empty = Empty

240− (One a) - (One b) = One (a - b)

241− (Junc a b) - (Junc c d) = Junc (a - c) (b - d)

242− (Split a b) - (Split c d) = Split (a - c) (b - d)

243− x@(Split _ _) - y@(Junc _ _) = x - abideJS y

244− x@(Junc _ _) - y@(Split _ _) = abideJS x - y

245−

246− Empty * Empty = Empty

247− (One a) * (One b) = One (a * b)

248− (Junc a b) * (Junc c d) = Junc (a * c) (b * d)

249− (Split a b) * (Split c d) = Split (a * c) (b * d)

250− x@(Split _ _) * y@(Junc _ _) = x * abideJS y

251− x@(Junc _ _) * y@(Split _ _) = abideJS x * y

252−

253− abs Empty = Empty

254− abs (One a) = One (abs a)

255− abs (Junc a b) = Junc (abs a) (abs b)

256− abs (Split a b) = Split (abs a) (abs b)

257−

258− signum Empty = Empty

259− signum (One a) = One (signum a)

260− signum (Junc a b) = Junc (signum a) (signum b)

261− signum (Split a b) = Split (signum a) (signum b)

272+ signum (One a) = One (signum a)

273+ signum (Join a b) = Join (signum a) (signum b)

274+ signum (Fork a b) = Fork (signum a) (signum b)

262275

263276instance Ord e => Ord (Matrix e cols rows) where

264− Empty <= Empty = True

265− (One a) <= (One b) = a <= b

266− (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)

267− (Split a b) <= (Split c d) = (a <= c) && (b <= d)

268− x@(Split _ _) <= y@(Junc _ _) = x <= abideJS y

269− x@(Junc _ _) <= y@(Split _ _) = abideJS x <= y

277+ (One a) <= (One b) = a <= b

278+ (Join a b) <= (Join c d) = (a <= c) && (b <= d)

279+ (Fork a b) <= (Fork c d) = (a <= c) && (b <= d)

280+ x@(Fork _ _) <= y@(Join _ _) = x <= abideJF y

281+ x@(Join _ _) <= y@(Fork _ _) = abideJF x <= y

270282

271283-- Primitives

272284

273−-- | Empty matrix constructor

274−empty :: Matrix e Void Void

275−empty = Empty

276−

277285-- | Unit matrix constructor

278286one :: e -> Matrix e () ()

279287one = One

280288

281−-- | Matrix 'Junc' constructor

282−junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

283−junc = Junc

289+-- | Matrix 'Join' constructor

290+join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

291+join = Join

284292

285293infixl 3 |||

286294

287−-- | Matrix 'Junc' constructor

295+-- | Matrix 'Join' constructor

288296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

289−(|||) = Junc

297+(|||) = Join

290298

291−-- | Matrix 'Split' constructor

292−split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

293−split = Split

299+-- | Matrix 'Fork' constructor

300+fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

301+fork = Fork

294302

295303infixl 2 ===

296304

297−-- | Matrix 'Split' constructor

305+-- | Matrix 'Fork' constructor

298306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

299−(===) = Split

307+(===) = Fork

300308

301309-- Construction

302310

303311-- | Type class for defining the 'fromList' conversion function.

304312--

305313-- Given that it is not possible to branch on types at the term level type

306−-- classes are needed bery much like an inductive definition but on types.

307−class FromLists e cols rows where

314+-- classes are needed very much like an inductive definition but on types.

315+class FromLists cols rows where

308316 -- | Build a matrix out of a list of list of elements. Throws a runtime

309317 -- error if the dimensions do not match.

310318 fromLists :: [[e]] -> Matrix e cols rows

311319

312−instance FromLists e Void Void where

313− fromLists [] = Empty

314− fromLists _ = error "Wrong dimensions"

315−

316−instance {-# OVERLAPPING #-} FromLists e () () where

320+instance FromLists () () where

317321 fromLists [[e]] = One e

318322 fromLists _ = error "Wrong dimensions"

319323

320−instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where

321− fromLists [h : t] = Junc (One h) (fromLists [t])

324+instance (FromLists cols ()) => FromLists (Either () cols) () where

325+ fromLists [h : t] = Join (One h) (fromLists [t])

322326 fromLists _ = error "Wrong dimensions"

323327

324−instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), KnownNat (Count a)) => FromLists e (Either a b) () where

325− fromLists [l] =

328+instance {-# OVERLAPPABLE #-}

329+ (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where

330+ fromLists [l] =

326331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

327− in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])

332+ in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])

328333 fromLists _ = error "Wrong dimensions"

329334

330−instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where

331− fromLists ([h] : t) = Split (One h) (fromLists t)

335+instance (FromLists () rows) => FromLists () (Either () rows) where

336+ fromLists ([h] : t) = Fork (One h) (fromLists t)

332337 fromLists _ = error "Wrong dimensions"

333338

334−instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, KnownNat (Count a)) => FromLists e () (Either a b) where

335− fromLists l@([_] : _) =

339+instance {-# OVERLAPPABLE #-}

340+ (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where

341+ fromLists l@([_] : _) =

336342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

337− in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))

343+ in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))

338344 fromLists _ = error "Wrong dimensions"

339345

340−instance {-# OVERLAPPABLE #-} (FromLists e (Either a b) c, FromLists e (Either a b) d, KnownNat (Count c)) => FromLists e (Either a b) (Either c d) where

346+instance (FromLists (Either a b) c, FromLists (Either a b) d, Countable c)

347+ => FromLists (Either a b) (Either c d) where

341348 fromLists l@(h : t) =

342349 let lh = length h

343350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))

344− condition = all (== lh) (map length t)

351+ condition = all ((== lh) . length) t

345352 in if lh > 0 && condition

346− then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))

353+ then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))

347354 else error "Not all rows have the same length"

348355

349356-- | Matrix builder function. Constructs a matrix provided with

350−-- a construction function.

351−matrixBuilder ::

357+-- a construction function that operates with indices.

358+matrixBuilder' ::

352359 forall e cols rows.

353− ( FromLists e cols rows,

354− KnownNat (Count cols),

355− KnownNat (Count rows)

356− ) =>

357− ((Int, Int) -> e) ->

358− Matrix e cols rows

359−matrixBuilder f =

360+ ( FL cols rows,

361+ CountableDims cols rows

362+ ) => ((Int, Int) -> e) -> Matrix e cols rows

363+matrixBuilder' f =

360364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

361365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))

362366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]

363367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions

364368

369+-- | Matrix builder function. Constructs a matrix provided with

370+-- a construction function that operates with arbitrary types.

371+matrixBuilder ::

372+ forall e a b.

373+ ( FLN a b,

374+ Enum a,

375+ Enum b,

376+ Bounded a,

377+ Bounded b,

378+ Eq a,

379+ Countable b

380+ ) => ((a, b) -> e) -> Matrix e (Normalize a) (Normalize b)

381+matrixBuilder f =

382+ let r = fromInteger $ natVal (Proxy :: Proxy (Count b))

383+ positions = [(a, b) | a <- [minBound .. maxBound], b <- [minBound .. maxBound]]

384+ in fromLists . map (map f) . transpose . buildList r $ positions

385+ where

386+ buildList _ [] = []

387+ buildList r l = take r l : buildList r (drop r l)

388+

365389-- | Constructs a column vector matrix

366−col :: (FromLists e () rows) => [e] -> Matrix e () rows

390+col :: (FL () rows) => [e] -> Matrix e () rows

367391col = fromLists . map (: [])

368392

369393-- | Constructs a row vector matrix

370−row :: (FromLists e cols ()) => [e] -> Matrix e cols ()

394+row :: (FL cols ()) => [e] -> Matrix e cols ()

371395row = fromLists . (: [])

372396

373397-- | Lifts functions to matrices with arbitrary dimensions.

374398--

375399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

376400-- types @a@ or @b@ allows.

377−fromF ::

401+fromF' ::

378402 forall a b cols rows e.

379− ( Bounded a,

380− Bounded b,

381− Enum a,

382− Enum b,

383− Eq b,

384− Num e,

385− Ord e,

386− KnownNat (Count cols),

387− KnownNat (Count rows),

388− FromLists e rows cols

403+ ( Liftable e a b,

404+ CountableDims cols rows,

405+ FL rows cols

389406 ) =>

390407 (a -> b) ->

391408 Matrix e cols rows

392−fromF f =

409+fromF' f =

393410 let minA = minBound @a

394411 maxA = maxBound @a

395412 minB = minBound @b

@@ -399,8 +416,8 @@ fromF f =

399416 elementsA = take ccols [minA .. maxA]

400417 elementsB = take rrows [minB .. maxB]

401418 combinations = (,) <$> elementsA <*> elementsB

402− combAp = map snd . sort . map (\(a, b) -> if f a == b

403− then ((fromEnum a, fromEnum b), 1)

419+ combAp = map snd . sort . map (\(a, b) -> if f a == b

420+ then ((fromEnum a, fromEnum b), 1)

404421 else ((fromEnum a, fromEnum b), 0)) $ combinations

405422 mList = buildList combAp rrows

406423 in tr $ fromLists mList

@@ -410,48 +427,23 @@ fromF f =

410427

411428-- | Lifts functions to matrices with dimensions matching @a@ and @b@

412429-- cardinality's.

413−fromF' ::

430+fromF ::

414431 forall a b e.

415− ( Bounded a,

416− Bounded b,

417− Enum a,

418− Enum b,

419− Eq b,

420− Num e,

421− Ord e,

422− KnownNat (Count (Normalize a)),

423− KnownNat (Count (Normalize b)),

424− FromLists e (Normalize b) (Normalize a)

432+ ( Liftable e a b,

433+ CountableDimsN a b,

434+ FLN b a

425435 ) =>

426436 (a -> b) ->

427437 Matrix e (Normalize a) (Normalize b)

428−fromF' f =

429− let minA = minBound @a

430− maxA = maxBound @a

431− minB = minBound @b

432− maxB = maxBound @b

433− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))

434− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))

435− elementsA = take ccols [minA .. maxA]

436− elementsB = take rrows [minB .. maxB]

437− combinations = (,) <$> elementsA <*> elementsB

438− combAp = map snd . sort . map (\(a, b) -> if f a == b

439− then ((fromEnum a, fromEnum b), 1)

440− else ((fromEnum a, fromEnum b), 0)) $ combinations

441− mList = buildList combAp rrows

442− in tr $ fromLists mList

443− where

444− buildList [] _ = []

445− buildList l r = take r l : buildList (drop r l) r

438+fromF = fromF'

446439

447440-- Conversion

448441

449442-- | Converts a matrix to a list of lists of elements.

450443toLists :: Matrix e cols rows -> [[e]]

451−toLists Empty = []

452444toLists (One e) = [[e]]

453−toLists (Split l r) = toLists l ++ toLists r

454−toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)

445+toLists (Fork l r) = toLists l ++ toLists r

446+toLists (Join l r) = zipWith (++) (toLists l) (toLists r)

455447

456448-- | Converts a matrix to a list of elements.

457449toList :: Matrix e cols rows -> [e]

@@ -460,37 +452,38 @@ toList = concat . toLists

460452-- Zeros Matrix

461453

462454-- | The zero matrix. A matrix wholly filled with zeros.

463−zeros :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows

464−zeros = matrixBuilder (const 0)

455+zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows

456+zeros = matrixBuilder' (const 0)

465457

466458-- Ones Matrix

467459

468460-- | The ones matrix. A matrix wholly filled with ones.

469461--

470462-- Also known as T (Top) matrix.

471−ones :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows

472−ones = matrixBuilder (const 1)

463+ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows

464+ones = matrixBuilder' (const 1)

473465

474466-- Const Matrix

475467

476468-- | The constant matrix constructor. A matrix wholly filled with a given

477469-- value.

478−constant :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => e -> Matrix e cols rows

479−constant e = matrixBuilder (const e)

470+constant :: (Num e, FL cols rows, CountableDims cols rows) => e -> Matrix e cols rows

471+constant e = matrixBuilder' (const e)

480472

481473-- Bang Matrix

482474

483475-- | The T (Top) row vector matrix.

484−bang :: forall e cols. (Num e, Enum e, FromLists e cols (), KnownNat (Count cols)) => Matrix e cols ()

476+bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()

485477bang =

486478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

487479 in fromLists [take c [1, 1 ..]]

488480

489−-- Identity Matrix

481+-- iden Matrix

490482

491−-- | Identity matrix.

492−identity :: (Num e, FromLists e cols cols, KnownNat (Count cols)) => Matrix e cols cols

493−identity = matrixBuilder (bool 0 1 . uncurry (==))

483+-- | iden matrix.

484+iden :: forall e cols . (Num e, FL cols cols, Countable cols) => Matrix e cols cols

485+iden = matrixBuilder' (bool 0 1 . uncurry (==))

486+{-# NOINLINE iden #-}

494487

495488-- Matrix composition (MMM)

496489

@@ -499,60 +492,90 @@ identity = matrixBuilder (bool 0 1 . uncurry (==))

499492-- This definition takes advantage of divide-and-conquer and fusion laws

500493-- from LAoP.

501494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

502−comp Empty Empty = Empty

503−comp (One a) (One b) = One (a * b)

504−comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law

505−comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law

506−comp c (Junc a b) = Junc (comp c a) (comp c b) -- Junc fusion law

495+comp (One a) (One b) = One (a * b)

496+comp (Join a b) (Fork c d) = comp a c + comp b d -- Divide-and-conquer law

497+comp (Fork a b) c = Fork (comp a c) (comp b c) -- Fork fusion law

498+comp c (Join a b) = Join (comp c a) (comp c b) -- Join fusion law

499+{-# NOINLINE comp #-}

500+{-# RULES

501+ "comp/iden1" forall m. comp m iden = m ;

502+ "comp/iden2" forall m. comp iden m = m

503+#-}

504+

505+-- Scalar multiplication of matrices

506+

507+infixl 7 .|

508+-- | Scalar multiplication of matrices.

509+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows

510+(.|) e (One a) = One (e * a)

511+(.|) e (Join a b) = Join (e .| a) (e .| b)

512+(.|) e (Fork a b) = Fork (e .| a) (e .| b)

513+

514+-- Scalar division of matrices

515+

516+infixl 7 ./

517+-- | Scalar multiplication of matrices.

518+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows

519+(./) (One a) e = One (a / e)

520+(./) (Join a b) e = Join (a ./ e) (b ./ e)

521+(./) (Fork a b) e = Fork (a ./ e) (b ./ e)

507522

508523-- Projections

509524

510525-- | Biproduct first component projection

511−p1 :: forall e m n. (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e (Either m n) m

512−p1 =

513− let iden = identity :: Matrix e m m

514− zero = zeros :: Matrix e n m

515− in junc iden zero

526+p1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e (Either m n) m

527+p1 = join id zeros

516528

517529-- | Biproduct second component projection

518−p2 :: forall e m n. (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e (Either m n) n

519−p2 =

520− let iden = identity :: Matrix e n n

521− zero = zeros :: Matrix e m n

522− in junc zero iden

530+p2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e (Either m n) n

531+p2 = join zeros id

523532

524533-- Injections

525534

526535-- | Biproduct first component injection

527−i1 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e m (Either m n)

536+i1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e m (Either m n)

528537i1 = tr p1

529538

530539-- | Biproduct second component injection

531−i2 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e n (Either m n)

540+i2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e n (Either m n)

532541i2 = tr p2

533542

534543-- Dimensions

535544

536545-- | Obtain the number of rows.

537546--

538−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

539−-- dimensions in constant time.

540−--

541−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

542−-- exchange for performance.

543−rows :: forall e cols rows. (KnownNat (Count rows)) => Matrix e cols rows -> Int

547+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

548+-- dimensions in constant time. For a version that doesn't require the

549+-- constraint see 'rows''.

550+rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int

544551rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))

545552

553+-- | Obtain the number of rows in an inefficient manner, but without any

554+-- constraints.

555+--

556+-- For a more efficient version see 'rows'.

557+rows' :: Matrix e cols rows -> Int

558+rows' (One _) = 1

559+rows' (Join lhs _) = rows' lhs

560+rows' (Fork top bottom) = rows' top + rows' bottom

561+

546562-- | Obtain the number of columns.

547−--

548−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

549−-- dimensions in constant time.

550563--

551−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

552−-- exchange for performance.

553−columns :: forall e cols rows. (KnownNat (Count cols)) => Matrix e cols rows -> Int

564+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

565+-- dimensions in constant time. For a version that doesn't require the

566+-- constraint see 'columns''.

567+columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int

554568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))

555569

570+-- | Obtain the number of columns in an inefficient manner, but without any

571+-- constraints.

572+--

573+-- For a more efficient version see 'columns'.

574+columns' :: Matrix e cols rows -> Int

575+columns' (One _) = 1

576+columns' (Join lhs rhs) = columns' lhs + columns' rhs

577+columns' (Fork top _) = columns' top

578+

556579-- Coproduct Bifunctor

557580

558581infixl 5 -|-

@@ -561,157 +584,172 @@ infixl 5 -|-

561584(-|-) ::

562585 forall e n k m j.

563586 ( Num e,

564− KnownNat (Count j),

565− KnownNat (Count k),

566− FromLists e k k,

567− FromLists e j k,

568− FromLists e k j,

569− FromLists e j j

587+ CountableDims j k,

588+ FL k k,

589+ FL j k,

590+ FL k j,

591+ FL j j

570592 ) =>

571593 Matrix e n k ->

572594 Matrix e m j ->

573595 Matrix e (Either n m) (Either k j)

574−(-|-) a b = Junc (i1 . a) (i2 . b)

596+(-|-) a b = Join (i1 . a) (i2 . b)

575597

576598-- Khatri Rao Product and projections

577599

578600-- | Khatri Rao product first component projection matrix.

579−kp1 ::

601+fstM ::

580602 forall e m k .

581603 ( Num e,

582− KnownNat (Count k),

583− FromLists e (FromNat (Count m * Count k)) m,

584− KnownNat (Count m),

585− KnownNat (Count (Normalize (m, k)))

604+ CountableDims k m,

605+ FL (Normalize (m, k)) m,

606+ CountableN (m, k)

586607 ) => Matrix e (Normalize (m, k)) m

587−kp1 = matrixBuilder f

608+fstM = matrixBuilder' f

588609 where

589610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

590611 f (x, y)

591612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1

592− | otherwise = 0

613+ | otherwise = 0

593614

594615-- | Khatri Rao product second component projection matrix.

595−kp2 ::

616+sndM ::

596617 forall e m k .

597618 ( Num e,

598− KnownNat (Count k),

599− FromLists e (FromNat (Count m * Count k)) k,

600− KnownNat (Count m),

601− KnownNat (Count (Normalize (m, k)))

619+ CountableDims k m,

620+ FL (Normalize (m, k)) k,

621+ CountableN (m, k)

602622 ) => Matrix e (Normalize (m, k)) k

603−kp2 = matrixBuilder f

623+sndM = matrixBuilder' f

604624 where

605625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

606626 f (x, y)

607627 | x == y || mod (y - x) offset == 0 = 1

608− | otherwise = 0

628+ | otherwise = 0

609629

610630-- | Khatri Rao Matrix product also known as matrix pairing.

611631--

612632-- NOTE: That this is not a true categorical product, see for instance:

613−--

633+--

614634-- @

615−-- | kp1 . khatri a b == a

616−-- khatri a b ==> |

617−-- | kp2 . khatri a b == b

635+-- | fstM . kr a b == a

636+-- kr a b ==> |

637+-- | sndM . kr a b == b

618638-- @

619639--

620640-- __Emphasis__ on the implication symbol.

621−khatri ::

622− forall e cols a b.

641+kr ::

642+ forall e cols a b.

