master..2eb21d5 · laop

git.iko.soy/laop

93 files changed+5957−2350

unifiedsplit

.ghc.environment.x86_64-linux-8.6.5new file+17

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

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.ymldeleted−43

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

1−name: CI

2−

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]

8−

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+2−54

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

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

251.stack-work/

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

2+laop.cabal

3+*~

.travis.ymldeleted−47

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

1−sudo: true

2−language: haskell

3−

4−git:

5− depth: 5

6−

7−cabal: "3.0"

8−

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.mddeleted−17

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

1−# Changelog

2−

3−`laop` uses [PVP Versioning][1].

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

5−

6−## 0.1.1.1

7−

8−* Bump base version to work with cabal

9−

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.mdnew file+3

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

1+# Changelog for laop

3+## Unreleased changes

LICENSE+26−17

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

1−MIT License

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:

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

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

117

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

13−copies or substantial portions of the Software.

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

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

1410

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.

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.

README.md+24−107

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

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)

8−

93The LAoP discipline generalises relations and functions treating them as

104Boolean matrices and in turn consider these as arrows.

115

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

1610This module offers many of the combinators mentioned in the work of

1711[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).

1812

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.

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

2214

2315## Features

2416

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).

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.

5140

5241## Notes

5342

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

129118rainning :: Matrix Double (Bool, (Bool, Bool)) One

130119rainning = row [0,1] . kp2 . kp2

131120

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−

199121main :: IO ()

200122main = do

201123 putStrLn "Monty Hall Problem solution:"

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

210132 prettyPrint (grass_wet . state)

211133 putStrLn "\n Probability of rain:"

212134 prettyPrint (rainning . state)

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

214− print (inv bankStateR)

215135```

216136

217137```Shell

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

270190┌ ┐

271191│ 0.2 │

272192└ ┘

273−

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

275−False

276193```

Setup.hsnew file+2

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

1+import Distribution.Simple

2+main = defaultMain

test/Examples/Readme.hs → app/Main.hsrenamed+33−61

@@ -1,13 +1,10 @@

1−{-# LANGUAGE GADTs #-}

21{-# LANGUAGE DeriveGeneric #-}

32{-# LANGUAGE FlexibleContexts #-}

3+{-# LANGUAGE TypeOperators #-}

44{-# LANGUAGE TypeApplications #-}

55{-# LANGUAGE DataKinds #-}

7−module Examples.Readme

8− ( exec

9− )

10− where

7+module Main where

118

129import LAoP.Matrix.Type

1310import qualified LAoP.Relation as R

@@ -15,12 +12,13 @@ import LAoP.Utils

1512import LAoP.Dist

1613import GHC.TypeLits

1714import Data.Coerce

18−import qualified GHC.Generics as G

19−import Prelude hiding (id, (.))

15+import GHC.Generics

16+import Control.Category hiding (id)

17+import Prelude hiding ((.))

2018

2119-- Monty Hall Problem

2220data Outcome = Win | Lose

23− deriving (Bounded, Enum, Eq, Show, G.Generic)

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

2422

2523switch :: Outcome -> Outcome

2624switch Win = Lose

@@ -30,7 +28,7 @@ firstChoice :: Matrix Double () Outcome

3028firstChoice = col [1/3, 2/3]

3129

3230secondChoice :: Matrix Double Outcome Outcome

33−secondChoice = fromF switch

31+secondChoice = fromF' switch

3432

3533-- Dice sum

3634

@@ -39,7 +37,7 @@ type SS = Natural 1 6 -- Sample Space

3937sumSS :: SS -> SS -> Natural 2 12

4038sumSS = coerceNat (+)

4139

42−sumSSM = fromF (uncurry sumSS)

40+sumSSM = fromF' (uncurry sumSS)

4341

4442condition :: (Int, Int) -> Int -> Int

4543condition (fst, snd) thrd = if fst == snd

@@ -49,64 +47,38 @@ condition (fst, snd) thrd = if fst == snd

4947conditionSS :: (SS, SS) -> SS -> Natural 3 18

5048conditionSS = coerceNat2 condition

5149

52−conditionalThrows = fromF (uncurry conditionSS) . kr (kr die die) die

50+conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die

5351

5452die :: Matrix Double () SS

5553die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]

5654

5755-- 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)

6356

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

57+rain :: Matrix Double () Bool

58+rain = col [0.8, 0.2]

7259

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

60+sprinkler :: Matrix Double Bool Bool

61+sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]

9262

93−tag f = kr f id

63+grass :: Matrix Double (Bool, Bool) Bool

64+grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]

9465

95−state g s r = tag g . tag s . r

66+state :: Matrix Double () (Bool, (Bool, Bool))

67+state = khatri grass identity . khatri sprinkler identity . rain

9668

97−grass_wet :: Matrix Double () (G, (S, R)) -> Matrix Double One One

98−grass_wet state = row [0,1] . fstM . state

69+grass_wet :: Matrix Double (Bool, (Bool, Bool)) One

70+grass_wet = row [0,1] . kp1

9971

100−rainning :: Matrix Double (G, (S, R)) One

101−rainning = row [0,1] . sndM . sndM

72+rainning :: Matrix Double (Bool, (Bool, Bool)) One

73+rainning = row [0,1] . kp2 . kp2

10274

10375-- Alcuin Puzzle

10476

10577data Being = Farmer | Fox | Goose | Beans

106− deriving (Bounded, Enum, Eq, Show, G.Generic)

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

10779

10880data Bank = LeftB | RightB

109− deriving (Bounded, Enum, Eq, Show, G.Generic)

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

11082

11183eats :: Being -> Being -> Bool

11284eats Fox Goose = True

@@ -121,21 +93,21 @@ cross LeftB = RightB

12193cross RightB = LeftB

12294

12395crossR :: R.Relation Bank Bank

124−crossR = R.fromF cross

96+crossR = R.fromF' cross

12597

12698-- | Initial state, everyone in the left bank

12799locationLeft :: Being -> Bank

128100locationLeft _ = LeftB

129101

130102locationLeftR :: R.Relation Being Bank

131−locationLeftR = R.fromF locationLeft

103+locationLeftR = R.fromF' locationLeft

132104

133105-- | Initial state, everyone in the right bank

134106locationRight :: Being -> Bank

135107locationRight _ = RightB

136108

137109locationRightR :: R.Relation Being Bank

138−locationRightR = R.fromF locationRight

110+locationRightR = R.fromF' locationRight

139111

140112-- Properties

141113

@@ -152,7 +124,7 @@ inv :: R.Relation Being Bank -> Bool

152124inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)

153125 where

154126 farmer :: R.Relation Being Being

155− farmer = R.fromF (const Farmer)

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

156128

157129-- Arbitrary state

158130bankState :: Being -> Bank -> Bool

@@ -166,19 +138,19 @@ bankStateR :: R.Relation Being Bank

166138bankStateR = R.toRel bankState

167139

168140-- Main

169−exec :: IO ()

170−exec = do

141+main :: IO ()

142+main = do

171143 putStrLn "Monty Hall Problem solution:"

172144 prettyPrint (secondChoice . firstChoice)

173145 putStrLn "\n Sum of dices probability:"

174− prettyPrint (sumSSM . kr die die)

146+ prettyPrint (sumSSM `comp` khatri die die)

175147 putStrLn "\n Conditional dice throw:"

176148 prettyPrint conditionalThrows

177149 putStrLn "\n Checking that the last result is indeed a distribution: "

178− prettyPrint (bang . sumSSM . kr die die)

150+ prettyPrint (bang . sumSSM . khatri die die)

179151 putStrLn "\n Probability of grass being wet:"

180− prettyPrint (grass_wet (state grass sprinkler rain))

152+ prettyPrint (grass_wet . state)

181153 putStrLn "\n Probability of rain:"

182− prettyPrint (rainning . state grass sprinkler rain)

154+ prettyPrint (rainning . state)

183155 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"

184156 print (inv bankStateR)

appveyor.ymldeleted−34

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

1−clone_folder: "c:\\WORK"

2−clone_depth: 5

3−

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

5−skip_branch_with_pr: true

6−

7−platform:

8− - x86_64

9−

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.hsdeleted−175

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

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.hsnew file+50

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

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

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.hnew file+143

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

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

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/confignew file

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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.htmlnew file+15

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

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.htmlnew file+36

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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#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.htmlnew file+52

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

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 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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" 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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.jsonnew file+1

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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 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1+-- Hoogle documentation, generated by Haddock

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

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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203+ padding: 0.2em 0.5em;

204+}

205+

206+.keyword { font-weight: normal; }

207+.def { font-weight: bold; }

208+

209+@media print {

210+ #footer { display: none; }

211+}

212+

213+/* @end */

214+

215+/* @group Page Structure */

216+

217+#content {

218+ margin: 0 auto;

219+ padding: 0 2em 6em;

220+}

221+

222+#package-header {

223+ background: rgb(41,56,69);

224+ border-top: 5px solid rgb(78,98,114);

225+ color: #ddd;

226+ padding: 0.2em;

227+ position: relative;

228+ text-align: left;

229+}

230+

231+#package-header .caption {

232+ background: url(hslogo-16.png) no-repeat 0em;

233+ color: white;

234+ margin: 0 2em;

235+ font-weight: normal;

236+ font-style: normal;

237+ padding-left: 2em;

238+}

239+

240+#package-header a:link, #package-header a:visited { color: white; }

241+#package-header a:hover { background: rgb(78,98,114); }

242+

243+#module-header .caption {

244+ color: rgb(78,98,114);

245+ font-weight: bold;

246+ border-bottom: 1px solid #ddd;

247+}

248+

249+table.info {

250+ float: right;

251+ padding: 0.5em 1em;

252+ border: 1px solid #ddd;

253+ color: rgb(78,98,114);

254+ background-color: #fff;

255+ max-width: 40%;

256+ border-spacing: 0;

257+ position: relative;

258+ top: -0.5em;

259+ margin: 0 0 0 2em;

260+}

261+

262+.info th {

263+ padding: 0 1em 0 0;

264+}

265+

266+div#style-menu-holder {

267+ position: relative;

268+ z-index: 2;

269+ display: inline;

270+}

271+

272+#style-menu {

273+ position: absolute;

274+ z-index: 1;

275+ overflow: visible;

276+ background: #374c5e;

277+ margin: 0;

278+ text-align: center;

279+ right: 0;

280+ padding: 0;

281+ top: 1.25em;

282+}

283+

284+#style-menu li {

285+ display: list-item;

286+ border-style: none;

287+ margin: 0;

288+ padding: 0;

289+ color: #000;

290+ list-style-type: none;

291+}

292+

293+#style-menu li + li {

294+ border-top: 1px solid #919191;

295+}

296+

297+#style-menu a {

298+ width: 6em;

299+ padding: 3px;

300+ display: block;

301+}

302+

303+#footer {

304+ background: #ddd;

305+ border-top: 1px solid #aaa;

306+ padding: 0.5em 0;

307+ color: #666;

308+ text-align: center;

309+ position: absolute;

310+ bottom: 0;

311+ width: 100%;

312+ height: 3em;

313+}

314+

315+/* @end */

316+

317+/* @group Front Matter */

318+

319+#table-of-contents {

320+ float: right;

321+ clear: right;

322+ background: #faf9dc;

323+ border: 1px solid #d8d7ad;

324+ padding: 0.5em 1em;

325+ max-width: 20em;

326+ margin: 0.5em 0 1em 1em;

327+}

328+

329+#table-of-contents .caption {

330+ text-align: center;

331+ margin: 0;

332+}

333+

334+#table-of-contents ul {

335+ list-style: none;

336+ margin: 0;

337+}

338+

339+#table-of-contents ul ul {

340+ margin-left: 2em;

341+}

342+

343+#description .caption {

344+ display: none;

345+}

346+

347+#synopsis {

348+ display: block;

349+ position: fixed;

350+ right: 0;

351+ height: 80%;

352+ top: 10%;

353+ padding: 0;

354+ max-width: 75%;

355+ /* Ensure that synopsis covers everything (including MathJAX markup) */

356+ z-index: 1;

357+}

358+

359+#synopsis summary {

360+ display: block;

361+ float: left;

362+ width: 29px;

363+ color: rgba(255,255,255,0);

364+ height: 110px;

365+ margin: 0;

366+ font-size: 1px;

367+ padding: 0;

368+ background: url(synopsis.png) no-repeat 0px -8px;

369+}

370+

371+#synopsis details[open] > summary {

372+ background: url(synopsis.png) no-repeat -64px -8px;

373+}

374+

375+#synopsis ul {

376+ height: 100%;

377+ overflow: auto;

378+ padding: 0.5em;

379+ margin: 0;

380+}

381+

382+#synopsis ul ul {

383+ overflow: hidden;

384+}

385+

386+#synopsis ul,

387+#synopsis ul li.src {

388+ background-color: #faf9dc;

389+ white-space: nowrap;

390+ list-style: none;

391+ margin-left: 0;

392+}

393+

394+/* @end */

395+

396+/* @group Main Content */

397+

398+#interface div.top { margin: 2em 0; }

399+#interface h1 + div.top,

400+#interface h2 + div.top,

401+#interface h3 + div.top,

402+#interface h4 + div.top,

403+#interface h5 + div.top {

404+ margin-top: 1em;

405+}

406+#interface .src .selflink,

407+#interface .src .link {

408+ float: right;

409+ color: #919191;

410+ background: #f0f0f0;

411+ padding: 0 0.5em 0.2em;

412+ margin: 0 -0.5em 0 0;

413+ -moz-user-select: none;

414+}

415+#interface .src .selflink {

416+ border-left: 1px solid #919191;

417+ margin: 0 -0.5em 0 0.5em;

418+}

419+

420+#interface span.fixity {

421+ color: #919191;

422+ border-left: 1px solid #919191;

423+ padding: 0.2em 0.5em 0.2em 0.5em;

424+ margin: 0 -1em 0 1em;

425+}

426+

427+#interface span.rightedge {

428+ border-left: 1px solid #919191;

429+ padding: 0.2em 0 0.2em 0;

430+ margin: 0 0 0 1em;

431+}

432+

433+#interface table { border-spacing: 2px; }

434+#interface td {

435+ vertical-align: top;

436+ padding-left: 0.5em;

437+}

438+

439+#interface td.doc p {

440+ margin: 0;

441+}

442+#interface td.doc p + p {

443+ margin-top: 0.8em;

444+}

445+

446+.doc table {

447+ border-collapse: collapse;

448+ border-spacing: 0px;

449+}

450+

451+.doc th,

452+.doc td {

453+ padding: 5px;

454+ border: 1px solid #ddd;

455+}

456+

457+.doc th {

458+ background-color: #f0f0f0;

