a78e6db...master · laop

git.iko.soy/laop

35 files changed+1644−2573

since splitsnapshots

unifiedsplit

.ghc.environment.x86_64-linux-8.6.5−17 .github/workflows/ci.yml+43 .gitignore+54−2 .travis.yml+47 CHANGELOG.md+17 ChangeLog.md−3 LICENSE+17−26 README.md+35−27 Setup.hs−2 app/Main.hs−7 app/Spreadsheets.hs−61 appveyor.yml+34 app/Benchmarks.hs → benchmark/Main.hs+31−24 hackage.sh−10 laop-0.1.0.0-docs.tar.gz laop.cabal+89 package.yaml−57 report.html−1169 report.pdf src/LAoP/Dist.hs+4−2 src/LAoP/Dist/Internal.hs+67−28 src/LAoP/Matrix/Internal.hs+396−396 src/LAoP/Matrix/Nat.hs+139−124 src/LAoP/Matrix/Type.hs+231−175 src/LAoP/Relation.hs+8−11 src/LAoP/Relation/Internal.hs+250−254 src/LAoP/Utils.hs+5−2 src/LAoP/Utils/Internal.hs+30−74 stack-8.8.3.yaml+1 stack.yaml+1−66 stack.yaml.lock+4−4 test/Examples/Quantum.hs+31 app/Examples.hs → test/Examples/Readme.hs+56−31 test/Examples/Spreadsheets.hs+47 test/Spec.hs+7−1

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

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

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

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

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

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

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

6−--

7−clear-package-db

8−global-package-db

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

10−package-db packagedb/ghc-8.6.5

11−package-id laop-0.1.0.0-inplace

12−package-id base-4.12.0.0

13−package-id ghc-prim-0.5.3

14−package-id rts

15−package-id integer-gmp-1.0.2.0

16−package-id deepseq-1.4.4.0

17−package-id array-0.5.3.0

.github/workflows/ci.ymlnew file+43

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

1+name: CI

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

4+on:

5+ pull_request:

6+ push:

7+ branches: [master]

9+jobs:

10+ build:

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

12+ runs-on: ubuntu-16.04

13+ strategy:

14+ matrix:

15+ cabal: ["3.0"]

16+ ghc:

17+ - "8.6.5"

18+ - "8.8.3"

19+

20+ steps:

21+ - uses: actions/checkout@v2

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

23+

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

25+ name: Setup Haskell

26+ with:

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

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

29+

30+ - uses: actions/cache@v1

31+ name: Cache ~/.cabal/store

32+ with:

33+ path: ~/.cabal/store

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

35+

36+ - name: Build

37+ run: |

38+ cabal v2-update

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

40+

41+ - name: Test

42+ run: |

43+ cabal v2-test --enable-tests

.gitignore+54−2

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

1+### Haskell

2+dist

3+dist-*

4+cabal-dev

5+*.o

6+*.hi

7+*.chi

8+*.chs.h

9+*.dyn_o

10+*.dyn_hi

11+*.prof

12+*.aux

13+*.hp

14+*.eventlog

15+.virtualenv

16+.hsenv

17+.hpc

18+.cabal-sandbox/

19+cabal.sandbox.config

20+cabal.config

21+cabal.project.local

22+.ghc.environment.*

23+.HTF/

24+# Stack

125.stack-work/

2−laop.cabal

3−*~

26+stack.yaml.lock

27+

28+### IDE/support

29+# Vim

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

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

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

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

34+*~

35+tags

36+

37+# IntellijIDEA

38+.idea/

39+.ideaHaskellLib/

40+*.iml

41+

42+# Atom

43+.haskell-ghc-mod.json

44+

45+# VS

46+.vscode/

47+

48+# Emacs

49+*#

50+.dir-locals.el

51+TAGS

52+

53+# other

54+.DS_Store

55+hackage.sh

.travis.ymlnew file+47

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

1+sudo: true

2+language: haskell

4+git:

5+ depth: 5

7+cabal: "3.0"

9+cache:

10+ directories:

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

12+ - "$HOME/.stack"

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

14+

15+matrix:

16+ include:

17+ - ghc: 8.6.5

18+ - ghc: 8.8.3

19+

20+ - ghc: 8.8.3

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

22+

23+ - ghc: 8.6.5

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

25+

26+

27+install:

28+ - |

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

30+ cabal v2-update

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

32+ else

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

34+ stack --version

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

36+ fi

37+

38+script:

39+ - |

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

41+ cabal v2-test --enable-tests

42+ else

43+ stack test --system-ghc

44+ fi

45+

46+notifications:

47+ email: false

CHANGELOG.mdnew file+17

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

1+# Changelog

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

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

6+## 0.1.1.1

8+* Bump base version to work with cabal

10+## 0.1.1.0

11+

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

13+* Repository is more organized and clean

14+* Added CI

15+

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

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

ChangeLog.mddeleted−3

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

1−# Changelog for laop

2−

3−## Unreleased changes

LICENSE+17−26

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

1+MIT License

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

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

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

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

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

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

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

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

711

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

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

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

13+copies or substantial portions of the Software.

1014

11− * Redistributions in binary form must reproduce the above

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

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

14− with the distribution.

15−

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

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

18− from this software without specific prior written permission.

19−

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

21+SOFTWARE.

README.md+35−27

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

11# laop - Linear Algebra of Programming library

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

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

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

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

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

39The LAoP discipline generalises relations and functions treating them as

410Boolean matrices and in turn consider these as arrows.

511

@@ -10,36 +16,38 @@ motivation behind the library, the underlying theory, and implementation details

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

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

1218

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

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

20+

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

1422

1523## Features

1624

17−This library offers:

18−

19−- An internal module 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−- An interfacing module that uses generalised types for the dimensions and is just a newtype wrapper

22− around the internal matrix data type.

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

24− around the internal matrix data type.

25−- An interfacing module that restricts the matrix contents to being only 0's or 1's (True

26− and False). This module is called 'Relation' and offers many of the combinators seen in

27− AoP and Relational Calculus/Algebra.

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

29− make it easier to deal with sample space.

30−

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

32−

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

34−- Statically typed dimensions;

35−- Polymorphic data type dimensions;

36−- Polymorphic matrix content;

37−- Awesome dimension type inference;

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

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

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

41−

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

25+This library offers 3 main matrix programming modules:

26+

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

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

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

30+

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

32+distributions.

33+

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

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

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

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

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

39+and transformation functions.

40+

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

42+

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

44+

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

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

47+

48+## Known issues

49+

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

4351

4452## Notes

4553

Setup.hsdeleted−2

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

1−import Distribution.Simple

2−main = defaultMain

app/Main.hsdeleted−7

@@ -1,7 +0,0 @@

1−module Main where

2−

3−import Examples

4−import Benchmarks

5−

6−main :: IO ()

7−main = benchmark

app/Spreadsheets.hsdeleted−61

@@ -1,61 +0,0 @@

1−{-# LANGUAGE DataKinds #-}

2−{-# LANGUAGE GADTs #-}

3−{-# LANGUAGE FlexibleContexts #-}

4−{-# LANGUAGE DeriveGeneric #-}

5−module Spreadsheets

6− (

7− )

8− where

9−

10−import LAoP.Matrix.Type

11−import qualified LAoP.Matrix.Internal as I

12−import GHC.Generics

13−

14−data Student = Student1 | Student2 | Student3 | Student4

15− deriving (Eq, Show, Enum, Bounded, Generic)

16−

17−data Question = Question1 | Question2 | Question3 | Question4

18− deriving (Eq, Show, Enum, Bounded, Generic)

19−

20−data Results = Test | Exam | Final

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

22−

23−test :: Matrix Float One Results

24−test = point Test

25−

26−exam :: Matrix Float One Results

27−exam = point Exam

28−

29−final :: Matrix Float One Results

30−final = point Final

31−

32−m :: Matrix Float Question Student

33−m = fromLists [[95, 90, 100, 40], [20, 90, 90, 0], [30, 20, 95, 0], [50, 80, 100, 30]]

34−

35−w :: Matrix Float Question One

36−w = fromLists [[0.2, 0.3, 0.2, 0.3]]

37−

38−xls :: Matrix Float Question One

39− -> Matrix Float Question Student

40− -> Matrix Float Student One

41− -> Matrix Float (Either Question Results) (Either One Student)

42−xls w m t = junc (split w m) (split zeros r)

43− where

44− rExam = m `comp` tr w

45− rTest = tr t

46− rFinal = rTest `maxPP` rExam

47− rAux = junc rTest (junc rExam rFinal)

48− r = tr (converter `comp` tr rAux)

49− converter = junc test (junc exam final)

50−

51−-- | Overloaded, point-wise 'max' function

52−maxPP_ :: Ord e => I.Matrix e a b -> I.Matrix e a b -> I.Matrix e a b

53−maxPP_ I.Empty I.Empty = I.Empty

54−maxPP_ (I.One a) (I.One b) = I.One (a `max` b)

55−maxPP_ (I.Junc a b) (I.Junc c d) = I.Junc (maxPP_ a c) (maxPP_ b d)

56−maxPP_ (I.Split a b) (I.Split c d) = I.Split (maxPP_ a c) (maxPP_ b d)