623643 ( Num e,

624− KnownNat (Count a),

625− KnownNat (Count b),

626− KnownNat (Count (Normalize (a, b))),

627− FromLists e (Normalize (a, b)) a,

628− FromLists e (Normalize (a, b)) b

644+ CountableDims a b,

645+ CountableN (a, b),

646+ FL (Normalize (a, b)) a,

647+ FL (Normalize (a, b)) b

629648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))

630−khatri a b =

631− let kp1' = kp1 @e @a @b

632− kp2' = kp2 @e @a @b

633− in (tr kp1') . a * (tr kp2') . b

649+kr a b =

650+ let fstM' = fstM @e @a @b

651+ sndM' = sndM @e @a @b

652+ in (tr fstM' . a) * (tr sndM' . b)

634653

635654-- Product Bifunctor (Kronecker)

636655

637656infixl 4 ><

638657

639658-- | Matrix product functor also known as kronecker product

640−(><) ::

641− forall e m p n q.

659+(><) ::

660+ forall e m p n q.

642661 ( Num e,

643− KnownNat (Count m),

644− KnownNat (Count n),

645− KnownNat (Count p),

646− KnownNat (Count q),

647− KnownNat (Count (Normalize (m, n))),

648− FromLists e (Normalize (m, n)) m,

649− FromLists e (Normalize (m, n)) n,

650− KnownNat (Count (Normalize (p, q))),

651− FromLists e (Normalize (p, q)) p,

652− FromLists e (Normalize (p, q)) q

653− )

662+ CountableDims m n,

663+ CountableDims p q,

664+ CountableDimsN (m, n) (p, q),

665+ FL (Normalize (m, n)) m,

666+ FL (Normalize (m, n)) n,

667+ FL (Normalize (p, q)) p,

668+ FL (Normalize (p, q)) q

669+ )

654670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))

655671(><) a b =

656− let kp1' = kp1 @e @m @n

657− kp2' = kp2 @e @m @n

658− in khatri (a . kp1') (b . kp2')

672+ let fstM' = fstM @e @m @n

673+ sndM' = sndM @e @m @n

674+ in kr (a . fstM') (b . sndM')

659675

660−-- Matrix abide Junc Split

676+-- Matrix abide Join Fork

661677

662−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

663−--

678+-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.

679+--

664680-- Law:

665681--

666682-- @

667−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

683+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)

668684-- @

669−abideJS :: Matrix e cols rows -> Matrix e cols rows

670−abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law

671−abideJS Empty = Empty

672−abideJS (One e) = One e

673−abideJS (Junc a b) = Junc (abideJS a) (abideJS b)

674−abideJS (Split a b) = Split (abideJS a) (abideJS b)

685+abideJF :: Matrix e cols rows -> Matrix e cols rows

686+abideJF (Join (Fork a c) (Fork b d)) = Fork (Join (abideJF a) (abideJF b)) (Join (abideJF c) (abideJF d)) -- Join-Fork abide law

687+abideJF (One e) = One e

688+abideJF (Join a b) = Join (abideJF a) (abideJF b)

689+abideJF (Fork a b) = Fork (abideJF a) (abideJF b)

675690

676−-- Matrix abide Split Junc

691+-- Matrix abide Fork Join

677692

678−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

679−--

693+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.

694+--

680695-- @

681−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

696+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)

682697-- @

683−abideSJ :: Matrix e cols rows -> Matrix e cols rows

684−abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law

685−abideSJ Empty = Empty

686−abideSJ (One e) = One e

687−abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)

688−abideSJ (Split a b) = Split (abideSJ a) (abideSJ b)

698+abideFJ :: Matrix e cols rows -> Matrix e cols rows

699+abideFJ (Fork (Join a b) (Join c d)) = Join (Fork (abideFJ a) (abideFJ c)) (Fork (abideFJ b) (abideFJ d)) -- Fork-Join abide law

700+abideFJ (One e) = One e

701+abideFJ (Join a b) = Join (abideFJ a) (abideFJ b)

702+abideFJ (Fork a b) = Fork (abideFJ a) (abideFJ b)

689703

690704-- Matrix transposition

691705

692706-- | Matrix transposition.

693707tr :: Matrix e cols rows -> Matrix e rows cols

694−tr Empty = Empty

695−tr (One e) = One e

696−tr (Junc a b) = Split (tr a) (tr b)

697−tr (Split a b) = Junc (tr a) (tr b)

708+tr (One e) = One e

709+tr (Join a b) = Fork (tr a) (tr b)

710+tr (Fork a b) = Join (tr a) (tr b)

698711

699712-- Selective 'select' operator

700713

701714-- | Selective functors 'select' operator equivalent inspired by the

702715-- ArrowMonad solution presented in the paper.

703−select :: (Num e, FromLists e b b, KnownNat (Count b)) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

704−select m y = junc y identity . m

716+select :: (Num e, FL b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

717+select (Fork a b) y = y . a + b -- Divide-and-conquer law

718+select (Join (Fork a c) (Fork b d)) y = join (y . a + c) (y . b + d) -- Pattern matching + DnC law

719+select m y = join y id . m

720+

721+branch ::

722+ ( Num e,

723+ CountableDims a b,

724+ CountableDims c (Either b c),

725+ FL c b,

726+ FL a b,

727+ FL a a,

728+ FL b b,

729+ FL c c,

730+ FL b a,

731+ FL b c,

732+ FL (Either b c) b,

733+ FL (Either b c) c

734+ )

735+ => Matrix e cols (Either a b) -> Matrix e a c -> Matrix e b c -> Matrix e cols c

736+branch x l r = f x `select` g l `select` r

737+ where

738+ f :: (Num e, Countable a, CountableDims b c, FL a b, FL c b, FL b b, FL b a, FL a a)

739+ => Matrix e cols (Either a b) -> Matrix e cols (Either a (Either b c))

740+ f m = fork (tr i1) (i1 . tr i2) . m

741+ g :: (Num e, CountableDims b c, FL b c, FL c c) => Matrix e a c -> Matrix e a (Either b c)

742+ g m = i2 . m

705743

706744-- McCarthy's Conditional

707745

708746-- | McCarthy's Conditional expresses probabilistic choice.

709747cond ::

710− ( cols ~ FromNat (Count cols),

711− KnownNat (Count cols),

712− FromLists e () cols,

713− FromLists e cols (),

714− FromLists e cols cols,

748+ ( Trivial cols,

749+ Countable cols,

750+ FL () cols,

751+ FL cols (),

752+ FL cols cols,

715753 Bounded a,

716754 Enum a,

717755 Num e,

@@ -719,14 +757,14 @@ cond ::

719757 )

720758 =>

721759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows

722−cond p f g = junc f g . grd p

723−

724−grd ::

725− ( q ~ FromNat (Count q),

726− KnownNat (Count q),

727− FromLists e () q,

728− FromLists e q (),

729− FromLists e q q,

760+cond p f g = join f g . grd p

761+

762+grd ::

763+ ( Trivial q,

764+ Countable q,

765+ FL () q,

766+ FL q (),

767+ FL q q,

730768 Bounded a,

731769 Enum a,

732770 Num e,

@@ -734,63 +772,86 @@ grd ::

734772 )

735773 =>

736774 (a -> Bool) -> Matrix e q (Either q q)

737−grd f = split (corr f) (corr (not . f))

738−

739−corr ::

740− forall e a q .

741− ( q ~ FromNat (Count q),

742− KnownNat (Count q),

743− FromLists e () q,

744− FromLists e q (),

745− FromLists e q q,

746− Bounded a,

747− Enum a,

748− Num e,

749− Ord e

750− )

775+grd f = fork (corr f) (corr (not . f))

776+

777+corr ::

778+ forall e a q .

779+ ( Trivial q,

780+ Countable q,

781+ FL () q,

782+ FL q (),

783+ FL q q,

784+ Liftable e a Bool

785+ )

751786 => (a -> Bool) -> Matrix e q q

752−corr p = let f = fromF p :: Matrix e q ()

753− in khatri f (identity :: Matrix e q q)

787+corr p = let f = fromF' p :: Matrix e q ()

788+ in kr f (id :: Matrix e q q)

754789

755790-- Pretty print

756791

757792prettyAux :: Show e => [[e]] -> [[e]] -> String

758793prettyAux [] _ = ""

759−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"

794+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"

760795 where

761796 v = fmap show m

762797 widest = maximum $ fmap length v

763798 fill str = replicate (widest - length str - 2) ' ' ++ str

764−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"

799+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"

765800 where

766− v = fmap show m

767− widest = maximum $ fmap length v

801+ v = fmap show m

802+ widest = maximum $ fmap length v

768803 fill str = replicate (widest - length str - 2) ' ' ++ str

769−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++

804+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++

770805 prettyAux t l

771806 where

772− v = fmap show l

773− widest = maximum $ fmap length v

807+ v = fmap show l

808+ widest = maximum $ fmap length v

774809 fill str = replicate (widest - length str - 2) ' ' ++ str

775810

776811-- | Matrix pretty printer

777−pretty :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> String

778−pretty m = "┌ " ++ unwords (replicate (columns m) blank) ++ " ┐\n" ++

779− prettyAux (toLists m) (toLists m) ++

780− "└ " ++ unwords (replicate (columns m) blank) ++ " ┘"

781− where

782− v = fmap show (toList m)

783− widest = maximum $ fmap length v

812+pretty :: (CountableDims cols rows, Show e) => Matrix e cols rows -> String

813+pretty m = concat

814+ [ "┌ ", unwords (replicate (columns m) blank), " ┐\n"

815+ , unlines

816+ [ "│ " ++ unwords (fmap (\j -> fill $ show $ getElem i j m) [1..columns m]) ++ " │" | i <- [1..rows m] ]

817+ , "└ ", unwords (replicate (columns m) blank), " ┘"

818+ ]

819+ where

820+ strings = map show (toList m)

821+ widest = maximum $ map length strings

784822 fill str = replicate (widest - length str) ' ' ++ str

785− blank = fill ""

823+ blank = fill ""

824+ safeGet i j m

825+ | i > rows m || j > columns m || i < 1 || j < 1 = Nothing

826+ | otherwise = Just $ unsafeGet i j m (toList m)

827+ unsafeGet i j m l = l !! encode (columns m) (i,j)

828+ encode m (i,j) = (i-1)*m + j - 1

829+ getElem i j m =

830+ fromMaybe

831+ (error $

832+ "getElem: Trying to get the "

833+ ++ show (i, j)

834+ ++ " element from a "

835+ ++ show (rows m) ++ "x" ++ show (columns m)

836+ ++ " matrix."

837+ )

838+ (safeGet i j m)

786839

787840-- | Matrix pretty printer

788−prettyPrint :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> IO ()

841+prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()

789842prettyPrint = putStrLn . pretty

790843

844+-- | Zip two matrices with a given binary function

845+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows

846+zipWithM f (One a) (One b) = One (f a b)

847+zipWithM f (Join a b) (Join c d) = Join (zipWithM f a c) (zipWithM f b d)

848+zipWithM f (Fork a b) (Fork c d) = Fork (zipWithM f a c) (zipWithM f b d)

849+zipWithM f x@(Fork _ _) y@(Join _ _) = zipWithM f x (abideJF y)

850+zipWithM f x@(Join _ _) y@(Fork _ _) = zipWithM f (abideJF x) y

851+

791852-- Relational operators functions

792853

793−type Boolean = Natural 0 1

854+type Boolean = Natural 0 1

794855type Relation a b = Matrix Boolean a b

795856

796857-- | Helper conversion function

@@ -806,54 +867,47 @@ fromBool False = nat 0

806867

807868-- | Relational negation

808869negateM :: Relation cols rows -> Relation cols rows

809−negateM Empty = Empty

810−negateM (One (Nat 0)) = One (Nat 1)

811−negateM (One (Nat 1)) = One (Nat 0)

812−negateM (Junc a b) = Junc (negateM a) (negateM b)

813−negateM (Split a b) = Split (negateM a) (negateM b)

870+negateM (One (Nat p)) = One (Nat (negate p))

871+negateM (Join a b) = Join (negateM a) (negateM b)

872+negateM (Fork a b) = Fork (negateM a) (negateM b)

814873

815874-- | Relational addition

816875orM :: Relation cols rows -> Relation cols rows -> Relation cols rows

817−orM Empty Empty = Empty

818−orM (One a) (One b) = One (fromBool (toBool a || toBool b))

819−orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)

820−orM (Split a b) (Split c d) = Split (orM a c) (orM b d)

821−orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)

822−orM x@(Junc _ _) y@(Split _ _) = orM (abideJS x) y

876+orM (One a) (One b) = One (fromBool (toBool a || toBool b))

877+orM (Join a b) (Join c d) = Join (orM a c) (orM b d)

878+orM (Fork a b) (Fork c d) = Fork (orM a c) (orM b d)

879+orM x@(Fork _ _) y@(Join _ _) = orM x (abideJF y)

880+orM x@(Join _ _) y@(Fork _ _) = orM (abideJF x) y

823881

824882-- | Relational multiplication

825883andM :: Relation cols rows -> Relation cols rows -> Relation cols rows

826−andM Empty Empty = Empty

827−andM (One a) (One b) = One (fromBool (toBool a && toBool b))

828−andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)

829−andM (Split a b) (Split c d) = Split (andM a c) (andM b d)

830−andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)

831−andM x@(Junc _ _) y@(Split _ _) = andM (abideJS x) y

884+andM (One a) (One b) = One (fromBool (toBool a && toBool b))

885+andM (Join a b) (Join c d) = Join (andM a c) (andM b d)

886+andM (Fork a b) (Fork c d) = Fork (andM a c) (andM b d)

887+andM x@(Fork _ _) y@(Join _ _) = andM x (abideJF y)

888+andM x@(Join _ _) y@(Fork _ _) = andM (abideJF x) y

832889

833890-- | Relational subtraction

834891subM :: Relation cols rows -> Relation cols rows -> Relation cols rows

835−subM Empty Empty = Empty

836−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)

837−subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)

838−subM (Split a b) (Split c d) = Split (subM a c) (subM b d)

839−subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)

840−subM x@(Junc _ _) y@(Split _ _) = subM (abideJS x) y

892+subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)

893+subM (Join a b) (Join c d) = Join (subM a c) (subM b d)

894+subM (Fork a b) (Fork c d) = Fork (subM a c) (subM b d)

895+subM x@(Fork _ _) y@(Join _ _) = subM x (abideJF y)

896+subM x@(Join _ _) y@(Fork _ _) = subM (abideJF x) y

841897

842898-- | Matrix relational composition.

843899compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows

844−compRel Empty Empty = Empty

845−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))

846−compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law

847−compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law

848−compRel c (Junc a b) = Junc (compRel c a) (compRel c b) -- Junc fusion law

900+compRel (One a) (One b) = One (fromBool (toBool a && toBool b))

901+compRel (Join a b) (Fork c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law

902+compRel (Fork a b) c = Fork (compRel a c) (compRel b c) -- Fork fusion law

903+compRel c (Join a b) = Join (compRel c a) (compRel c b) -- Join fusion law

849904

850905-- | Matrix relational right division

851906divR :: Relation b c -> Relation b a -> Relation a c

852−divR Empty Empty = Empty

853907divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a

854−divR (Junc a b) (Junc c d) = andM (divR a c) (divR b d)

855−divR (Split a b) c = Split (divR a c) (divR b c)

856−divR c (Split a b) = Junc (divR c a) (divR c b)

908+divR (Join a b) (Join c d) = andM (divR a c) (divR b d)

909+divR (Fork a b) c = Fork (divR a c) (divR b c)

910+divR c (Fork a b) = Join (divR c a) (divR c b)

857911

858912-- | Matrix relational left division

859913divL :: Relation c b -> Relation a b -> Relation a c

@@ -869,97 +923,52 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)

869923--

870924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of

871925-- types @a@ or @b@ allows.

872−fromFRel ::

926+fromFRel' ::

873927 forall a b cols rows.

874− ( Bounded a,

875− Bounded b,

876− Enum a,

877− Enum b,

878− Eq b,

879− KnownNat (Count cols),

880− KnownNat (Count rows),

881− FromLists (Natural 0 1) rows cols

928+ ( Liftable Boolean a b,

929+ CountableDims cols rows,

930+ FL rows cols

882931 ) =>

883932 (a -> b) ->

884933 Relation cols rows

885−fromFRel f =

886− let minA = minBound @a

887− maxA = maxBound @a

888− minB = minBound @b

889− maxB = maxBound @b

890− ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))

891− rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))

892− elementsA = take ccols [minA .. maxA]

893− elementsB = take rrows [minB .. maxB]

894− combinations = (,) <$> elementsA <*> elementsB

895− combAp = map snd . sort . map (\(a, b) -> if f a == b

896− then ((fromEnum a, fromEnum b), nat 1)

897− else ((fromEnum a, fromEnum b), nat 0)) $ combinations

898− mList = buildList combAp rrows

899− in tr $ fromLists mList

900− where

901− buildList [] _ = []

902− buildList l r = take r l : buildList (drop r l) r

934+fromFRel' = fromF'

903935

904936-- | Lifts functions to relations with dimensions matching @a@ and @b@

905937-- cardinality's.

906−fromFRel' ::

938+fromFRel ::

907939 forall a b.

908− ( Bounded a,

909− Bounded b,

910− Enum a,

911− Enum b,

912− Eq b,

913− KnownNat (Count (Normalize a)),

914− KnownNat (Count (Normalize b)),

915− FromLists (Natural 0 1) (Normalize b) (Normalize a)

940+ ( Liftable Boolean a b,

941+ CountableDimsN a b,

942+ FLN b a

916943 ) =>

917944 (a -> b) ->

918945 Relation (Normalize a) (Normalize b)

919−fromFRel' f =

920− let minA = minBound @a

921− maxA = maxBound @a

922− minB = minBound @b

923− maxB = maxBound @b

924− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))

925− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))

926− elementsA = take ccols [minA .. maxA]

927− elementsB = take rrows [minB .. maxB]

928− combinations = (,) <$> elementsA <*> elementsB

929− combAp = map snd . sort . map (\(a, b) -> if f a == b

930− then ((fromEnum a, fromEnum b), nat 1)

931− else ((fromEnum a, fromEnum b), nat 0)) $ combinations

932− mList = buildList combAp rrows

933− in tr $ fromLists mList

934− where

935− buildList [] _ = []

936− buildList l r = take r l : buildList (drop r l) r

946+fromFRel = fromFRel'

937947

938948-- | Lifts a relation function to a Boolean Matrix

939−toRel ::

949+toRel ::

940950 forall a b.