459+}

460+

461+.clearfix:after {

462+ clear: both;

463+ content: " ";

464+ display: block;

465+ height: 0;

466+ visibility: hidden;

467+}

468+

469+.subs ul {

470+ list-style: none;

471+ display: table;

472+ margin: 0;

473+}

474+

475+.subs ul li {

476+ display: table-row;

477+}

478+

479+.subs ul li dfn {

480+ display: table-cell;

481+ font-style: normal;

482+ font-weight: bold;

483+ margin: 1px 0;

484+ white-space: nowrap;

485+}

486+

487+.subs ul li > .doc {

488+ display: table-cell;

489+ padding-left: 0.5em;

490+ margin-bottom: 0.5em;

491+}

492+

493+.subs ul li > .doc p {

494+ margin: 0;

495+}

496+

497+/* Render short-style data instances */

498+.inst ul {

499+ height: 100%;

500+ padding: 0.5em;

501+ margin: 0;

502+}

503+

504+.inst, .inst li {

505+ list-style: none;

506+ margin-left: 1em;

507+}

508+

509+/* Workaround for bug in Firefox (issue #384) */

510+.inst-left {

511+ float: left;

512+}

513+

514+.top p.src {

515+ border-top: 1px solid #ccc;

516+}

517+

518+.subs, .doc {

519+ /* use this selector for one level of indent */

520+ padding-left: 2em;

521+}

522+

523+.warning {

524+ color: red;

525+}

526+

527+.arguments {

528+ margin-top: -0.4em;

529+}

530+.arguments .caption {

531+ display: none;

532+}

533+

534+.fields { padding-left: 1em; }

535+

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

537+

538+.fields p { margin: 0 0; }

539+

540+/* this seems bulky to me

541+.methods, .constructors {

542+ background: #f8f8f8;

543+ border: 1px solid #eee;

544+}

545+*/

546+

547+/* @end */

548+

549+/* @group Auxillary Pages */

550+

551+

552+.extension-list {

553+ list-style-type: none;

554+ margin-left: 0;

555+}

556+

557+#mini {

558+ margin: 0 auto;

559+ padding: 0 1em 1em;

560+}

561+

562+#mini > * {

563+ font-size: 93%; /* 12pt */

564+}

565+

566+#mini #module-list .caption,

567+#mini #module-header .caption {

568+ font-size: 125%; /* 15pt */

569+}

570+

571+#mini #interface h1,

572+#mini #interface h2,

573+#mini #interface h3,

574+#mini #interface h4 {

575+ font-size: 109%; /* 13pt */

576+ margin: 1em 0 0;

577+}

578+

579+#mini #interface .top,

580+#mini #interface .src {

581+ margin: 0;

582+}

583+

584+#mini #module-list ul {

585+ list-style: none;

586+ margin: 0;

587+}

588+

589+#alphabet ul {

590+ list-style: none;

591+ padding: 0;

592+ margin: 0.5em 0 0;

593+ text-align: center;

594+}

595+

596+#alphabet li {

597+ display: inline;

598+ margin: 0 0.25em;

599+}

600+

601+#alphabet a {

602+ font-weight: bold;

603+}

604+

605+#index .caption,

606+#module-list .caption { font-size: 131%; /* 17pt */ }

607+

608+#index table {

609+ margin-left: 2em;

610+}

611+

612+#index .src {

613+ font-weight: bold;

614+}

615+#index .alt {

616+ font-size: 77%; /* 10pt */

617+ font-style: italic;

618+ padding-left: 2em;

619+}

620+

621+#index td + td {

622+ padding-left: 1em;

623+}

624+

625+#module-list ul {

626+ list-style: none;

627+ margin: 0 0 0 2em;

628+}

629+

630+#module-list li {

631+ clear: right;

632+}

633+

634+#module-list span.collapser,

635+#module-list span.expander {

636+ background-position: 0 0.3em;

637+}

638+

639+#module-list .package {

640+ float: right;

641+}

642+

643+:target {

644+ background-color: #ffff00;

645+}

646+

647+/* @end */

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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/quick-jump.cssnew file+164

@@ -0,0 +1,164 @@

1+/* @group Search box layout */

3+#search {

4+ position: fixed;

5+ top: 3.2em;

6+ bottom: 0;

7+ left: calc(50% - 22em);

8+ width: 44em;

9+ z-index: 1000;

10+ pointer-events: none;

11+ overflow-y: auto;

12+}

13+

14+#search.hidden {

15+ display: none;

16+}

17+

18+#search-form, #search-results {

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

20+ pointer-events: all;

21+}

22+

23+#search-form input {

24+ font-size: 1.25em; line-height: 2.3em; height: 2.4em;

25+ display: block;

26+ box-sizing: border-box;

27+ width: 100%;

28+ margin: 0;

29+ padding: 0 0.75em;

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

31+}

32+

33+#search input:focus {

34+ outline: none;

35+}

36+

37+#search p.error {

38+ color: rgb(107, 24, 24);

39+ font-weight: bold;

40+}

41+

42+#search-results {

43+ box-sizing: border-box;

44+ border: 0.05em solid #b2d5fb;

45+ background: #e8f3ff;

46+}

47+

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

49+ border-top: none;

50+ top: 3em;

51+ max-height: calc(100% - 3em);

52+}

53+

54+/* @end */

55+

56+/* @group search results */

57+

58+#search-results > ul {

59+ margin: 0;

60+ list-style: none;

61+}

62+

63+#search-results > ul > li,

64+#search-results > p,

65+#search-results > table {

66+ padding: 0.5em 1em;

67+ margin: 0;

68+}

69+

70+#search-results > ul > li {

71+ border-bottom: 1px solid #b2d5fb;

72+}

73+

74+#search-results > ul > li > ul {

75+ list-style: none;

76+}

77+

78+.search-module h4 {

79+ margin: 0;

80+}

81+

82+.search-module > ul {

83+ margin: 0.5em 0 0.5em 2em;

84+}

85+

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

87+ display: block;

88+ color: inherit;

89+ padding: 0.25em 0.5em;

90+}

91+

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

93+ background: #faf9dc;

94+}

95+

96+.search-module a[href]:hover {

97+ text-decoration: none;

98+}

99+

100+.search-result a a {

101+ pointer-events: none;

102+}

103+

104+.search-result ul.subs {

105+ display: inline-block;

106+ margin: 0; padding: 0;

107+}

108+

109+.search-result ul.subs li {

110+ display: none;

111+}

112+

113+.search-result ul.subs::after {

114+ display: inline-block;

115+ content: "...";

116+ color: rgb(78,98,114);

117+ margin: 0 0.25em;

118+}

119+

120+.more-results {

121+ color: rgb(99, 141, 173);

122+ position: relative;

123+}

124+

125+.more-results::before {

126+ content: "+";

127+ display: inline-block;

128+ color: #b2d5fb;

129+ font-weight: bold;

130+ font-size: 1.25em; line-height: inherit;

131+ position: absolute;

132+ left: -1em;

133+}

134+

135+/* @end */

136+

137+/* @group Keyboard shortcuts table */

138+

139+.keyboard-shortcuts {

140+ line-height: 1.6em;

141+}

142+

143+.keyboard-shortcuts th {

144+ color: rgb(78,98,114);

145+}

146+

147+.keyboard-shortcuts td:first-child,

148+.keyboard-shortcuts th:first-child {

149+ text-align: right;

150+ padding-right: 0.6em;

151+}

152+

153+.key {

154+ display: inline-block;

155+ font-size: 0.9em;

156+ min-width: 0.8em; line-height: 1.2em;

157+ text-align: center;

158+ background: #b2d5fb;

159+ border: 1px solid #74a3d6;

160+ padding: 0 0.2em;

161+ margin: 0 0.1em;

162+}

163+

164+/* @end */

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/quick-jump.min.jsnew file+2

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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.htmlnew file+174

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

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.htmlnew file+70

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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 : 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.htmlnew file+903

@@ -0,0 +1,903 @@

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.htmlnew file+484

@@ -0,0 +1,484 @@

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.htmlnew file+648

@@ -0,0 +1,648 @@

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.htmlnew file+51

@@ -0,0 +1,51 @@

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.htmlnew file+177

@@ -0,0 +1,177 @@

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.htmlnew file+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>

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/highlight.jsnew file+27

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23+ for (var i = 0; i < links.length; i++) {

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25+ links[i].onmouseout = highlight(false);

26+ }

27+};

build/x86_64-linux/ghc-8.6.5/laop-0.1.0.0/doc/html/laop-0.1.0.0-docs/src/style.cssnew file+55

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1+body {

2+ background-color: #fdf6e3;

3+}

5+.hs-identifier {

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7+}

9+.hs-identifier.hs-var {

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30+}

31+

32+.hs-glyph, .hs-special {

33+ color: #dc322f;

34+}

35+

36+.hs-comment {

37+ color: #8a8a8a;

38+}

39+

40+.hs-pragma {

41+ color: #2aa198;

42+}

43+

44+.hs-cpp {

45+ color: #859900;

46+}

47+

48+a:link, a:visited {

49+ text-decoration: none;

50+ border-bottom: 1px solid #eee8d5;

51+}

52+

53+a:hover, a.hover-highlight {

54+ background-color: #eee8d5;

55+}

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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":"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/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/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/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/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/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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dist-newstyle/build/x86_64-linux/ghc-8.6.5/fake-package-0/x/script/build/script/autogen/Paths_fake_package.hsnew file+50

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

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

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)

dist-newstyle/build/x86_64-linux/ghc-8.6.5/fake-package-0/x/script/build/script/autogen/cabal_macros.hnew file+118

@@ -0,0 +1,118 @@

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

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.shnew file+10

@@ -0,0 +1,10 @@

1+#!/bin/sh

2+set -e

4+dir=$(mktemp -d dist-docs.XXXXXX)

5+trap 'rm -r "$dir"' EXIT

7+# assumes cabal 2.4 or later

8+cabal v2-haddock --builddir="$dir" --haddock-for-hackage --enable-doc

10+cabal upload -d $dir/*-docs.tar.gz

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laop.cabaldeleted−89

@@ -1,89 +0,0 @@

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.yamlnew file+48

@@ -0,0 +1,48 @@

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"

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

@@ -36,14 +36,6 @@ 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−

4739 -- * Functor instance equivalent functions

4840 fmapD,

4941

@@ -53,8 +45,6 @@ module LAoP.Dist

5345

5446 -- * Selective equivalent functions

5547 selectD,

56− branchD,

57− ifD,

5848

5949 -- * Monad equivalent functions

6050 returnD,

src/LAoP/Dist/Internal.hs+36−87

@@ -1,5 +1,3 @@

1−{-# LANGUAGE DerivingVia #-}

2−{-# LANGUAGE ConstraintKinds #-}

31{-# LANGUAGE GADTs #-}

42{-# LANGUAGE AllowAmbiguousTypes #-}

53{-# LANGUAGE TypeApplications #-}

@@ -10,27 +8,18 @@

108{-# LANGUAGE RankNTypes #-}

119

1210module LAoP.Dist.Internal

13− (

11+ (

1412 Dist(..),

1513 Prob,

1614

17− Countable,

18− CountableN,

19− CountableDimsN,

20− FLN,

21− Liftable,

22− TrivialP,

23−

2415 fmapD,

2516 unitD,

2617 multD,

2718 selectD,

28− branchD,

29− ifD,

3019 returnD,

3120 bindD,

3221 (??),

33−

22+

3423 choose,

3524 shape,

3625 linear,

@@ -43,15 +32,11 @@ module LAoP.Dist.Internal

4332 )

4433 where

4534

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

35+import LAoP.Matrix.Type

4936import LAoP.Utils

5037import GHC.TypeLits

5138import Data.Proxy

5239import Data.List (sortBy)

53−import Control.DeepSeq

54−import Data.Bool

5540

5641-- | Type synonym for probability value

5742type Prob = Double

@@ -59,21 +44,17 @@ type Prob = Double

5944-- | Type synonym for column vector matrices. This represents a probability

6045-- distribution.

6146newtype 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)

7147

7248-- | Functor instance

7349fmapD ::

74− ( Liftable a b,

75− CountableDimsN a b,

76− FLN b a

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)

7758 )

7859 =>

7960 (a -> b) -> Dist a -> Dist b

@@ -85,58 +66,26 @@ unitD = D (one 1)

8566

8667-- | Applicative/Monoidal instance 'mult' function

8768multD ::

88− ( CountableDimsN a b,

89− CountableN (a, b),

90− FLN (a, b) a,

91− FLN (a, b) b,

92− TrivialP a b

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)

9375 ) => Dist a -> Dist b -> Dist (a, b)

94−multD (D a) (D b) = D (kr a b)

76+multD (D a) (D b) = D (khatri a b)

9577

9678-- | Selective instance function

9779selectD ::

98− ( FLN b b,

99− CountableN b

80+ ( Normalize (Either a b) ~ Either (Normalize a) (Normalize b),

81+ FromLists Prob (Normalize b) (Normalize b),

82+ KnownNat (Count (Normalize b))

10083 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b

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

84+selectD (D d) m = D (junc m identity `comp` d)

85+

13786

13887-- | Monad instance 'return' function

139−returnD :: forall a . (Enum a, FLN () a, Countable a) => a -> Dist a

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

14089returnD a = D (col l)

14190 where

14291 i = fromInteger $ natVal (Proxy :: Proxy (Count a))

@@ -150,8 +99,8 @@ bindD (D d) m = D (m `comp` d)

15099-- | Extract probabilities given an Event.

151100(??) ::

152101 ( Enum a,

153− Countable a,

154− FLN () a

102+ KnownNat (Count a),

103+ FromLists Prob () (Normalize a)

155104 ) => (a -> Bool) -> Dist a -> Prob

156105(??) p d =

157106 let l = toValues d

@@ -161,11 +110,11 @@ bindD (D d) m = D (m `comp` d)

161110-- Distribution Construction

162111

163112-- | Constructs a Bernoulli distribution

164−choose :: (FLN () a) => Prob -> Dist a

113+choose :: (FromLists Prob () (Normalize a)) => Prob -> Dist a

165114choose prob = D (col [prob, 1 - prob])

166115

167116-- | Creates a distribution given a shape function

168−shape :: (FLN () a) => (Prob -> Prob) -> [a] -> Dist a

117+shape :: (FromLists Prob () (Normalize a)) => (Prob -> Prob) -> [a] -> Dist a

169118shape _ [] = error "Probability.shape: empty list"

170119shape f xs =

171120 let incr = 1 / fromIntegral (length xs - 1)

@@ -173,23 +122,23 @@ shape f xs =

173122 in fromFreqs (zip xs ps)

174123

175124-- | Constructs a Linear distribution

176−linear :: (FLN () a) => [a] -> Dist a

125+linear :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

177126linear = shape id

178127

179128-- | Constructs an Uniform distribution

180−uniform :: (FLN () a) => [a] -> Dist a

129+uniform :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

181130uniform = shape (const 1)

182131

183132-- | Constructs an Negative Exponential distribution

184−negExp :: (FLN () a) => [a] -> Dist a

133+negExp :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

185134negExp = shape (\x -> exp (-x))

186135

187136-- | Constructs an Normal distribution

188−normal :: (FLN () a) => [a] -> Dist a

137+normal :: (FromLists Prob () (Normalize a)) => [a] -> Dist a

189138normal = shape (normalCurve 0.5 0.5)

190139

191140-- | Transforms a 'Dist' into a list of pairs.