57−maxPP_ x@(I.Split _ _) y@(I.Junc _ _) = maxPP_ x (I.abideJS y)

58−maxPP_ x@(I.Junc _ _) y@(I.Split _ _) = maxPP_ (I.abideJS x) y

59−

60−maxPP :: Ord e => Matrix e a b -> Matrix e a b -> Matrix e a b

61−maxPP (M a) (M b) = M (maxPP_ a b)

appveyor.ymlnew file+34

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

1+clone_folder: "c:\\WORK"

2+clone_depth: 5

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

5+skip_branch_with_pr: true

7+platform:

8+ - x86_64

10+cache:

11+ - "C:\\SR"

12+ - dist-newstyle

13+

14+environment:

15+ global:

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

17+

18+ matrix:

19+ - GHCVER: 8.6.5

20+

21+install:

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

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

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

25+ - refreshenv

26+

27+before_build:

28+ - cabal --version

29+ - ghc --version

30+ - cabal %CABOPTS% v2-update

31+

32+build_script:

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

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

app/Benchmarks.hs → benchmark/Main.hsrenamed+31−24

@@ -8,35 +8,36 @@

88{-# LANGUAGE AllowAmbiguousTypes #-}

99{-# LANGUAGE ScopedTypeVariables #-}

1010

11−module Benchmarks

12− ( benchmark

13− )

14− where

11+module Main (main) where

1512

1613import LAoP.Dist.Internal

1714import LAoP.Utils

18−import LAoP.Matrix.Type hiding (CountableN, Countable, FromListsN)

19−import qualified LAoP.Matrix.Type as T (FromListsN)

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

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

2017import Criterion.Main

2118import Control.DeepSeq

2219import Test.QuickCheck

2320import GHC.TypeLits

2421import Data.Proxy

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

2523

2624selectM2 ::

2725 ( Num e,

28− T.FromListsN e b b,

26+ T.FLN b b,

2927 CountableN b

3028 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

31−selectM2 m y = junc y identity `comp` m

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

3230

3331selectD2 ::

34− ( FromListsN b b,

32+ ( FLN b b,

3533 CountableN b

3634 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b

37−selectD2 (D d) m = D (junc m identity `comp` d)

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

36+

37+-- Composition with iden

38+compId m = comp iden m

3839

39−randomDist :: forall a . (Countable a, FromListsN () a) => Gen (Dist a)

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

4041randomDist = do

4142 let size = fromInteger (natVal (Proxy :: Proxy (Count a)))

4243 l <- vectorOf size (arbitrary :: Gen Prob)

@@ -45,7 +46,7 @@ randomDist = do

4546 m = fromLists lr

4647 return (D m)

4748

48−randomMatrix :: forall a b . (CountableDimensions a b, FromListsN a b) => Gen (Matrix Prob a b)

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

4950randomMatrix = do

5051 let cols = fromInteger (natVal (Proxy :: Proxy (Count a)))

5152 rows = fromInteger (natVal (Proxy :: Proxy (Count b)))

@@ -104,12 +105,14 @@ normalize3 as l = let s = sum l

104105 probs = map (/ s) l

105106 in zip as probs

106107

107−randomDistF :: forall a b . (CoArbitrary a, Arbitrary b) => Gen (Dist' (a -> b))

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

108109randomDistF = do

109− let size = 200

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

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

112+ size = a * b

110113 l <- vectorOf size (arbitrary :: Gen Prob)

111114 l2 <- vectorOf size (arbitrary :: Gen (a -> b))

112− let ln = normalize3 l2 (map P l)

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

113116 return (D' ln)

114117

115118instance CoArbitrary (Natural a b)

@@ -134,15 +137,15 @@ setupEnv = do

134137

135138setupEnv2 = do

136139 m21 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))

137− m40 <- generate (resize 1 (randomMatrix @(Natural 0 300) @(Either (Natural 0 100) (Natural 0 100) )))

138− dist <- generate (resize 1 (randomDist @(Either (Natural 0 1000) (Natural 0 1000)) ))

139− dist2 <- generate (resize 1 (randomMatrix @(Natural 0 1000) @(Natural 0 1000) ))

140− distList1 <- generate (resize 1 (randomDist2 @(Either (Natural 0 1000) (Natural 0 1000))))

141− distList2 <- generate (resize 1 (randomDistF @(Natural 0 1000) @(Natural 0 1000)))

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)))

142145 return (m21, m40, dist, dist2, distList1, distList2)

143146

144−benchmark :: IO ()

145−benchmark = defaultMain [

147+main :: IO ()

148+main = defaultMain [

146149 env setupEnv $ \ ~(m11, m12, m21, m22, m31, m32) -> bgroup "Matrix Composition" [

147150 bgroup "10x10" [

148151 bench "WHNF - 10x10" $ whnf (comp m11) m12

@@ -155,14 +158,18 @@ benchmark = defaultMain [

155158 bgroup "200x200" [

156159 bench "WHNF - 200x200" $ whnf (comp m31) m32

157160 , 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

158165 ] ],

159166 env setupEnv2 $ \ ~(m21, m40, dist, dist2, dl1, dl2) -> bgroup "Matrix vs List - `select`" [

160− bgroup "Distribution" [

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

161168 bench "List Distribution - Applicative version" $ nf (select dl1) dl2

162169 , bench "Matrix Distribution - Applicative version" $ nf (selectD2 dist) dist2

163170 , bench "Matrix Distribution - Selective version" $ nf (selectD dist) dist2

164171 ],

165− bgroup "Matrix `select`" [

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

166173 bench "Applicative version" $ nf (selectM2 m40) m21

167174 , bench "Selective version" $ nf (selectM m40) m21

168175 ] ] ]

hackage.shdeleted−10

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

1−#!/bin/sh

2−set -e

3−

4−dir=$(mktemp -d dist-docs.XXXXXX)

5−trap 'rm -r "$dir"' EXIT

6−

7−# assumes cabal 2.4 or later

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

9−

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

laop-0.1.0.0-docs.tar.gzdeleted

Binary file not shown.

laop.cabalnew file+89

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

1+cabal-version: 2.4

2+name: laop

3+version: 0.1.1.1

4+synopsis: Matrix programming library

5+description: Matrix programming library that uses a type-safe inductive definition

6+ inspired by the Linear Algebra of Programming

7+homepage: https://github.com/bolt12/laop

8+bug-reports: https://github.com/bolt12/laop/issues

9+license: MIT

10+license-file: LICENSE

11+author: Armando Santos

12+maintainer: Armando Santos <armandoifsantos@gmail.com>

13+copyright: 2020 Armando Santos

14+category: Data

15+build-type: Simple

16+extra-doc-files: README.md

17+ CHANGELOG.md

18+tested-with: GHC == 8.6.5

19+ GHC == 8.8.3

20+

21+source-repository head

22+ type: git

23+ location: https://github.com/bolt12/laop.git

24+

25+common common-options

26+ build-depends: base >= 4.12.0.0 && < 5,

27+ deepseq

28+

29+ ghc-options: -Wall

30+ -Wcompat

31+ -Widentities

32+ -Wincomplete-uni-patterns

33+ -Wincomplete-record-updates

34+ if impl(ghc >= 8.0)

35+ ghc-options: -Wredundant-constraints

36+ if impl(ghc >= 8.2)

37+ ghc-options: -fhide-source-paths

38+ if impl(ghc >= 8.4)

39+ ghc-options: -Wmissing-export-lists

40+ -Wpartial-fields

41+ if impl(ghc >= 8.8)

42+ ghc-options: -Wmissing-deriving-strategies

43+

44+ default-language: Haskell2010

45+

46+library

47+ import: common-options

48+ hs-source-dirs: src

49+ exposed-modules: LAoP.Dist

50+ LAoP.Dist.Internal

51+ LAoP.Matrix.Internal

52+ LAoP.Matrix.Nat

53+ LAoP.Matrix.Type

54+ LAoP.Relation

55+ LAoP.Relation.Internal

56+ LAoP.Utils

57+ LAoP.Utils.Internal

58+

59+test-suite laop-test

60+ import: common-options

61+ type: exitcode-stdio-1.0

62+ hs-source-dirs: test

63+ other-modules: Examples.Readme

64+ main-is: Spec.hs

65+ build-depends: laop

66+ ghc-options: -threaded

67+ -rtsopts

68+ -with-rtsopts=-N

69+ -funfolding-use-threshold=16

70+ -fexcess-precision

71+ -optc-O3

72+ -optc-ffast-math

73+

74+benchmark laop-benchmark

75+ import: common-options

76+ type: exitcode-stdio-1.0

77+ hs-source-dirs: benchmark

78+ main-is: Main.hs

79+ build-depends: criterion

80+ , QuickCheck

81+ , deepseq

82+ , laop

83+ ghc-options: -threaded

84+ -rtsopts

85+ -with-rtsopts=-N

86+ -funfolding-use-threshold=16

87+ -fexcess-precision

88+ -optc-O3

89+ -optc-ffast-math

package.yamldeleted−57

@@ -1,57 +0,0 @@

1−name: laop

2−version: 0.1.0.3

3−github: "bolt12/laop"

4−license: BSD3

5−author: "Armando Santos"

6−maintainer: "armandoifsantos@gmail.com"

7−copyright: "MIT"

8−

9−extra-source-files:

10−- README.md

11−- ChangeLog.md

12−

13−# Metadata used when publishing your package

14−# synopsis: Short description of your package

15−category: Data

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−- deepseq

25−

26−library:

27− source-dirs: src

28−

29−executables:

30− laop-exe:

31− main: Main.hs

32− source-dirs: app

33− ghc-options:

34− - -threaded

35− - -rtsopts

36− - -with-rtsopts=-N

37− - -funfolding-use-threshold=16

38− - -fexcess-precision # Good for doubles

39− - -optc-O3 # Good for doubles

40− - -optc-ffast-math # Good for doubles

41− - -O # Optimize

42− dependencies:

43− - laop

44− - criterion

45− - deepseq

46− - QuickCheck

47−

48−tests:

49− laop-test:

50− main: Spec.hs

51− source-dirs: test

52− ghc-options:

53− - -threaded

54− - -rtsopts

55− - -with-rtsopts=-N

56− dependencies:

57− - laop

report.htmldeleted−1169

@@ -1,1169 +0,0 @@

1−<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">

2−<html>

3− <head>

4− <meta http-equiv="Content-Type" content="text/html; charset=utf-8">

5− <title>criterion report</title>

6− <script language="javascript" type="text/javascript">

7− /*! jQuery v3.3.1 | (c) JS Foundation and other contributors | jquery.org/license */

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xmin=topSentry,ymin=topSentry,xmax=bottomSentry,ymax=bottomSentry;for(j=0;j<points.length;j+=ps){if(points[j]==null)continue;for(m=0;m<ps;++m){val=points[j+m];f=format[m];if(!f||f.autoscale===false||val==fakeInfinity||val==-fakeInfinity)continue;if(f.x){if(val<xmin)xmin=val;if(val>xmax)xmax=val}if(f.y){if(val<ymin)ymin=val;if(val>ymax)ymax=val}}}if(s.bars.show){var delta;switch(s.bars.align){case"left":delta=0;break;case"right":delta=-s.bars.barWidth;break;default:delta=-s.bars.barWidth/2}if(s.bars.horizontal){ymin+=delta;ymax+=delta+s.bars.barWidth}else{xmin+=delta;xmax+=delta+s.bars.barWidth}}updateAxis(s.xaxis,xmin,xmax);updateAxis(s.yaxis,ymin,ymax)}$.each(allAxes(),function(_,axis){if(axis.datamin==topSentry)axis.datamin=null;if(axis.datamax==bottomSentry)axis.datamax=null})}function setupCanvases(){placeholder.css("padding",0).children().filter(function(){return!$(this).hasClass("flot-overlay")&&!$(this).hasClass("flot-base")}).remove();if(placeholder.css("position")=="static")placeholder.css("position","relative");surface=new Canvas("flot-base",placeholder);overlay=new Canvas("flot-overlay",placeholder);ctx=surface.context;octx=overlay.context;eventHolder=$(overlay.element).unbind();var existing=placeholder.data("plot");if(existing){existing.shutdown();overlay.clear()}placeholder.data("plot",plot)}function bindEvents(){if(options.grid.hoverable){eventHolder.mousemove(onMouseMove);eventHolder.bind("mouseleave",onMouseLeave)}if(options.grid.clickable)eventHolder.click(onClick);executeHooks(hooks.bindEvents,[eventHolder])}function shutdown(){if(redrawTimeout)clearTimeout(redrawTimeout);eventHolder.unbind("mousemove",onMouseMove);eventHolder.unbind("mouseleave",onMouseLeave);eventHolder.unbind("click",onClick);executeHooks(hooks.shutdown,[eventHolder])}function setTransformationHelpers(axis){function identity(x){return x}var s,m,t=axis.options.transform||identity,it=axis.options.inverseTransform;if(axis.direction=="x"){s=axis.scale=plotWidth/Math.abs(t(axis.max)-t(axis.min));m=Math.min(t(axis.max),t(axis.min))}else{s=axis.scale=plotHeight/Math.abs(t(axis.max)-t(axis.min));s=-s;m=Math.max(t(axis.max),t(axis.min))}if(t==identity)axis.p2c=function(p){return(p-m)*s};else axis.p2c=function(p){return(t(p)-m)*s};if(!it)axis.c2p=function(c){return m+c/s};else axis.c2p=function(c){return it(m+c/s)}}function measureTickLabels(axis){var opts=axis.options,ticks=axis.ticks||[],labelWidth=opts.labelWidth||0,labelHeight=opts.labelHeight||0,maxWidth=labelWidth||(axis.direction=="x"?Math.floor(surface.width/(ticks.length||1)):null),legacyStyles=axis.direction+"Axis "+axis.direction+axis.n+"Axis",layer="flot-"+axis.direction+"-axis flot-"+axis.direction+axis.n+"-axis "+legacyStyles,font=opts.font||"flot-tick-label tickLabel";for(var i=0;i<ticks.length;++i){var t=ticks[i];if(!t.label)continue;var info=surface.getTextInfo(layer,t.label,font,null,maxWidth);labelWidth=Math.max(labelWidth,info.width);labelHeight=Math.max(labelHeight,info.height)}axis.labelWidth=opts.labelWidth||labelWidth;axis.labelHeight=opts.labelHeight||labelHeight}function allocateAxisBoxFirstPhase(axis){var lw=axis.labelWidth,lh=axis.labelHeight,pos=axis.options.position,isXAxis=axis.direction==="x",tickLength=axis.options.tickLength,axisMargin=options.grid.axisMargin,padding=options.grid.labelMargin,innermost=true,outermost=true,first=true,found=false;$.each(isXAxis?xaxes:yaxes,function(i,a){if(a&&(a.show||a.reserveSpace)){if(a===axis){found=true}else if(a.options.position===pos){if(found){outermost=false}else{innermost=false}}if(!found){first=false}}});if(outermost){axisMargin=0}if(tickLength==null){tickLength=first?"full":5}if(!isNaN(+tickLength))padding+=+tickLength;if(isXAxis){lh+=padding;if(pos=="bottom"){plotOffset.bottom+=lh+axisMargin;axis.box={top:surface.height-plotOffset.bottom,height:lh}}else{axis.box={top:plotOffset.top+axisMargin,height:lh};plotOffset.top+=lh+axisMargin}}else{lw+=padding;if(pos=="left"){axis.box={left:plotOffset.left+axisMargin,width:lw};plotOffset.left+=lw+axisMargin}else{plotOffset.right+=lw+axisMargin;axis.box={left:surface.width-plotOffset.right,width:lw}}}axis.position=pos;axis.tickLength=tickLength;axis.box.padding=padding;axis.innermost=innermost}function allocateAxisBoxSecondPhase(axis){if(axis.direction=="x"){axis.box.left=plotOffset.left-axis.labelWidth/2;axis.box.width=surface.width-plotOffset.left-plotOffset.right+axis.labelWidth}else{axis.box.top=plotOffset.top-axis.labelHeight/2;axis.box.height=surface.height-plotOffset.bottom-plotOffset.top+axis.labelHeight}}function adjustLayoutForThingsStickingOut(){var minMargin=options.grid.minBorderMargin,axis,i;if(minMargin==null){minMargin=0;for(i=0;i<series.length;++i)minMargin=Math.max(minMargin,2*(series[i].points.radius+series[i].points.lineWidth/2))}var margins={left:minMargin,right:minMargin,top:minMargin,bottom:minMargin};$.each(allAxes(),function(_,axis){if(axis.reserveSpace&&axis.ticks&&axis.ticks.length){if(axis.direction==="x"){margins.left=Math.max(margins.left,axis.labelWidth/2);margins.right=Math.max(margins.right,axis.labelWidth/2)}else{margins.bottom=Math.max(margins.bottom,axis.labelHeight/2);margins.top=Math.max(margins.top,axis.labelHeight/2)}}});plotOffset.left=Math.ceil(Math.max(margins.left,plotOffset.left));plotOffset.right=Math.ceil(Math.max(margins.right,plotOffset.right));plotOffset.top=Math.ceil(Math.max(margins.top,plotOffset.top));plotOffset.bottom=Math.ceil(Math.max(margins.bottom,plotOffset.bottom))}function setupGrid(){var i,axes=allAxes(),showGrid=options.grid.show;for(var a in plotOffset){var margin=options.grid.margin||0;plotOffset[a]=typeof margin=="number"?margin:margin[a]||0}executeHooks(hooks.processOffset,[plotOffset]);for(var a in plotOffset){if(typeof options.grid.borderWidth=="object"){plotOffset[a]+=showGrid?options.grid.borderWidth[a]:0}else{plotOffset[a]+=showGrid?options.grid.borderWidth:0}}$.each(axes,function(_,axis){var axisOpts=axis.options;axis.show=axisOpts.show==null?axis.used:axisOpts.show;axis.reserveSpace=axisOpts.reserveSpace==null?axis.show:axisOpts.reserveSpace;setRange(axis)});if(showGrid){var allocatedAxes=$.grep(axes,function(axis){return axis.show||axis.reserveSpace});$.each(allocatedAxes,function(_,axis){setupTickGeneration(axis);setTicks(axis);snapRangeToTicks(axis,axis.ticks);measureTickLabels(axis)});for(i=allocatedAxes.length-1;i>=0;--i)allocateAxisBoxFirstPhase(allocatedAxes[i]);adjustLayoutForThingsStickingOut();$.each(allocatedAxes,function(_,axis){allocateAxisBoxSecondPhase(axis)})}plotWidth=surface.width-plotOffset.left-plotOffset.right;plotHeight=surface.height-plotOffset.bottom-plotOffset.top;$.each(axes,function(_,axis){setTransformationHelpers(axis)});if(showGrid){drawAxisLabels()}insertLegend()}function setRange(axis){var opts=axis.options,min=+(opts.min!=null?opts.min:axis.datamin),max=+(opts.max!=null?opts.max:axis.datamax),delta=max-min;if(delta==0){var widen=max==0?1:.01;if(opts.min==null)min-=widen;if(opts.max==null||opts.min!=null)max+=widen}else{var margin=opts.autoscaleMargin;if(margin!=null){if(opts.min==null){min-=delta*margin;if(min<0&&axis.datamin!=null&&axis.datamin>=0)min=0}if(opts.max==null){max+=delta*margin;if(max>0&&axis.datamax!=null&&axis.datamax<=0)max=0}}}axis.min=min;axis.max=max}function setupTickGeneration(axis){var opts=axis.options;var noTicks;if(typeof opts.ticks=="number"&&opts.ticks>0)noTicks=opts.ticks;else noTicks=.3*Math.sqrt(axis.direction=="x"?surface.width:surface.height);var delta=(axis.max-axis.min)/noTicks,dec=-Math.floor(Math.log(delta)/Math.LN10),maxDec=opts.tickDecimals;if(maxDec!=null&&dec>maxDec){dec=maxDec}var magn=Math.pow(10,-dec),norm=delta/magn,size;if(norm<1.5){size=1}else if(norm<3){size=2;if(norm>2.25&&(maxDec==null||dec+1<=maxDec)){size=2.5;++dec}}else if(norm<7.5){size=5}else{size=10}size*=magn;if(opts.minTickSize!=null&&size<opts.minTickSize){size=opts.minTickSize}axis.delta=delta;axis.tickDecimals=Math.max(0,maxDec!=null?maxDec:dec);axis.tickSize=opts.tickSize||size;if(opts.mode=="time"&&!axis.tickGenerator){throw new Error("Time mode requires the flot.time plugin.")}if(!axis.tickGenerator){axis.tickGenerator=function(axis){var ticks=[],start=floorInBase(axis.min,axis.tickSize),i=0,v=Number.NaN,prev;do{prev=v;v=start+i*axis.tickSize;ticks.push(v);++i}while(v<axis.max&&v!=prev);return ticks};axis.tickFormatter=function(value,axis){var factor=axis.tickDecimals?Math.pow(10,axis.tickDecimals):1;var formatted=""+Math.round(value*factor)/factor;if(axis.tickDecimals!=null){var decimal=formatted.indexOf(".");var precision=decimal==-1?0:formatted.length-decimal-1;if(precision<axis.tickDecimals){return(precision?formatted:formatted+".")+(""+factor).substr(1,axis.tickDecimals-precision)}}return formatted}}if($.isFunction(opts.tickFormatter))axis.tickFormatter=function(v,axis){return""+opts.tickFormatter(v,axis)};if(opts.alignTicksWithAxis!=null){var otherAxis=(axis.direction=="x"?xaxes:yaxes)[opts.alignTicksWithAxis-1];if(otherAxis&&otherAxis.used&&otherAxis!=axis){var niceTicks=axis.tickGenerator(axis);if(niceTicks.length>0){if(opts.min==null)axis.min=Math.min(axis.min,niceTicks[0]);if(opts.max==null&&niceTicks.length>1)axis.max=Math.max(axis.max,niceTicks[niceTicks.length-1])}axis.tickGenerator=function(axis){var ticks=[],v,i;for(i=0;i<otherAxis.ticks.length;++i){v=(otherAxis.ticks[i].v-otherAxis.min)/(otherAxis.max-otherAxis.min);v=axis.min+v*(axis.max-axis.min);ticks.push(v)}return ticks};if(!axis.mode&&opts.tickDecimals==null){var extraDec=Math.max(0,-Math.floor(Math.log(axis.delta)/Math.LN10)+1),ts=axis.tickGenerator(axis);if(!(ts.length>1&&/\..*0$/.test((ts[1]-ts[0]).toFixed(extraDec))))axis.tickDecimals=extraDec}}}}function setTicks(axis){var oticks=axis.options.ticks,ticks=[];if(oticks==null||typeof oticks=="number"&&oticks>0)ticks=axis.tickGenerator(axis);else if(oticks){if($.isFunction(oticks))ticks=oticks(axis);else ticks=oticks}var i,v;axis.ticks=[];for(i=0;i<ticks.length;++i){var label=null;var t=ticks[i];if(typeof t=="object"){v=+t[0];if(t.length>1)label=t[1]}else v=+t;if(label==null)label=axis.tickFormatter(v,axis);if(!isNaN(v))axis.ticks.push({v:v,label:label})}}function snapRangeToTicks(axis,ticks){if(axis.options.autoscaleMargin&&ticks.length>0){if(axis.options.min==null)axis.min=Math.min(axis.min,ticks[0].v);if(axis.options.max==null&&ticks.length>1)axis.max=Math.max(axis.max,ticks[ticks.length-1].v)}}function draw(){surface.clear();executeHooks(hooks.drawBackground,[ctx]);var grid=options.grid;if(grid.show&&grid.backgroundColor)drawBackground();if(grid.show&&!grid.aboveData){drawGrid()}for(var i=0;i<series.length;++i){executeHooks(hooks.drawSeries,[ctx,series[i]]);drawSeries(series[i])}executeHooks(hooks.draw,[ctx]);if(grid.show&&grid.aboveData){drawGrid()}surface.render();triggerRedrawOverlay()}function extractRange(ranges,coord){var axis,from,to,key,axes=allAxes();for(var i=0;i<axes.length;++i){axis=axes[i];if(axis.direction==coord){key=coord+axis.n+"axis";if(!ranges[key]&&axis.n==1)key=coord+"axis";if(ranges[key]){from=ranges[key].from;to=ranges[key].to;break}}}if(!ranges[key]){axis=coord=="x"?xaxes[0]:yaxes[0];from=ranges[coord+"1"];to=ranges[coord+"2"]}if(from!=null&&to!=null&&from>to){var tmp=from;from=to;to=tmp}return{from:from,to:to,axis:axis}}function drawBackground(){ctx.save();ctx.translate(plotOffset.left,plotOffset.top);ctx.fillStyle=getColorOrGradient(options.grid.backgroundColor,plotHeight,0,"rgba(255, 255, 255, 0)");ctx.fillRect(0,0,plotWidth,plotHeight);ctx.restore()}function drawGrid(){var i,axes,bw,bc;ctx.save();ctx.translate(plotOffset.left,plotOffset.top);var markings=options.grid.markings;if(markings){if($.isFunction(markings)){axes=plot.getAxes();axes.xmin=axes.xaxis.min;axes.xmax=axes.xaxis.max;axes.ymin=axes.yaxis.min;axes.ymax=axes.yaxis.max;markings=markings(axes)}for(i=0;i<markings.length;++i){var m=markings[i],xrange=extractRange(m,"x"),yrange=extractRange(m,"y");if(xrange.from==null)xrange.from=xrange.axis.min;if(xrange.to==null)xrange.to=xrange.axis.max;