941951 ( Bounded a,

942952 Bounded b,

943953 Enum a,

944954 Enum b,

945955 Eq b,

946− KnownNat (Count (Normalize a)),

947− KnownNat (Count (Normalize b)),

948− FromLists (Natural 0 1) (Normalize b) (Normalize a)

949− )

956+ CountableDims a b,

957+ FLN b a

958+ )

950959 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)

951960toRel f =

952961 let minA = minBound @a

953962 maxA = maxBound @a

954963 minB = minBound @b

955964 maxB = maxBound @b

956− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))

957− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))

965+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))

966+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))

958967 elementsA = take ccols [minA .. maxA]

959968 elementsB = take rrows [minB .. maxB]

960969 combinations = (,) <$> elementsA <*> elementsB

961− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)

962− then ((fromEnum a, fromEnum b), nat 1)

970+ combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)

971+ then ((fromEnum a, fromEnum b), nat 1)

963972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations

964973 mList = buildList combAp rrows

965974 in tr $ fromLists mList

src/LAoP/Matrix/Nat.hs+194−187

@@ -1,3 +1,4 @@

1+{-# LANGUAGE ConstraintKinds #-}

12{-# LANGUAGE AllowAmbiguousTypes #-}

23{-# LANGUAGE DataKinds #-}

34{-# LANGUAGE DerivingVia #-}

@@ -53,11 +54,22 @@ module LAoP.Matrix.Nat

5354 -- * Type safe matrix representation

5455 Matrix (..),

5556

57+ -- * Constraint type synonyms

58+ Countable,

59+ CountableDims,

60+ CountableN,

61+ CountableNz,

62+ CountableDimsN,

63+ FLN,

64+ FLNz,

65+ Liftable,

66+ TrivialE,

67+ TrivialP,

68+

5669 -- * Primitives

57− empty,

5870 one,

59− junc,

60− split,

71+ join,

72+ fork,

6173

6274 -- * Auxiliary type families

6375 I.FromNat,

@@ -65,11 +77,11 @@ module LAoP.Matrix.Nat

6577 I.Normalize,

6678

6779 -- * Matrix construction and conversion

68− I.FromLists,

80+ I.FL,

6981 fromLists,

7082 toLists,

7183 toList,

72− matrixBuilder,

84+ matrixBuilder',

7385 row,

7486 col,

7587 zeros,

@@ -85,6 +97,10 @@ module LAoP.Matrix.Nat

8597 -- ** Matrix Transposition

8698 tr,

8799

100+ -- ** Scalar multiplication/division of matrices

101+ (.|),

102+ (./),

103+

88104 -- ** Selective operator

89105 select,

90106

@@ -92,16 +108,19 @@ module LAoP.Matrix.Nat

92108 cond,

93109

94110 -- ** Matrix "abiding"

95− abideJS,

96− abideSJ,

111+ abideJF,

112+ abideFJ,

113+

114+ -- ** Zip Matrices

115+ zipWithM,

97116

98117 -- * Biproduct approach

99− -- ** Split

118+ -- ** Fork

100119 (===),

101120 -- *** Projections

102121 p1,

103122 p2,

104− -- ** Junc

123+ -- ** Join

105124 (|||),

106125 -- *** Injections

107126 i1,

@@ -115,11 +134,11 @@ module LAoP.Matrix.Nat

115134 -- | Note that given the restrictions imposed it is not possible to

116135 -- implement the standard type classes present in standard Haskell.

117136 -- *** Matrix pairing projections

118− kp1,

119− kp2,

137+ fstM,

138+ sndM,

120139

121140 -- *** Matrix pairing

122− khatri,

141+ kr,

123142

124143 -- * Matrix composition and lifting

125144

@@ -127,10 +146,10 @@ module LAoP.Matrix.Nat

127146

128147 -- | Note that given the restrictions imposed it is not possible to

129148 -- implement the standard type classes present in standard Haskell.

130− identity,

149+ iden,

131150 comp,

132− fromF,

133151 fromF',

152+ fromF,

134153

135154 -- * Matrix printing

136155 pretty,

@@ -141,107 +160,95 @@ where

141160import Data.Proxy

142161import GHC.TypeLits

143162import Control.DeepSeq

144−import qualified Control.Category as C

145163import qualified LAoP.Matrix.Internal as I

146164

147165newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))

148166 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))

149167

150−-- | It isn't possible to implement the 'id' function so it's

151−-- implementation is 'undefined'. However 'comp' can be and this partial

152−-- class implementation exists just to make the code more readable.

153−--

154−-- Please use 'identity' instead.

155−instance (Num e) => C.Category (Matrix e) where

156− id = undefined

157− (.) = comp

168+-- | Constraint type synonyms to keep the type signatures less convoluted

169+type Countable a = KnownNat (I.Count a)

170+type CountableDims a b = (Countable a, Countable b)

171+type CountableN a = KnownNat (I.Count (I.FromNat a))

172+type CountableNz a = KnownNat (I.Count (I.Normalize a))

173+type CountableDimsN a b = (CountableN a, CountableN b)

174+type FLN e a b = I.FL (I.FromNat a) (I.FromNat b)

175+type FLNz e a b = I.FL (I.Normalize a) (I.Normalize b)

176+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

177+type TrivialE a b = I.FromNat (a + b) ~ Either (I.FromNat a) (I.FromNat b)

178+type TrivialP a b = I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))

158179

159180-- Primitives

160181

161−empty :: Matrix e 0 0

162−empty = M I.Empty

163−

164182one :: e -> Matrix e 1 1

165183one = M . I.One

166184

167−junc ::

168− (I.FromNat cols3 ~ Either (I.FromNat cols1) (I.FromNat cols2)) =>

169− Matrix e cols1 rows ->

170− Matrix e cols2 rows ->

171− Matrix e cols3 rows

172−junc (M a) (M b) = M (I.Junc a b)

185+join ::

186+ (TrivialE a b) =>

187+ Matrix e a rows ->

188+ Matrix e b rows ->

189+ Matrix e (a + b) rows

190+join (M a) (M b) = M (I.Join a b)

173191

174192infixl 3 |||

175193(|||) ::

176− (I.FromNat cols3 ~ Either (I.FromNat cols1) (I.FromNat cols2)) =>

177− Matrix e cols1 rows ->

178− Matrix e cols2 rows ->

179− Matrix e cols3 rows

180−(|||) = junc

181−

182−split ::

183− (I.FromNat rows3 ~ Either (I.FromNat rows1) (I.FromNat rows2)) =>

184− Matrix e cols rows1 ->

185− Matrix e cols rows2 ->

186− Matrix e cols rows3

187−split (M a) (M b) = M (I.Split a b)

194+ (TrivialE a b) =>

195+ Matrix e a rows ->

196+ Matrix e b rows ->

197+ Matrix e (a + b) rows

198+(|||) = join

199+

200+fork ::

201+ (TrivialE a b) =>

202+ Matrix e cols a ->

203+ Matrix e cols b ->

204+ Matrix e cols (a + b)

205+fork (M a) (M b) = M (I.Fork a b)

188206

189207infixl 2 ===

190208(===) ::

191− (I.FromNat rows3 ~ Either (I.FromNat rows1) (I.FromNat rows2)) =>

192− Matrix e cols rows1 ->

193− Matrix e cols rows2 ->

194− Matrix e cols rows3

195−(===) = split

209+ (TrivialE a b) =>

210+ Matrix e cols a ->

211+ Matrix e cols b ->

212+ Matrix e cols (a + b)

213+(===) = fork

196214

197215-- Construction

198216

199−fromLists :: (I.FromLists e (I.FromNat cols) (I.FromNat rows)) => [[e]] -> Matrix e cols rows

217+fromLists :: (FLN e cols rows) => [[e]] -> Matrix e cols rows

200218fromLists = M . I.fromLists

201219

202−matrixBuilder ::

203− (I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>

204− ((Int, Int) -> e) ->

205− Matrix e cols rows

206−matrixBuilder = M . I.matrixBuilder

220+-- | Matrix builder function. Constructs a matrix provided with

221+-- a construction function that operates with indices.

222+matrixBuilder' ::

223+ (FLN e cols rows, CountableN cols, CountableN rows)

224+ => ((Int, Int) -> e) -> Matrix e cols rows

225+matrixBuilder' = M . I.matrixBuilder'

207226

208−col :: (I.FromLists e () (I.FromNat rows)) => [e] -> Matrix e 1 rows

227+col :: (I.FL () (I.FromNat rows)) => [e] -> Matrix e 1 rows

209228col = M . I.col

210229

211−row :: (I.FromLists e (I.FromNat cols) ()) => [e] -> Matrix e cols 1

230+row :: (I.FL (I.FromNat cols) ()) => [e] -> Matrix e cols 1

212231row = M . I.row

213232

214−fromF ::

215− ( Bounded a,

216− Bounded b,

217− Enum a,

218− Enum b,

219− Eq b,

220− Num e,

221− Ord e,

222− KnownNat (I.Count (I.FromNat cols)),

223− KnownNat (I.Count (I.FromNat rows)),

224− I.FromLists e (I.FromNat rows) (I.FromNat cols)

233+fromF' ::

234+ ( Liftable e a b,

235+ CountableN cols,

236+ CountableN rows,

237+ FLN e rows cols

225238 ) =>

226239 (a -> b) ->

227240 Matrix e cols rows

228−fromF = M . I.fromF

241+fromF' = M . I.fromF'

229242

230−fromF' ::

231− ( Bounded a,

232− Bounded b,

233− Enum a,

234− Enum b,

235− Eq b,

236− Num e,

237− Ord e,

238− KnownNat (I.Count (I.Normalize a)),

239− KnownNat (I.Count (I.Normalize b)),

240− I.FromLists e (I.Normalize b) (I.Normalize a)

243+fromF ::

244+ ( Liftable e a b,

245+ CountableNz a,

246+ CountableNz b,

247+ FLNz e b a

241248 ) =>

242249 (a -> b) ->

243250 Matrix e (I.Count a) (I.Count b)

244−fromF' = M . I.fromF'

251+fromF = undefined -- M . I.fromF

245252

246253-- Conversion

247254

@@ -254,21 +261,21 @@ toList (M m) = I.toList m

254261-- Zeros Matrix

255262

256263zeros ::

257− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>

264+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

258265 Matrix e cols rows

259266zeros = M I.zeros

260267

261268-- Ones Matrix

262269

263270ones ::

264− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>

271+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

265272 Matrix e cols rows

266273ones = M I.ones

267274

268275-- Const Matrix

269276

270277constant ::

271− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>

278+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

272279 e ->

273280 Matrix e cols rows

274281constant = M . I.constant

@@ -277,40 +284,52 @@ constant = M . I.constant

277284

278285bang ::

279286 forall e cols.

280− (Num e, Enum e, I.FromLists e (I.FromNat cols) (), KnownNat (I.Count (I.FromNat cols))) =>

287+ (Num e, Enum e, I.FL (I.FromNat cols) (), CountableN cols) =>

281288 Matrix e cols 1

282289bang = M I.bang

283290

284−-- Identity Matrix

291+-- iden Matrix

285292

286−identity ::

287− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat cols), KnownNat (I.Count (I.FromNat cols))) =>

293+iden ::

294+ (Num e, FLN e cols cols, CountableN cols) =>

288295 Matrix e cols cols

289−identity = M I.identity

296+iden = M I.iden

290297

291298-- Matrix composition (MMM)

292299

293300comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

294301comp (M a) (M b) = M (I.comp a b)

295302

303+-- Scalar multiplication of matrices

304+

305+infixl 7 .|

306+-- | Scalar multiplication of matrices.

307+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows

308+(.|) e (M m) = M (e I..| m)

309+

310+-- Scalar division of matrices

311+

312+infixl 7 ./

313+-- | Scalar multiplication of matrices.

314+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows

315+(./) (M m) e = M (m I../ e)

316+

296317p1 ::

297318 ( Num e,

298− KnownNat (I.Count (I.FromNat n)),

299− KnownNat (I.Count (I.FromNat m)),

300− I.FromLists e (I.FromNat n) (I.FromNat m),

301− I.FromLists e (I.FromNat m) (I.FromNat m),

302− I.FromNat (m + n) ~ Either (I.FromNat m) (I.FromNat n)

319+ CountableDimsN n m,

320+ FLN e n m,

321+ FLN e m m,

322+ TrivialE m n

303323 ) =>

304324 Matrix e (m + n) m

305325p1 = M I.p1

306326

307327p2 ::

308328 ( Num e,

309− KnownNat (I.Count (I.FromNat n)),

310− KnownNat (I.Count (I.FromNat m)),

311− I.FromLists e (I.FromNat m) (I.FromNat n),

312− I.FromLists e (I.FromNat n) (I.FromNat n),

313− I.FromNat (m + n) ~ Either (I.FromNat m) (I.FromNat n)

329+ CountableDimsN n m,

330+ FLN e m n,

331+ FLN e n n,

332+ TrivialE m n

314333 ) =>

315334 Matrix e (m + n) n

316335p2 = M I.p2

@@ -319,138 +338,125 @@ p2 = M I.p2

319338

320339i1 ::

321340 ( Num e,

322− KnownNat (I.Count (I.FromNat n)),

323− KnownNat (I.Count (I.FromNat rows)),

324− I.FromLists e (I.FromNat n) (I.FromNat rows),

325− I.FromLists e (I.FromNat rows) (I.FromNat rows),

326− I.FromNat (rows + n) ~ Either (I.FromNat rows) (I.FromNat n)

341+ CountableDimsN n rows,

342+ FLN e n rows,

343+ FLN e rows rows,

344+ TrivialE rows n

327345 ) =>

328346 Matrix e rows (rows + n)

329347i1 = tr p1

330348

331349i2 ::

332350 ( Num e,

333− KnownNat (I.Count (I.FromNat rows)),

334− KnownNat (I.Count (I.FromNat m)),

335− I.FromLists e (I.FromNat m) (I.FromNat rows),

336− I.FromLists e (I.FromNat rows) (I.FromNat rows),

337− I.FromNat (m + rows) ~ Either (I.FromNat m) (I.FromNat rows)

351+ CountableDimsN rows m,

352+ FLN e m rows,

353+ FLN e rows rows,

354+ TrivialE m rows

338355 ) =>

339356 Matrix e rows (m + rows)

340357i2 = tr p2

341358

342359-- Dimensions

343360

344−rows :: (KnownNat (I.Count (I.FromNat rows))) => Matrix e cols rows -> Int

361+rows :: (CountableN rows) => Matrix e cols rows -> Int

345362rows (M m) = I.rows m

346363

347−columns :: (KnownNat (I.Count (I.FromNat cols))) => Matrix e cols rows -> Int

364+columns :: (CountableN cols) => Matrix e cols rows -> Int

348365columns (M m) = I.columns m

349366

350−-- Coproduct Bifunctor

351−

352367infixl 5 -|-

353368

369+-- | Coproduct Bifunctor (Direct sum)

354370(-|-) ::

355371 ( Num e,

356− KnownNat (I.Count (I.FromNat j)),

357− KnownNat (I.Count (I.FromNat k)),

358− I.FromLists e (I.FromNat k) (I.FromNat k),

359− I.FromLists e (I.FromNat j) (I.FromNat k),

360− I.FromLists e (I.FromNat k) (I.FromNat j),

361− I.FromLists e (I.FromNat j) (I.FromNat j),

362− I.FromNat (n + m) ~ Either (I.FromNat n) (I.FromNat m),

363− I.FromNat (k + j) ~ Either (I.FromNat k) (I.FromNat j)

372+ CountableDimsN j k,

373+ FLN e k k,

374+ FLN e j k,

375+ FLN e k j,

376+ FLN e j j,

377+ TrivialE n m,

378+ TrivialE k j

364379 ) =>

365380 Matrix e n k ->

366381 Matrix e m j ->

367382 Matrix e (n + m) (k + j)

368383(-|-) (M a) (M b) = M ((I.-|-) a b)

369384

370−-- Khatri Rao Product and projections

371−

372−kp1 ::

385+-- | Khatri Rao Product first projection

386+fstM ::

373387 forall e m k .

374388 ( Num e,

375− KnownNat (I.Count (I.FromNat m)),

376− KnownNat (I.Count (I.FromNat k)),

377− KnownNat (I.Count (I.FromNat (m * k))),

378− I.FromLists e (I.FromNat (m * k)) (I.FromNat m),

379− I.FromNat (m * k) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat k))

389+ CountableDimsN m k,

390+ CountableN (m * k),

391+ FLN e (m * k) m,

392+ TrivialP m k

380393 ) => Matrix e (m * k) m

381−kp1 = M (I.kp1 @e @(I.FromNat m) @(I.FromNat k))

394+fstM = M (I.fstM @e @(I.FromNat m) @(I.FromNat k))

382395

383−kp2 ::

396+-- | Khatri Rao Product second projection

397+sndM ::

384398 forall e m k.

385399 ( Num e,

386− KnownNat (I.Count (I.FromNat k)),

387− I.FromLists e (I.FromNat (m * k)) (I.FromNat k),

388− KnownNat (I.Count (I.FromNat m)),

389− KnownNat (I.Count (I.FromNat (m * k))),

390− I.FromNat (m * k) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat k))

400+ CountableDimsN k m,

401+ FLN e (m * k) k,

402+ CountableN (m * k),

403+ TrivialP m k

391404 ) => Matrix e (m * k) k

392−kp2 = M (I.kp2 @e @(I.FromNat m) @(I.FromNat k))

405+sndM = M (I.sndM @e @(I.FromNat m) @(I.FromNat k))

393406

394−khatri ::

407+-- | Khatri Rao Product

408+kr ::

395409 forall e cols a b.

396410 ( Num e,

397− KnownNat (I.Count (I.FromNat a)),

398− KnownNat (I.Count (I.FromNat b)),

399− KnownNat (I.Count (I.FromNat (a * b))),

400− I.FromLists e (I.FromNat (a * b)) (I.FromNat a),

401− I.FromLists e (I.FromNat (a * b)) (I.FromNat b),

402− I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))

411+ CountableDimsN a b,

412+ CountableN (a * b),

413+ FLN e (a * b) a,

414+ FLN e (a * b) b,

415+ TrivialP a b

403416 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)

404−khatri a b =

405− let kp1' = kp1 @e @a @b

406− kp2' = kp2 @e @a @b

407− in comp (tr kp1') a * comp (tr kp2') b

408−

409−-- Product Bifunctor (Kronecker)

417+kr a b =

418+ let fstM' = fstM @e @a @b

419+ sndM' = sndM @e @a @b

420+ in comp (tr fstM') a * comp (tr sndM') b

410421

422+-- | Product Bifunctor (Kronecker)

411423infixl 4 ><

412424

413425(><) ::

414426 forall e m p n q.