192−toValues :: forall a . (Enum a, Countable a, FLN () a) => Dist a -> [(a, Prob)]

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

193142toValues (D d) =

194143 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))

195144 probs = toList d

@@ -197,7 +146,7 @@ toValues (D d) =

197146 in res

198147

199148-- | Pretty a distribution

200−prettyDist :: forall a. (Show a, Enum a, Countable a, FLN () a) => Dist a -> String

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

201150prettyDist d =

202151 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)

203152 w = maximum (map (length . show . fst) values)

@@ -209,12 +158,12 @@ prettyDist d =

209158 showR n x = show x ++ " "

210159

211160-- | Pretty Print a distribution

212−prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FLN () a) => Dist a -> IO ()

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

213162prettyPrintDist = putStrLn . prettyDist @a

214163

215164-- Auxiliary functions

216165

217−fromFreqs :: (FLN () a) => [(a,Prob)] -> Dist a

166+fromFreqs :: (FromLists Prob () (Normalize a)) => [(a,Prob)] -> Dist a

218167fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))

219168 where q = sum $ map snd xs

220169

src/LAoP/Matrix/Internal.hs+436−445

@@ -1,6 +1,4 @@

1−{-# LANGUAGE StrictData #-}

21{-# LANGUAGE TypeFamilyDependencies #-}

3−{-# LANGUAGE ConstraintKinds #-}

42{-# LANGUAGE AllowAmbiguousTypes #-}

53{-# LANGUAGE DataKinds #-}

64{-# LANGUAGE FlexibleContexts #-}

@@ -30,7 +28,7 @@

3028-- motivation behind the library, the underlying theory, and implementation details.

3129--

3230-- This module offers many of the combinators mentioned in the work of

33−-- Macedo (2012) and Oliveira (2012).

31+-- Macedo (2012) and Oliveira (2012).

3432--

3533-- This is an Internal module and it is no supposed to be imported.

3634--

@@ -43,26 +41,17 @@ module LAoP.Matrix.Internal

4341 --

4442 -- There exists two type families that make it easier to write

4543 -- matrix dimensions: 'FromNat' and 'Count'. This approach

46− -- leads to a very straightforward implementation

47− -- of LAoP combinators.

44+ -- leads to a very straightforward implementation

45+ -- of LAoP combinators.

4846

4947 -- * Type safe matrix representation

5048 Matrix (..),

5149

52− -- * Constraint type aliases

53− Countable,

54− CountableDims,

55− CountableN,

56− CountableDimsN,

57− FL,

58− FLN,

59− Liftable,

60− Trivial,

61−

6250 -- * Primitives

51+ empty,

6352 one,

64− join,

65− fork,

53+ junc,

54+ split,

6655

6756 -- * Auxiliary type families

6857 FromNat,

@@ -75,7 +64,6 @@ module LAoP.Matrix.Internal

7564 toLists,

7665 toList,

7766 matrixBuilder,

78− matrixBuilder',

7967 row,

8068 col,

8169 zeros,

@@ -86,38 +74,28 @@ module LAoP.Matrix.Internal

8674 -- * Misc

8775 -- ** Get dimensions

8876 columns,

89− columns',

9077 rows,

91− rows',

9278

9379 -- ** Matrix Transposition

9480 tr,

9581

96− -- ** Scalar multiplication/division of matrices

97− (.|),

98− (./),

99−

10082 -- ** Selective operator

101− select,

102− branch,

83+ select,

10384

10485 -- ** McCarthy's Conditional

10586 cond,

10687

10788 -- ** Matrix "abiding"

108− abideJF,

109− abideFJ,

110−

111− -- ** Zip matrices

112− zipWithM,

89+ abideJS,

90+ abideSJ,

11391

11492 -- * Biproduct approach

115− -- ** Fork

93+ -- ** Split

11694 (===),

11795 -- *** Projections

11896 p1,

11997 p2,

120− -- ** Join

98+ -- ** Junc

12199 (|||),

122100 -- *** Injections

123101 i1,

@@ -130,13 +108,12 @@ module LAoP.Matrix.Internal

130108

131109 -- | Note that given the restrictions imposed it is not possible to

132110 -- implement the standard type classes present in standard Haskell.

133−

134111 -- *** Matrix pairing projections

135− fstM,

136− sndM,

112+ kp1,

113+ kp2,

137114

138115 -- *** Matrix pairing

139− kr,

116+ khatri,

140117

141118 -- * Matrix composition and lifting

142119

@@ -144,10 +121,10 @@ module LAoP.Matrix.Internal

144121

145122 -- | Note that given the restrictions imposed it is not possible to

146123 -- implement the standard type classes present in standard Haskell.

147− iden,

124+ identity,

148125 comp,

149− fromF',

150126 fromF,

127+ fromF',

151128

152129 -- * Matrix printing

153130 pretty,

@@ -174,22 +151,21 @@ import LAoP.Utils.Internal

174151import Data.Bool

175152import Data.Kind

176153import Data.List

177−import Data.Maybe

178154import Data.Proxy

179−import Data.Typeable

180155import Data.Void

181156import GHC.TypeLits

182157import Data.Type.Equality

183158import GHC.Generics

184159import Control.DeepSeq

185−import Prelude hiding (id, (.))

186−import qualified Prelude (id, (.))

160+import Control.Category

161+import Prelude hiding ((.))

187162

188163-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.

189164data Matrix e cols rows where

165+ Empty :: Matrix e Void Void

190166 One :: e -> Matrix e () ()

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)

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)

193169

194170deriving instance (Show e) => Show (Matrix e cols rows)

195171

@@ -199,214 +175,221 @@ deriving instance (Show e) => Show (Matrix e cols rows)

199175-- 'Generic' instance.

200176type family Count (d :: Type) :: Nat where

201177 Count (Natural n m) = (m - n) + 1

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)

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)

206181 -- Generics

207− Count (M1 _ _ f p) = Count (f p)

208− Count (K1 _ _ _) = 1

209− Count (V1 _) = 0

210− Count (U1 _) = 1

182+ Count (M1 _ _ f p) = Count (f p)

183+ Count (K1 _ _ _) = 1

184+ Count (V1 _) = 0

185+ Count (U1 _) = 1

211186 Count ((:*:) a b p) = Count (a p) * Count (b p)

212187 Count ((:+:) a b p) = Count (a p) + Count (b p)

213− Count d = Count (Rep d R)

188+ Count d = Count (Rep d R)

214189

215190-- | Type family that computes of a given type dimension from a given natural

216191--

217192-- Thanks to Li-Yao Xia this type family is super fast.

218193type family FromNat (n :: Nat) :: Type where

194+ FromNat 0 = Void

219195 FromNat 1 = ()

220196 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))

221197

222198type family FromNat' (b :: Bool) (m :: Type) :: Type where

223− FromNat' 'True m = Either m m

199+ FromNat' 'True m = Either m m

224200 FromNat' 'False m = Either () (Either m m)

225201

226202-- | Type family that normalizes the representation of a given data

227203-- structure

228204type family Normalize (d :: Type) :: Type where

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.

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.

244212instance (Num e) => Category (Matrix e) where

245− type Object (Matrix e) a = (FL a a, Countable a)

246− id = iden

247− (.) = comp

213+ id = undefined

214+ (.) = comp

248215

249216instance NFData e => NFData (Matrix e cols rows) where

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

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

253221

254222instance Eq e => Eq (Matrix e cols rows) where

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

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

260229

261230instance 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)

271231

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)

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)

275262

276263instance Ord e => Ord (Matrix e cols rows) where

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

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

282270

283271-- Primitives

284272

273+-- | Empty matrix constructor

274+empty :: Matrix e Void Void

275+empty = Empty

276+

285277-- | Unit matrix constructor

286278one :: e -> Matrix e () ()

287279one = One

288280

289−-- | Matrix 'Join' constructor

290−join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

291−join = Join

281+-- | Matrix 'Junc' constructor

282+junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

283+junc = Junc

292284

293285infixl 3 |||

294286

295−-- | Matrix 'Join' constructor

287+-- | Matrix 'Junc' constructor

296288(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

297−(|||) = Join

289+(|||) = Junc

298290

299−-- | Matrix 'Fork' constructor

300−fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

301−fork = Fork

291+-- | Matrix 'Split' constructor

292+split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

293+split = Split

302294

303295infixl 2 ===

304296

305−-- | Matrix 'Fork' constructor

297+-- | Matrix 'Split' constructor

306298(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

307−(===) = Fork

299+(===) = Split

308300

309301-- Construction

310302

311303-- | Type class for defining the 'fromList' conversion function.

312304--

313305-- Given that it is not possible to branch on types at the term level type

314−-- classes are needed very much like an inductive definition but on types.

315−class FromLists cols rows where

306+-- classes are needed bery much like an inductive definition but on types.

307+class FromLists e cols rows where

316308 -- | Build a matrix out of a list of list of elements. Throws a runtime

317309 -- error if the dimensions do not match.

318310 fromLists :: [[e]] -> Matrix e cols rows

319311

320−instance FromLists () () where

312+instance FromLists e Void Void where

313+ fromLists [] = Empty

314+ fromLists _ = error "Wrong dimensions"

315+

316+instance {-# OVERLAPPING #-} FromLists e () () where

321317 fromLists [[e]] = One e

322318 fromLists _ = error "Wrong dimensions"

323319

324−instance (FromLists cols ()) => FromLists (Either () cols) () where

325− fromLists [h : t] = Join (One h) (fromLists [t])

320+instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where

321+ fromLists [h : t] = Junc (One h) (fromLists [t])

326322 fromLists _ = error "Wrong dimensions"

327323

328−instance {-# OVERLAPPABLE #-}

329− (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where

330− fromLists [l] =

324+instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), KnownNat (Count a)) => FromLists e (Either a b) () where

325+ fromLists [l] =

331326 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

332− in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])

327+ in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])

333328 fromLists _ = error "Wrong dimensions"

334329

335−instance (FromLists () rows) => FromLists () (Either () rows) where

336− fromLists ([h] : t) = Fork (One h) (fromLists t)

330+instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where

331+ fromLists ([h] : t) = Split (One h) (fromLists t)

337332 fromLists _ = error "Wrong dimensions"

338333

339−instance {-# OVERLAPPABLE #-}

340− (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where

341− fromLists l@([_] : _) =

334+instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, KnownNat (Count a)) => FromLists e () (Either a b) where

335+ fromLists l@([_] : _) =

342336 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

343− in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))

337+ in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))

344338 fromLists _ = error "Wrong dimensions"

345339

346−instance (FromLists (Either a b) c, FromLists (Either a b) d, Countable c)

347− => FromLists (Either a b) (Either c d) where

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

348341 fromLists l@(h : t) =

349342 let lh = length h

350343 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))

351− condition = all ((== lh) . length) t

344+ condition = all (== lh) (map length t)

352345 in if lh > 0 && condition

353− then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))

346+ then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))

354347 else error "Not all rows have the same length"

355348

356349-- | Matrix builder function. Constructs a matrix provided with

357−-- a construction function that operates with indices.

358−matrixBuilder' ::

350+-- a construction function.

351+matrixBuilder ::

359352 forall e cols rows.

360− ( FL cols rows,

361− CountableDims cols rows

362− ) => ((Int, Int) -> e) -> Matrix e cols rows

363−matrixBuilder' f =

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 =

364360 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

365361 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))

366362 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]

367363 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions

368364

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−

389365-- | Constructs a column vector matrix

390−col :: (FL () rows) => [e] -> Matrix e () rows

366+col :: (FromLists e () rows) => [e] -> Matrix e () rows

391367col = fromLists . map (: [])

392368

393369-- | Constructs a row vector matrix

394−row :: (FL cols ()) => [e] -> Matrix e cols ()

370+row :: (FromLists e cols ()) => [e] -> Matrix e cols ()

395371row = fromLists . (: [])

396372

397373-- | Lifts functions to matrices with arbitrary dimensions.

398374--

399375-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

400376-- types @a@ or @b@ allows.

401−fromF' ::

377+fromF ::

402378 forall a b cols rows e.

403− ( Liftable e a b,

404− CountableDims cols rows,

405− FL rows cols

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

406389 ) =>

407390 (a -> b) ->

408391 Matrix e cols rows

409−fromF' f =

392+fromF f =

410393 let minA = minBound @a

411394 maxA = maxBound @a

412395 minB = minBound @b

@@ -416,8 +399,8 @@ fromF' f =

416399 elementsA = take ccols [minA .. maxA]

417400 elementsB = take rrows [minB .. maxB]

418401 combinations = (,) <$> elementsA <*> elementsB

419− combAp = map snd . sort . map (\(a, b) -> if f a == b

420− then ((fromEnum a, fromEnum b), 1)

402+ combAp = map snd . sort . map (\(a, b) -> if f a == b

403+ then ((fromEnum a, fromEnum b), 1)

421404 else ((fromEnum a, fromEnum b), 0)) $ combinations

422405 mList = buildList combAp rrows

423406 in tr $ fromLists mList

@@ -427,23 +410,48 @@ fromF' f =

427410

428411-- | Lifts functions to matrices with dimensions matching @a@ and @b@

429412-- cardinality's.

430−fromF ::

413+fromF' ::

431414 forall a b e.

432− ( Liftable e a b,

433− CountableDimsN a b,

434− FLN b a

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)

435425 ) =>

436426 (a -> b) ->

437427 Matrix e (Normalize a) (Normalize b)

438−fromF = fromF'

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

439446

440447-- Conversion

441448

442449-- | Converts a matrix to a list of lists of elements.

443450toLists :: Matrix e cols rows -> [[e]]

451+toLists Empty = []

444452toLists (One e) = [[e]]

445−toLists (Fork l r) = toLists l ++ toLists r

446−toLists (Join l r) = zipWith (++) (toLists l) (toLists r)

453+toLists (Split l r) = toLists l ++ toLists r

454+toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)

447455

448456-- | Converts a matrix to a list of elements.