19−if(yrange.from==null)yrange.from=yrange.axis.min;if(yrange.to==null)yrange.to=yrange.axis.max;if(xrange.to<xrange.axis.min||xrange.from>xrange.axis.max||yrange.to<yrange.axis.min||yrange.from>yrange.axis.max)continue;xrange.from=Math.max(xrange.from,xrange.axis.min);xrange.to=Math.min(xrange.to,xrange.axis.max);yrange.from=Math.max(yrange.from,yrange.axis.min);yrange.to=Math.min(yrange.to,yrange.axis.max);var xequal=xrange.from===xrange.to,yequal=yrange.from===yrange.to;if(xequal&&yequal){continue}xrange.from=Math.floor(xrange.axis.p2c(xrange.from));xrange.to=Math.floor(xrange.axis.p2c(xrange.to));yrange.from=Math.floor(yrange.axis.p2c(yrange.from));yrange.to=Math.floor(yrange.axis.p2c(yrange.to));if(xequal||yequal){var lineWidth=m.lineWidth||options.grid.markingsLineWidth,subPixel=lineWidth%2?.5:0;ctx.beginPath();ctx.strokeStyle=m.color||options.grid.markingsColor;ctx.lineWidth=lineWidth;if(xequal){ctx.moveTo(xrange.to+subPixel,yrange.from);ctx.lineTo(xrange.to+subPixel,yrange.to)}else{ctx.moveTo(xrange.from,yrange.to+subPixel);ctx.lineTo(xrange.to,yrange.to+subPixel)}ctx.stroke()}else{ctx.fillStyle=m.color||options.grid.markingsColor;ctx.fillRect(xrange.from,yrange.to,xrange.to-xrange.from,yrange.from-yrange.to)}}}axes=allAxes();bw=options.grid.borderWidth;for(var j=0;j<axes.length;++j){var axis=axes[j],box=axis.box,t=axis.tickLength,x,y,xoff,yoff;if(!axis.show||axis.ticks.length==0)continue;ctx.lineWidth=1;if(axis.direction=="x"){x=0;if(t=="full")y=axis.position=="top"?0:plotHeight;else y=box.top-plotOffset.top+(axis.position=="top"?box.height:0)}else{y=0;if(t=="full")x=axis.position=="left"?0:plotWidth;else x=box.left-plotOffset.left+(axis.position=="left"?box.width:0)}if(!axis.innermost){ctx.strokeStyle=axis.options.color;ctx.beginPath();xoff=yoff=0;if(axis.direction=="x")xoff=plotWidth+1;else yoff=plotHeight+1;if(ctx.lineWidth==1){if(axis.direction=="x"){y=Math.floor(y)+.5}else{x=Math.floor(x)+.5}}ctx.moveTo(x,y);ctx.lineTo(x+xoff,y+yoff);ctx.stroke()}ctx.strokeStyle=axis.options.tickColor;ctx.beginPath();for(i=0;i<axis.ticks.length;++i){var v=axis.ticks[i].v;xoff=yoff=0;if(isNaN(v)||v<axis.min||v>axis.max||t=="full"&&(typeof bw=="object"&&bw[axis.position]>0||bw>0)&&(v==axis.min||v==axis.max))continue;if(axis.direction=="x"){x=axis.p2c(v);yoff=t=="full"?-plotHeight:t;if(axis.position=="top")yoff=-yoff}else{y=axis.p2c(v);xoff=t=="full"?-plotWidth:t;if(axis.position=="left")xoff=-xoff}if(ctx.lineWidth==1){if(axis.direction=="x")x=Math.floor(x)+.5;else y=Math.floor(y)+.5}ctx.moveTo(x,y);ctx.lineTo(x+xoff,y+yoff)}ctx.stroke()}if(bw){bc=options.grid.borderColor;if(typeof bw=="object"||typeof bc=="object"){if(typeof bw!=="object"){bw={top:bw,right:bw,bottom:bw,left:bw}}if(typeof bc!=="object"){bc={top:bc,right:bc,bottom:bc,left:bc}}if(bw.top>0){ctx.strokeStyle=bc.top;ctx.lineWidth=bw.top;ctx.beginPath();ctx.moveTo(0-bw.left,0-bw.top/2);ctx.lineTo(plotWidth,0-bw.top/2);ctx.stroke()}if(bw.right>0){ctx.strokeStyle=bc.right;ctx.lineWidth=bw.right;ctx.beginPath();ctx.moveTo(plotWidth+bw.right/2,0-bw.top);ctx.lineTo(plotWidth+bw.right/2,plotHeight);ctx.stroke()}if(bw.bottom>0){ctx.strokeStyle=bc.bottom;ctx.lineWidth=bw.bottom;ctx.beginPath();ctx.moveTo(plotWidth+bw.right,plotHeight+bw.bottom/2);ctx.lineTo(0,plotHeight+bw.bottom/2);ctx.stroke()}if(bw.left>0){ctx.strokeStyle=bc.left;ctx.lineWidth=bw.left;ctx.beginPath();ctx.moveTo(0-bw.left/2,plotHeight+bw.bottom);ctx.lineTo(0-bw.left/2,0);ctx.stroke()}}else{ctx.lineWidth=bw;ctx.strokeStyle=options.grid.borderColor;ctx.strokeRect(-bw/2,-bw/2,plotWidth+bw,plotHeight+bw)}}ctx.restore()}function drawAxisLabels(){$.each(allAxes(),function(_,axis){var box=axis.box,legacyStyles=axis.direction+"Axis "+axis.direction+axis.n+"Axis",layer="flot-"+axis.direction+"-axis flot-"+axis.direction+axis.n+"-axis "+legacyStyles,font=axis.options.font||"flot-tick-label tickLabel",tick,x,y,halign,valign;surface.removeText(layer);if(!axis.show||axis.ticks.length==0)return;for(var i=0;i<axis.ticks.length;++i){tick=axis.ticks[i];if(!tick.label||tick.v<axis.min||tick.v>axis.max)continue;if(axis.direction=="x"){halign="center";x=plotOffset.left+axis.p2c(tick.v);if(axis.position=="bottom"){y=box.top+box.padding}else{y=box.top+box.height-box.padding;valign="bottom"}}else{valign="middle";y=plotOffset.top+axis.p2c(tick.v);if(axis.position=="left"){x=box.left+box.width-box.padding;halign="right"}else{x=box.left+box.padding}}surface.addText(layer,x,y,tick.label,font,null,null,halign,valign)}})}function drawSeries(series){if(series.lines.show)drawSeriesLines(series);if(series.bars.show)drawSeriesBars(series);if(series.points.show)drawSeriesPoints(series)}function drawSeriesLines(series){function plotLine(datapoints,xoffset,yoffset,axisx,axisy){var points=datapoints.points,ps=datapoints.pointsize,prevx=null,prevy=null;ctx.beginPath();for(var i=ps;i<points.length;i+=ps){var x1=points[i-ps],y1=points[i-ps+1],x2=points[i],y2=points[i+1];if(x1==null||x2==null)continue;if(y1<=y2&&y1<axisy.min){if(y2<axisy.min)continue;x1=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.min}else if(y2<=y1&&y2<axisy.min){if(y1<axisy.min)continue;x2=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.min}if(y1>=y2&&y1>axisy.max){if(y2>axisy.max)continue;x1=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.max}else if(y2>=y1&&y2>axisy.max){if(y1>axisy.max)continue;x2=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.max}if(x1<=x2&&x1<axisx.min){if(x2<axisx.min)continue;y1=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.min}else if(x2<=x1&&x2<axisx.min){if(x1<axisx.min)continue;y2=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.min}if(x1>=x2&&x1>axisx.max){if(x2>axisx.max)continue;y1=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.max}else if(x2>=x1&&x2>axisx.max){if(x1>axisx.max)continue;y2=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.max}if(x1!=prevx||y1!=prevy)ctx.moveTo(axisx.p2c(x1)+xoffset,axisy.p2c(y1)+yoffset);prevx=x2;prevy=y2;ctx.lineTo(axisx.p2c(x2)+xoffset,axisy.p2c(y2)+yoffset)}ctx.stroke()}function plotLineArea(datapoints,axisx,axisy){var points=datapoints.points,ps=datapoints.pointsize,bottom=Math.min(Math.max(0,axisy.min),axisy.max),i=0,top,areaOpen=false,ypos=1,segmentStart=0,segmentEnd=0;while(true){if(ps>0&&i>points.length+ps)break;i+=ps;var x1=points[i-ps],y1=points[i-ps+ypos],x2=points[i],y2=points[i+ypos];if(areaOpen){if(ps>0&&x1!