415427 ( Num e,

416− KnownNat (I.Count (I.FromNat m)),

417− KnownNat (I.Count (I.FromNat n)),

418− KnownNat (I.Count (I.FromNat p)),

419− KnownNat (I.Count (I.FromNat q)),

420− KnownNat (I.Count (I.FromNat (m * n))),

421− KnownNat (I.Count (I.FromNat (p * q))),

422− I.FromLists e (I.FromNat (m * n)) (I.FromNat m),

423− I.FromLists e (I.FromNat (m * n)) (I.FromNat n),

424− I.FromLists e (I.FromNat (p * q)) (I.FromNat p),

425− I.FromLists e (I.FromNat (p * q)) (I.FromNat q),

426− I.FromNat (m * n) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat n)),

427− I.FromNat (p * q) ~ I.FromNat (I.Count (I.FromNat p) * I.Count (I.FromNat q))

428+ CountableDimsN m n,

429+ CountableDimsN p q,

430+ CountableDimsN (m * n) (p * q),

431+ FLN e (m * n) m,

432+ FLN e (m * n) n,

433+ FLN e (p * q) p,

434+ FLN e (p * q) q,

435+ TrivialP m n,

436+ TrivialP p q

428437 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)

429438(><) a b =

430− let kp1' = kp1 @e @m @n

431− kp2' = kp2 @e @m @n

432− in khatri (comp a kp1') (comp b kp2')

433−

434−-- Matrix abide Junc Split

439+ let fstM' = fstM @e @m @n

440+ sndM' = sndM @e @m @n

441+ in kr (comp a fstM') (comp b sndM')

435442

436−abideJS :: Matrix e cols rows -> Matrix e cols rows

437−abideJS (M m) = M (I.abideJS m)

443+-- | Matrix abide Join Fork

444+abideJF :: Matrix e cols rows -> Matrix e cols rows

445+abideJF (M m) = M (I.abideJF m)

438446

439−-- Matrix abide Split Junc

440−

441−abideSJ :: Matrix e cols rows -> Matrix e cols rows

442−abideSJ (M m) = M (I.abideSJ m)

443−

444−-- Matrix transposition

447+-- | Matrix abide Fork Join

448+abideFJ :: Matrix e cols rows -> Matrix e cols rows

449+abideFJ (M m) = M (I.abideFJ m)

445450

451+-- | Matrix transposition

446452tr :: Matrix e cols rows -> Matrix e rows cols

447453tr (M m) = M (I.tr m)

448454

449455-- Selective 'select' operator

450456select ::

451457 ( Num e,

452− I.FromLists e (I.FromNat rows1) (I.FromNat rows1),

453− KnownNat (I.Count (I.FromNat rows1)),

458+ FLN e rows1 rows1,

459+ CountableN rows1,

454460 I.FromNat rows2 ~ I.FromNat rows1,

455461 I.FromNat cols1 ~ I.FromNat cols2,

456462 I.FromNat rows3 ~ Either (I.FromNat cols3) (I.FromNat rows1)

@@ -461,14 +467,11 @@ select (M m) (M y) = M (I.select m y)

461467

462468cond ::

463469 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,

464− KnownNat (I.Count (I.FromNat cols)),

465− I.FromLists e () (I.FromNat cols),

466− I.FromLists e (I.FromNat cols) (),

467− I.FromLists e (I.FromNat cols) (I.FromNat cols),

468− Bounded a,

469− Enum a,

470− Num e,

471− Ord e

470+ CountableN cols,

471+ I.FL () (I.FromNat cols),

472+ I.FL (I.FromNat cols) (),

473+ FLN e cols cols,

474+ Liftable e a Bool

472475 )

473476 =>

474477 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows

@@ -476,8 +479,12 @@ cond p (M a) (M b) = M (I.cond p a b)

476479

477480-- Pretty print

478481

479−pretty :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> String

482+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String

480483pretty (M m) = I.pretty m

481484

482−prettyPrint :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> IO ()

485+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()

483486prettyPrint (M m) = I.prettyPrint m

487+

488+-- | Zip two matrices with a given binary function

489+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows

490+zipWithM f (M a) (M b) = M (I.zipWithM f a b)

src/LAoP/Matrix/Type.hs+283−240

@@ -1,10 +1,11 @@

1+{-# LANGUAGE ConstraintKinds #-}

2+{-# LANGUAGE TypeFamilies #-}

13{-# LANGUAGE AllowAmbiguousTypes #-}

24{-# LANGUAGE DataKinds #-}

35{-# LANGUAGE DerivingVia #-}

46{-# LANGUAGE FlexibleContexts #-}

57{-# LANGUAGE FlexibleInstances #-}

68{-# LANGUAGE GADTs #-}

7−{-# LANGUAGE KindSignatures #-}

89{-# LANGUAGE MultiParamTypeClasses #-}

910{-# LANGUAGE NoStarIsType #-}

1011{-# LANGUAGE RankNTypes #-}

@@ -55,15 +56,24 @@ module LAoP.Matrix.Type

5556 -- * Type safe matrix representation

5657 Matrix (..),

5758

59+ -- * Constraint type aliases

60+ Countable,

61+ CountableDims,

62+ CountableN,

63+ CountableDimsN,

64+ FLN,

65+ Liftable,

66+ Trivial,

67+ TrivialP,

68+

5869 -- * Type aliases

5970 Zero,

6071 One,

6172

6273 -- * Primitives

63− empty,

6474 one,

65− junc,

66− split,

75+ join,

76+ fork,

6777

6878 -- * Auxiliary type families

6979 I.FromNat,

@@ -75,24 +85,27 @@ module LAoP.Matrix.Type

7585 fromLists,

7686 toLists,

7787 toList,

88+ matrixBuilder',

7889 matrixBuilder,

7990 row,

8091 col,

8192 zeros,

8293 ones,

8394 bang,

95+ point,

8496 constant,

8597

8698 -- * Functor instance equivalent function

8799 fmapM,

100+ bimapM,

88101

89102 -- * Applicative/Monoidal instance equivalent functions

90103 unitM,

91104 multM,

92105

93106 -- * Selective equivalent instance function

94− selectM,

95−

107+ selectM,

108+

96109 -- * Monad equivalent instance function

97110 returnM,

98111 bindM,

@@ -100,25 +113,34 @@ module LAoP.Matrix.Type

100113 -- * Misc

101114 -- ** Get dimensions

102115 columns,

116+ columns',

103117 rows,

118+ rows',

104119

105120 -- ** Matrix Transposition

106121 tr,

107122

123+ -- ** Scalar multiplication/division of matrices

124+ (.|),

125+ (./),

126+

108127 -- ** McCarthy's Conditional

109128 cond,

110129

111130 -- ** Matrix "abiding"

112− abideJS,

113− abideSJ,

131+ abideJF,

132+ abideFJ,

133+

134+ -- ** Zip Matrices

135+ zipWithM,

114136

115137 -- * Biproduct approach

116− -- ** Split

138+ -- ** Fork

117139 (===),

118140 -- *** Projections

119141 p1,

120142 p2,

121− -- ** Junc

143+ -- ** Join

122144 (|||),

123145 -- *** Injections

124146 i1,

@@ -132,11 +154,11 @@ module LAoP.Matrix.Type

132154 -- | Note that given the restrictions imposed it is not possible to

133155 -- implement the standard type classes present in standard Haskell.

134156 -- *** Matrix pairing projections

135− kp1,

136− kp2,

157+ fstM,

158+ sndM,

137159

138160 -- *** Matrix pairing

139− khatri,

161+ kr,

140162

141163 -- * Matrix composition and lifting

142164

@@ -144,10 +166,10 @@ module LAoP.Matrix.Type

144166

145167 -- | Note that given the restrictions imposed it is not possible to

146168 -- implement the standard type classes present in standard Haskell.

147− identity,

169+ iden,

148170 comp,

149− fromF,

150171 fromF',

172+ fromF,

151173

152174 -- * Relation

153175 toRel,

@@ -161,23 +183,44 @@ where

161183import Data.Void

162184import Data.Proxy

163185import Data.Kind

164−import GHC.TypeLits

186+import GHC.TypeLits

165187import Control.DeepSeq

166188import LAoP.Utils

167−import qualified Control.Category as C

168189import qualified LAoP.Matrix.Internal as I

190+import Prelude hiding (id, (.))

169191

170192newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))

171193 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))

172194

173−-- | It isn't possible to implement the 'id' function so it's

174−-- implementation is 'undefined'. However 'comp' can be and this partial

175−-- class implementation exists just to make the code more readable.

176−--

177−-- Please use 'identity' instead.

178−instance (Num e) => C.Category (Matrix e) where

179− id = undefined

180− (.) = comp

195+-- | Constraint type synonyms to keep the type signatures less convoluted

196+type Countable a = KnownNat (I.Count a)

197+type CountableDims a b = (Countable a, Countable b)

198+type CountableN a = KnownNat (I.Count (I.Normalize a))

199+type CountableDimsN a b = (CountableN a, CountableN b)

200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)

201+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

202+type Trivial a = I.Normalize (I.Normalize a) ~ I.Normalize (I.Normalize (I.Normalize a))

203+type Trivial2 a = I.Normalize a ~ I.Normalize (I.Normalize a)

204+type Trivial3 a = I.FromNat (I.Count (I.Normalize (I.Normalize a))) ~ I.Normalize (I.Normalize a)

205+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

206+

207+-- | It is possible to implement a constrained version of the category type

208+-- class.

209+instance (Num e) => Category (Matrix e) where

210+ type Object (Matrix e) a = (FLN a a, CountableN a)

211+ id = iden

212+ (.) = comp

213+

214+-- | Bifunctor equivalent function

215+bimapM ::

216+ ( Liftable e a b,

217+ Liftable e c d,

218+ CountableDimsN a c,

219+ CountableDimsN b d,

220+ FLN d c,

221+ FLN b a

222+ ) => (a -> b) -> (c -> d) -> Matrix e a c -> Matrix e b d

223+bimapM f g m = fromF g . m . tr (fromF f)

181224

182225-- | Zero type alias

183226type Zero = Void

@@ -187,92 +230,75 @@ type One = ()

187230

188231-- Primitives

189232

190−-- | Empty matrix constructor

191−empty :: Matrix e Zero Zero

192−empty = M I.Empty

193−

194233-- | Unit matrix constructor

195234one :: e -> Matrix e One One

196235one = M . I.One

197236

198−-- | Matrix 'Junc' constructor

199−junc ::

200− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

237+-- | Matrix 'Join' constructor

238+join ::

201239 Matrix e a rows ->

202240 Matrix e b rows ->

203241 Matrix e (Either a b) rows

204−junc (M a) (M b) = M (I.Junc a b)

242+join (M a) (M b) = M (I.Join a b)

205243

206244infixl 3 |||

207−-- | Matrix 'Junc' constructor

245+-- | Matrix 'Join' constructor

208246(|||) ::

209− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

210247 Matrix e a rows ->

211248 Matrix e b rows ->

212249 Matrix e (Either a b) rows

213−(|||) = junc

250+(|||) = join

214251

215−-- | Matrix 'Split' constructor

216−split ::

217− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

252+-- | Matrix 'Fork' constructor

253+fork ::

218254 Matrix e cols a ->

219255 Matrix e cols b ->

220256 Matrix e cols (Either a b)

221−split (M a) (M b) = M (I.Split a b)

257+fork (M a) (M b) = M (I.Fork a b)

222258

223259infixl 2 ===

224−-- | Matrix 'Split' constructor

260+-- | Matrix 'Fork' constructor

225261(===) ::

226− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

227262 Matrix e cols a ->

228263 Matrix e cols b ->

229264 Matrix e cols (Either a b)

230−(===) = split

265+(===) = fork

231266

232267-- Functor hierarchy

233268

234269-- | Functor instance equivalent function

235−fmapM ::

236− ( Bounded a,

237− Bounded b,

238− Enum a,

239− Enum b,

240− Eq b,

241− Num e,

242− Ord e,

243− KnownNat (I.Count (I.Normalize a)),

244− KnownNat (I.Count (I.Normalize b)),

245− I.FromLists e (I.Normalize b) (I.Normalize a)

270+fmapM ::

271+ ( Liftable e a b,

272+ CountableDimsN a b,

273+ FLN b a

246274 )

247275 =>

248276 (a -> b) -> Matrix e c a -> Matrix e c b

249−fmapM f m = fromF' f `comp` m

277+fmapM f m = fromF f . m

250278

251279-- | Applicative instance equivalent 'unit' function,

252280unitM :: (Num e) => Matrix e () ()

253281unitM = one 1

254282

255283-- | Applicative instance equivalent 'unit' function,

256−multM ::

257− ( KnownNat (I.Count (I.Normalize a)),

258− KnownNat (I.Count (I.Normalize b)),

259− KnownNat (I.Count (I.Normalize (a, b))),

284+multM ::

285+ ( CountableDimsN a b,

286+ CountableN (a, b),

260287 Num e,

261− I.FromLists e (I.Normalize (a, b)) (I.Normalize a),

262− I.FromLists e (I.Normalize (a, b)) (I.Normalize b),

263− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

288+ FLN (a, b) a,

289+ FLN (a, b) b,

290+ TrivialP a b

264291 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)

265−multM a b = khatri a b

266−

292+multM = kr

267293

268294-- | Monad instance equivalent 'return' function,

269−returnM ::

270− forall e a .

271− ( Num e,

272− Enum e,

273− Enum a,

274− I.FromLists e () (I.Normalize a),

275− KnownNat (I.Count a)

295+returnM ::

296+ forall e a .

297+ ( Num e,

298+ Enum e,

299+ Enum a,

300+ FLN () a,

301+ Countable a

276302 ) => a -> Matrix e One a

277303returnM a = col l

278304 where

@@ -288,73 +314,64 @@ bindM = flip comp

288314

289315-- | Build a matrix out of a list of list of elements. Throws a runtime

290316-- error if the dimensions do not match.

291−fromLists :: (I.FromLists e (I.Normalize cols) (I.Normalize rows)) => [[e]] -> Matrix e cols rows

317+fromLists :: (FLN cols rows) => [[e]] -> Matrix e cols rows

292318fromLists = M . I.fromLists

293319

294320-- | Matrix builder function. Constructs a matrix provided with

295−-- a construction function.

321+-- a construction function that operates with indices.

322+matrixBuilder' ::

323+ (FLN cols rows, CountableDimsN cols rows )

324+ => ((Int, Int) -> e) -> Matrix e cols rows

325+matrixBuilder' = M . I.matrixBuilder'

326+

327+-- | Matrix builder function. Constructs a matrix provided with

328+-- a construction function that operates with arbitrary types.

296329matrixBuilder ::

297− (I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>

298− ((Int, Int) -> e) ->

299− Matrix e cols rows

300−matrixBuilder = M . I.matrixBuilder

330+ ( FLN a b,

331+ Enum a,

332+ Enum b,

333+ Bounded a,

334+ Bounded b,

335+ Eq a,

336+ Countable b

337+ ) => ((a, b) -> e) -> Matrix e a b

338+matrixBuilder f = M (I.matrixBuilder f)

301339

302340-- | Constructs a column vector matrix

303−col :: (I.FromLists e () (I.Normalize rows)) => [e] -> Matrix e One rows

341+col :: (FLN () rows) => [e] -> Matrix e One rows

304342col = M . I.col

305343

306344-- | Constructs a row vector matrix

307−row :: (I.FromLists e (I.Normalize cols) ()) => [e] -> Matrix e cols One

345+row :: (FLN cols ()) => [e] -> Matrix e cols One

308346row = M . I.row

309347

310348-- | Lifts functions to matrices with arbitrary dimensions.

311349--

312350-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

313351-- types @a@ or @b@ allows.

314−fromF ::

315− ( Bounded a,

316− Bounded b,

317− Enum a,

318− Enum b,

319− Eq b,

320− Num e,

321− Ord e,

322− KnownNat (I.Count (I.Normalize cols)),

323− KnownNat (I.Count (I.Normalize rows)),

324− I.FromLists e (I.Normalize rows) (I.Normalize cols)

352+fromF' ::

353+ ( Liftable e a b,

354+ CountableDimsN cols rows,

355+ FLN rows cols

325356 ) =>

326− (a -> b) ->

327− Matrix e cols rows

328−fromF = M . I.fromF

357+ (a -> b) -> Matrix e cols rows

358+fromF' = M . I.fromF'

329359

330360-- | Lifts functions to matrices with dimensions matching @a@ and @b@

331361-- cardinality's.

332−fromF' ::

333− ( Bounded a,

334− Bounded b,

335− Enum a,

336− Enum b,

337− Eq b,

338− Num e,

339− Ord e,

340− KnownNat (I.Count (I.Normalize a)),

341− KnownNat (I.Count (I.Normalize b)),

342− I.FromLists e (I.Normalize b) (I.Normalize a)

362+fromF ::

363+ ( Liftable e a b,

364+ CountableDimsN a b,

365+ FLN b a

343366 ) =>

344− (a -> b) ->

345− Matrix e a b

346−fromF' = M . I.fromF'

367+ (a -> b) -> Matrix e a b

368+fromF = M . I.fromF

347369

348370-- | Lifts relation functions to Boolean Matrix

349371toRel ::

350− ( Bounded a,

351− Bounded b,

352− Enum a,

353− Enum b,

354− Eq b,

355− KnownNat (I.Count (I.Normalize a)),

356− KnownNat (I.Count (I.Normalize b)),

357− I.FromLists (Natural 0 1) (I.Normalize b) (I.Normalize a)

372+ ( Liftable (Natural 0 1) a b,

373+ CountableDims a b,

374+ FLN b a

358375 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b

359376toRel = M . I.toRel

360377

@@ -372,8 +389,8 @@ toList (M m) = I.toList m

372389

373390-- | The zero matrix. A matrix wholly filled with zeros.

374391zeros ::

375− (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>

376− Matrix e cols rows

392+ (Num e, FLN cols rows, CountableDimsN cols rows)

393+ => Matrix e cols rows

377394zeros = M I.zeros

378395

379396-- Ones Matrix

@@ -382,8 +399,8 @@ zeros = M I.zeros

382399--

383400-- Also known as T (Top) matrix.

384401ones ::

385− (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>

386− Matrix e cols rows

402+ (Num e, FLN cols rows, CountableDimsN cols rows)

403+ => Matrix e cols rows

387404ones = M I.ones

388405

389406-- Const Matrix

@@ -391,9 +408,8 @@ ones = M I.ones

391408-- | The constant matrix constructor. A matrix wholly filled with a given

392409-- value.

393410constant ::

394− (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>

395− e ->

396− Matrix e cols rows

411+ (Num e, FLN cols rows, CountableDimsN cols rows)

412+ => e -> Matrix e cols rows

397413constant = M . I.constant

398414

399415-- Bang Matrix

@@ -401,17 +417,30 @@ constant = M . I.constant

401417-- | The T (Top) row vector matrix.

402418bang ::

403419 forall e cols.

404− (Num e, Enum e, I.FromLists e (I.Normalize cols) (), KnownNat (I.Count (I.Normalize cols))) =>

420+ (Num e, Enum e, FLN cols (), CountableN cols) =>

405421 Matrix e cols One

406422bang = M I.bang

407423

408−-- Identity Matrix

424+-- | Point constant relation

425+point ::

426+ ( Bounded a,

427+ Enum a,

428+ Eq a,

429+ Num e,

430+ Ord e,

431+ CountableN a,

432+ FLN a One

433+ ) => a -> Matrix e One a

434+point = fromF . const

435+

436+-- iden Matrix

409437

410−-- | Identity matrix

411−identity ::

412− (Num e, I.FromLists e (I.Normalize a) (I.Normalize a), KnownNat (I.Count (I.Normalize a))) =>

438+-- | iden matrix

439+iden ::

440+ (Num e, FLN a a, CountableN a) =>

413441 Matrix e a a

414−identity = M I.identity

442+iden = M I.iden

443+{-# NOINLINE iden #-}

415444

416445-- Matrix composition (MMM)

417446

@@ -421,15 +450,32 @@ identity = M I.identity

421450-- from LAoP.

422451comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

423452comp (M a) (M b) = M (I.comp a b)

453+{-# NOINLINE comp #-}

454+{-# RULES

455+ "comp/iden1" forall m. comp m iden = m ;

456+ "comp/iden2" forall m. comp iden m = m

457+#-}

458+

459+-- Scalar multiplication of matrices

460+

461+infixl 7 .|

462+-- | Scalar multiplication of matrices.

463+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows

464+(.|) e (M m) = M (e I..| m)

465+

466+-- Scalar division of matrices

467+

468+infixl 7 ./

469+-- | Scalar multiplication of matrices.

470+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows

471+(./) (M m) e = M (m I../ e)

424472

425473-- | Biproduct first component projection

426474p1 ::

427475 ( Num e,

428− KnownNat (I.Count (I.Normalize n)),

429− KnownNat (I.Count (I.Normalize m)),

430− I.FromLists e (I.Normalize n) (I.Normalize m),

431− I.FromLists e (I.Normalize m) (I.Normalize m),

432− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)

476+ CountableDimsN n m,

477+ FLN n m,

478+ FLN m m

433479 ) =>

434480 Matrix e (Either m n) m

435481p1 = M I.p1

@@ -437,11 +483,9 @@ p1 = M I.p1

437483-- | Biproduct second component projection

438484p2 ::

439485 ( Num e,

440− KnownNat (I.Count (I.Normalize n)),

441− KnownNat (I.Count (I.Normalize m)),

442− I.FromLists e (I.Normalize m) (I.Normalize n),

443− I.FromLists e (I.Normalize n) (I.Normalize n),

444− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)

486+ CountableDimsN n m,

487+ FLN m n,

488+ FLN n n

445489 ) =>

446490 Matrix e (Either m n) n

447491p2 = M I.p2

@@ -451,11 +495,9 @@ p2 = M I.p2

451495-- | Biproduct first component injection

452496i1 ::

453497 ( Num e,

454− KnownNat (I.Count (I.Normalize n)),

455− KnownNat (I.Count (I.Normalize m)),

456− I.FromLists e (I.Normalize n) (I.Normalize m),

457− I.FromLists e (I.Normalize m) (I.Normalize m),

458− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)

498+ CountableDimsN n m,

499+ FLN n m,

500+ FLN m m

459501 ) =>

460502 Matrix e m (Either m n)

461503i1 = tr p1

@@ -463,11 +505,9 @@ i1 = tr p1

463505-- | Biproduct second component injection

464506i2 ::

465507 ( Num e,

466− KnownNat (I.Count (I.Normalize n)),

467− KnownNat (I.Count (I.Normalize m)),

468− I.FromLists e (I.Normalize m) (I.Normalize n),

469− I.FromLists e (I.Normalize n) (I.Normalize n),

470− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)

508+ CountableDimsN n m,

509+ FLN m n,

510+ FLN n n

471511 ) =>

472512 Matrix e n (Either m n)

473513i2 = tr p2

@@ -476,24 +516,34 @@ i2 = tr p2

476516

477517-- | Obtain the number of rows.