449457toList :: Matrix e cols rows -> [e]

@@ -452,38 +460,37 @@ toList = concat . toLists

452460-- Zeros Matrix

453461

454462-- | The zero matrix. A matrix wholly filled with zeros.

455−zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows

456−zeros = matrixBuilder' (const 0)

463+zeros :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows

464+zeros = matrixBuilder (const 0)

457465

458466-- Ones Matrix

459467

460468-- | The ones matrix. A matrix wholly filled with ones.

461469--

462470-- Also known as T (Top) matrix.

463−ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows

464−ones = matrixBuilder' (const 1)

471+ones :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows

472+ones = matrixBuilder (const 1)

465473

466474-- Const Matrix

467475

468476-- | The constant matrix constructor. A matrix wholly filled with a given

469477-- value.

470−constant :: (Num e, FL cols rows, CountableDims cols rows) => e -> Matrix e cols rows

471−constant e = matrixBuilder' (const e)

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)

472480

473481-- Bang Matrix

474482

475483-- | The T (Top) row vector matrix.

476−bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()

484+bang :: forall e cols. (Num e, Enum e, FromLists e cols (), KnownNat (Count cols)) => Matrix e cols ()

477485bang =

478486 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

479487 in fromLists [take c [1, 1 ..]]

480488

481−-- iden Matrix

489+-- Identity Matrix

482490

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 #-}

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 (==))

487494

488495-- Matrix composition (MMM)

489496

@@ -492,90 +499,60 @@ iden = matrixBuilder' (bool 0 1 . uncurry (==))

492499-- This definition takes advantage of divide-and-conquer and fusion laws

493500-- from LAoP.

494501comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

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)

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

522507

523508-- Projections

524509

525510-- | Biproduct first component projection

526−p1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e (Either m n) m

527−p1 = join id zeros

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

528516

529517-- | Biproduct second component projection

530−p2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e (Either m n) n

531−p2 = join zeros id

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

532523

533524-- Injections

534525

535526-- | Biproduct first component injection

536−i1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e m (Either m n)

527+i1 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e m (Either m n)

537528i1 = tr p1

538529

539530-- | Biproduct second component injection

540−i2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e n (Either m n)

531+i2 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e n (Either m n)

541532i2 = tr p2

542533

543534-- Dimensions

544535

545536-- | Obtain the number of rows.

546537--

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

551−rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))

552−

553−-- | Obtain the number of rows in an inefficient manner, but without any

554−-- constraints.

538+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

539+-- dimensions in constant time.

555540--

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

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

544+rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))

561545

562546-- | Obtain the number of columns.

547+--

548+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

549+-- dimensions in constant time.

563550--

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

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

568554columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))

569555

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−

579556-- Coproduct Bifunctor

580557

581558infixl 5 -|-

@@ -584,172 +561,157 @@ infixl 5 -|-

584561(-|-) ::

585562 forall e n k m j.

586563 ( Num e,

587− CountableDims j k,

588− FL k k,

589− FL j k,

590− FL k j,

591− FL j j

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

592570 ) =>

593571 Matrix e n k ->

594572 Matrix e m j ->

595573 Matrix e (Either n m) (Either k j)

596−(-|-) a b = Join (i1 . a) (i2 . b)

574+(-|-) a b = Junc (i1 . a) (i2 . b)

597575

598576-- Khatri Rao Product and projections

599577

600578-- | Khatri Rao product first component projection matrix.

601−fstM ::

579+kp1 ::

602580 forall e m k .

603581 ( Num e,

604− CountableDims k m,

605− FL (Normalize (m, k)) m,

606− CountableN (m, k)

582+ KnownNat (Count k),

583+ FromLists e (FromNat (Count m * Count k)) m,

584+ KnownNat (Count m),

585+ KnownNat (Count (Normalize (m, k)))

607586 ) => Matrix e (Normalize (m, k)) m

608−fstM = matrixBuilder' f

587+kp1 = matrixBuilder f

609588 where

610589 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

611590 f (x, y)

612591 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1

613− | otherwise = 0

592+ | otherwise = 0

614593

615594-- | Khatri Rao product second component projection matrix.

616−sndM ::

595+kp2 ::

617596 forall e m k .

618597 ( Num e,

619− CountableDims k m,

620− FL (Normalize (m, k)) k,

621− CountableN (m, k)

598+ KnownNat (Count k),

599+ FromLists e (FromNat (Count m * Count k)) k,

600+ KnownNat (Count m),

601+ KnownNat (Count (Normalize (m, k)))

622602 ) => Matrix e (Normalize (m, k)) k

623−sndM = matrixBuilder' f

603+kp2 = matrixBuilder f

624604 where

625605 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

626606 f (x, y)

627607 | x == y || mod (y - x) offset == 0 = 1

628− | otherwise = 0

608+ | otherwise = 0

629609

630610-- | Khatri Rao Matrix product also known as matrix pairing.

631611--

632612-- NOTE: That this is not a true categorical product, see for instance:

633−--

613+--

634614-- @

635−-- | fstM . kr a b == a

636−-- kr a b ==> |

637−-- | sndM . kr a b == b

615+-- | kp1 . khatri a b == a

616+-- khatri a b ==> |

617+-- | kp2 . khatri a b == b

638618-- @

639619--

640620-- __Emphasis__ on the implication symbol.

641−kr ::

642− forall e cols a b.

621+khatri ::

622+ forall e cols a b.

643623 ( Num e,

644− CountableDims a b,

645− CountableN (a, b),

646− FL (Normalize (a, b)) a,

647− FL (Normalize (a, b)) b

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

648629 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, 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)

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

653634

654635-- Product Bifunctor (Kronecker)

655636

656637infixl 4 ><

657638

658639-- | Matrix product functor also known as kronecker product

659−(><) ::

660− forall e m p n q.

640+(><) ::

641+ forall e m p n q.

661642 ( Num e,

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− )

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+ )

670654 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))

671655(><) a b =

672− let fstM' = fstM @e @m @n

673− sndM' = sndM @e @m @n

674− in kr (a . fstM') (b . sndM')

656+ let kp1' = kp1 @e @m @n

657+ kp2' = kp2 @e @m @n

658+ in khatri (a . kp1') (b . kp2')

675659

676−-- Matrix abide Join Fork

660+-- Matrix abide Junc Split

677661

678−-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.

679−--

662+-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

663+--

680664-- Law:

681665--

682666-- @

683−-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)

667+-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

684668-- @

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)

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)

690675

691−-- Matrix abide Fork Join

676+-- Matrix abide Split Junc

692677

693−-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.

694−--

678+-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

679+--

695680-- @

696−-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)

681+-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

697682-- @

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)

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)

703689

704690-- Matrix transposition

705691

706692-- | Matrix transposition.

707693tr :: Matrix e cols rows -> Matrix e rows cols

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)

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)

711698

712699-- Selective 'select' operator

713700

714701-- | Selective functors 'select' operator equivalent inspired by the

715702-- ArrowMonad solution presented in the paper.

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

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

743705

744706-- McCarthy's Conditional

745707

746708-- | McCarthy's Conditional expresses probabilistic choice.

747709cond ::

748− ( Trivial cols,

749− Countable cols,

750− FL () cols,

751− FL cols (),

752− FL cols cols,

710+ ( cols ~ FromNat (Count cols),

711+ KnownNat (Count cols),

712+ FromLists e () cols,

713+ FromLists e cols (),

714+ FromLists e cols cols,

753715 Bounded a,

754716 Enum a,

755717 Num e,

@@ -757,14 +719,14 @@ cond ::

757719 )

758720 =>

759721 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows

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,

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,

768730 Bounded a,

769731 Enum a,

770732 Num e,

@@ -772,86 +734,63 @@ grd ::

772734 )

773735 =>

774736 (a -> Bool) -> Matrix e q (Either q q)

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− )

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+ )

786751 => (a -> Bool) -> Matrix e q q

787−corr p = let f = fromF' p :: Matrix e q ()

788− in kr f (id :: Matrix e q q)

752+corr p = let f = fromF p :: Matrix e q ()

753+ in khatri f (identity :: Matrix e q q)

789754

790755-- Pretty print

791756

792757prettyAux :: Show e => [[e]] -> [[e]] -> String

793758prettyAux [] _ = ""

794−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"

759+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"

795760 where

796761 v = fmap show m

797762 widest = maximum $ fmap length v

798763 fill str = replicate (widest - length str - 2) ' ' ++ str

799−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"

764+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"

800765 where

801− v = fmap show m

802− widest = maximum $ fmap length v

766+ v = fmap show m

767+ widest = maximum $ fmap length v

803768 fill str = replicate (widest - length str - 2) ' ' ++ str

804−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++

769+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++

805770 prettyAux t l

806771 where

807− v = fmap show l

808− widest = maximum $ fmap length v

772+ v = fmap show l

773+ widest = maximum $ fmap length v

809774 fill str = replicate (widest - length str - 2) ' ' ++ str

810775

811776-- | Matrix pretty printer

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

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

822784 fill str = replicate (widest - length str) ' ' ++ str

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)

785+ blank = fill ""

839786

840787-- | Matrix pretty printer

841−prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()

788+prettyPrint :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> IO ()

842789prettyPrint = putStrLn . pretty

843790

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−

852791-- Relational operators functions

853792

854−type Boolean = Natural 0 1

793+type Boolean = Natural 0 1

855794type Relation a b = Matrix Boolean a b

856795

857796-- | Helper conversion function

@@ -867,47 +806,54 @@ fromBool False = nat 0

867806

868807-- | Relational negation

869808negateM :: Relation cols rows -> Relation cols rows

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)

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)

873814

874815-- | Relational addition

875816orM :: Relation cols rows -> Relation cols rows -> Relation cols rows

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

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

881823

882824-- | Relational multiplication

883825andM :: Relation cols rows -> Relation cols rows -> Relation cols rows

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

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

889832

890833-- | Relational subtraction

891834subM :: Relation cols rows -> Relation cols rows -> Relation cols rows

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

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

897841

898842-- | Matrix relational composition.

899843compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows

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

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

904849

905850-- | Matrix relational right division

906851divR :: Relation b c -> Relation b a -> Relation a c

852+divR Empty Empty = Empty

907853divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a

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)

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)

911857

912858-- | Matrix relational left division

913859divL :: Relation c b -> Relation a b -> Relation a c

@@ -923,52 +869,97 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)

923869--

924870-- NOTE: Be careful to not ask for a relation bigger than the cardinality of

925871-- types @a@ or @b@ allows.

926−fromFRel' ::

872+fromFRel ::

927873 forall a b cols rows.

928− ( Liftable Boolean a b,

929− CountableDims cols rows,

930− FL rows cols

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

931882 ) =>

932883 (a -> b) ->

933884 Relation cols rows

934−fromFRel' = fromF'

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

935903

936904-- | Lifts functions to relations with dimensions matching @a@ and @b@

937905-- cardinality's.

938−fromFRel ::

906+fromFRel' ::

939907 forall a b.

940− ( Liftable Boolean a b,

941− CountableDimsN a b,

942− FLN b a

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)

943916 ) =>

944917 (a -> b) ->

945918 Relation (Normalize a) (Normalize b)

946−fromFRel = fromFRel'

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

947937

948938-- | Lifts a relation function to a Boolean Matrix

949−toRel ::

939+toRel ::

950940 forall a b.

951941 ( Bounded a,

952942 Bounded b,

953943 Enum a,

954944 Enum b,

955945 Eq b,

956− CountableDims a b,

957− FLN b a

958− )

946+ KnownNat (Count (Normalize a)),

947+ KnownNat (Count (Normalize b)),

948+ FromLists (Natural 0 1) (Normalize b) (Normalize a)

949+ )

959950 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)

960951toRel f =

961952 let minA = minBound @a

962953 maxA = maxBound @a

963954 minB = minBound @b

964955 maxB = maxBound @b

965− ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))

966− rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))

956+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))

957+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))

967958 elementsA = take ccols [minA .. maxA]

968959 elementsB = take rrows [minB .. maxB]

969960 combinations = (,) <$> elementsA <*> elementsB

970− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)

971− then ((fromEnum a, fromEnum b), nat 1)

961+ combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)

962+ then ((fromEnum a, fromEnum b), nat 1)

972963 else ((fromEnum a, fromEnum b), nat 0)) $ combinations

973964 mList = buildList combAp rrows

974965 in tr $ fromLists mList

src/LAoP/Matrix/Nat.hs+187−194

@@ -1,4 +1,3 @@

1−{-# LANGUAGE ConstraintKinds #-}

21{-# LANGUAGE AllowAmbiguousTypes #-}

32{-# LANGUAGE DataKinds #-}

43{-# LANGUAGE DerivingVia #-}

@@ -54,22 +53,11 @@ module LAoP.Matrix.Nat

5453 -- * Type safe matrix representation

5554 Matrix (..),

5655

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−

6956 -- * Primitives

57+ empty,

7058 one,

71− join,

72− fork,

59+ junc,

60+ split,

7361

7462 -- * Auxiliary type families

7563 I.FromNat,

@@ -77,11 +65,11 @@ module LAoP.Matrix.Nat

7765 I.Normalize,

7866

7967 -- * Matrix construction and conversion

80− I.FL,

68+ I.FromLists,

8169 fromLists,

8270 toLists,

8371 toList,

84− matrixBuilder',

72+ matrixBuilder,

8573 row,

8674 col,

8775 zeros,

@@ -97,10 +85,6 @@ module LAoP.Matrix.Nat

9785 -- ** Matrix Transposition

9886 tr,

9987

100− -- ** Scalar multiplication/division of matrices

101− (.|),

102− (./),

103−

10488 -- ** Selective operator

10589 select,

10690

@@ -108,19 +92,16 @@ module LAoP.Matrix.Nat

10892 cond,

10993

11094 -- ** Matrix "abiding"

111− abideJF,

112− abideFJ,

113−

114− -- ** Zip Matrices

115− zipWithM,

95+ abideJS,

96+ abideSJ,

11697

11798 -- * Biproduct approach

118− -- ** Fork

99+ -- ** Split

119100 (===),

120101 -- *** Projections

121102 p1,

122103 p2,

123− -- ** Join

104+ -- ** Junc

124105 (|||),

125106 -- *** Injections

126107 i1,

@@ -134,11 +115,11 @@ module LAoP.Matrix.Nat

134115 -- | Note that given the restrictions imposed it is not possible to

135116 -- implement the standard type classes present in standard Haskell.

136117 -- *** Matrix pairing projections

137− fstM,

138− sndM,

118+ kp1,

119+ kp2,

139120

140121 -- *** Matrix pairing

141− kr,

122+ khatri,

142123

143124 -- * Matrix composition and lifting

144125

@@ -146,10 +127,10 @@ module LAoP.Matrix.Nat

146127

147128 -- | Note that given the restrictions imposed it is not possible to

148129 -- implement the standard type classes present in standard Haskell.