=null&&x2==null){segmentEnd=i;ps=-ps;ypos=2;continue}if(ps<0&&i==segmentStart+ps){ctx.fill();areaOpen=false;ps=-ps;ypos=1;i=segmentStart=segmentEnd+ps;continue}}if(x1==null||x2==null)continue;if(x1<=x2&&x1<axisx.min){if(x2<axisx.min)continue;y1=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.min}else if(x2<=x1&&x2<axisx.min){if(x1<axisx.min)continue;y2=(axisx.min-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.min}if(x1>=x2&&x1>axisx.max){if(x2>axisx.max)continue;y1=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x1=axisx.max}else if(x2>=x1&&x2>axisx.max){if(x1>axisx.max)continue;y2=(axisx.max-x1)/(x2-x1)*(y2-y1)+y1;x2=axisx.max}if(!areaOpen){ctx.beginPath();ctx.moveTo(axisx.p2c(x1),axisy.p2c(bottom));areaOpen=true}if(y1>=axisy.max&&y2>=axisy.max){ctx.lineTo(axisx.p2c(x1),axisy.p2c(axisy.max));ctx.lineTo(axisx.p2c(x2),axisy.p2c(axisy.max));continue}else if(y1<=axisy.min&&y2<=axisy.min){ctx.lineTo(axisx.p2c(x1),axisy.p2c(axisy.min));ctx.lineTo(axisx.p2c(x2),axisy.p2c(axisy.min));continue}var x1old=x1,x2old=x2;if(y1<=y2&&y1<axisy.min&&y2>=axisy.min){x1=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.min}else if(y2<=y1&&y2<axisy.min&&y1>=axisy.min){x2=(axisy.min-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.min}if(y1>=y2&&y1>axisy.max&&y2<=axisy.max){x1=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y1=axisy.max}else if(y2>=y1&&y2>axisy.max&&y1<=axisy.max){x2=(axisy.max-y1)/(y2-y1)*(x2-x1)+x1;y2=axisy.max}if(x1!=x1old){ctx.lineTo(axisx.p2c(x1old),axisy.p2c(y1))}ctx.lineTo(axisx.p2c(x1),axisy.p2c(y1));ctx.lineTo(axisx.p2c(x2),axisy.p2c(y2));if(x2!=x2old){ctx.lineTo(axisx.p2c(x2),axisy.p2c(y2));ctx.lineTo(axisx.p2c(x2old),axisy.p2c(y2))}}}ctx.save();ctx.translate(plotOffset.left,plotOffset.top);ctx.lineJoin="round";var lw=series.lines.lineWidth,sw=series.shadowSize;if(lw>0&&sw>0){ctx.lineWidth=sw;ctx.strokeStyle="rgba(0,0,0,0.1)";var angle=Math.PI/18;plotLine(series.datapoints,Math.sin(angle)*(lw/2+sw/2),Math.cos(angle)*(lw/2+sw/2),series.xaxis,series.yaxis);ctx.lineWidth=sw/2;plotLine(series.datapoints,Math.sin(angle)*(lw/2+sw/4),Math.cos(angle)*(lw/2+sw/4),series.xaxis,series.yaxis)}ctx.lineWidth=lw;ctx.strokeStyle=series.color;var fillStyle=getFillStyle(series.lines,series.color,0,plotHeight);if(fillStyle){ctx.fillStyle=fillStyle;plotLineArea(series.datapoints,series.xaxis,series.yaxis)}if(lw>0)plotLine(series.datapoints,0,0,series.xaxis,series.yaxis);ctx.restore()}function drawSeriesPoints(series){function plotPoints(datapoints,radius,fillStyle,offset,shadow,axisx,axisy,symbol){var points=datapoints.points,ps=datapoints.pointsize;for(var i=0;i<points.length;i+=ps){var x=points[i],y=points[i+1];if(x==null||x<axisx.min||x>axisx.max||y<axisy.min||y>axisy.max)continue;ctx.beginPath();x=axisx.p2c(x);y=axisy.p2c(y)+offset;if(symbol=="circle")ctx.arc(x,y,radius,0,shadow?Math.PI:Math.PI*2,false);else symbol(ctx,x,y,radius,shadow);ctx.closePath();if(fillStyle){ctx.fillStyle=fillStyle;ctx.fill()}ctx.stroke()}}ctx.save();ctx.translate(plotOffset.left,plotOffset.top);var lw=series.points.lineWidth,sw=series.shadowSize,radius=series.points.radius,symbol=series.points.symbol;if(lw==0)lw=1e-4;if(lw>0&&sw>0){var w=sw/2;ctx.lineWidth=w;ctx.strokeStyle="rgba(0,0,0,0.1)";plotPoints(series.datapoints,radius,null,w+w/2,true,series.xaxis,series.yaxis,symbol);ctx.strokeStyle="rgba(0,0,0,0.2)";plotPoints(series.datapoints,radius,null,w/2,true,series.xaxis,series.yaxis,symbol)}ctx.lineWidth=lw;ctx.strokeStyle=series.color;plotPoints(series.datapoints,radius,getFillStyle(series.points,series.color),0,false,series.xaxis,series.yaxis,symbol);ctx.restore()}function drawBar(x,y,b,barLeft,barRight,fillStyleCallback,axisx,axisy,c,horizontal,lineWidth){var left,right,bottom,top,drawLeft,drawRight,drawTop,drawBottom,tmp;if(horizontal){drawBottom=drawRight=drawTop=true;drawLeft=false;left=b;right=x;top=y+barLeft;bottom=y+barRight;if(right<left){tmp=right;right=left;left=tmp;drawLeft=true;drawRight=false}}else{drawLeft=drawRight=drawTop=true;drawBottom=false;left=x+barLeft;right=x+barRight;bottom=b;top=y;if(top<bottom){tmp=top;top=bottom;bottom=tmp;drawBottom=true;drawTop=false}}if(right<axisx.min||left>axisx.max||top<axisy.min||bottom>axisy.max)return;if(left<axisx.min){left=axisx.min;drawLeft=false}if(right>axisx.max){right=axisx.max;drawRight=false}if(bottom<axisy.min){bottom=axisy.min;drawBottom=false}if(top>axisy.max){top=axisy.max;drawTop=false}left=axisx.p2c(left);bottom=axisy.p2c(bottom);right=axisx.p2c(right);top=axisy.p2c(top);if(fillStyleCallback){c.fillStyle=fillStyleCallback(bottom,top);c.fillRect(left,top,right-left,bottom-top)}if(lineWidth>0&&(drawLeft||drawRight||drawTop||drawBottom)){c.beginPath();c.moveTo(left,bottom);if(drawLeft)c.lineTo(left,top);else c.moveTo(left,top);if(drawTop)c.lineTo(right,top);else c.moveTo(right,top);if(drawRight)c.lineTo(right,bottom);else c.moveTo(right,bottom);if(drawBottom)c.lineTo(left,bottom);else c.moveTo(left,bottom);c.stroke()}}function drawSeriesBars(series){function plotBars(datapoints,barLeft,barRight,fillStyleCallback,axisx,axisy){var points=datapoints.points,ps=datapoints.pointsize;for(var i=0;i<points.length;i+=ps){if(points[i]==null)continue;drawBar(points[i],points[i+1],points[i+2],barLeft,barRight,fillStyleCallback,axisx,axisy,ctx,series.bars.horizontal,series.bars.lineWidth)}}ctx.save();ctx.translate(plotOffset.left,plotOffset.top);ctx.lineWidth=series.bars.lineWidth;ctx.strokeStyle=series.color;var barLeft;switch(series.bars.align){case"left":barLeft=0;break;case"right":barLeft=-series.bars.barWidth;break;default:barLeft=-series.bars.barWidth/2}var fillStyleCallback=series.bars.fill?function(bottom,top){return getFillStyle(series.bars,series.color,bottom,top)}:null;plotBars(series.datapoints,barLeft,barLeft+series.bars.barWidth,fillStyleCallback,series.xaxis,series.yaxis);ctx.restore()}function getFillStyle(filloptions,seriesColor,bottom,top){var fill=filloptions.fill;if(!fill)return null;if(filloptions.fillColor)return getColorOrGradient(filloptions.fillColor,bottom,top,seriesColor);var c=$.color.parse(seriesColor);c.a=typeof fill=="number"?fill:.4;c.normalize();return c.toString()}function insertLegend(){if(options.legend.container!=null){$(options.legend.container).html("")}else{placeholder.find(".legend").remove()}if(!options.legend.show){return}var fragments=[],entries=[],rowStarted=false,lf=options.legend.labelFormatter,s,label;for(var i=0;i<series.length;++i){s=series[i];if(s.label){label=lf?lf(s.label,s):s.label;if(label){entries.push({label:label,color:s.color})}}}if(options.legend.sorted){if($.isFunction(options.legend.sorted)){entries.sort(options.legend.sorted)}else 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pos="",p=options.legend.position,m=options.legend.margin;if(m[0]==null)m=[m,m];if(p.charAt(0)=="n")pos+="top:"+(m[1]+plotOffset.top)+"px;";else if(p.charAt(0)=="s")pos+="bottom:"+(m[1]+plotOffset.bottom)+"px;";if(p.charAt(1)=="e")pos+="right:"+(m[0]+plotOffset.right)+"px;";else if(p.charAt(1)=="w")pos+="left:"+(m[0]+plotOffset.left)+"px;";var legend=$('<div class="legend">'+table.replace('style="','style="position:absolute;'+pos+";")+"</div>").appendTo(placeholder);if(options.legend.backgroundOpacity!