478518--

479−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

480−-- dimensions in constant time.

481−--

482−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

483−-- exchange for performance.

484−rows :: (KnownNat (I.Count (I.Normalize rows))) => Matrix e cols rows -> Int

519+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

520+-- dimensions in constant time. For a version that doesn't require the

521+-- constraint see 'rows''.

522+rows :: (CountableN rows) => Matrix e cols rows -> Int

485523rows (M m) = I.rows m

486524

525+-- | Obtain the number of rows in an inefficient manner, but without any

526+-- constraints.

527+--

528+-- For a more efficient version see 'rows'.

529+rows' :: Matrix e cols rows -> Int

530+rows' (M m) = I.rows' m

531+

487532-- | Obtain the number of columns.

488−--

489−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

490−-- dimensions in constant time.

491533--

492−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

493−-- exchange for performance.

494−columns :: (KnownNat (I.Count (I.Normalize cols))) => Matrix e cols rows -> Int

534+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the

535+-- dimensions in constant time. For a version that doesn't require the

536+-- constraint see 'columns''.

537+columns :: (CountableN cols) => Matrix e cols rows -> Int

495538columns (M m) = I.columns m

496539

540+-- | Obtain the number of columns in an inefficient manner, but without any

541+-- constraints.

542+--

543+-- For a more efficient version see 'columns'.

544+columns' :: Matrix e cols rows -> Int

545+columns' (M m) = I.columns' m

546+

497547-- Coproduct Bifunctor

498548

499549infixl 5 -|-

@@ -501,14 +551,11 @@ infixl 5 -|-

501551-- | Matrix coproduct functor also known as matrix direct sum.

502552(-|-) ::

503553 ( Num e,

504− KnownNat (I.Count (I.Normalize j)),

505− KnownNat (I.Count (I.Normalize k)),

506− I.FromLists e (I.Normalize k) (I.Normalize k),

507− I.FromLists e (I.Normalize j) (I.Normalize k),

508− I.FromLists e (I.Normalize k) (I.Normalize j),

509− I.FromLists e (I.Normalize j) (I.Normalize j),

510− I.Normalize (Either n m) ~ Either (I.Normalize n) (I.Normalize m),

511− I.Normalize (Either k j) ~ Either (I.Normalize k) (I.Normalize j)

554+ CountableDimsN j k,

555+ FLN k k,

556+ FLN j k,

557+ FLN k j,

558+ FLN j j

512559 ) =>

513560 Matrix e n k ->

514561 Matrix e m j ->

@@ -518,104 +565,98 @@ infixl 5 -|-

518565-- Khatri Rao Product and projections

519566

520567-- | Khatri Rao product first component projection matrix.

521−kp1 ::

568+fstM ::

522569 forall e m k .

523570 ( Num e,

524− KnownNat (I.Count (I.Normalize m)),

525− KnownNat (I.Count (I.Normalize k)),

526− KnownNat (I.Count (I.Normalize (m, k))),

527− I.FromLists e (I.Normalize (m, k)) (I.Normalize m),

528− I.Normalize (m, k) ~ I.Normalize (I.Normalize m, I.Normalize k)

571+ CountableDimsN m k,

572+ CountableN (m, k),

573+ FLN (m, k) m,

574+ TrivialP m k

529575 ) => Matrix e (m, k) m

530−kp1 = M (I.kp1 @e @(I.Normalize m) @(I.Normalize k))

576+fstM = M (I.fstM @e @(I.Normalize m) @(I.Normalize k))

531577

532578-- | Khatri Rao product second component projection matrix.

533−kp2 ::

579+sndM ::

534580 forall e m k.

535581 ( Num e,

536− KnownNat (I.Count (I.Normalize k)),

537− KnownNat (I.Count (I.Normalize m)),

538− KnownNat (I.Count (I.Normalize (m, k))),

539− I.FromLists e (I.Normalize (m, k)) (I.Normalize k),

540− I.Normalize (m, k) ~ I.Normalize (I.Normalize m, I.Normalize k)

582+ CountableDimsN k m,

583+ CountableN (m, k),

584+ FLN (m, k) k,

585+ TrivialP m k

541586 ) => Matrix e (m, k) k

542−kp2 = M (I.kp2 @e @(I.Normalize m) @(I.Normalize k))

587+sndM = M (I.sndM @e @(I.Normalize m) @(I.Normalize k))

543588

544589-- | Khatri Rao Matrix product also known as matrix pairing.

545590--

546591-- NOTE: That this is not a true categorical product, see for instance:

547−--

592+--

548593-- @

549−-- | kp1 `comp` khatri a b == a

550−-- khatri a b ==> |

551−-- | kp2 `comp` khatri a b == b

594+-- | fstM . kr a b == a

595+-- kr a b ==> |

596+-- | sndM . kr a b == b

552597-- @

553598--

554599-- __Emphasis__ on the implication symbol.

555−khatri ::

600+kr ::

556601 forall e cols a b.

557602 ( Num e,

558− KnownNat (I.Count (I.Normalize a)),

559− KnownNat (I.Count (I.Normalize b)),

560− KnownNat (I.Count (I.Normalize (a, b))),

561− I.FromLists e (I.Normalize (a, b)) (I.Normalize a),

562− I.FromLists e (I.Normalize (a, b)) (I.Normalize b),

563− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

603+ CountableDimsN a b,

604+ CountableN (a, b),

605+ FLN (a, b) a,

606+ FLN (a, b) b,

607+ TrivialP a b

564608 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)

565−khatri a b =

566− let kp1' = kp1 @e @a @b

567− kp2' = kp2 @e @a @b

568− in comp (tr kp1') a * comp (tr kp2') b

609+kr a b =

610+ let fstM' = fstM @e @a @b

611+ sndM' = sndM @e @a @b

612+ in comp (tr fstM') a * comp (tr sndM') b

569613

570614-- Product Bifunctor (Kronecker)

571615

572616infixl 4 ><

573617

574−-- | Matrix product functor also known as kronecker product

618+-- | Matrix product functor also known as Kronecker product

575619(><) ::

576620 forall e m p n q.

577621 ( Num e,

578− KnownNat (I.Count (I.Normalize m)),

579− KnownNat (I.Count (I.Normalize n)),

580− KnownNat (I.Count (I.Normalize p)),

581− KnownNat (I.Count (I.Normalize q)),

582− KnownNat (I.Count (I.Normalize (m, n))),

583− KnownNat (I.Count (I.Normalize (p, q))),

584− I.FromLists e (I.Normalize (m, n)) (I.Normalize m),

585− I.FromLists e (I.Normalize (m, n)) (I.Normalize n),

586− I.FromLists e (I.Normalize (p, q)) (I.Normalize p),

587− I.FromLists e (I.Normalize (p, q)) (I.Normalize q),

588− I.Normalize (m, n) ~ I.Normalize (I.Normalize m, I.Normalize n),

589− I.Normalize (p, q) ~ I.Normalize (I.Normalize p, I.Normalize q)

622+ CountableDimsN m n,

623+ CountableDimsN p q,

624+ CountableDimsN (m, n) (p, q),

625+ FLN (m, n) m,

626+ FLN (m, n) n,

627+ FLN (p, q) p,

628+ FLN (p, q) q,

629+ TrivialP m n,

630+ TrivialP p q

590631 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)

591632(><) a b =

592− let kp1' = kp1 @e @m @n

593− kp2' = kp2 @e @m @n

594− in khatri (comp a kp1') (comp b kp2')

633+ let fstM' = fstM @e @m @n

634+ sndM' = sndM @e @m @n

635+ in kr (comp a fstM') (comp b sndM')

595636

596−-- Matrix abide Junc Split

637+-- Matrix abide Join Fork

597638

598−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

599−--

639+-- | Matrix "abiding" followin the 'Join'-'Fork' abide law.

640+--

600641-- Law:

601642--

602643-- @

603−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

644+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)

604645-- @

605−abideJS :: Matrix e cols rows -> Matrix e cols rows

606−abideJS (M m) = M (I.abideJS m)

646+abideJF :: Matrix e cols rows -> Matrix e cols rows

647+abideJF (M m) = M (I.abideJF m)

607648

608−-- Matrix abide Split Junc

649+-- Matrix abide Fork Join

609650

610−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

611−--

651+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.

652+--

612653-- Law:

613654--

614655-- @

615−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

656+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)

616657-- @

617−abideSJ :: Matrix e cols rows -> Matrix e cols rows

618−abideSJ (M m) = M (I.abideSJ m)

658+abideFJ :: Matrix e cols rows -> Matrix e cols rows

659+abideFJ (M m) = M (I.abideFJ m)

619660

620661-- Matrix transposition

621662

@@ -627,11 +668,10 @@ tr (M m) = M (I.tr m)

627668

628669-- | Selective functors 'select' operator equivalent inspired by the

629670-- ArrowMonad solution presented in the paper.

630−selectM ::

671+selectM ::

631672 ( Num e,

632− I.FromLists e (I.Normalize b) (I.Normalize b),

633− KnownNat (I.Count (I.Normalize b)),

634− I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)

673+ FLN b b,

674+ CountableN b

635675 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

636676selectM (M m) (M y) = M (I.select m y)

637677

@@ -639,15 +679,14 @@ selectM (M m) (M y) = M (I.select m y)

639679

640680-- | McCarthy's Conditional expresses probabilistic choice.

641681cond ::

642− ( I.Normalize (I.Normalize a) ~ I.Normalize a,

643− KnownNat (I.Count (I.Normalize a)),

644− I.FromLists e () (I.Normalize a),

645− I.FromLists e (I.Normalize a) (),

646− I.FromLists e (I.Normalize a) (I.Normalize a),

647− Bounded a,

648− Enum a,

649− Num e,

650− Ord e

682+ ( Trivial a,

683+ Trivial2 a,

684+ Trivial3 a,

685+ CountableN a,

686+ FLN () a,

687+ FLN a (),

688+ FLN a a,

689+ Liftable e a Bool

651690 )

652691 =>

653692 (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b

@@ -656,9 +695,13 @@ cond p (M a) (M b) = M (I.cond p a b)

656695-- Pretty print

657696

658697-- | Matrix pretty printer

659−pretty :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> String

698+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String

660699pretty (M m) = I.pretty m

661700

662701-- | Matrix pretty printer

663−prettyPrint :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> IO ()

702+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()

664703prettyPrint (M m) = I.prettyPrint m

704+

705+-- | Zip two matrices with a given binary function

706+zipWithM :: (e -> f -> g) -> Matrix e a b -> Matrix f a b -> Matrix g a b

707+zipWithM f (M a) (M b) = M (I.zipWithM f a b)

src/LAoP/Relation.hs+34−7

@@ -30,12 +30,21 @@ module LAoP.Relation

3030 Relation (..),

3131 Boolean,

3232

33+ -- * Constraint type synonyms

34+ Countable,

35+ CountableDims,

36+ CountableN,

37+ CountableDimsN,

38+ FLN,

39+ Liftable,

40+ Trivial,

41+ TrivialP,

42+

3343 -- * Primitives

34− empty,

3544 one,

36− junc,

45+ join,

3746 (|||),

38− split,

47+ fork,

3948 (===),

4049

4150 -- * Auxiliary type families

@@ -53,6 +62,7 @@ module LAoP.Relation

5362 toList,

5463 toBool,

5564 pt,

65+ belongs,

5666 relationBuilder,

5767 zeros,

5868 ones,

@@ -80,15 +90,25 @@ module LAoP.Relation

8090 injection,

8191 surjection,

8292 bijection,

93+ domain,

94+ range,

8395

8496 -- * Function division

8597 divisionF,

8698

99+ -- * Relation division

100+ divR,

101+ divL,

102+ divS,

103+ shrunkBy,

104+ overriddenBy,

105+

87106 -- * Relational pairing

88107 splitR,

108+

89109 -- ** Projections

90− p1,

91− p2,

110+ fstR,

111+ sndR,

92112 -- ** Bifunctor

93113 (><),

94114 -- * Relational coproduct

@@ -116,18 +136,25 @@ module LAoP.Relation

116136 linearOrder,

117137 equivalence,

118138 partialEquivalence,

139+ difunctional,

140+

141+ -- * Conditionals

142+ equalizer,

119143

120144 -- ** McCarthy's Conditional

121− -- cond,

145+ predR,

146+ guard,

147+ cond,

122148

123149 -- * Relational composition and lifting

124− identity,

150+ iden,

125151 comp,

126152 fromF,

127153 fromF',

128154

129155 -- ** Relational application

130156 pointAp,

157+ pointApBool,

131158

132159 -- * Matrix printing

133160 pretty,

src/LAoP/Relation/Internal.hs+486−317

@@ -1,3 +1,5 @@

1+{-# LANGUAGE TypeFamilies #-}

2+{-# LANGUAGE ConstraintKinds #-}

13{-# LANGUAGE GADTs #-}

24{-# LANGUAGE UndecidableInstances #-}

35{-# LANGUAGE FlexibleContexts #-}

@@ -39,12 +41,21 @@ module LAoP.Relation.Internal

3941 Relation (..),

4042 Boolean,

4143

44+ -- * Constraint type synonyms

45+ Countable,

46+ CountableDims,

47+ CountableN,

48+ CountableDimsN,

49+ FLN,

50+ Liftable,

51+ Trivial,

52+ TrivialP,

53+

4254 -- * Primitives

43− empty,

4455 one,

45− junc,

56+ join,

4657 (|||),

47− split,

58+ fork,

4859 (===),

4960

5061 -- * Auxiliary type families

@@ -55,13 +66,15 @@ module LAoP.Relation.Internal

5566 -- * Matrix construction and conversion

5667 I.FromLists,

5768 fromLists,

58− fromF,

5969 fromF',

70+ fromF,

6071 toRel,

6172 toLists,

6273 toList,

6374 toBool,

6475 pt,

76+ belongs,

77+ relationBuilder',

6578 relationBuilder,

6679 zeros,

6780 ones,

@@ -89,6 +102,8 @@ module LAoP.Relation.Internal

89102 injection,

90103 surjection,

91104 bijection,

105+ domain,

106+ range,

92107

93108 -- * Function division

94109 divisionF,

@@ -104,8 +119,8 @@ module LAoP.Relation.Internal

104119 splitR,

105120

106121 -- ** Projections

107− p1,

108− p2,

122+ fstR,

123+ sndR,

109124 -- ** Bifunctor

110125 (><),

111126 -- * Relational coproduct

@@ -133,6 +148,7 @@ module LAoP.Relation.Internal

133148 linearOrder,

134149 equivalence,

135150 partialEquivalence,

151+ difunctional,

136152

137153 -- * Conditionals

138154 equalizer,

@@ -143,10 +159,10 @@ module LAoP.Relation.Internal

143159 cond,

144160

145161 -- * Relational composition and lifting

146− identity,

162+ iden,

147163 comp,

148− fromF,

149164 fromF',

165+ fromF,

150166

151167 -- ** Relational application

152168 pointAp,

@@ -162,12 +178,13 @@ import Data.Void

162178import qualified LAoP.Matrix.Internal as I

163179import LAoP.Utils.Internal

164180import Control.DeepSeq

165−import qualified Control.Category as C

166181import Data.Bool

167182import GHC.TypeLits

183+import Prelude hiding (id, (.))

168184

169185-- | Boolean type synonym for working with boolean matrices

170186type Boolean = Natural 0 1

187+type Powerset a = List (List a)

171188

172189-- | Relation data type.

173190newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))

@@ -175,25 +192,34 @@ newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))

175192

176193deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)

177194

178−-- | It isn't possible to implement the 'id' function so it's

179−-- implementation is 'undefined'. However 'comp' can be and this partial

180−-- class implementation exists just to make the code more readable.

181−--

182−-- Please use 'identity' instead.

183−instance C.Category Relation where

184− id = undefined

185− (.) = comp

195+-- | Constraint type synonyms to keep the type signatures less convoluted

196+type Countable a = KnownNat (I.Count a)

197+type CountableDims a b = (Countable a, Countable b)

198+type CountableN a = KnownNat (I.Count (I.Normalize a))

199+type CountableDimsN a b = (CountableN a, CountableN b)

200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)

201+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Boolean, Ord Boolean)

202+type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)

203+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

204+

205+-- | It is possible to implement a constrained version of the category type

206+-- class.

207+instance Category Relation where

208+ type Object Relation a = (FLN a a, CountableN a)

209+ id = iden

210+ (.) = comp

186211

187212instance Num (Relation a b) where

188− -- | Matrix addition becomes Boolean matrix disjunction

213+ -- | Matrix addition becomes Boolean matrix disjointion

189214 (R a) + (R b) = R (I.orM a b)

190215

191216 -- | Matrix subtraction becomes Relational subtraction

192217 (R a) - (R b) = R (I.subM a b)

193218

194− -- | Matrix multiplication becomes Boolean matrix conjunction

219+ -- | Matrix multiplication becomes Boolean matrix conjointion

195220 (R a) * (R b) = R (I.andM a b)

196221

222+ -- | Matrix negation becomes Boolean matrix negation

197223 negate (R a) = R (I.negateM a)

198224

199225-- Type alias

@@ -202,105 +228,107 @@ type One = ()

202228

203229-- Primitives

204230

205−-- | Empty matrix constructor

206−empty :: Relation Zero Zero

207−empty = R I.Empty

208−

209231-- | Unit matrix constructor

210232one :: Boolean -> Relation One One

211233one = R . I.One

212234

213−-- | Boolean Matrix 'Junc' constructor, also known as relational coproduct.

235+-- | Boolean Matrix 'Join' constructor, also known as relational coproduct.

214236--

215237-- See 'eitherR'.

216−junc :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))

217− => Relation a c -> Relation b c -> Relation (Either a b) c

218−junc (R a) (R b) = R (I.Junc a b)

238+join :: Relation a c -> Relation b c -> Relation (Either a b) c

239+join (R a) (R b) = R (I.Join a b)

219240

220241infixl 3 |||

221−-- | Boolean Matrix 'Junc' constructor

242+-- | Boolean Matrix 'Join' constructor

243+--

244+-- See 'eitherR'.

222245(|||) ::

223− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

224246 Relation a c ->

225247 Relation b c ->

226248 Relation (Either a b) c

227−(|||) = junc

249+(|||) = join

228250

229−-- | Boolean Matrix 'Split' constructor, also known as relational product.

230−split :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))

231− => Relation c a -> Relation c b -> Relation c (Either a b)

232−split (R a) (R b) = R (I.Split a b)

251+-- | Boolean Matrix 'Fork' constructor, also known as relational product.

252+fork :: Relation c a -> Relation c b -> Relation c (Either a b)

253+fork (R a) (R b) = R (I.Fork a b)

233254

234255infixl 2 ===

235−-- | Boolean Matrix 'Split' constructor

256+-- | Boolean Matrix 'Fork' constructor

236257(===) ::

237− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

238258 Relation c a ->

239259 Relation c b ->

240260 Relation c (Either a b)

241−(===) = split

261+(===) = fork

242262

243263-- Construction

244264

245265-- | Build a matrix out of a list of list of elements. Throws a runtime

246266-- error if the dimensions do not match.

247−fromLists :: (I.FromLists Boolean (I.Normalize a) (I.Normalize b)) => [[Boolean]] -> Relation a b

267+fromLists :: (FLN a b) => [[Boolean]] -> Relation a b

248268fromLists = R . I.fromLists

249269

250−-- | Matrix builder function. Constructs a matrix provided with

251−-- a construction function.