149− iden,

130+ identity,

150131 comp,

151− fromF',

152132 fromF,

133+ fromF',

153134

154135 -- * Matrix printing

155136 pretty,

@@ -160,95 +141,107 @@ where

160141import Data.Proxy

161142import GHC.TypeLits

162143import Control.DeepSeq

144+import qualified Control.Category as C

163145import qualified LAoP.Matrix.Internal as I

164146

165147newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))

166148 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))

167149

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))

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

179158

180159-- Primitives

181160

161+empty :: Matrix e 0 0

162+empty = M I.Empty

163+

182164one :: e -> Matrix e 1 1

183165one = M . I.One

184166

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)

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)

191173

192174infixl 3 |||

193175(|||) ::

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)

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)

206188

207189infixl 2 ===

208190(===) ::

209− (TrivialE a b) =>

210− Matrix e cols a ->

211− Matrix e cols b ->

212− Matrix e cols (a + b)

213−(===) = fork

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

214196

215197-- Construction

216198

217−fromLists :: (FLN e cols rows) => [[e]] -> Matrix e cols rows

199+fromLists :: (I.FromLists e (I.FromNat cols) (I.FromNat rows)) => [[e]] -> Matrix e cols rows

218200fromLists = M . I.fromLists

219201

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'

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

226207

227−col :: (I.FL () (I.FromNat rows)) => [e] -> Matrix e 1 rows

208+col :: (I.FromLists e () (I.FromNat rows)) => [e] -> Matrix e 1 rows

228209col = M . I.col

229210

230−row :: (I.FL (I.FromNat cols) ()) => [e] -> Matrix e cols 1

211+row :: (I.FromLists e (I.FromNat cols) ()) => [e] -> Matrix e cols 1

231212row = M . I.row

232213

233−fromF' ::

234− ( Liftable e a b,

235− CountableN cols,

236− CountableN rows,

237− FLN e rows cols

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)

238225 ) =>

239226 (a -> b) ->

240227 Matrix e cols rows

241−fromF' = M . I.fromF'

228+fromF = M . I.fromF

242229

243−fromF ::

244− ( Liftable e a b,

245− CountableNz a,

246− CountableNz b,

247− FLNz e b a

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)

248241 ) =>

249242 (a -> b) ->

250243 Matrix e (I.Count a) (I.Count b)

251−fromF = undefined -- M . I.fromF

244+fromF' = M . I.fromF'

252245

253246-- Conversion

254247

@@ -261,21 +254,21 @@ toList (M m) = I.toList m

261254-- Zeros Matrix

262255

263256zeros ::

264− (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

257+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>

265258 Matrix e cols rows

266259zeros = M I.zeros

267260

268261-- Ones Matrix

269262

270263ones ::

271− (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

264+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>

272265 Matrix e cols rows

273266ones = M I.ones

274267

275268-- Const Matrix

276269

277270constant ::

278− (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>

271+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>

279272 e ->

280273 Matrix e cols rows

281274constant = M . I.constant

@@ -284,52 +277,40 @@ constant = M . I.constant

284277

285278bang ::

286279 forall e cols.

287− (Num e, Enum e, I.FL (I.FromNat cols) (), CountableN cols) =>

280+ (Num e, Enum e, I.FromLists e (I.FromNat cols) (), KnownNat (I.Count (I.FromNat cols))) =>

288281 Matrix e cols 1

289282bang = M I.bang

290283

291−-- iden Matrix

284+-- Identity Matrix

292285

293−iden ::

294− (Num e, FLN e cols cols, CountableN cols) =>

286+identity ::

287+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat cols), KnownNat (I.Count (I.FromNat cols))) =>

295288 Matrix e cols cols

296−iden = M I.iden

289+identity = M I.identity

297290

298291-- Matrix composition (MMM)

299292

300293comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

301294comp (M a) (M b) = M (I.comp a b)

302295

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−

317296p1 ::

318297 ( Num e,

319− CountableDimsN n m,

320− FLN e n m,

321− FLN e m m,

322− TrivialE m n

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)

323303 ) =>

324304 Matrix e (m + n) m

325305p1 = M I.p1

326306

327307p2 ::

328308 ( Num e,

329− CountableDimsN n m,

330− FLN e m n,

331− FLN e n n,

332− TrivialE m n

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)

333314 ) =>

334315 Matrix e (m + n) n

335316p2 = M I.p2

@@ -338,125 +319,138 @@ p2 = M I.p2

338319

339320i1 ::

340321 ( Num e,

341− CountableDimsN n rows,

342− FLN e n rows,

343− FLN e rows rows,

344− TrivialE rows n

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)

345327 ) =>

346328 Matrix e rows (rows + n)

347329i1 = tr p1

348330

349331i2 ::

350332 ( Num e,

351− CountableDimsN rows m,

352− FLN e m rows,

353− FLN e rows rows,

354− TrivialE m rows

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)

355338 ) =>

356339 Matrix e rows (m + rows)

357340i2 = tr p2

358341

359342-- Dimensions

360343

361−rows :: (CountableN rows) => Matrix e cols rows -> Int

344+rows :: (KnownNat (I.Count (I.FromNat rows))) => Matrix e cols rows -> Int

362345rows (M m) = I.rows m

363346

364−columns :: (CountableN cols) => Matrix e cols rows -> Int

347+columns :: (KnownNat (I.Count (I.FromNat cols))) => Matrix e cols rows -> Int

365348columns (M m) = I.columns m

366349

350+-- Coproduct Bifunctor

351+

367352infixl 5 -|-

368353

369−-- | Coproduct Bifunctor (Direct sum)

370354(-|-) ::

371355 ( Num e,

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

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)

379364 ) =>

380365 Matrix e n k ->

381366 Matrix e m j ->

382367 Matrix e (n + m) (k + j)

383368(-|-) (M a) (M b) = M ((I.-|-) a b)

384369

385−-- | Khatri Rao Product first projection

386−fstM ::

370+-- Khatri Rao Product and projections

371+

372+kp1 ::

387373 forall e m k .

388374 ( Num e,

389− CountableDimsN m k,

390− CountableN (m * k),

391− FLN e (m * k) m,

392− TrivialP m k

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))

393380 ) => Matrix e (m * k) m

394−fstM = M (I.fstM @e @(I.FromNat m) @(I.FromNat k))

381+kp1 = M (I.kp1 @e @(I.FromNat m) @(I.FromNat k))

395382

396−-- | Khatri Rao Product second projection

397−sndM ::

383+kp2 ::

398384 forall e m k.

399385 ( Num e,

400− CountableDimsN k m,

401− FLN e (m * k) k,

402− CountableN (m * k),

403− TrivialP m k

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))

404391 ) => Matrix e (m * k) k

405−sndM = M (I.sndM @e @(I.FromNat m) @(I.FromNat k))

392+kp2 = M (I.kp2 @e @(I.FromNat m) @(I.FromNat k))

406393

407−-- | Khatri Rao Product

408−kr ::

394+khatri ::

409395 forall e cols a b.

410396 ( Num e,

411− CountableDimsN a b,

412− CountableN (a * b),

413− FLN e (a * b) a,

414− FLN e (a * b) b,

415− TrivialP a b

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))

416403 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)

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

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)

421410

422−-- | Product Bifunctor (Kronecker)

423411infixl 4 ><

424412

425413(><) ::

426414 forall e m p n q.

427415 ( Num e,

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

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))

437428 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)

438429(><) a b =

439− let fstM' = fstM @e @m @n

440− sndM' = sndM @e @m @n

441− in kr (comp a fstM') (comp b sndM')

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

442435

443−-- | Matrix abide Join Fork

444−abideJF :: Matrix e cols rows -> Matrix e cols rows

445−abideJF (M m) = M (I.abideJF m)

436+abideJS :: Matrix e cols rows -> Matrix e cols rows

437+abideJS (M m) = M (I.abideJS m)

446438

447−-- | Matrix abide Fork Join

448−abideFJ :: Matrix e cols rows -> Matrix e cols rows

449−abideFJ (M m) = M (I.abideFJ m)

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

450445

451−-- | Matrix transposition

452446tr :: Matrix e cols rows -> Matrix e rows cols

453447tr (M m) = M (I.tr m)

454448

455449-- Selective 'select' operator

456450select ::

457451 ( Num e,

458− FLN e rows1 rows1,

459− CountableN rows1,

452+ I.FromLists e (I.FromNat rows1) (I.FromNat rows1),

453+ KnownNat (I.Count (I.FromNat rows1)),

460454 I.FromNat rows2 ~ I.FromNat rows1,

461455 I.FromNat cols1 ~ I.FromNat cols2,

462456 I.FromNat rows3 ~ Either (I.FromNat cols3) (I.FromNat rows1)

@@ -467,11 +461,14 @@ select (M m) (M y) = M (I.select m y)

467461

468462cond ::

469463 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,

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

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

475472 )

476473 =>

477474 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows

@@ -479,12 +476,8 @@ cond p (M a) (M b) = M (I.cond p a b)

479476

480477-- Pretty print

481478

482−pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String

479+pretty :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> String

483480pretty (M m) = I.pretty m

484481

485−prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()

482+prettyPrint :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> IO ()

486483prettyPrint (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+240−283

@@ -1,11 +1,10 @@

1−{-# LANGUAGE ConstraintKinds #-}

2−{-# LANGUAGE TypeFamilies #-}

31{-# LANGUAGE AllowAmbiguousTypes #-}

42{-# LANGUAGE DataKinds #-}

53{-# LANGUAGE DerivingVia #-}

64{-# LANGUAGE FlexibleContexts #-}

75{-# LANGUAGE FlexibleInstances #-}

86{-# LANGUAGE GADTs #-}

7+{-# LANGUAGE KindSignatures #-}

98{-# LANGUAGE MultiParamTypeClasses #-}

109{-# LANGUAGE NoStarIsType #-}

1110{-# LANGUAGE RankNTypes #-}

@@ -56,24 +55,15 @@ module LAoP.Matrix.Type

5655 -- * Type safe matrix representation

5756 Matrix (..),

5857

59− -- * Constraint type aliases

60− Countable,

61− CountableDims,

62− CountableN,

63− CountableDimsN,

64− FLN,

65− Liftable,

66− Trivial,

67− TrivialP,

68−

6958 -- * Type aliases

7059 Zero,

7160 One,

7261

7362 -- * Primitives

63+ empty,

7464 one,

75− join,

76− fork,

65+ junc,

66+ split,

7767

7868 -- * Auxiliary type families

7969 I.FromNat,

@@ -85,27 +75,24 @@ module LAoP.Matrix.Type

8575 fromLists,

8676 toLists,

8777 toList,

88− matrixBuilder',

8978 matrixBuilder,

9079 row,

9180 col,

9281 zeros,

9382 ones,

9483 bang,

95− point,

9684 constant,

9785

9886 -- * Functor instance equivalent function

9987 fmapM,

100− bimapM,

10188

10289 -- * Applicative/Monoidal instance equivalent functions

10390 unitM,

10491 multM,

10592

10693 -- * Selective equivalent instance function

107− selectM,

108−

94+ selectM,

95+

10996 -- * Monad equivalent instance function

11097 returnM,

11198 bindM,

@@ -113,34 +100,25 @@ module LAoP.Matrix.Type

113100 -- * Misc

114101 -- ** Get dimensions

115102 columns,

116− columns',

117103 rows,

118− rows',

119104

120105 -- ** Matrix Transposition

121106 tr,

122107

123− -- ** Scalar multiplication/division of matrices

124− (.|),

125− (./),

126−

127108 -- ** McCarthy's Conditional

128109 cond,

129110

130111 -- ** Matrix "abiding"

131− abideJF,

132− abideFJ,

133−

134− -- ** Zip Matrices

135− zipWithM,

112+ abideJS,

113+ abideSJ,

136114

137115 -- * Biproduct approach

138− -- ** Fork

116+ -- ** Split

139117 (===),

140118 -- *** Projections

141119 p1,

142120 p2,

143− -- ** Join

121+ -- ** Junc

144122 (|||),

145123 -- *** Injections

146124 i1,

@@ -154,11 +132,11 @@ module LAoP.Matrix.Type

154132 -- | Note that given the restrictions imposed it is not possible to

155133 -- implement the standard type classes present in standard Haskell.

156134 -- *** Matrix pairing projections

157− fstM,

158− sndM,

135+ kp1,

136+ kp2,

159137

160138 -- *** Matrix pairing

161− kr,

139+ khatri,

162140

163141 -- * Matrix composition and lifting

164142

@@ -166,10 +144,10 @@ module LAoP.Matrix.Type

166144

167145 -- | Note that given the restrictions imposed it is not possible to

168146 -- implement the standard type classes present in standard Haskell.

169− iden,

147+ identity,

170148 comp,

171− fromF',

172149 fromF,

150+ fromF',

173151

174152 -- * Relation

175153 toRel,

@@ -183,44 +161,23 @@ where

183161import Data.Void

184162import Data.Proxy

185163import Data.Kind

186−import GHC.TypeLits

164+import GHC.TypeLits

187165import Control.DeepSeq

188166import LAoP.Utils

167+import qualified Control.Category as C

189168import qualified LAoP.Matrix.Internal as I

190−import Prelude hiding (id, (.))