=0){var c=options.legend.backgroundColor;if(c==null){c=options.grid.backgroundColor;if(c&&typeof c=="string")c=$.color.parse(c);else c=$.color.extract(legend,"background-color");c.a=1;c=c.toString()}var div=legend.children();$('<div style="position:absolute;width:'+div.width()+"px;height:"+div.height()+"px;"+pos+"background-color:"+c+';"> </div>').prependTo(legend).css("opacity",options.legend.backgroundOpacity)}}}var highlights=[],redrawTimeout=null;function findNearbyItem(mouseX,mouseY,seriesFilter){var maxDistance=options.grid.mouseActiveRadius,smallestDistance=maxDistance*maxDistance+1,item=null,foundPoint=false,i,j,ps;for(i=series.length-1;i>=0;--i){if(!seriesFilter(series[i]))continue;var s=series[i],axisx=s.xaxis,axisy=s.yaxis,points=s.datapoints.points,mx=axisx.c2p(mouseX),my=axisy.c2p(mouseY),maxx=maxDistance/axisx.scale,maxy=maxDistance/axisy.scale;ps=s.datapoints.pointsize;if(axisx.options.inverseTransform)maxx=Number.MAX_VALUE;if(axisy.options.inverseTransform)maxy=Number.MAX_VALUE;if(s.lines.show||s.points.show){for(j=0;j<points.length;j+=ps){var x=points[j],y=points[j+1];if(x==null)continue;if(x-mx>maxx||x-mx<-maxx||y-my>maxy||y-my<-maxy)continue;var dx=Math.abs(axisx.p2c(x)-mouseX),dy=Math.abs(axisy.p2c(y)-mouseY),dist=dx*dx+dy*dy;if(dist<smallestDistance){smallestDistance=dist;item=[i,j/ps]}}}if(s.bars.show&&!item){var barLeft,barRight;switch(s.bars.align){case"left":barLeft=0;break;case"right":barLeft=-s.bars.barWidth;break;default:barLeft=-s.bars.barWidth/2}barRight=barLeft+s.bars.barWidth;for(j=0;j<points.length;j+=ps){var x=points[j],y=points[j+1],b=points[j+2];if(x==null)continue;if(series[i].bars.horizontal?mx<=Math.max(b,x)&&mx>=Math.min(b,x)&&my>=y+barLeft&&my<=y+barRight:mx>=x+barLeft&&mx<=x+barRight&&my>=Math.min(b,y)&&my<=Math.max(b,y))item=[i,j/ps]}}}if(item){i=item[0];j=item[1];ps=series[i].datapoints.pointsize;return{datapoint:series[i].datapoints.points.slice(j*ps,(j+1)*ps),dataIndex:j,series:series[i],seriesIndex:i}}return null}function onMouseMove(e){if(options.grid.hoverable)triggerClickHoverEvent("plothover",e,function(s){return s["hoverable"]!=false})}function onMouseLeave(e){if(options.grid.hoverable)triggerClickHoverEvent("plothover",e,function(s){return false})}function onClick(e){triggerClickHoverEvent("plotclick",e,function(s){return s["clickable"]!=false})}function triggerClickHoverEvent(eventname,event,seriesFilter){var offset=eventHolder.offset(),canvasX=event.pageX-offset.left-plotOffset.left,canvasY=event.pageY-offset.top-plotOffset.top,pos=canvasToAxisCoords({left:canvasX,top:canvasY});pos.pageX=event.pageX;pos.pageY=event.pageY;var item=findNearbyItem(canvasX,canvasY,seriesFilter);if(item){item.pageX=parseInt(item.series.xaxis.p2c(item.datapoint[0])+offset.left+plotOffset.left,10);item.pageY=parseInt(item.series.yaxis.p2c(item.datapoint[1])+offset.top+plotOffset.top,10)}if(options.grid.autoHighlight){for(var i=0;i<highlights.length;++i){var h=highlights[i];if(h.auto==eventname&&!(item&&h.series==item.series&&h.point[0]==item.datapoint[0]&&h.point[1]==item.datapoint[1]))unhighlight(h.series,h.point)}if(item)highlight(item.series,item.datapoint,eventname)}placeholder.trigger(eventname,[pos,item])}function triggerRedrawOverlay(){var t=options.interaction.redrawOverlayInterval;if(t==-1){drawOverlay();return}if(!redrawTimeout)redrawTimeout=setTimeout(drawOverlay,t)}function drawOverlay(){redrawTimeout=null;octx.save();overlay.clear();octx.translate(plotOffset.left,plotOffset.top);var i,hi;for(i=0;i<highlights.length;++i){hi=highlights[i];if(hi.series.bars.show)drawBarHighlight(hi.series,hi.point);else drawPointHighlight(hi.series,hi.point)}octx.restore();executeHooks(hooks.drawOverlay,[octx])}function highlight(s,point,auto){if(typeof s=="number")s=series[s];if(typeof point=="number"){var ps=s.datapoints.pointsize;point=s.datapoints.points.slice(ps*point,ps*(point+1))}var i=indexOfHighlight(s,point);if(i==-1){highlights.push({series:s,point:point,auto:auto});triggerRedrawOverlay()}else if(!auto)highlights[i].auto=false}function unhighlight(s,point){if(s==null&&point==null){highlights=[];triggerRedrawOverlay();return}if(typeof s=="number")s=series[s];if(typeof point=="number"){var ps=s.datapoints.pointsize;point=s.datapoints.points.slice(ps*point,ps*(point+1))}var i=indexOfHighlight(s,point);if(i!=-1){highlights.splice(i,1);triggerRedrawOverlay()}}function indexOfHighlight(s,p){for(var i=0;i<highlights.length;++i){var h=highlights[i];if(h.series==s&&h.point[0]==p[0]&&h.point[1]==p[1])return i}return-1}function drawPointHighlight(series,point){var x=point[0],y=point[1],axisx=series.xaxis,axisy=series.yaxis,highlightColor=typeof series.highlightColor==="string"?series.highlightColor:$.color.parse(series.color).scale("a",.5).toString();if(x<axisx.min||x>axisx.max||y<axisy.min||y>axisy.max)return;var pointRadius=series.points.radius+series.points.lineWidth/2;octx.lineWidth=pointRadius;octx.strokeStyle=highlightColor;var radius=1.5*pointRadius;x=axisx.p2c(x);y=axisy.p2c(y);octx.beginPath();if(series.points.symbol=="circle")octx.arc(x,y,radius,0,2*Math.PI,false);else series.points.symbol(octx,x,y,radius,false);octx.closePath();octx.stroke()}function drawBarHighlight(series,point){var highlightColor=typeof series.highlightColor==="string"?series.highlightColor:$.color.parse(series.color).scale("a",.5).toString(),fillStyle=highlightColor,barLeft;switch(series.bars.align){case"left":barLeft=0;break;case"right":barLeft=-series.bars.barWidth;break;default:barLeft=-series.bars.barWidth/2}octx.lineWidth=series.bars.lineWidth;octx.strokeStyle=highlightColor;drawBar(point[0],point[1],point[2]||0,barLeft,barLeft+series.bars.barWidth,function(){return fillStyle},series.xaxis,series.yaxis,octx,series.bars.horizontal,series.bars.lineWidth)}function getColorOrGradient(spec,bottom,top,defaultColor){if(typeof spec=="string")return spec;else{var gradient=ctx.createLinearGradient(0,top,0,bottom);for(var i=0,l=spec.colors.length;i<l;++i){var c=spec.colors[i];if(typeof c!="string"){var co=$.color.parse(defaultColor);if(c.brightness!=null)co=co.scale("rgb",c.brightness);if(c.opacity!=null)co.a*=c.opacity;c=co.toString()}gradient.addColorStop(i/(l-1),c)}return gradient}}}$.plot=function(placeholder,data,options){var plot=new Plot($(placeholder),data,options,$.plot.plugins);return plot};$.plot.version="0.8.3";$.plot.plugins=[];$.fn.plot=function(data,options){return this.each(function(){$.plot(this,data,options)})};function floorInBase(n,base){return base*Math.floor(n/base)}})(jQuery);