270+-- | Relation builder function. Constructs a relation provided with

271+-- a construction function that operates with indices.

272+relationBuilder' ::

273+ (FLN a b, CountableDimsN a b) =>

274+ ((Int, Int) -> Boolean) -> Relation a b

275+relationBuilder' = R . I.matrixBuilder'

276+

277+-- | Relation builder function. Constructs a relation provided with

278+-- a construction function that operates with arbitrary types.

252279relationBuilder ::

253− (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>

254− ((Int, Int) -> Boolean) ->

255− Relation a b

280+ ( FLN a b,

281+ Enum a,

282+ Enum b,

283+ Bounded a,

284+ Bounded b,

285+ Eq a,

286+ CountableDims a b

287+ ) => ((a, b) -> Boolean) -> Relation a b

256288relationBuilder = R . I.matrixBuilder

257289

258290-- | Lifts functions to matrices with arbitrary dimensions.

259291--

260292-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

261293-- types @a@ or @b@ allows.

262−fromF ::

263− ( Bounded a,

264− Bounded b,

265− Enum a,

266− Enum b,

267− Eq b,

268− KnownNat (I.Count (I.Normalize c)),

269− KnownNat (I.Count (I.Normalize d)),

270− I.FromLists Boolean (I.Normalize d) (I.Normalize c)

294+fromF' ::

295+ ( Liftable a b,

296+ CountableDimsN c d,

297+ FLN d c

271298 )

272299 => (a -> b) -> Relation c d

273−fromF f = R (I.fromFRel f)

300+fromF' f = R (I.fromFRel' f)

274301

275302-- | Lifts functions to matrices with dimensions matching @a@ and @b@

276303-- cardinality's.

277−fromF' ::

278− ( Bounded b,

279− Bounded a,

280− Enum a,

281− Enum b,

282− Eq b,

283− KnownNat (I.Count (I.Normalize a)),

284− KnownNat (I.Count (I.Normalize b)),

285− I.FromLists Boolean (I.Normalize b) (I.Normalize a)

304+fromF ::

305+ ( Liftable a b,

306+ CountableDimsN a b,

307+ FLN b a

286308 )

287309 => (a -> b) -> Relation a b

288−fromF' f = R (I.fromFRel' f)

310+fromF f = R (I.fromFRel f)

289311

290312-- | Lifts relation functions to 'Relation'

291313toRel ::

292− ( Bounded b,

293− Bounded a,

294− Enum a,

295− Enum b,

296− Eq b,

297− KnownNat (I.Count (I.Normalize a)),

298− KnownNat (I.Count (I.Normalize b)),

299− I.FromLists Boolean (I.Normalize b) (I.Normalize a)

314+ ( Liftable a b,

315+ CountableDims a b,

316+ FLN b a

300317 )

301− => (a -> b -> Bool) -> Relation a b

318+ => (a -> b -> Bool) -> Relation a b

302319toRel = R . I.toRel

303320

321+-- | Lowers a 'Relation' to a function

322+fromRel ::

323+ ( Liftable a b,

324+ Eq a,

325+ CountableDimsN a b,

326+ FLN a One,

327+ FLN b One

328+ )

329+ => Relation a b -> (a -> b -> Bool)

330+fromRel r a b = pointApBool a b r

331+

304332-- Conversion

305333

306334-- | Converts a matrix to a list of lists of elements.

@@ -311,93 +339,135 @@ toLists (R m) = I.toLists m

311339toList :: Relation a b -> [Boolean]

312340toList (R m) = I.toList m

313341

314−-- | Converts a well typed @'Relation'@ to @'Bool'@.

342+-- | Converts a well typed 'Relation' to 'Bool'.

315343toBool :: Relation One One -> Bool

316344toBool r = case toList r of

317345 [Nat 0] -> False

318346 _ -> True

319347

320−-- | Power transpose

321−pt ::

322− ( Bounded a,

323− Bounded b,

324− Enum a,

325− Enum b,

348+-- | Power transpose.

349+--

350+-- Maps a relation to a set valued function.

351+pt ::

352+ ( Liftable a b,

326353 Eq a,

327− Eq b,

328− KnownNat (I.Count (I.Normalize a)),

329− KnownNat (I.Count (I.Normalize b)),

330− I.FromLists Boolean (I.Normalize a) One,

331− I.FromLists Boolean (I.Normalize b) One

354+ CountableDimsN a b,

355+ FLN a One,

356+ FLN b One

332357 )

333− => Relation a b -> (a -> [b])

358+ => Relation a b -> (a -> List b)

334359pt r a =

335− let lb = [minBound .. maxBound]

336− in [ b | b <- lb, toBool (pointAp a b r) ]

360+ let (L lb) = maxBound

361+ in L [ b | b <- lb, toBool (pointAp a b r) ]

362+

363+-- | Belongs relation

364+belongs ::

365+ ( Bounded a,

366+ Enum a,

367+ Eq a,

368+ CountableDims (List a) a,

369+ FLN a (List a)

370+ )

371+ => Relation (List a) a

372+belongs = toRel elemR

373+ where

374+ elemR (L l) x = x `elem` l

337375

338376-- Zeros Matrix

339377

340−-- | The zero matrix. A matrix wholly filled with zeros.

378+-- | The zero relation. A relation where no element of type @a@ relates

379+-- with elements of type @b@.

380+--

381+-- Also known as ⊥ (Bottom) Relation.

382+--

383+-- @

384+-- r `.` ⊥ == ⊥ `.` r == ⊥

385+-- ⊥ ``sse`` R && R ``sse`` T == True

386+-- @

341387zeros ::

342− (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>

388+ (FLN a b, CountableDimsN a b) =>

343389 Relation a b

344−zeros = relationBuilder (const (nat 0))

390+zeros = relationBuilder' (const (nat 0))

345391

346392-- Ones Matrix

347393

348−-- | The ones matrix. A matrix wholly filled with ones.

394+-- | The ones relation. A relation where every element of type @a@ relates

395+-- with every element of type @b@.

396+--

397+-- Also known as T (Top) Relation or universal Relation.

349398--

350−-- Also known as T (Top) matrix.

399+-- @

400+-- ⊥ ``sse`` R && R ``sse`` T == True

401+-- @

351402ones ::

352− (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>

403+ (FLN a b, CountableDimsN a b) =>

353404 Relation a b

354−ones = relationBuilder (const (nat 1))

405+ones = relationBuilder' (const (nat 1))

355406

356407-- Bang Matrix

357408

358−-- | The T (Top) row vector matrix.

409+-- | The T (Top) row vector relation.

359410bang ::

360− (I.FromLists Boolean (I.Normalize a) One, KnownNat (I.Count (I.Normalize a))) =>

411+ (FLN a One, CountableN a) =>

361412 Relation a One

362413bang = ones

363414

364415-- | Point constant relation

365−point ::

416+point ::

366417 ( Bounded a,

367418 Enum a,

368419 Eq a,

369− KnownNat (I.Count (I.Normalize a)),

370− I.FromLists Boolean (I.Normalize a) One

420+ CountableN a,

421+ FLN a One

371422 ) => a -> Relation One a

372−point = fromF' . const

423+point = fromF . const

373424

374−-- Identity Matrix

425+-- iden Matrix

375426

376−-- | Identity matrix

377−identity ::

378− (I.FromLists Boolean (I.Normalize a) (I.Normalize a), KnownNat (I.Count (I.Normalize a))) =>

379− Relation a a

380−identity = relationBuilder (bool (nat 0) (nat 1) . uncurry (==))

427+-- | iden matrix

428+--

429+-- @

430+-- 'iden' `.` r == r == r `.` 'iden'

431+-- @

432+iden ::

433+ (FLN a a, CountableN a) => Relation a a

434+iden = relationBuilder' (bool (nat 0) (nat 1) . uncurry (==))

381435

382436-- | Relational composition

437+--

438+-- @

439+-- r `.` (s `.` p) = (r `.` s) `.` p

440+-- @

383441comp :: Relation b c -> Relation a b -> Relation a c

384442comp (R a) (R b) = R (I.compRel a b)

385443

386444-- | Relational right division

445+--

446+-- @'divR' x y@ is the largest relation @z@ which,

447+-- pre-composed with @y@, approximates @x@.

387448divR :: Relation b c -> Relation b a -> Relation a c

388449divR (R x) (R y) = R (I.divR x y)

389450

390451-- | Relational left division

452+--

453+-- The dual division operator:

454+--

455+-- @

456+-- 'divL' y x == 'conv' ('divR' ('conv' x) ('conv' y)

457+-- @

391458divL :: Relation c b -> Relation a b -> Relation a c

392459divL (R x) (R y) = R (I.divL x y)

393460

394461-- | Relational symmetric division

462+--

463+-- @'pointAp' c b ('divS' s r)@ means that @b@ and @c@

464+-- are related to exactly the same outputs by @r@ and by @s@.

395465divS :: Relation c a -> Relation b a -> Relation c b

396466divS (R x) (R y) = R (I.divS x y)

397467

398468-- | Relational shrinking.

399469--

400−-- @r `'shrunkBy'` s@ is the largest part of @r@ such that,

470+-- @r ``shrunkBy`` s@ is the largest part of @r@ such that,

401471-- if it yields an output for an input @x@, it must be a maximum,

402472-- with respect to @s@, among all possible outputs of @x@ by @r@.

403473shrunkBy :: Relation b a -> Relation a a -> Relation b a

@@ -405,51 +475,52 @@ shrunkBy r s = r `intersection` divR s (conv r)

405475

406476-- | Relational overriding.

407477--

408−-- @r `'overriddenBy'` s@ yields the relation which contains the

478+-- @r ``overriddenBy`` s@ yields the relation which contains the

409479-- whole of @s@ and that part of @r@ where @s@ is undefined.

410480--

411481-- @

412−-- 'zeros' `'overriddenBy'` s == s

413−-- r `'overriddenBy'` 'zeros' == r

414−-- r `'overriddenBy'` r == r

482+-- 'zeros' ``overriddenBy`` s == s

483+-- r ``overriddenBy`` 'zeros' == r

484+-- r ``overriddenBy`` r == r

415485-- @

416486overriddenBy ::

417− ( I.FromLists Boolean (I.Normalize b) (I.Normalize b),

418− KnownNat (I.Count (I.Normalize b))

487+ ( FLN b b,

488+ CountableN b

419489 ) => Relation a b -> Relation a b -> Relation a b

420490overriddenBy r s = s `union` r `intersection` divR zeros (conv s)

421491

422−-- | Relational application

492+-- | Relational application.

493+--

494+-- If @a@ and @b@ are related by 'Relation' @r@

495+-- then @'pointAp' a b r == 'one' ('nat' 1)@

423496pointAp ::

424− ( Bounded a,

425− Bounded b,

426− Enum a,

427− Enum b,

497+ ( Liftable a b,

428498 Eq a,

429− Eq b,

430− KnownNat (I.Count (I.Normalize a)),

431− KnownNat (I.Count (I.Normalize b)),

432− I.FromLists Boolean (I.Normalize a) One,

433− I.FromLists Boolean (I.Normalize b) One

499+ CountableDimsN a b,

500+ FLN a One,

501+ FLN b One

434502 ) => a -> b -> Relation a b -> Relation One One

435−pointAp a b r = conv (point b) `comp` r `comp` point a

503+pointAp a b r = conv (point b) . r . point a

436504

437505-- | Relational application

506+--

507+-- The same as 'pointAp' but converts 'Boolean' to 'Bool'

438508pointApBool ::

439− ( Bounded a,

440− Bounded b,

441− Enum a,

442− Enum b,

509+ ( Liftable a b,

443510 Eq a,

444− Eq b,

445− KnownNat (I.Count (I.Normalize a)),

446− KnownNat (I.Count (I.Normalize b)),

447− I.FromLists Boolean (I.Normalize a) One,

448− I.FromLists Boolean (I.Normalize b) One

511+ CountableDimsN a b,

512+ FLN a One,

513+ FLN b One

449514 ) => a -> b -> Relation a b -> Bool

450−pointApBool a b r = toBool $ conv (point b) `comp` r `comp` point a

515+pointApBool a b r = toBool $ conv (point b) . r . point a

451516

452517-- | Relational converse

518+--

519+-- Given binary 'Relation' r, writing @'pointAp' a b r@

520+-- (read: “@b@ is related to @a@ by @r@”) means the same as

521+-- @'pointAp' b a ('conv' r)@, where @'conv' r@ is said to be

522+-- the converse of @r@.

523+-- In terms of grammar, @'conv' r@ corresponds to the passive voice

453524conv :: Relation a b -> Relation b a

454525conv (R a) = R (I.tr a)

455526

@@ -467,165 +538,187 @@ iff r s = r == s

467538

468539-- | Relational intersection

469540--

470−-- Lifts pointwise conjunction.

541+-- Lifts pointwise conjointion.

542+--

543+-- @

544+-- (r ``intersection`` s) ``intersection`` t == r ``intersection`` (s ``intersection`` t)

545+-- x ``sse`` r ``intersection`` s == x ``intersection`` r && x ``intersection`` s

546+-- @

471547intersection :: Relation a b -> Relation a b -> Relation a b

472548intersection a b = a * b

473549

474550-- | Relational union

475551--

476−-- Lifts pointwise disjunction.

552+-- Lifts pointwise disjointion.

553+--

554+-- @

555+-- (r ``union`` s) ``union`` t == r `'union' (s ``union`` t)

556+-- r ``union`` s ``sse`` x == r ``sse`` x && s ``sse`` x

557+-- r `.` (s ``union`` t) == (r `.` s) ``union`` (r `.` t)

558+-- (s ``union`` t) `.` r == (s `.` r) ``union`` (t `.` r)

559+-- @

477560union :: Relation a b -> Relation a b -> Relation a b

478561union a b = a + b

479562

480563-- | Relation Kernel

564+--

565+-- @

566+-- 'ker' r == 'conv' r `.` r

567+-- 'ker' r == 'img' ('conv' r)

568+-- @

481569ker :: Relation a b -> Relation a a

482−ker r = conv r `comp` r

570+ker r = conv r . r

483571

484572-- | Relation Image

573+--

574+-- @

575+-- 'img' r == r `.` conv r

576+-- 'img' r == 'ker' ('conv' r)

577+-- @

485578img :: Relation a b -> Relation b b

486−img r = r `comp` conv r

579+img r = r . conv r

487580

488−-- | Function division.

581+-- | Function division. Special case of 'divS'.

489582--

490583-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and

491584-- 'entire'.

492585--

493−-- @division f g == conv g `'comp'` f@

586+-- @'divisionF' f g == 'conv' g `.` f@

494587divisionF :: Relation a c -> Relation b c -> Relation a b

495−divisionF f g = conv g `comp` f

588+divisionF f g = conv g . f

496589

497590-- Taxonomy of binary relations

498591

499−-- | A 'Relation' @r@ is simple iff @'coreflexive' ('img' r)@

500−simple :: (KnownNat (I.Count (I.Normalize b)), I.FromLists Boolean (I.Normalize b) (I.Normalize b)) => Relation a b -> Bool

592+-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@

593+simple :: (CountableN b, FLN b b) => Relation a b -> Bool

501594simple = coreflexive . img

502595

503−-- | A 'Relation' @r@ is injective iff @'coreflexive' ('ker' r)@

504−injective :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a b -> Bool

596+-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@

597+injective :: (CountableN a, FLN a a) => Relation a b -> Bool

505598injective = coreflexive . ker

506599

507−-- | A 'Relation' @r@ is entire iff @'reflexive' ('ker' r)@

508−entire :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a b -> Bool

600+-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@

601+entire :: (CountableN a, FLN a a) => Relation a b -> Bool

509602entire = reflexive . ker

510603

511−-- | A 'Relation' @r@ is surjective iff @'reflexive' ('img' r)@

512−surjective :: (KnownNat (I.Count (I.Normalize b)), I.FromLists Boolean (I.Normalize b) (I.Normalize b)) => Relation a b -> Bool

604+-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@

605+surjective :: (CountableN b, FLN b b) => Relation a b -> Bool

513606surjective = reflexive . img

514607

515−-- | A 'Relation' @r@ is a function iff @'simple' r && 'entire' r@

608+-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@

516609--

517610-- A 'function' @f@ enjoys the following properties, where @r@ and @s@ are binary

518611-- relations:

519612--

520613-- @

521−-- f `'comp'` r `'sse'` s == r `'sse'` f `'comp'` s

522−-- r `'comp'` f `'sse'` s == r `'sse'` s `'comp'` f

614+-- f `.` r ``sse`` s == r ``sse`` f `.` s

615+-- r `.` f ``sse`` s == r ``sse`` s `.` f

523616-- @

524617function ::

525− ( KnownNat (I.Count (I.Normalize b)),

526− KnownNat (I.Count (I.Normalize a)),

527− I.FromLists Boolean (I.Normalize a) (I.Normalize a),

528− I.FromLists Boolean (I.Normalize b) (I.Normalize b)

618+ ( CountableDimsN a b,

619+ FLN a a,

620+ FLN b b

529621 )

530622 => Relation a b -> Bool

531623function r = simple r && entire r

532624

533−-- | A 'Relation' @r@ is a representation iff @'injective' r && 'entire' r@

625+-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@

534626representation ::

535− ( KnownNat (I.Count (I.Normalize a)),

536− I.FromLists Boolean (I.Normalize a) (I.Normalize a)

627+ ( CountableN a,

628+ FLN a a

537629 )

538630 => Relation a b -> Bool

539631representation r = injective r && entire r

540632

541−-- | A 'Relation' @r@ is an abstraction iff @'surjective' r && 'simple' r@

633+-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@

542634abstraction ::

543− ( KnownNat (I.Count (I.Normalize b)),

544− I.FromLists Boolean (I.Normalize b) (I.Normalize b)

635+ ( CountableN b,

636+ FLN b b

545637 )

546638 => Relation a b -> Bool

547639abstraction r = surjective r && simple r

548640

549−-- | A 'Relation' @r@ is a surjection iff @'function' r && 'abstraction' r@

641+-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@

550642surjection ::

551− ( KnownNat (I.Count (I.Normalize b)),

552− KnownNat (I.Count (I.Normalize a)),

553− I.FromLists Boolean (I.Normalize a) (I.Normalize a),

554− I.FromLists Boolean (I.Normalize b) (I.Normalize b)

643+ ( CountableDimsN a b,

644+ FLN a a,

645+ FLN b b

555646 )

556647 => Relation a b -> Bool

557648surjection r = function r && abstraction r

558649

559−-- | A 'Relation' @r@ is a injection iff @'function' r && 'representation' r@

650+-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@

560651injection ::

561− ( KnownNat (I.Count (I.Normalize b)),

562− KnownNat (I.Count (I.Normalize a)),

563− I.FromLists Boolean (I.Normalize a) (I.Normalize a),

564− I.FromLists Boolean (I.Normalize b) (I.Normalize b)

652+ ( CountableDimsN a b,

653+ FLN a a,

654+ FLN b b

565655 )

566656 => Relation a b -> Bool

567657injection r = function r && representation r

568658

569−-- | A 'Relation' @r@ is an bijection iff @'injection' r && 'surjection' r@

659+-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@

570660bijection ::

571− ( KnownNat (I.Count (I.Normalize b)),

572− KnownNat (I.Count (I.Normalize a)),

573− I.FromLists Boolean (I.Normalize b) (I.Normalize b),

574− I.FromLists Boolean (I.Normalize a) (I.Normalize a)