191169

192170newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))

193171 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))

194172

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)

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

224181

225182-- | Zero type alias

226183type Zero = Void

@@ -230,75 +187,92 @@ type One = ()

230187

231188-- Primitives

232189

190+-- | Empty matrix constructor

191+empty :: Matrix e Zero Zero

192+empty = M I.Empty

193+

233194-- | Unit matrix constructor

234195one :: e -> Matrix e One One

235196one = M . I.One

236197

237−-- | Matrix 'Join' constructor

238−join ::

198+-- | Matrix 'Junc' constructor

199+junc ::

200+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

239201 Matrix e a rows ->

240202 Matrix e b rows ->

241203 Matrix e (Either a b) rows

242−join (M a) (M b) = M (I.Join a b)

204+junc (M a) (M b) = M (I.Junc a b)

243205

244206infixl 3 |||

245−-- | Matrix 'Join' constructor

207+-- | Matrix 'Junc' constructor

246208(|||) ::

209+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

247210 Matrix e a rows ->

248211 Matrix e b rows ->

249212 Matrix e (Either a b) rows

250−(|||) = join

213+(|||) = junc

251214

252−-- | Matrix 'Fork' constructor

253−fork ::

215+-- | Matrix 'Split' constructor

216+split ::

217+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

254218 Matrix e cols a ->

255219 Matrix e cols b ->

256220 Matrix e cols (Either a b)

257−fork (M a) (M b) = M (I.Fork a b)

221+split (M a) (M b) = M (I.Split a b)

258222

259223infixl 2 ===

260−-- | Matrix 'Fork' constructor

224+-- | Matrix 'Split' constructor

261225(===) ::

226+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

262227 Matrix e cols a ->

263228 Matrix e cols b ->

264229 Matrix e cols (Either a b)

265−(===) = fork

230+(===) = split

266231

267232-- Functor hierarchy

268233

269234-- | Functor instance equivalent function

270−fmapM ::

271− ( Liftable e a b,

272− CountableDimsN a b,

273− FLN b a

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)

274246 )

275247 =>

276248 (a -> b) -> Matrix e c a -> Matrix e c b

277−fmapM f m = fromF f . m

249+fmapM f m = fromF' f `comp` m

278250

279251-- | Applicative instance equivalent 'unit' function,

280252unitM :: (Num e) => Matrix e () ()

281253unitM = one 1

282254

283255-- | Applicative instance equivalent 'unit' function,

284−multM ::

285− ( CountableDimsN a b,

286− CountableN (a, b),

256+multM ::

257+ ( KnownNat (I.Count (I.Normalize a)),

258+ KnownNat (I.Count (I.Normalize b)),

259+ KnownNat (I.Count (I.Normalize (a, b))),

287260 Num e,

288− FLN (a, b) a,

289− FLN (a, b) b,

290− TrivialP a b

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)

291264 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)

292−multM = kr

265+multM a b = khatri a b

266+

293267

294268-- | Monad instance equivalent 'return' function,

295−returnM ::

296− forall e a .

297− ( Num e,

298− Enum e,

299− Enum a,

300− FLN () a,

301− Countable a

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)

302276 ) => a -> Matrix e One a

303277returnM a = col l

304278 where

@@ -314,64 +288,73 @@ bindM = flip comp

314288

315289-- | Build a matrix out of a list of list of elements. Throws a runtime

316290-- error if the dimensions do not match.

317−fromLists :: (FLN cols rows) => [[e]] -> Matrix e cols rows

291+fromLists :: (I.FromLists e (I.Normalize cols) (I.Normalize rows)) => [[e]] -> Matrix e cols rows

318292fromLists = M . I.fromLists

319293

320294-- | Matrix builder function. Constructs a matrix provided with

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.

295+-- a construction function.

329296matrixBuilder ::

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)

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

339301

340302-- | Constructs a column vector matrix

341−col :: (FLN () rows) => [e] -> Matrix e One rows

303+col :: (I.FromLists e () (I.Normalize rows)) => [e] -> Matrix e One rows

342304col = M . I.col

343305

344306-- | Constructs a row vector matrix

345−row :: (FLN cols ()) => [e] -> Matrix e cols One

307+row :: (I.FromLists e (I.Normalize cols) ()) => [e] -> Matrix e cols One

346308row = M . I.row

347309

348310-- | Lifts functions to matrices with arbitrary dimensions.

349311--

350312-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

351313-- types @a@ or @b@ allows.

352−fromF' ::

353− ( Liftable e a b,

354− CountableDimsN cols rows,

355− FLN rows cols

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)

356325 ) =>

357− (a -> b) -> Matrix e cols rows

358−fromF' = M . I.fromF'

326+ (a -> b) ->

327+ Matrix e cols rows

328+fromF = M . I.fromF

359329

360330-- | Lifts functions to matrices with dimensions matching @a@ and @b@

361331-- cardinality's.

362−fromF ::

363− ( Liftable e a b,

364− CountableDimsN a b,

365− FLN b a

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)

366343 ) =>

367− (a -> b) -> Matrix e a b

368−fromF = M . I.fromF

344+ (a -> b) ->

345+ Matrix e a b

346+fromF' = M . I.fromF'

369347

370348-- | Lifts relation functions to Boolean Matrix

371349toRel ::

372− ( Liftable (Natural 0 1) a b,

373− CountableDims a b,

374− FLN b a

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)

375358 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b

376359toRel = M . I.toRel

377360

@@ -389,8 +372,8 @@ toList (M m) = I.toList m

389372

390373-- | The zero matrix. A matrix wholly filled with zeros.

391374zeros ::

392− (Num e, FLN cols rows, CountableDimsN cols rows)

393− => Matrix e cols rows

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

394377zeros = M I.zeros

395378

396379-- Ones Matrix

@@ -399,8 +382,8 @@ zeros = M I.zeros

399382--

400383-- Also known as T (Top) matrix.

401384ones ::

402− (Num e, FLN cols rows, CountableDimsN cols rows)

403− => Matrix e cols rows

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

404387ones = M I.ones

405388

406389-- Const Matrix

@@ -408,8 +391,9 @@ ones = M I.ones

408391-- | The constant matrix constructor. A matrix wholly filled with a given

409392-- value.

410393constant ::

411− (Num e, FLN cols rows, CountableDimsN cols rows)

412− => e -> Matrix e cols rows

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

413397constant = M . I.constant

414398

415399-- Bang Matrix

@@ -417,30 +401,17 @@ constant = M . I.constant

417401-- | The T (Top) row vector matrix.

418402bang ::

419403 forall e cols.

420− (Num e, Enum e, FLN cols (), CountableN cols) =>

404+ (Num e, Enum e, I.FromLists e (I.Normalize cols) (), KnownNat (I.Count (I.Normalize cols))) =>

421405 Matrix e cols One

422406bang = M I.bang

423407

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

408+-- Identity Matrix

437409

438−-- | iden matrix

439−iden ::

440− (Num e, FLN a a, CountableN a) =>

410+-- | Identity matrix

411+identity ::

412+ (Num e, I.FromLists e (I.Normalize a) (I.Normalize a), KnownNat (I.Count (I.Normalize a))) =>

441413 Matrix e a a

442−iden = M I.iden

443−{-# NOINLINE iden #-}

414+identity = M I.identity

444415

445416-- Matrix composition (MMM)

446417

@@ -450,32 +421,15 @@ iden = M I.iden

450421-- from LAoP.

451422comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

452423comp (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)

472424

473425-- | Biproduct first component projection

474426p1 ::

475427 ( Num e,

476− CountableDimsN n m,

477− FLN n m,

478− FLN m m

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)

479433 ) =>

480434 Matrix e (Either m n) m

481435p1 = M I.p1

@@ -483,9 +437,11 @@ p1 = M I.p1

483437-- | Biproduct second component projection

484438p2 ::

485439 ( Num e,

486− CountableDimsN n m,

487− FLN m n,

488− FLN n n

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)

489445 ) =>

490446 Matrix e (Either m n) n

491447p2 = M I.p2

@@ -495,9 +451,11 @@ p2 = M I.p2

495451-- | Biproduct first component injection

496452i1 ::

497453 ( Num e,

498− CountableDimsN n m,

499− FLN n m,

500− FLN m m

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)

501459 ) =>

502460 Matrix e m (Either m n)

503461i1 = tr p1

@@ -505,9 +463,11 @@ i1 = tr p1

505463-- | Biproduct second component injection

506464i2 ::

507465 ( Num e,

508− CountableDimsN n m,

509− FLN m n,

510− FLN n n

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)

511471 ) =>

512472 Matrix e n (Either m n)

513473i2 = tr p2

@@ -516,34 +476,24 @@ i2 = tr p2

516476

517477-- | Obtain the number of rows.

518478--

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

523−rows (M m) = I.rows m

524−

525−-- | Obtain the number of rows in an inefficient manner, but without any

526−-- constraints.

479+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

480+-- dimensions in constant time.

527481--

528−-- For a more efficient version see 'rows'.

529−rows' :: Matrix e cols rows -> Int

530−rows' (M m) = I.rows' m

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

485+rows (M m) = I.rows m

531486

532487-- | Obtain the number of columns.

488+--

489+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

490+-- dimensions in constant time.

533491--

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

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

538495columns (M m) = I.columns m

539496

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−

547497-- Coproduct Bifunctor

548498

549499infixl 5 -|-

@@ -551,11 +501,14 @@ infixl 5 -|-

551501-- | Matrix coproduct functor also known as matrix direct sum.

552502(-|-) ::

553503 ( Num e,

554− CountableDimsN j k,

555− FLN k k,

556− FLN j k,

557− FLN k j,

558− FLN j j

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)

559512 ) =>

560513 Matrix e n k ->

561514 Matrix e m j ->

@@ -565,98 +518,104 @@ infixl 5 -|-

565518-- Khatri Rao Product and projections

566519

567520-- | Khatri Rao product first component projection matrix.

568−fstM ::

521+kp1 ::

569522 forall e m k .

570523 ( Num e,

571− CountableDimsN m k,

572− CountableN (m, k),

573− FLN (m, k) m,

574− TrivialP m k

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)

575529 ) => Matrix e (m, k) m

576−fstM = M (I.fstM @e @(I.Normalize m) @(I.Normalize k))

530+kp1 = M (I.kp1 @e @(I.Normalize m) @(I.Normalize k))

577531

578532-- | Khatri Rao product second component projection matrix.

579−sndM ::

533+kp2 ::

580534 forall e m k.

581535 ( Num e,

582− CountableDimsN k m,

583− CountableN (m, k),

584− FLN (m, k) k,

585− TrivialP m k

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)

586541 ) => Matrix e (m, k) k

587−sndM = M (I.sndM @e @(I.Normalize m) @(I.Normalize k))

542+kp2 = M (I.kp2 @e @(I.Normalize m) @(I.Normalize k))

588543

589544-- | Khatri Rao Matrix product also known as matrix pairing.

590545--

591546-- NOTE: That this is not a true categorical product, see for instance:

592−--

547+--

593548-- @

594−-- | fstM . kr a b == a

595−-- kr a b ==> |

596−-- | sndM . kr a b == b

549+-- | kp1 `comp` khatri a b == a

550+-- khatri a b ==> |

551+-- | kp2 `comp` khatri a b == b

597552-- @

598553--

599554-- __Emphasis__ on the implication symbol.

600−kr ::

555+khatri ::

601556 forall e cols a b.

602557 ( Num e,

603− CountableDimsN a b,

604− CountableN (a, b),

605− FLN (a, b) a,

606− FLN (a, b) b,

607− TrivialP a b

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)

608564 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, 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

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

613569

614570-- Product Bifunctor (Kronecker)

615571

616572infixl 4 ><

617573

618−-- | Matrix product functor also known as Kronecker product

574+-- | Matrix product functor also known as kronecker product

619575(><) ::

620576 forall e m p n q.

621577 ( Num e,

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

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)

631590 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)

632591(><) a b =

633− let fstM' = fstM @e @m @n

634− sndM' = sndM @e @m @n

635− in kr (comp a fstM') (comp b sndM')

592+ let kp1' = kp1 @e @m @n

593+ kp2' = kp2 @e @m @n

594+ in khatri (comp a kp1') (comp b kp2')

636595

637−-- Matrix abide Join Fork

596+-- Matrix abide Junc Split

638597

639−-- | Matrix "abiding" followin the 'Join'-'Fork' abide law.

640−--

598+-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.

599+--

641600-- Law:

642601--

643602-- @

644−-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)

603+-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

645604-- @

646−abideJF :: Matrix e cols rows -> Matrix e cols rows

647−abideJF (M m) = M (I.abideJF m)

605+abideJS :: Matrix e cols rows -> Matrix e cols rows

606+abideJS (M m) = M (I.abideJS m)

648607

649−-- Matrix abide Fork Join

608+-- Matrix abide Split Junc

650609

651−-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.

652−--

610+-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.

611+--

653612-- Law:

654613--

655614-- @

656−-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)

615+-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

657616-- @

658−abideFJ :: Matrix e cols rows -> Matrix e cols rows

659−abideFJ (M m) = M (I.abideFJ m)

617+abideSJ :: Matrix e cols rows -> Matrix e cols rows

618+abideSJ (M m) = M (I.abideSJ m)

660619

661620-- Matrix transposition

662621

@@ -668,10 +627,11 @@ tr (M m) = M (I.tr m)

668627

669628-- | Selective functors 'select' operator equivalent inspired by the

670629-- ArrowMonad solution presented in the paper.

671−selectM ::

630+selectM ::

672631 ( Num e,

673− FLN b b,

674− CountableN b

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)

675635 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

676636selectM (M m) (M y) = M (I.select m y)

677637

@@ -679,14 +639,15 @@ selectM (M m) (M y) = M (I.select m y)

679639

680640-- | McCarthy's Conditional expresses probabilistic choice.

681641cond ::

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

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

690651 )

691652 =>

692653 (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b

@@ -695,13 +656,9 @@ cond p (M a) (M b) = M (I.cond p a b)

695656-- Pretty print

696657

697658-- | Matrix pretty printer

698−pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String

659+pretty :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> String

699660pretty (M m) = I.pretty m

700661

701662-- | Matrix pretty printer

702−prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()

663+prettyPrint :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> IO ()