20− </script>

21− <script language="javascript" type="text/javascript">

22− /* Javascript plotting library for jQuery, version 0.8.3.

23−

25−Licensed under the MIT license.

26−

27−*/

28−(function($){var options={series:{points:{errorbars:null,xerr:{err:"x",show:null,asymmetric:null,upperCap:null,lowerCap:null,color:null,radius:null},yerr:{err:"y",show:null,asymmetric:null,upperCap:null,lowerCap:null,color:null,radius:null}}}};function processRawData(plot,series,data,datapoints){if(!series.points.errorbars)return;var format=[{x:true,number:true,required:true},{y:true,number:true,required:true}];var errors=series.points.errorbars;if(errors=="x"||errors=="xy"){if(series.points.xerr.asymmetric){format.push({x:true,number:true,required:true});format.push({x:true,number:true,required:true})}else format.push({x:true,number:true,required:true})}if(errors=="y"||errors=="xy"){if(series.points.yerr.asymmetric){format.push({y:true,number:true,required:true});format.push({y:true,number:true,required:true})}else format.push({y:true,number:true,required:true})}datapoints.format=format}function parseErrors(series,i){var points=series.datapoints.points;var exl=null,exu=null,eyl=null,eyu=null;var xerr=series.points.xerr,yerr=series.points.yerr;var eb=series.points.errorbars;if(eb=="x"||eb=="xy"){if(xerr.asymmetric){exl=points[i+2];exu=points[i+3];if(eb=="xy")if(yerr.asymmetric){eyl=points[i+4];eyu=points[i+5]}else eyl=points[i+4]}else{exl=points[i+2];if(eb=="xy")if(yerr.asymmetric){eyl=points[i+3];eyu=points[i+4]}else eyl=points[i+3]}}else if(eb=="y")if(yerr.asymmetric){eyl=points[i+2];eyu=points[i+3]}else eyl=points[i+2];if(exu==null)exu=exl;if(eyu==null)eyu=eyl;var errRanges=[exl,exu,eyl,eyu];if(!xerr.show){errRanges[0]=null;errRanges[1]=null}if(!yerr.show){errRanges[2]=null;errRanges[3]=null}return errRanges}function drawSeriesErrors(plot,ctx,s){var points=s.datapoints.points,ps=s.datapoints.pointsize,ax=[s.xaxis,s.yaxis],radius=s.points.radius,err=[s.points.xerr,s.points.yerr];var invertX=false;if(ax[0].p2c(ax[0].max)<ax[0].p2c(ax[0].min)){invertX=true;var tmp=err[0].lowerCap;err[0].lowerCap=err[0].upperCap;err[0].upperCap=tmp}var 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257− <![endif]-->

258− </head>

259− <body>

260− <div id="wrap">

261− <div id="main" class="body">

262− <h1>criterion performance measurements</h1>

263−

264−<h2>overview</h2>

265−

266−<p><a href="#grokularation">want to understand this report?</a></p>

267−

268−<div id="overview" class="ovchart" style="width:900px;height:100px;"></div>

269−

270−<h2><a name="b0">Matrix Composition/10x10/WHNF - 10x10</a></h2>

271− <table width="100%">

272− <tbody>

273− <tr>

274− <td><div id="kde0" class="kdechart"

275− style="width:450px;height:278px;"></div></td>

276− <td><div id="time0" class="timechart"