661+ ( CountableDimsN a b,

662+ FLN b b,

663+ FLN a a

575664 ) => Relation a b -> Bool

576665bijection r = injection r && surjection r

577666

578−

579667-- Properties of relations

580668

581−-- | A 'Relation' @r@ is reflexive iff @'identity' `'sse'` r@

582−reflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

583−reflexive r = identity <= r

669+-- | A 'Relation' @r@ is 'reflexive' 'iff' @'id' ``sse`` r@

670+reflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

671+reflexive r = id <= r

584672

585−-- | A 'Relation' @r@ is coreflexive iff @r `'sse'` 'identity'@

586−coreflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

587−coreflexive r = r <= identity

673+-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'id'@

674+coreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

675+coreflexive r = r <= id

588676

589−-- | A 'Relation' @r@ is transitive iff @(r `'comp'` r) `'sse'` r@

677+-- | A 'Relation' @r@ is 'transitive' 'iff' @(r `.` r) ``sse`` r@

590678transitive :: Relation a a -> Bool

591−transitive r = (r `comp` r) `sse` r

679+transitive r = (r . r) `sse` r

592680

593−-- | A 'Relation' @r@ is symmetric iff @r == conv r@

681+-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@

594682symmetric :: Relation a a -> Bool

595683symmetric r = r == conv r

596684

597−-- | A 'Relation' @r@ is anti-symmetric iff @(r `'intersection'` 'conv' r) `'sse'` 'identity'@

598−antiSymmetric :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

599−antiSymmetric r = (r `intersection` conv r) `sse` identity

685+-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'id'@

686+antiSymmetric :: (CountableN a, FLN a a) => Relation a a -> Bool

687+antiSymmetric r = (r `intersection` conv r) `sse` id

600688

601−-- | A 'Relation' @r@ is irreflexive iff @(r `'intersection'` identity) == 'zeros'@

602−irreflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

603−irreflexive r = (r `intersection` identity) == zeros

689+-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'id') == 'zeros'@

690+irreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

691+irreflexive r = (r `intersection` id) == zeros

604692

605−-- | A 'Relation' @r@ is connected iff @(r `'union'` 'conv' r) == 'ones'@

606−connected :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

693+-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@

694+connected :: (CountableN a, FLN a a) => Relation a a -> Bool

607695connected r = (r `union` conv r) == ones

608696

609−-- | A 'Relation @r@ is a preorder iff @'reflexive' r && 'transitive' r@

610−preorder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

697+-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@

698+preorder :: (CountableN a, FLN a a) => Relation a a -> Bool

611699preorder r = reflexive r && transitive r

612700

613−-- | A 'Relation @r@ is a partial order iff @'antiSymmetric' r && 'preorder' r@

614−partialOrder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

701+-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@

702+partialOrder :: (CountableN a, FLN a a) => Relation a a -> Bool

615703partialOrder r = antiSymmetric r && preorder r

616704

617−-- | A 'Relation @r@ is a linear order iff @'connected' r && 'partialOrder' r@

618−linearOrder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

705+-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@

706+linearOrder :: (CountableN a, FLN a a) => Relation a a -> Bool

619707linearOrder r = connected r && partialOrder r

620708

621−-- | A 'Relation @r@ is an equivalence iff @'symmetric' r && 'preorder' r@

622−equivalence :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

709+-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@

710+equivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

623711equivalence r = symmetric r && preorder r

624712

625−-- | A 'Relation @r@ is a partial equivalence iff @'partialOrder' r && 'equivalence' r@

626−partialEquivalence :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool

713+-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@

714+partialEquivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

627715partialEquivalence r = partialOrder r && equivalence r

628716

717+-- | A 'Relation' @r@ is 'difunctional' or regular wherever

718+-- @r `.` 'conv' r `.` r == r@

719+difunctional :: Relation a b -> Bool

720+difunctional r = r . conv r . r == r

721+

629722-- Relational pairing

630723

631724-- | Relational pairing.

@@ -633,64 +726,81 @@ partialEquivalence r = partialOrder r && equivalence r

633726-- NOTE: That this is not a true categorical product, see for instance:

634727--

635728-- @

636−-- | 'p1' `'comp'` 'splitR' a b `'sse'` a

637−-- 'splitR' a b <=> |

638−-- | 'p2' `'comp'` 'splitR' a b `'sse'` b

729+-- | 'fstR' `.` 'splitR' a b ``sse`` a

730+-- 'splitR' a b <=> |

731+-- | 'sndR' `.` 'splitR' a b ``sse`` b

639732-- @

640733--

641734-- __Emphasis__ on the 'sse'.

735+--

736+-- @

737+-- 'splitR' r s `.` f == 'splitR' (r `.` f) (s `.` f)

738+-- (R '><' S) `.` 'splitR' p q == 'splitR' (r `.` p) (s `.` q)

739+-- 'conv' ('splitR' r s) `.` 'splitR' x y == ('conv' r `.` x) ``intersection`` ('conv' s `.` y)

740+-- @

741+--

742+-- @

743+-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)

744+-- @

642745splitR ::

643− ( KnownNat (I.Count (I.Normalize a)),

644− KnownNat (I.Count (I.Normalize b)),

645− KnownNat (I.Count (I.Normalize (a, b))),

646− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize a),

647− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize b),

648− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

746+ ( CountableDimsN a b,

747+ CountableN (a, b),

748+ FLN (a, b) a,

749+ FLN (a, b) b,

750+ TrivialP a b

649751 )

650752 => Relation c a -> Relation c b -> Relation c (a, b)

651−splitR (R f) (R g) = R (I.khatri f g)

753+splitR (R f) (R g) = R (I.kr f g)

652754

653755-- | Relational pairing first component projection

654−p1 ::

756+--

757+-- @

758+-- 'fstR' `.` 'splitR' r s ``sse`` r

759+-- @

760+fstR ::

655761 forall a b .

656− ( KnownNat (I.Count (I.Normalize a)),

657− KnownNat (I.Count (I.Normalize b)),

658− KnownNat (I.Count (I.Normalize (a, b))),

659− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize a),

660− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

762+ ( CountableDimsN a b,

763+ CountableN (a, b),

764+ FLN (a, b) a,

765+ TrivialP a b

661766 )

662767 => Relation (a, b) a

663−p1 = R (I.kp1 @Boolean @(I.Normalize a) @(I.Normalize b))

768+fstR = R (I.fstM @Boolean @(I.Normalize a) @(I.Normalize b))

664769

665770-- | Relational pairing second component projection

666−p2 ::

771+--

772+-- @

773+-- 'sndR' `.` 'splitR' r s ``sse`` s

774+-- @

775+sndR ::

667776 forall a b .

668− ( KnownNat (I.Count (I.Normalize b)),

669− KnownNat (I.Count (I.Normalize a)),

670− KnownNat (I.Count (I.Normalize (a, b))),

671− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize b),

672− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

777+ ( CountableDimsN a b,

778+ CountableN (a, b),

779+ FLN (a, b) b,

780+ TrivialP a b

673781 )

674782 => Relation (a, b) b

675−p2 = R (I.kp2 @Boolean @(I.Normalize a) @(I.Normalize b))

783+sndR = R (I.sndM @Boolean @(I.Normalize a) @(I.Normalize b))

676784

677785-- Relational pairing functor

678786

679787infixl 4 ><

680788-- | Relational pairing functor

789+--

790+-- @

791+-- r '><' s == 'splitR' (r `.` fstR) (s `.` sndR)

792+-- (r '><' s) `.` (p '><' q) == (r `.` p) '><' (s `.` q)

793+-- @

681794(><) ::

682− ( KnownNat (I.Count (I.Normalize a)),

683− KnownNat (I.Count (I.Normalize b)),

684− KnownNat (I.Count (I.Normalize c)),

685− KnownNat (I.Count (I.Normalize d)),

686− KnownNat (I.Count (I.Normalize (a, c))),

687− KnownNat (I.Count (I.Normalize (b, d))),

688− I.FromLists Boolean (I.Normalize (a, c)) (I.Normalize a),

689− I.FromLists Boolean (I.Normalize (a, c)) (I.Normalize c),

690− I.FromLists Boolean (I.Normalize (b, d)) (I.Normalize b),

691− I.FromLists Boolean (I.Normalize (b, d)) (I.Normalize d),

692− I.Normalize (a, c) ~ I.Normalize (I.Normalize a, I.Normalize c),

693− I.Normalize (b, d) ~ I.Normalize (I.Normalize b, I.Normalize d)

795+ ( CountableDimsN a b,

796+ CountableDimsN c d,

797+ CountableDimsN (a, c) (b, d),

798+ FLN (a, c) a,

799+ FLN (a, c) c,

800+ FLN (b, d) b,

801+ FLN (b, d) d,

802+ TrivialP a c,

803+ TrivialP b d

694804 )

695805 => Relation a b -> Relation c d -> Relation (a, c) (b, d)

696806(><) (R a) (R b) = R ((I.><) a b)

@@ -698,28 +808,47 @@ infixl 4 ><

698808-- Relational co-products

699809

700810-- | Relational coproduct.

701−eitherR :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))

702− => Relation a c -> Relation b c -> Relation (Either a b) c

703−eitherR = junc

811+--

812+-- @

813+-- | 'eitherR' a b `.` 'i1' == a

814+-- 'eitherR' a b <=> |

815+-- | 'eitherR' a b `.` 'i2' == b

816+-- @

817+--

818+-- @

819+-- 'eitherR' r s `.` 'conv' ('eitherR' t u) == (r `.` 'conv' t) ``union`` (s `.` 'conv' u)

820+-- @

821+--

822+-- @

823+-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)

824+-- @

825+eitherR :: Relation a c -> Relation b c -> Relation (Either a b) c

826+eitherR = join

704827

705828-- | Relational coproduct first component injection

829+--

830+-- @

831+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

832+-- 'i1' `.` 'i2' = 'zeros'

833+-- @

706834i1 ::

707− ( KnownNat (I.Count (I.Normalize b)),

708− KnownNat (I.Count (I.Normalize a)),

709− I.FromLists Boolean (I.Normalize b) (I.Normalize a),

710− I.FromLists Boolean (I.Normalize a) (I.Normalize a),

711− I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)

835+ ( CountableDimsN a b,

836+ FLN b a,

837+ FLN a a

712838 )

713839 => Relation a (Either a b)

714840i1 = R I.i1

715841

716842-- | Relational coproduct second component injection

843+--

844+-- @

845+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

846+-- 'i1' `.` 'i2' = 'zeros'

847+-- @

717848i2 ::

718− ( KnownNat (I.Count (I.Normalize b)),

719− KnownNat (I.Count (I.Normalize a)),

720− I.FromLists Boolean (I.Normalize a) (I.Normalize b),

721− I.FromLists Boolean (I.Normalize b) (I.Normalize b),

722− I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)

849+ ( CountableDimsN a b,

850+ FLN a b,

851+ FLN b b

723852 )

724853 => Relation b (Either a b)

725854i2 = R I.i2

@@ -727,15 +856,16 @@ i2 = R I.i2

727856infixl 5 -|-

728857

729858-- | Relational coproduct functor.

859+--

860+-- @

861+-- r '-|-' s == 'eitherR' ('i1' `.` r) ('i2' `.` s)

862+-- @

730863(-|-) ::

731− ( KnownNat (I.Count (I.Normalize d)),

732− KnownNat (I.Count (I.Normalize b)),

733− I.FromLists Boolean (I.Normalize b) (I.Normalize b),

734− I.FromLists Boolean (I.Normalize d) (I.Normalize b),

735− I.FromLists Boolean (I.Normalize b) (I.Normalize d),

736− I.FromLists Boolean (I.Normalize d) (I.Normalize d),

737− I.Normalize (Either a c) ~ Either (I.Normalize a) (I.Normalize c),

738− I.Normalize (Either b d) ~ Either (I.Normalize b) (I.Normalize d)

864+ ( CountableDimsN b d,

865+ FLN b b,

866+ FLN d b,

867+ FLN b d,

868+ FLN d d

739869 )

740870 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)

741871(-|-) (R a) (R b) = R ((I.-|-) a b)

@@ -743,97 +873,136 @@ infixl 5 -|-

743873-- Relational "Currying"

744874

745875-- | Relational 'trans'

876+--

877+-- Every n-ary relation can be expressed as a binary relation through

878+-- 'trans'/'untrans';

879+-- more-over, where each particular attribute is placed (input/output) is irrelevant.

746880trans ::

747− ( KnownNat (I.Count (I.Normalize (a, b))),

748− KnownNat (I.Count (I.Normalize a)),

749− KnownNat (I.Count (I.Normalize (c, b))),

750− KnownNat (I.Count (I.Normalize b)),

751− KnownNat (I.Count (I.Normalize c)),

752− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count c)),

753− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count b)),

754− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count b)),

755− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count a)),

756− I.Normalize (a, b) ~ I.Normalize (I.Normalize (a, b)),

757− I.Normalize (c, b) ~ I.Normalize (I.Normalize (c, b)),

758− I.Normalize (I.Normalize a, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (a, b))),

759− I.Normalize (I.Normalize c, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (c, b)))

881+ ( CountableDimsN a b,

882+ CountableN c,

883+ CountableDimsN (a, b) (c, b),

884+ FLN (c, b) c,

885+ FLN (c, b) b,

886+ FLN (a, b) a,

887+ FLN (a, b) b,

888+ Trivial (a, b),

889+ Trivial (c, b),

890+ TrivialP a b,

891+ TrivialP c b

760892 )

761893 => Relation (a, b) c -> Relation a (c, b)

762−trans r = splitR r p2 `comp` conv p1

894+trans r = splitR r sndR . conv fstR

763895

764896-- | Relational 'untrans'

897+--

898+-- Every n-ary relation can be expressed as a binary relation through

899+-- 'trans'/'untrans';

900+-- more-over, where each particular attribute is placed (input/output) is irrelevant.

765901untrans ::

766− ( KnownNat (I.Count (I.Normalize (a, b))),

767− KnownNat (I.Count (I.Normalize a)),

768− KnownNat (I.Count (I.Normalize (c, b))),

769− KnownNat (I.Count (I.Normalize b)),

770− KnownNat (I.Count (I.Normalize c)),

771− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count c)),

772− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count b)),

773− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count b)),

774− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count a)),

775− I.Normalize (a, b) ~ I.Normalize (I.Normalize (a, b)),

776− I.Normalize (c, b) ~ I.Normalize (I.Normalize (c, b)),

777− I.Normalize (I.Normalize a, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (a, b))),

778− I.Normalize (I.Normalize c, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (c, b)))

902+ ( CountableDimsN a b,

903+ CountableN c,

904+ CountableDimsN (a, b) (c, b),

905+ FLN (c, b) c,

906+ FLN (c, b) b,

907+ FLN (a, b) b,

908+ FLN (a, b) a,

909+ Trivial (a, b),

910+ Trivial (c, b),

911+ TrivialP a b,

912+ TrivialP c b

779913 )

780914 => Relation a (c, b) -> Relation (a, b) c

781−untrans s = p1 `comp` conv (splitR (conv s) p2)

915+untrans s = fstR . conv (splitR (conv s) sndR)

782916

783917-- | Transforms predicate @p@ into a correflexive relation.

918+--

919+-- @

920+-- 'predR' ('const' True) == 'id'

921+-- 'predR' ('const' False) == 'zeros'

922+-- @

923+--

924+-- @

925+-- 'predR' q `.` 'predR' p == 'predR' q ``union`` 'predR' p

926+-- @

784927predR ::

785928 ( Bounded a,

786929 Enum a,

787− KnownNat (I.Count (I.Normalize a)),

788− I.FromLists Boolean (I.Normalize a) (I.Normalize a),

789− I.FromLists Boolean (I.Normalize Bool) (I.Normalize a)

930+ Countable a,

931+ CountableN a,

932+ FLN a a,

933+ FLN Bool a

790934 )

791− => (a -> Bool) -> Relation a a

792−predR p = identity `intersection` divisionF (fromF' (const True)) (fromF' p)

935+ => Relation a Bool -> Relation a a

936+predR p = id `intersection` divisionF (fromF (const True)) p

793937

794−-- | Equalizes functions @f@ and @g@.

938+-- | Equalizes functions @f@ and @g@.

795939-- That is, @'equalizer' f g@ is the largest coreflexive

796940-- that restricts @g@ so that @f@ and @g@ yield the same outputs.

941+--

942+-- @

943+-- 'equalizer' r r == 'id'

944+-- 'equalizer' ('point' True) ('point' False) == 'zeros'

945+-- @

797946equalizer ::

798− ( KnownNat (I.Count (I.Normalize a)),

799− I.FromLists Boolean (I.Normalize a) (I.Normalize a)

947+ ( CountableN a,

948+ FLN a a

800949 )

801950 => Relation a b -> Relation a b -> Relation a a

802−equalizer f g = identity `intersection` divisionF f g

951+equalizer f g = id `intersection` divisionF f g

803952

804953-- | Relational conditional guard.

805954--

806955-- @

807−-- 'guard' p = 'i2' `'overriddenBy'` 'i1' `'comp'` 'predR' p

956+-- 'guard' p = 'i2' ``overriddenBy`` 'i1' `.` 'predR' p

808957-- @

809−guard ::

958+guard ::

810959 ( Bounded b,

811960 Enum b,

812− KnownNat (I.Count (I.Normalize b)),

813− I.FromLists Boolean (I.Normalize b) (I.Normalize b),

814− I.FromLists Boolean (I.Normalize Bool) (I.Normalize b),

815− I.Normalize (Either b b) ~ Either (I.Normalize b) (I.Normalize b)

816− ) => (b -> Bool) -> Relation b (Either b b)

817−guard p = conv (eitherR (predR p) (predR (not . p)))

961+ CountableN b,

962+ Countable b,

963+ FLN b b,

964+ FLN Bool b

965+ ) => Relation b Bool -> Relation b (Either b b)

966+guard p = conv (eitherR (predR p) (predR (negate p)))

818967

819968-- | Relational McCarthy's conditional.

820−cond ::

969+cond ::

821970 ( Bounded b,

822971 Enum b,

823− KnownNat (I.Count (I.Normalize b)),

824− I.FromLists Boolean (I.Normalize b) (I.Normalize b),

825− I.FromLists Boolean (I.Normalize Bool) (I.Normalize b),

826− I.Normalize (Either b b) ~ Either (I.Normalize b) (I.Normalize b)

827− )

828− => (b -> Bool) -> Relation b c -> Relation b c -> Relation b c

829−cond p r s = eitherR r s `comp` guard p

972+ Countable b,

973+ CountableN b,

974+ FLN b b,

975+ FLN Bool b

976+ )

977+ => Relation b Bool -> Relation b c -> Relation b c -> Relation b c

978+cond p r s = eitherR r s . guard p

979+

980+-- | Relational domain.

981+--

982+-- For injective relations, 'domain' and 'ker'nel coincide,

983+-- since @'ker' r ``sse`` 'id'@ in such situations.

984+domain ::

985+ ( CountableN a,

986+ FLN a a

987+ ) => Relation a b -> Relation a a

988+domain r = ker r `intersection` id

989+

990+-- | Relational range.

991+--

992+-- For functions, 'range' and 'img' (image) coincide,

993+-- since @'img' f ``sse`` id@ for any @f@.

994+range ::

995+ ( CountableN b,

996+ FLN b b

997+ ) => Relation a b -> Relation b b

998+range r = img r `intersection` id

830999

8311000-- Relation pretty print

8321001

8331002-- | Relation pretty printing

834−pretty :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> String

1003+pretty :: (CountableDimsN a b) => Relation a b -> String

8351004pretty (R a) = I.pretty a

8361005

8371006-- | Relation pretty printing

838−prettyPrint :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> IO ()

1007+prettyPrint :: (CountableDimsN a b) => Relation a b -> IO ()

8391008prettyPrint (R a) = I.prettyPrint a

src/LAoP/Utils.hs+6

@@ -94,6 +94,12 @@ module LAoP.Utils

9494 coerceNat,

9595 coerceNat2,

9696 coerceNat3,

97+

98+ -- * List data type

99+ List (..),

100+

101+ -- * Category type-class

102+ Category(..)