703664prettyPrint (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+7−34

@@ -30,21 +30,12 @@ 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−

4333 -- * Primitives

34+ empty,

4435 one,

45− join,

36+ junc,

4637 (|||),

47− fork,

38+ split,

4839 (===),

4940

5041 -- * Auxiliary type families

@@ -62,7 +53,6 @@ module LAoP.Relation

6253 toList,

6354 toBool,

6455 pt,

65− belongs,

6656 relationBuilder,

6757 zeros,

6858 ones,

@@ -90,25 +80,15 @@ module LAoP.Relation

9080 injection,

9181 surjection,

9282 bijection,

93− domain,

94− range,

9583

9684 -- * Function division

9785 divisionF,

9886

99− -- * Relation division

100− divR,

101− divL,

102− divS,

103− shrunkBy,

104− overriddenBy,

105−

10687 -- * Relational pairing

10788 splitR,

108−

10989 -- ** Projections

110− fstR,

111− sndR,

90+ p1,

91+ p2,

11292 -- ** Bifunctor

11393 (><),

11494 -- * Relational coproduct

@@ -136,25 +116,18 @@ module LAoP.Relation

136116 linearOrder,

137117 equivalence,

138118 partialEquivalence,

139− difunctional,

140−

141− -- * Conditionals

142− equalizer,

143119

144120 -- ** McCarthy's Conditional

145− predR,

146− guard,

147− cond,

121+ -- cond,

148122

149123 -- * Relational composition and lifting

150− iden,

124+ identity,

151125 comp,

152126 fromF,

153127 fromF',

154128

155129 -- ** Relational application

156130 pointAp,

157− pointApBool,

158131

159132 -- * Matrix printing

160133 pretty,

src/LAoP/Relation/Internal.hs+317−486

@@ -1,5 +1,3 @@

1−{-# LANGUAGE TypeFamilies #-}

2−{-# LANGUAGE ConstraintKinds #-}

31{-# LANGUAGE GADTs #-}

42{-# LANGUAGE UndecidableInstances #-}

53{-# LANGUAGE FlexibleContexts #-}

@@ -41,21 +39,12 @@ module LAoP.Relation.Internal

4139 Relation (..),

4240 Boolean,

4341

44− -- * Constraint type synonyms

45− Countable,

46− CountableDims,

47− CountableN,

48− CountableDimsN,

49− FLN,

50− Liftable,

51− Trivial,

52− TrivialP,

53−

5442 -- * Primitives

43+ empty,

5544 one,

56− join,

45+ junc,

5746 (|||),

58− fork,

47+ split,

5948 (===),

6049

6150 -- * Auxiliary type families

@@ -66,15 +55,13 @@ module LAoP.Relation.Internal

6655 -- * Matrix construction and conversion

6756 I.FromLists,

6857 fromLists,

69− fromF',

7058 fromF,

59+ fromF',

7160 toRel,

7261 toLists,

7362 toList,

7463 toBool,

7564 pt,

76− belongs,

77− relationBuilder',

7865 relationBuilder,

7966 zeros,

8067 ones,

@@ -102,8 +89,6 @@ module LAoP.Relation.Internal

10289 injection,

10390 surjection,

10491 bijection,

105− domain,

106− range,

10792

10893 -- * Function division

10994 divisionF,

@@ -119,8 +104,8 @@ module LAoP.Relation.Internal

119104 splitR,

120105

121106 -- ** Projections

122− fstR,

123− sndR,

107+ p1,

108+ p2,

124109 -- ** Bifunctor

125110 (><),

126111 -- * Relational coproduct

@@ -148,7 +133,6 @@ module LAoP.Relation.Internal

148133 linearOrder,

149134 equivalence,

150135 partialEquivalence,

151− difunctional,

152136

153137 -- * Conditionals

154138 equalizer,

@@ -159,10 +143,10 @@ module LAoP.Relation.Internal

159143 cond,

160144

161145 -- * Relational composition and lifting

162− iden,

146+ identity,

163147 comp,

164− fromF',

165148 fromF,

149+ fromF',

166150

167151 -- ** Relational application

168152 pointAp,

@@ -178,13 +162,12 @@ import Data.Void

178162import qualified LAoP.Matrix.Internal as I

179163import LAoP.Utils.Internal

180164import Control.DeepSeq

165+import qualified Control.Category as C

181166import Data.Bool

182167import GHC.TypeLits

183−import Prelude hiding (id, (.))

184168

185169-- | Boolean type synonym for working with boolean matrices

186170type Boolean = Natural 0 1

187−type Powerset a = List (List a)

188171

189172-- | Relation data type.

190173newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))

@@ -192,34 +175,25 @@ newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))

192175

193176deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)

194177

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

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

211186

212187instance Num (Relation a b) where

213− -- | Matrix addition becomes Boolean matrix disjointion

188+ -- | Matrix addition becomes Boolean matrix disjunction

214189 (R a) + (R b) = R (I.orM a b)

215190

216191 -- | Matrix subtraction becomes Relational subtraction

217192 (R a) - (R b) = R (I.subM a b)

218193

219− -- | Matrix multiplication becomes Boolean matrix conjointion

194+ -- | Matrix multiplication becomes Boolean matrix conjunction

220195 (R a) * (R b) = R (I.andM a b)

221196

222− -- | Matrix negation becomes Boolean matrix negation

223197 negate (R a) = R (I.negateM a)

224198

225199-- Type alias

@@ -228,107 +202,105 @@ type One = ()

228202

229203-- Primitives

230204

205+-- | Empty matrix constructor

206+empty :: Relation Zero Zero

207+empty = R I.Empty

208+

231209-- | Unit matrix constructor

232210one :: Boolean -> Relation One One

233211one = R . I.One

234212

235−-- | Boolean Matrix 'Join' constructor, also known as relational coproduct.

213+-- | Boolean Matrix 'Junc' constructor, also known as relational coproduct.

236214--

237215-- See 'eitherR'.

238−join :: Relation a c -> Relation b c -> Relation (Either a b) c

239−join (R a) (R b) = R (I.Join a b)

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)

240219

241220infixl 3 |||

242−-- | Boolean Matrix 'Join' constructor

243−--

244−-- See 'eitherR'.

221+-- | Boolean Matrix 'Junc' constructor

245222(|||) ::

223+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

246224 Relation a c ->

247225 Relation b c ->

248226 Relation (Either a b) c

249−(|||) = join

227+(|||) = junc

250228

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)

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)

254233

255234infixl 2 ===

256−-- | Boolean Matrix 'Fork' constructor

235+-- | Boolean Matrix 'Split' constructor

257236(===) ::

237+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>

258238 Relation c a ->

259239 Relation c b ->

260240 Relation c (Either a b)

261−(===) = fork

241+(===) = split

262242

263243-- Construction

264244

265245-- | Build a matrix out of a list of list of elements. Throws a runtime

266246-- error if the dimensions do not match.

267−fromLists :: (FLN a b) => [[Boolean]] -> Relation a b

247+fromLists :: (I.FromLists Boolean (I.Normalize a) (I.Normalize b)) => [[Boolean]] -> Relation a b

268248fromLists = R . I.fromLists

269249

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.

250+-- | Matrix builder function. Constructs a matrix provided with

251+-- a construction function.

279252relationBuilder ::

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

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

288256relationBuilder = R . I.matrixBuilder

289257

290258-- | Lifts functions to matrices with arbitrary dimensions.

291259--

292260-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

293261-- types @a@ or @b@ allows.

294−fromF' ::

295− ( Liftable a b,

296− CountableDimsN c d,

297− FLN d c

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)

298271 )

299272 => (a -> b) -> Relation c d

300−fromF' f = R (I.fromFRel' f)

273+fromF f = R (I.fromFRel f)

301274

302275-- | Lifts functions to matrices with dimensions matching @a@ and @b@

303276-- cardinality's.

304−fromF ::

305− ( Liftable a b,

306− CountableDimsN a b,

307− FLN b a

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)

308286 )

309287 => (a -> b) -> Relation a b

310−fromF f = R (I.fromFRel f)

288+fromF' f = R (I.fromFRel' f)

311289

312290-- | Lifts relation functions to 'Relation'

313291toRel ::

314− ( Liftable a b,

315− CountableDims a b,

316− FLN b a

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)

317300 )

318− => (a -> b -> Bool) -> Relation a b

301+ => (a -> b -> Bool) -> Relation a b

319302toRel = R . I.toRel

320303

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−

332304-- Conversion

333305

334306-- | Converts a matrix to a list of lists of elements.

@@ -339,135 +311,93 @@ toLists (R m) = I.toLists m

339311toList :: Relation a b -> [Boolean]

340312toList (R m) = I.toList m

341313

342−-- | Converts a well typed 'Relation' to 'Bool'.

314+-- | Converts a well typed @'Relation'@ to @'Bool'@.

343315toBool :: Relation One One -> Bool

344316toBool r = case toList r of

345317 [Nat 0] -> False

346318 _ -> True

347319

348−-- | Power transpose.

349−--

350−-- Maps a relation to a set valued function.

351−pt ::

352− ( Liftable a b,

320+-- | Power transpose

321+pt ::

322+ ( Bounded a,

323+ Bounded b,

324+ Enum a,

325+ Enum b,

353326 Eq a,

354− CountableDimsN a b,

355− FLN a One,

356− FLN b One

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

357332 )

358− => Relation a b -> (a -> List b)

333+ => Relation a b -> (a -> [b])

359334pt r a =

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

335+ let lb = [minBound .. maxBound]

336+ in [ b | b <- lb, toBool (pointAp a b r) ]

375337

376338-- Zeros Matrix

377339

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−-- @

340+-- | The zero matrix. A matrix wholly filled with zeros.

387341zeros ::

388− (FLN a b, CountableDimsN a b) =>

342+ (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>

389343 Relation a b

390−zeros = relationBuilder' (const (nat 0))

344+zeros = relationBuilder (const (nat 0))

391345

392346-- Ones Matrix

393347

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.

348+-- | The ones matrix. A matrix wholly filled with ones.

398349--

399−-- @

400−-- ⊥ ``sse`` R && R ``sse`` T == True

401−-- @

350+-- Also known as T (Top) matrix.

402351ones ::

403− (FLN a b, CountableDimsN a b) =>

352+ (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>

404353 Relation a b

405−ones = relationBuilder' (const (nat 1))

354+ones = relationBuilder (const (nat 1))

406355

407356-- Bang Matrix

408357

409−-- | The T (Top) row vector relation.

358+-- | The T (Top) row vector matrix.

410359bang ::

411− (FLN a One, CountableN a) =>

360+ (I.FromLists Boolean (I.Normalize a) One, KnownNat (I.Count (I.Normalize a))) =>

412361 Relation a One

413362bang = ones

414363

415364-- | Point constant relation

416−point ::

365+point ::

417366 ( Bounded a,

418367 Enum a,

419368 Eq a,

420− CountableN a,

421− FLN a One

369+ KnownNat (I.Count (I.Normalize a)),

370+ I.FromLists Boolean (I.Normalize a) One

422371 ) => a -> Relation One a

423−point = fromF . const

372+point = fromF' . const

424373

425−-- iden Matrix

374+-- Identity Matrix

426375

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 (==))

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 (==))

435381

436382-- | Relational composition

437−--

438−-- @

439−-- r `.` (s `.` p) = (r `.` s) `.` p

440−-- @

441383comp :: Relation b c -> Relation a b -> Relation a c

442384comp (R a) (R b) = R (I.compRel a b)

443385

444386-- | Relational right division

445−--

446−-- @'divR' x y@ is the largest relation @z@ which,

447−-- pre-composed with @y@, approximates @x@.

448387divR :: Relation b c -> Relation b a -> Relation a c

449388divR (R x) (R y) = R (I.divR x y)

450389

451390-- | Relational left division

452−--

453−-- The dual division operator:

454−--

455−-- @

456−-- 'divL' y x == 'conv' ('divR' ('conv' x) ('conv' y)

457−-- @

458391divL :: Relation c b -> Relation a b -> Relation a c

459392divL (R x) (R y) = R (I.divL x y)

460393

461394-- | 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@.

465395divS :: Relation c a -> Relation b a -> Relation c b

466396divS (R x) (R y) = R (I.divS x y)

467397

468398-- | Relational shrinking.

469399--

470−-- @r ``shrunkBy`` s@ is the largest part of @r@ such that,

400+-- @r `'shrunkBy'` s@ is the largest part of @r@ such that,

471401-- if it yields an output for an input @x@, it must be a maximum,

472402-- with respect to @s@, among all possible outputs of @x@ by @r@.

473403shrunkBy :: Relation b a -> Relation a a -> Relation b a

@@ -475,52 +405,51 @@ shrunkBy r s = r `intersection` divR s (conv r)

475405

476406-- | Relational overriding.

477407--

478−-- @r ``overriddenBy`` s@ yields the relation which contains the

408+-- @r `'overriddenBy'` s@ yields the relation which contains the

479409-- whole of @s@ and that part of @r@ where @s@ is undefined.

480410--

481411-- @

482−-- 'zeros' ``overriddenBy`` s == s

483−-- r ``overriddenBy`` 'zeros' == r

484−-- r ``overriddenBy`` r == r

412+-- 'zeros' `'overriddenBy'` s == s

413+-- r `'overriddenBy'` 'zeros' == r

414+-- r `'overriddenBy'` r == r

485415-- @

486416overriddenBy ::

487− ( FLN b b,

488− CountableN b

417+ ( I.FromLists Boolean (I.Normalize b) (I.Normalize b),

418+ KnownNat (I.Count (I.Normalize b))

489419 ) => Relation a b -> Relation a b -> Relation a b

490420overriddenBy r s = s `union` r `intersection` divR zeros (conv s)

491421

492−-- | Relational application.

493−--

494−-- If @a@ and @b@ are related by 'Relation' @r@

495−-- then @'pointAp' a b r == 'one' ('nat' 1)@

422+-- | Relational application

496423pointAp ::

497− ( Liftable a b,

424+ ( Bounded a,

425+ Bounded b,

426+ Enum a,

427+ Enum b,

498428 Eq a,

499− CountableDimsN a b,

500− FLN a One,

501− FLN b One

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

502434 ) => a -> b -> Relation a b -> Relation One One

503−pointAp a b r = conv (point b) . r . point a

435+pointAp a b r = conv (point b) `comp` r `comp` point a

504436

505437-- | Relational application

506−--

507−-- The same as 'pointAp' but converts 'Boolean' to 'Bool'

508438pointApBool ::

509− ( Liftable a b,

439+ ( Bounded a,

440+ Bounded b,

441+ Enum a,

442+ Enum b,

510443 Eq a,

511− CountableDimsN a b,

512− FLN a One,

513− FLN b One

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

514449 ) => a -> b -> Relation a b -> Bool

515−pointApBool a b r = toBool $ conv (point b) . r . point a

450+pointApBool a b r = toBool $ conv (point b) `comp` r `comp` point a

516451

517452-- | 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

524453conv :: Relation a b -> Relation b a

525454conv (R a) = R (I.tr a)

526455

@@ -538,187 +467,165 @@ iff r s = r == s

538467

539468-- | Relational intersection

540469--

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−-- @

470+-- Lifts pointwise conjunction.

547471intersection :: Relation a b -> Relation a b -> Relation a b

548472intersection a b = a * b

549473

550474-- | Relational union

551475--

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−-- @

476+-- Lifts pointwise disjunction.

560477union :: Relation a b -> Relation a b -> Relation a b

561478union a b = a + b

562479

563480-- | Relation Kernel

564−--

565−-- @

566−-- 'ker' r == 'conv' r `.` r

567−-- 'ker' r == 'img' ('conv' r)

568−-- @

569481ker :: Relation a b -> Relation a a

570−ker r = conv r . r

482+ker r = conv r `comp` r

571483

572484-- | Relation Image

573−--

574−-- @

575−-- 'img' r == r `.` conv r

576−-- 'img' r == 'ker' ('conv' r)

577−-- @

578485img :: Relation a b -> Relation b b

579−img r = r . conv r

486+img r = r `comp` conv r

580487

581−-- | Function division. Special case of 'divS'.

488+-- | Function division.

582489--

583490-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and

584491-- 'entire'.

585492--

586−-- @'divisionF' f g == 'conv' g `.` f@

493+-- @division f g == conv g `'comp'` f@

587494divisionF :: Relation a c -> Relation b c -> Relation a b

588−divisionF f g = conv g . f

495+divisionF f g = conv g `comp` f

589496

590497-- Taxonomy of binary relations

591498

592−-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@

593−simple :: (CountableN b, FLN b b) => Relation a b -> Bool

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

594501simple = coreflexive . img

595502

596−-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@

597−injective :: (CountableN a, FLN a a) => Relation a b -> Bool

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

598505injective = coreflexive . ker

599506

600−-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@

601−entire :: (CountableN a, FLN a a) => Relation a b -> Bool

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

602509entire = reflexive . ker

603510

604−-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@

605−surjective :: (CountableN b, FLN b b) => Relation a b -> Bool

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

606513surjective = reflexive . img

607514

608−-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@

515+-- | A 'Relation' @r@ is a function iff @'simple' r && 'entire' r@

609516--

610517-- A 'function' @f@ enjoys the following properties, where @r@ and @s@ are binary

611518-- relations:

612519--

613520-- @

614−-- f `.` r ``sse`` s == r ``sse`` f `.` s

615−-- r `.` f ``sse`` s == r ``sse`` s `.` f

521+-- f `'comp'` r `'sse'` s == r `'sse'` f `'comp'` s

522+-- r `'comp'` f `'sse'` s == r `'sse'` s `'comp'` f

616523-- @

617524function ::

618− ( CountableDimsN a b,

619− FLN a a,

620− FLN b b

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)

621529 )

622530 => Relation a b -> Bool

623531function r = simple r && entire r

624532

625−-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@

533+-- | A 'Relation' @r@ is a representation iff @'injective' r && 'entire' r@

626534representation ::

627− ( CountableN a,

628− FLN a a

535+ ( KnownNat (I.Count (I.Normalize a)),

536+ I.FromLists Boolean (I.Normalize a) (I.Normalize a)

629537 )

630538 => Relation a b -> Bool

631539representation r = injective r && entire r

632540

633−-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@

541+-- | A 'Relation' @r@ is an abstraction iff @'surjective' r && 'simple' r@

634542abstraction ::

635− ( CountableN b,

636− FLN b b

543+ ( KnownNat (I.Count (I.Normalize b)),

544+ I.FromLists Boolean (I.Normalize b) (I.Normalize b)

637545 )

638546 => Relation a b -> Bool

639547abstraction r = surjective r && simple r

640548

641−-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@

549+-- | A 'Relation' @r@ is a surjection iff @'function' r && 'abstraction' r@

642550surjection ::

643− ( CountableDimsN a b,

644− FLN a a,

645− FLN b b

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)

646555 )

647556 => Relation a b -> Bool

648557surjection r = function r && abstraction r

649558

650−-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@

559+-- | A 'Relation' @r@ is a injection iff @'function' r && 'representation' r@

651560injection ::

652− ( CountableDimsN a b,

653− FLN a a,

654− FLN b b

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)

655565 )

656566 => Relation a b -> Bool

657567injection r = function r && representation r

658568

659−-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@

569+-- | A 'Relation' @r@ is an bijection iff @'injection' r && 'surjection' r@

660570bijection ::

661− ( CountableDimsN a b,

662− FLN b b,

663− FLN a a

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)

664575 ) => Relation a b -> Bool

665576bijection r = injection r && surjection r

666577

578+

667579-- Properties of relations

668580

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

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

672584

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

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

676588

677−-- | A 'Relation' @r@ is 'transitive' 'iff' @(r `.` r) ``sse`` r@

589+-- | A 'Relation' @r@ is transitive iff @(r `'comp'` r) `'sse'` r@

678590transitive :: Relation a a -> Bool

679−transitive r = (r . r) `sse` r

591+transitive r = (r `comp` r) `sse` r

680592

681−-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@

593+-- | A 'Relation' @r@ is symmetric iff @r == conv r@

682594symmetric :: Relation a a -> Bool

683595symmetric r = r == conv r

684596

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

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

688600

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

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

692604

693−-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@

694−connected :: (CountableN a, FLN a a) => Relation a a -> Bool

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

695607connected r = (r `union` conv r) == ones

696608

697−-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@

698−preorder :: (CountableN a, FLN a a) => Relation a a -> Bool

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

699611preorder r = reflexive r && transitive r

700612

701−-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@

702−partialOrder :: (CountableN a, FLN a a) => Relation a a -> Bool

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

703615partialOrder r = antiSymmetric r && preorder r

704616

705−-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@

706−linearOrder :: (CountableN a, FLN a a) => Relation a a -> Bool

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

707619linearOrder r = connected r && partialOrder r

708620

709−-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@

710−equivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

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

711623equivalence r = symmetric r && preorder r

712624

713−-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@

714−partialEquivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

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

715627partialEquivalence r = partialOrder r && equivalence r

716628

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−

722629-- Relational pairing

723630

724631-- | Relational pairing.

@@ -726,81 +633,64 @@ difunctional r = r . conv r . r == r

726633-- NOTE: That this is not a true categorical product, see for instance:

727634--

728635-- @

729−-- | 'fstR' `.` 'splitR' a b ``sse`` a

730−-- 'splitR' a b <=> |

731−-- | 'sndR' `.` 'splitR' a b ``sse`` b

636+-- | 'p1' `'comp'` 'splitR' a b `'sse'` a

637+-- 'splitR' a b <=> |

638+-- | 'p2' `'comp'` 'splitR' a b `'sse'` b

732639-- @

733640--

734641-- __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−-- @

745642splitR ::

746− ( CountableDimsN a b,

747− CountableN (a, b),

748− FLN (a, b) a,

749− FLN (a, b) b,

750− TrivialP a b

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)

751649 )

752650 => Relation c a -> Relation c b -> Relation c (a, b)

753−splitR (R f) (R g) = R (I.kr f g)

651+splitR (R f) (R g) = R (I.khatri f g)

754652

755653-- | Relational pairing first component projection

756−--

757−-- @

758−-- 'fstR' `.` 'splitR' r s ``sse`` r

759−-- @

760−fstR ::

654+p1 ::

761655 forall a b .

762− ( CountableDimsN a b,

763− CountableN (a, b),

764− FLN (a, b) a,

765− TrivialP 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)

766661 )

767662 => Relation (a, b) a

768−fstR = R (I.fstM @Boolean @(I.Normalize a) @(I.Normalize b))

663+p1 = R (I.kp1 @Boolean @(I.Normalize a) @(I.Normalize b))

769664

770665-- | Relational pairing second component projection

771−--

772−-- @

773−-- 'sndR' `.` 'splitR' r s ``sse`` s

774−-- @

775−sndR ::

666+p2 ::

776667 forall a b .

777− ( CountableDimsN a b,

778− CountableN (a, b),

779− FLN (a, b) b,

780− TrivialP 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)

781673 )

782674 => Relation (a, b) b

783−sndR = R (I.sndM @Boolean @(I.Normalize a) @(I.Normalize b))

675+p2 = R (I.kp2 @Boolean @(I.Normalize a) @(I.Normalize b))

784676

785677-- Relational pairing functor

786678

787679infixl 4 ><

788680-- | Relational pairing functor

789−--

790−-- @

791−-- r '><' s == 'splitR' (r `.` fstR) (s `.` sndR)

792−-- (r '><' s) `.` (p '><' q) == (r `.` p) '><' (s `.` q)

793−-- @

794681(><) ::

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

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)

804694 )

805695 => Relation a b -> Relation c d -> Relation (a, c) (b, d)

806696(><) (R a) (R b) = R ((I.><) a b)

@@ -808,47 +698,28 @@ infixl 4 ><

808698-- Relational co-products

809699

810700-- | Relational coproduct.

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

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

827704

828705-- | Relational coproduct first component injection

829−--

830−-- @

831−-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

832−-- 'i1' `.` 'i2' = 'zeros'

833−-- @

834706i1 ::

835− ( CountableDimsN a b,

836− FLN b a,

837− FLN a a

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)

838712 )

839713 => Relation a (Either a b)

840714i1 = R I.i1

841715

842716-- | Relational coproduct second component injection

843−--

844−-- @

845−-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

846−-- 'i1' `.` 'i2' = 'zeros'

847−-- @

848717i2 ::

849− ( CountableDimsN a b,

850− FLN a b,

851− FLN b b

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)

852723 )

853724 => Relation b (Either a b)

854725i2 = R I.i2

@@ -856,16 +727,15 @@ i2 = R I.i2

856727infixl 5 -|-

857728

858729-- | Relational coproduct functor.

859−--

860−-- @

861−-- r '-|-' s == 'eitherR' ('i1' `.` r) ('i2' `.` s)

862−-- @

863730(-|-) ::

864− ( CountableDimsN b d,

865− FLN b b,

866− FLN d b,

867− FLN b d,

868− FLN d d

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)

869739 )

870740 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)

871741(-|-) (R a) (R b) = R ((I.-|-) a b)

@@ -873,136 +743,97 @@ infixl 5 -|-

873743-- Relational "Currying"

874744

875745-- | 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.

880746trans ::

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

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)))

892760 )

893761 => Relation (a, b) c -> Relation a (c, b)

894−trans r = splitR r sndR . conv fstR

762+trans r = splitR r p2 `comp` conv p1

895763

896764-- | 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.

901765untrans ::

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

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)))

913779 )

914780 => Relation a (c, b) -> Relation (a, b) c

915−untrans s = fstR . conv (splitR (conv s) sndR)

781+untrans s = p1 `comp` conv (splitR (conv s) p2)

916782

917783-- | 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−-- @

927784predR ::

928785 ( Bounded a,

929786 Enum a,

930− Countable a,

931− CountableN a,

932− FLN a a,

933− FLN Bool 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)

934790 )

935− => Relation a Bool -> Relation a a

936−predR p = id `intersection` divisionF (fromF (const True)) p

791+ => (a -> Bool) -> Relation a a

792+predR p = identity `intersection` divisionF (fromF' (const True)) (fromF' p)

937793

938−-- | Equalizes functions @f@ and @g@.

794+-- | Equalizes functions @f@ and @g@.

939795-- That is, @'equalizer' f g@ is the largest coreflexive

940796-- 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−-- @

946797equalizer ::

947− ( CountableN a,

948− FLN a a

798+ ( KnownNat (I.Count (I.Normalize a)),

799+ I.FromLists Boolean (I.Normalize a) (I.Normalize a)

949800 )

950801 => Relation a b -> Relation a b -> Relation a a

951−equalizer f g = id `intersection` divisionF f g

802+equalizer f g = identity `intersection` divisionF f g

952803

953804-- | Relational conditional guard.

954805--

955806-- @

956−-- 'guard' p = 'i2' ``overriddenBy`` 'i1' `.` 'predR' p

807+-- 'guard' p = 'i2' `'overriddenBy'` 'i1' `'comp'` 'predR' p

957808-- @

958−guard ::

809+guard ::

959810 ( Bounded b,

960811 Enum b,

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)))

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)))

967818

968819-- | Relational McCarthy's conditional.

969−cond ::

820+cond ::

970821 ( Bounded b,

971822 Enum b,

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

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

999830

1000831-- Relation pretty print

1001832

1002833-- | Relation pretty printing

1003−pretty :: (CountableDimsN a b) => Relation a b -> String

834+pretty :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> String

1004835pretty (R a) = I.pretty a

1005836

1006837-- | Relation pretty printing

1007−prettyPrint :: (CountableDimsN a b) => Relation a b -> IO ()

838+prettyPrint :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> IO ()

1008839prettyPrint (R a) = I.prettyPrint a

src/LAoP/Utils.hs−6

@@ -94,12 +94,6 @@ module LAoP.Utils

9494 coerceNat,

9595 coerceNat2,

9696 coerceNat3,

97−

98− -- * List data type

99− List (..),

100−

101− -- * Category type-class

102− Category(..)

10397 )

10498where

10599

src/LAoP/Utils/Internal.hs+73−70

@@ -1,13 +1,82 @@

1−{-# LANGUAGE TypeFamilies #-}

2−{-# LANGUAGE DeriveGeneric #-}

31{-# LANGUAGE TypeApplications #-}

42{-# LANGUAGE DataKinds #-}

53{-# LANGUAGE FlexibleInstances #-}

64{-# LANGUAGE GeneralizedNewtypeDeriving #-}

5+{-# LANGUAGE KindSignatures #-}

76{-# LANGUAGE ScopedTypeVariables #-}

98module LAoP.Utils.Internal

10− (

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+ --

1180 -- * 'Natural' data type

1281 Natural(..),

1382 nat,

@@ -17,31 +86,19 @@ module LAoP.Utils.Internal

1786 coerceNat,

1887 coerceNat2,

1988 coerceNat3,

20−

21− -- * 'List' data type

22− List (..),

23−

24− -- * Category type class

25− Category(..)

2689 )

2790where

2891

2992import Data.Coerce

3093import Data.Proxy

31−import Data.List

32−import Data.Maybe

3394import GHC.TypeLits

3495import Control.DeepSeq

35−import GHC.Generics

36−import Data.Kind

37−import Prelude hiding (id, (.))

38−import qualified Prelude

3996

4097-- | Wrapper around 'Int's that have a restrictive semantic associated.

4198-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'

4299-- @i@ that's @n <= i <= m@.

43100newtype Natural (start :: Nat) (end :: Nat) = Nat Int

44− deriving (Show, Read, Eq, Ord, NFData, Generic)

101+ deriving (Show, Read, Eq, Ord, NFData)

45102

46103-- | Throws a runtime error if any of the operations overflows or

47104-- underflows.

@@ -140,57 +197,3 @@ instance

140197

141198 fromEnum (Left a) = fromEnum a

142199 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.yamldeleted−1

@@ -1 +0,0 @@

1−resolver: lts-15.5

stack.yaml+66−1

@@ -1 +1,66 @@

1−resolver: lts-14.3

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/

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

stack.yaml.lock+4−4

@@ -6,7 +6,7 @@

66packages: []

77snapshots:

88- completed:

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

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

test/Examples/Quantum.hsdeleted−31

@@ -1,31 +0,0 @@

1−{-# LANGUAGE StandaloneDeriving #-}

2−module Examples.Quantum where

3−

4−import LAoP.Utils

5−import LAoP.Matrix.Type

6−import Data.Complex

7−import Prelude hiding (id, (.))

8−

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)

test/Examples/Spreadsheets.hsdeleted−47

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

1−{-# LANGUAGE DeriveGeneric #-}

2−

3−module Examples.Spreadsheets

4−

5−import LAoP.Utils

6−import LAoP.Matrix.Type

7−import GHC.Generics

8−import Prelude hiding (id, (.))

9−

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+1−7

@@ -1,8 +1,2 @@

1−module Main (main) where

2−

3−import qualified Examples.Readme as RD

4−

51main :: IO ()

6−main = do

7− putStrLn ("Testing if README works" :: String)

8− RD.exec

2+main = putStrLn "Test suite not yet implemented"