97103 )

98104where

99105

src/LAoP/Utils/Internal.hs+70−73

@@ -1,82 +1,13 @@

1+{-# LANGUAGE TypeFamilies #-}

2+{-# LANGUAGE DeriveGeneric #-}

13{-# LANGUAGE TypeApplications #-}

24{-# LANGUAGE DataKinds #-}

35{-# LANGUAGE FlexibleInstances #-}

46{-# LANGUAGE GeneralizedNewtypeDeriving #-}

5−{-# LANGUAGE KindSignatures #-}

67{-# LANGUAGE ScopedTypeVariables #-}

89module LAoP.Utils.Internal

9− ( -- | This is an internal module and it is not meant to be imported.

10− --

11− -- Utility module that provides the 'Natural' data type.

12− -- The semantic associated with this data type is that

13− -- it's meant to be a restricted 'Int' value. For example

14− -- the type @Natural 6@ can only be instanciated with @Nat n@

15− -- where @0 <= n <= 6@. Why, You might ask, because with normal

16− -- 'Int's it is not possible to have a decent @Enum (Int, Int)@

17− -- instance. See the following probabilistic programming model as and

18− -- example:

19− --

20− -- We want to calculate the probability of the sum of two dice throws.

21− -- To do this we start by defining the sample space:

22− --

23− -- @

24− -- type SampleSpace = Int -- We think 'Int' are enough

25− --

26− -- die :: Dist Int 6

27− -- die = unifrom [1..6]

28− --

29− -- -- Promote 'Int' addition to a matrix

30− -- addM = fromF (uncurry (+)) -- Impossible

31− -- @

32− --

33− -- The last line is impossible because @(Int, Int)@ does not have

34− -- a good 'Enum' instance: @[(0, 1), (0, 2), .. (0, maxBound), (1, 0),

35− -- ..]@. And we'd like the addition matrix to be of 36 columns by 12

36− -- rows but limited to integers up to @6@!

37− --

38− -- One way to solve this issue is by defining and auxilary data type to

39− -- represent the sample space:

40− --

41− -- @

42− -- data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6

43− -- deriving (Show, Eq, Enum, Bounded) -- Enum and Bounded are

44− -- important

45− -- @

46− --

47− -- And write the sample space addition function:

48− --

49− -- @

50− -- ssAdd :: SampleSpace -> SampleSpace -> Int

51− -- ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)

52− -- @

53− --

54− -- And then promote that function to matrix and everything is alright:

55− --

56− -- @

57− -- ssAddM = fromF' (uncurry ssAdd)

58− --

59− -- dieSumProb = ssAddM `comp` (khatri die die)

60− -- @

61− --

62− -- This is a nice solution for small sample spaces. But for larger ones

63− -- it is not feasible to write a data type with hundreds of constructors

64− -- and then write manipulation functions that need to deal with them.

65− -- To mitigate this limitation the 'Natural' type comes a long way and

66− -- allows one to model the sample in an easier way. See for instance:

67− --

68− -- @

69− -- ssAdd :: Natural 6 -> Natural 6 -> Natural 12

70− -- ssAdd = coerceNat (+)

71− --

72− -- ssAddM = fromF' (uncurry sumSS)

73− --

74− -- die :: Dist (Natural 6) 6

75− -- die = uniform [nat @6 1 .. nat 6]

76− --

77− -- dieSumProb = ssAddM `comp` (khatri die die)

78− -- @

79− --

10+ (

8011 -- * 'Natural' data type

8112 Natural(..),

8213 nat,

@@ -86,19 +17,31 @@ module LAoP.Utils.Internal

8617 coerceNat,

8718 coerceNat2,

8819 coerceNat3,

20+

21+ -- * 'List' data type

22+ List (..),

23+

24+ -- * Category type class

25+ Category(..)

8926 )

9027where

9128

9229import Data.Coerce

9330import Data.Proxy

31+import Data.List

32+import Data.Maybe

9433import GHC.TypeLits

9534import Control.DeepSeq

35+import GHC.Generics

36+import Data.Kind

37+import Prelude hiding (id, (.))

38+import qualified Prelude

9639

9740-- | Wrapper around 'Int's that have a restrictive semantic associated.

9841-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'

9942-- @i@ that's @n <= i <= m@.

10043newtype Natural (start :: Nat) (end :: Nat) = Nat Int

101− deriving (Show, Read, Eq, Ord, NFData)

44+ deriving (Show, Read, Eq, Ord, NFData, Generic)

10245

10346-- | Throws a runtime error if any of the operations overflows or

10447-- underflows.

@@ -197,3 +140,57 @@ instance

197140

198141 fromEnum (Left a) = fromEnum a

199142 fromEnum (Right b) = fromEnum (maxBound :: a) + fromEnum b + 1

143+

144+-- | Powerset data type.

145+--

146+-- This data type is a newtype wrapper around '[]'. This exists in order to

147+-- implement an 'Enum' and 'Bounded' instance that cannot be harmful for the outside.

148+newtype List a = L [a]

149+ deriving (Eq, Show, Read)

150+

151+powerset :: [a] -> [[a]]

152+powerset [] = [[]]

153+powerset (x:xs) = powerset xs ++ [x:ps | ps <- powerset xs]

154+

155+instance

156+ ( Enum a,

157+ Bounded a

158+ ) => Bounded (List a)

159+ where

160+ minBound = L []

161+ maxBound = L [minBound .. maxBound]

162+

163+instance

164+ ( Bounded a,

165+ Enum a,

166+ Eq a

167+ ) => Enum (List a)

168+ where

169+ toEnum i =

170+ let as = [minBound .. maxBound]

171+ in L (powerset as !! i)

172+

173+ fromEnum (L []) = 0

174+ fromEnum (L x) =

175+ let as = [minBound .. maxBound]

176+ in fromMaybe (error "Does not exist") $ elemIndex x (powerset as)

177+

178+-- | Constrained category instance

179+class Category k where

180+ type Object k o :: Constraint

181+ type Object k o = ()

182+ id :: Object k a => k a a

183+ (.) :: k b c -> k a b -> k a c

184+

185+{-# RULES

186+"identity/left" forall p .

187+ id . p = p

188+"identity/right" forall p .

189+ p . id = p

190+"association" forall p q r .

191+ (p . q) . r = p . (q . r)

192+ #-}

193+

194+instance Category (->) where

195+ id = Prelude.id

196+ (.) = (Prelude..)

stack-8.8.3.yamlnew file+1

@@ -0,0 +1 @@

1+resolver: lts-15.5

stack.yaml+1−66

@@ -1,66 +1 @@

1−# This file was automatically generated by 'stack init'

2−#

3−# Some commonly used options have been documented as comments in this file.

4−# For advanced use and comprehensive documentation of the format, please see:

5−# https://docs.haskellstack.org/en/stable/yaml_configuration/

6−

7−# Resolver to choose a 'specific' stackage snapshot or a compiler version.

8−# A snapshot resolver dictates the compiler version and the set of packages

9−# to be used for project dependencies. For example:

10−#

11−# resolver: lts-3.5

12−# resolver: nightly-2015-09-21

13−# resolver: ghc-7.10.2

14−#

15−# The location of a snapshot can be provided as a file or url. Stack assumes

16−# a snapshot provided as a file might change, whereas a url resource does not.

17−#

18−# resolver: ./custom-snapshot.yaml

19−# resolver: https://example.com/snapshots/2018-01-01.yaml

20−resolver: lts-14.18

21−

22−# User packages to be built.

23−# Various formats can be used as shown in the example below.

24−#

25−# packages:

26−# - some-directory

27−# - https://example.com/foo/bar/baz-0.0.2.tar.gz

28−# subdirs:

29−# - auto-update

30−# - wai

31−packages:

32−- .

33−# Dependency packages to be pulled from upstream that are not in the resolver.

34−# These entries can reference officially published versions as well as

35−# forks / in-progress versions pinned to a git hash. For example:

36−#

37−# extra-deps:

38−# - acme-missiles-0.3

39−# - git: https://github.com/commercialhaskell/stack.git

40−# commit: e7b331f14bcffb8367cd58fbfc8b40ec7642100a

41−#

42−# extra-deps: []

43−

44−# Override default flag values for local packages and extra-deps

45−# flags: {}

46−

47−# Extra package databases containing global packages

48−# extra-package-dbs: []

49−

50−# Control whether we use the GHC we find on the path

51−# system-ghc: true

52−#

53−# Require a specific version of stack, using version ranges

54−# require-stack-version: -any # Default

55−# require-stack-version: ">=2.1"

56−#

57−# Override the architecture used by stack, especially useful on Windows

58−# arch: i386

59−# arch: x86_64

60−#

61−# Extra directories used by stack for building

62−# extra-include-dirs: [/path/to/dir]

63−# extra-lib-dirs: [/path/to/dir]

64−#

65−# Allow a newer minor version of GHC than the snapshot specifies

66−# compiler-check: newer-minor

1+resolver: lts-14.3

stack.yaml.lock+4−4

@@ -6,7 +6,7 @@

66packages: []

77snapshots:

88- completed:

9− size: 524789

10− url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/18.yaml

11− sha256: 646be71223e08234131c6989912e6011e01b9767bc447b6d466a35e14360bdf2

12− original: lts-14.18

9+ size: 523878

10+ url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/3.yaml

11+ sha256: 470c46c27746a48c7c50f829efc0cf00112787a7804ee4ac7a27754658f6d92c

12+ original: lts-14.3

test/Examples/Quantum.hsnew file+31

@@ -0,0 +1,31 @@

1+{-# LANGUAGE StandaloneDeriving #-}

2+module Examples.Quantum where

4+import LAoP.Utils

5+import LAoP.Matrix.Type

6+import Data.Complex

7+import Prelude hiding (id, (.))

9+deriving instance Ord a => Ord (Complex a)

10+

11+xor :: (Bool, Bool) -> Bool

12+xor (False, b) = b

13+xor (True, b) = not b

14+

15+ker :: Num e => Matrix e a b -> Matrix e a a

16+ker m = tr m . m

17+

18+cnot :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)

19+cnot = kr fstM (fromF xor)

20+

21+ccnot :: (Num e, Ord e) => Matrix e ((Bool, Bool), Bool) ((Bool, Bool), Bool)

22+ccnot = kr fstM (fromF f)

23+ where

24+ f = xor . tp (uncurry (&&)) id

25+ tp f g (a,b) = (f a, g b)

26+

27+had :: Matrix (Complex Double) Bool Bool

28+had = (1/sqrt 2) .| fromLists [[1, 1], [1, -1]]

29+

30+bell :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)

31+bell = cnot . (had >< iden)

app/Main.hs → test/Examples/Readme.hsrenamed+61−33

@@ -1,10 +1,13 @@

1+{-# LANGUAGE GADTs #-}

12{-# LANGUAGE DeriveGeneric #-}

23{-# LANGUAGE FlexibleContexts #-}

3−{-# LANGUAGE TypeOperators #-}

44{-# LANGUAGE TypeApplications #-}

55{-# LANGUAGE DataKinds #-}

7−module Main where

7+module Examples.Readme

8+ ( exec

9+ )

10+ where

811

912import LAoP.Matrix.Type

1013import qualified LAoP.Relation as R

@@ -12,13 +15,12 @@ import LAoP.Utils

1215import LAoP.Dist

1316import GHC.TypeLits

1417import Data.Coerce

15−import GHC.Generics

16−import Control.Category hiding (id)

17−import Prelude hiding ((.))

18+import qualified GHC.Generics as G

19+import Prelude hiding (id, (.))

1820

1921-- Monty Hall Problem

2022data Outcome = Win | Lose

21− deriving (Bounded, Enum, Eq, Show, Generic)

23+ deriving (Bounded, Enum, Eq, Show, G.Generic)

2224

2325switch :: Outcome -> Outcome

2426switch Win = Lose

@@ -28,7 +30,7 @@ firstChoice :: Matrix Double () Outcome

2830firstChoice = col [1/3, 2/3]

2931

3032secondChoice :: Matrix Double Outcome Outcome

31−secondChoice = fromF' switch

33+secondChoice = fromF switch

3234

3335-- Dice sum

3436

@@ -37,7 +39,7 @@ type SS = Natural 1 6 -- Sample Space

3739sumSS :: SS -> SS -> Natural 2 12

3840sumSS = coerceNat (+)

3941

40−sumSSM = fromF' (uncurry sumSS)

42+sumSSM = fromF (uncurry sumSS)

4143

4244condition :: (Int, Int) -> Int -> Int

4345condition (fst, snd) thrd = if fst == snd

@@ -47,38 +49,64 @@ condition (fst, snd) thrd = if fst == snd

4749conditionSS :: (SS, SS) -> SS -> Natural 3 18

4850conditionSS = coerceNat2 condition

4951

50−conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die

52+conditionalThrows = fromF (uncurry conditionSS) . kr (kr die die) die

5153

5254die :: Matrix Double () SS

5355die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]

5456

5557-- Sprinkler

58+data G = Dry | Wet

59+ deriving (Bounded, Enum, Eq, Show, G.Generic)

60+

61+data S = Off | On

62+ deriving (Bounded, Enum, Eq, Show, G.Generic)

5663

57−rain :: Matrix Double () Bool

58−rain = col [0.8, 0.2]

64+data R = No | Yes

65+ deriving (Bounded, Enum, Eq, Show, G.Generic)

66+

67+rain :: Matrix Double () R

68+rain = matrixBuilder gen

69+ where

70+ gen (_, No) = 0.8

71+ gen (_, Yes) = 0.2

5972

60−sprinkler :: Matrix Double Bool Bool

61−sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]

73+sprinkler :: Matrix Double R S

74+sprinkler = matrixBuilder gen

75+ where

76+ gen (No, Off) = 0.6

77+ gen (No, On) = 0.4

78+ gen (Yes, Off) = 0.99

79+ gen (Yes, On) = 0.01

80+

81+grass :: Matrix Double (S, R) G

82+grass = matrixBuilder gen

83+ where

84+ gen ((Off, No), Dry) = 1

85+ gen ((Off, Yes), Dry) = 0.2

86+ gen ((On, No), Dry) = 0.1

87+ gen ((On, Yes), Dry) = 0.01

88+ gen ((Off, No), Wet) = 0

89+ gen ((Off, Yes), Wet) = 0.8

90+ gen ((On, No), Wet) = 0.9

91+ gen ((On, Yes), Wet) = 0.99

6292

63−grass :: Matrix Double (Bool, Bool) Bool

64−grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]

93+tag f = kr f id

6594

66−state :: Matrix Double () (Bool, (Bool, Bool))

67−state = khatri grass identity . khatri sprinkler identity . rain

95+state g s r = tag g . tag s . r

6896

69−grass_wet :: Matrix Double (Bool, (Bool, Bool)) One

70−grass_wet = row [0,1] . kp1

97+grass_wet :: Matrix Double () (G, (S, R)) -> Matrix Double One One

98+grass_wet state = row [0,1] . fstM . state

7199

72−rainning :: Matrix Double (Bool, (Bool, Bool)) One

73−rainning = row [0,1] . kp2 . kp2

100+rainning :: Matrix Double (G, (S, R)) One

101+rainning = row [0,1] . sndM . sndM

74102

75103-- Alcuin Puzzle

76104

77105data Being = Farmer | Fox | Goose | Beans

78− deriving (Bounded, Enum, Eq, Show, Generic)

106+ deriving (Bounded, Enum, Eq, Show, G.Generic)

79107

80108data Bank = LeftB | RightB

81− deriving (Bounded, Enum, Eq, Show, Generic)

109+ deriving (Bounded, Enum, Eq, Show, G.Generic)

82110

83111eats :: Being -> Being -> Bool

84112eats Fox Goose = True

@@ -93,21 +121,21 @@ cross LeftB = RightB

93121cross RightB = LeftB

94122

95123crossR :: R.Relation Bank Bank

96−crossR = R.fromF' cross

124+crossR = R.fromF cross

97125

98126-- | Initial state, everyone in the left bank

99127locationLeft :: Being -> Bank

100128locationLeft _ = LeftB

101129

102130locationLeftR :: R.Relation Being Bank

103−locationLeftR = R.fromF' locationLeft

131+locationLeftR = R.fromF locationLeft

104132

105133-- | Initial state, everyone in the right bank

106134locationRight :: Being -> Bank

107135locationRight _ = RightB

108136

109137locationRightR :: R.Relation Being Bank

110−locationRightR = R.fromF' locationRight

138+locationRightR = R.fromF locationRight

111139

112140-- Properties

113141

@@ -124,7 +152,7 @@ inv :: R.Relation Being Bank -> Bool

124152inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)

125153 where

126154 farmer :: R.Relation Being Being

127− farmer = R.fromF' (const Farmer)

155+ farmer = R.fromF (const Farmer)

128156

129157-- Arbitrary state

130158bankState :: Being -> Bank -> Bool

@@ -138,19 +166,19 @@ bankStateR :: R.Relation Being Bank

138166bankStateR = R.toRel bankState

139167

140168-- Main

141−main :: IO ()

142−main = do

169+exec :: IO ()

170+exec = do

143171 putStrLn "Monty Hall Problem solution:"

144172 prettyPrint (secondChoice . firstChoice)

145173 putStrLn "\n Sum of dices probability:"

146− prettyPrint (sumSSM `comp` khatri die die)

174+ prettyPrint (sumSSM . kr die die)

147175 putStrLn "\n Conditional dice throw:"

148176 prettyPrint conditionalThrows

149177 putStrLn "\n Checking that the last result is indeed a distribution: "

150− prettyPrint (bang . sumSSM . khatri die die)

178+ prettyPrint (bang . sumSSM . kr die die)

151179 putStrLn "\n Probability of grass being wet:"

152− prettyPrint (grass_wet . state)

180+ prettyPrint (grass_wet (state grass sprinkler rain))

153181 putStrLn "\n Probability of rain:"

154− prettyPrint (rainning . state)

182+ prettyPrint (rainning . state grass sprinkler rain)

155183 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"

156184 print (inv bankStateR)

test/Examples/Spreadsheets.hsnew file+47

@@ -0,0 +1,47 @@

1+{-# LANGUAGE DeriveGeneric #-}

3+module Examples.Spreadsheets

5+import LAoP.Utils

6+import LAoP.Matrix.Type

7+import GHC.Generics

8+import Prelude hiding (id, (.))

10+data Student = Student1 | Student2 | Student3 | Student4

11+ deriving (Eq, Show, Enum, Bounded, Generic)

12+

13+data Question = Question1 | Question2 | Question3 | Question4

14+ deriving (Eq, Show, Enum, Bounded, Generic)

15+

16+data Results = Exam | Test | Final

17+ deriving (Eq, Show, Enum, Bounded, Generic)

18+

19+test :: Matrix Float One Results

20+test = point Test

21+

22+exam :: Matrix Float One Results

23+exam = point Exam

24+

25+final :: Matrix Float One Results

26+final = point Final

27+

28+m :: Matrix Float Question Student

29+m = fromLists [[95, 90, 100, 40], [20, 90, 90, 0], [30, 20, 95, 0], [50, 80, 100, 30]]

30+

31+w :: Matrix Float Question One

32+w = fromLists [[0.2, 0.3, 0.2, 0.3]]

33+

34+xls :: Matrix Float Question One

35+ -> Matrix Float Question Student

36+ -> Matrix Float Student One

37+ -> Matrix Float (Either Question Results) (Either One Student)

38+xls w m t = join (fork w m) (fork zeros r)

39+ where

40+ rExam = m . tr w

41+ rTest = tr t

42+ rFinal = rTest `maxPW` rExam

43+ r = (rExam . tr exam) + (rTest . tr test) + (rFinal . tr final)

44+

45+-- | Overloaded, point-wise 'max' function

46+maxPW :: Ord e => Matrix e a b -> Matrix e a b -> Matrix e a b

47+maxPW = zipWithM max

test/Spec.hs+7−1

@@ -1,2 +1,8 @@

1+module Main (main) where

3+import qualified Examples.Readme as RD

15main :: IO ()

2−main = putStrLn "Test suite not yet implemented"

6+main = do

7+ putStrLn ("Testing if README works" :: String)

8+ RD.exec