277− style="width:450px;height:278px;"></div></td>

278−<!--

279− <td><div id="cycle0" class="cyclechart"

280− style="width:300px;height:278px;"></div></td>

281−-->

282− </tr>

283− </tbody>

284− </table>

285−

286− <table>

287− <thead class="analysis">

288− <th></th>

289− <th class="cibound"

290− title="0.95 confidence level">lower bound</th>

291− <th>estimate</th>

292− <th class="cibound"

293− title="0.95 confidence level">upper bound</th>

294− </thead>

295− <tbody>

296− <tr>

297− <td>OLS regression</td>

298− <td><span class="confinterval olstimelb0">xxx</span></td>

299− <td><span class="olstimept0">xxx</span></td>

300− <td><span class="confinterval olstimeub0">xxx</span></td>

301− </tr>

302− <tr>

303− <td>R² goodness-of-fit</td>

304− <td><span class="confinterval olsr2lb0">xxx</span></td>

305− <td><span class="olsr2pt0">xxx</span></td>

306− <td><span class="confinterval olsr2ub0">xxx</span></td>

307− </tr>

308− <tr>

309− <td>Mean execution time</td>

310− <td><span class="confinterval citime">1.2049433368049536e-4</span></td>

311− <td><span class="time">1.2273010647022493e-4</span></td>

312− <td><span class="confinterval citime">1.2778054268834733e-4</span></td>

313− </tr>

314− <tr>

315− <td>Standard deviation</td>

316− <td><span class="confinterval citime">5.116451922891949e-6</span></td>

317− <td><span class="time">1.0291105469156734e-5</span></td>

318− <td><span class="confinterval citime">1.899929874113666e-5</span></td>

319− </tr>

320− </tbody>

321− </table>

322−

323− <span class="outliers">

324− <p>Outlying measurements have severe

325− (<span class="percent">0.749671001723266</span>%)

326− effect on estimated standard deviation.</p>

327− </span>

328−<h2><a name="b1">Matrix Composition/10x10/NF - 10x10</a></h2>

329− <table width="100%">

330− <tbody>

331− <tr>

332− <td><div id="kde1" class="kdechart"

333− style="width:450px;height:278px;"></div></td>

334− <td><div id="time1" class="timechart"

335− style="width:450px;height:278px;"></div></td>

336−<!--

337− <td><div id="cycle1" class="cyclechart"

338− style="width:300px;height:278px;"></div></td>

339−-->

340− </tr>

341− </tbody>

342− </table>

343−

344− <table>

345− <thead class="analysis">

346− <th></th>

347− <th class="cibound"

348− title="0.95 confidence level">lower bound</th>

349− <th>estimate</th>

350− <th class="cibound"

351− title="0.95 confidence level">upper bound</th>

352− </thead>

353− <tbody>

354− <tr>

355− <td>OLS regression</td>

356− <td><span class="confinterval olstimelb1">xxx</span></td>

357− <td><span class="olstimept1">xxx</span></td>

358− <td><span class="confinterval olstimeub1">xxx</span></td>

359− </tr>

360− <tr>

361− <td>R² goodness-of-fit</td>

362− <td><span class="confinterval olsr2lb1">xxx</span></td>

363− <td><span class="olsr2pt1">xxx</span></td>

364− <td><span class="confinterval olsr2ub1">xxx</span></td>

365− </tr>

366− <tr>

367− <td>Mean execution time</td>

368− <td><span class="confinterval citime">1.219807334985906e-4</span></td>

369− <td><span class="time">1.2343493665093922e-4</span></td>

370− <td><span class="confinterval citime">1.2633720476730494e-4</span></td>

371− </tr>

372− <tr>

373− <td>Standard deviation</td>

374− <td><span class="confinterval citime">4.876323665872892e-6</span></td>

375− <td><span class="time">7.497583796499309e-6</span></td>

376− <td><span class="confinterval citime">1.2305448428855409e-5</span></td>

377− </tr>

378− </tbody>

379− </table>

380−

381− <span class="outliers">

382− <p>Outlying measurements have severe

383− (<span class="percent">0.6135229039528359</span>%)

384− effect on estimated standard deviation.</p>

385− </span>

386−<h2><a name="b2">Matrix Composition/100x100/WHNF - 100x100</a></h2>

387− <table width="100%">

388− <tbody>

389− <tr>

390− <td><div id="kde2" class="kdechart"

391− style="width:450px;height:278px;"></div></td>

392− <td><div id="time2" class="timechart"

393− style="width:450px;height:278px;"></div></td>

394−<!--

395− <td><div id="cycle2" class="cyclechart"

396− style="width:300px;height:278px;"></div></td>

397−-->

398− </tr>

399− </tbody>

400− </table>

401−

402− <table>

403− <thead class="analysis">

404− <th></th>

405− <th class="cibound"

406− title="0.95 confidence level">lower bound</th>

407− <th>estimate</th>

408− <th class="cibound"

409− title="0.95 confidence level">upper bound</th>

410− </thead>

411− <tbody>

412− <tr>

413− <td>OLS regression</td>

414− <td><span class="confinterval olstimelb2">xxx</span></td>

415− <td><span class="olstimept2">xxx</span></td>

416− <td><span class="confinterval olstimeub2">xxx</span></td>

417− </tr>

418− <tr>

419− <td>R² goodness-of-fit</td>

420− <td><span class="confinterval olsr2lb2">xxx</span></td>

421− <td><span class="olsr2pt2">xxx</span></td>

422− <td><span class="confinterval olsr2ub2">xxx</span></td>

423− </tr>

424− <tr>

425− <td>Mean execution time</td>

426− <td><span class="confinterval citime">0.10670925100021122</span></td>

427− <td><span class="time">0.11259088430757033</span></td>

428− <td><span class="confinterval citime">0.12810259287901513</span></td>

429− </tr>

430− <tr>

431− <td>Standard deviation</td>

432− <td><span class="confinterval citime">2.794985106828957e-3</span></td>

433− <td><span class="time">1.4513815882466386e-2</span></td>

434− <td><span class="confinterval citime">2.275626878296965e-2</span></td>

435− </tr>

436− </tbody>

437− </table>

438−

439− <span class="outliers">

440− <p>Outlying measurements have moderate

441− (<span class="percent">0.3584714096581915</span>%)

442− effect on estimated standard deviation.</p>

443− </span>

444−<h2><a name="b3">Matrix Composition/100x100/NF - 100x100</a></h2>

445− <table width="100%">

446− <tbody>

447− <tr>

448− <td><div id="kde3" class="kdechart"

449− style="width:450px;height:278px;"></div></td>

450− <td><div id="time3" class="timechart"

451− style="width:450px;height:278px;"></div></td>

452−<!--

453− <td><div id="cycle3" class="cyclechart"

454− style="width:300px;height:278px;"></div></td>

455−-->

456− </tr>

457− </tbody>

458− </table>

459−

460− <table>

461− <thead class="analysis">

462− <th></th>

463− <th class="cibound"

464− title="0.95 confidence level">lower bound</th>

465− <th>estimate</th>

466− <th class="cibound"

467− title="0.95 confidence level">upper bound</th>

468− </thead>

469− <tbody>

470− <tr>

471− <td>OLS regression</td>

472− <td><span class="confinterval olstimelb3">xxx</span></td>

473− <td><span class="olstimept3">xxx</span></td>

474− <td><span class="confinterval olstimeub3">xxx</span></td>

475− </tr>

476− <tr>

477− <td>R² goodness-of-fit</td>

478− <td><span class="confinterval olsr2lb3">xxx</span></td>

479− <td><span class="olsr2pt3">xxx</span></td>

480− <td><span class="confinterval olsr2ub3">xxx</span></td>

481− </tr>

482− <tr>

483− <td>Mean execution time</td>

484− <td><span class="confinterval citime">0.10572328594183394</span></td>

485− <td><span class="time">0.11065191628694417</span></td>

486− <td><span class="confinterval citime">0.1190700679846183</span></td>

487− </tr>

488− <tr>

489− <td>Standard deviation</td>

490− <td><span class="confinterval citime">1.5396112567768426e-3</span></td>

491− <td><span class="time">9.633081000847341e-3</span></td>

492− <td><span class="confinterval citime">1.3776160319998043e-2</span></td>

493− </tr>

494− </tbody>

495− </table>

496−

497− <span class="outliers">

498− <p>Outlying measurements have moderate

499− (<span class="percent">0.2338936359506027</span>%)

500− effect on estimated standard deviation.</p>

501− </span>

502−<h2><a name="b4">Matrix Composition/200x200/WHNF - 200x200</a></h2>

503− <table width="100%">

504− <tbody>

505− <tr>

506− <td><div id="kde4" class="kdechart"

507− style="width:450px;height:278px;"></div></td>

508− <td><div id="time4" class="timechart"

509− style="width:450px;height:278px;"></div></td>

510−<!--

511− <td><div id="cycle4" class="cyclechart"

512− style="width:300px;height:278px;"></div></td>

513−-->

514− </tr>

515− </tbody>

516− </table>

517−

518− <table>

519− <thead class="analysis">

520− <th></th>

521− <th class="cibound"

522− title="0.95 confidence level">lower bound</th>

523− <th>estimate</th>

524− <th class="cibound"

525− title="0.95 confidence level">upper bound</th>

526− </thead>

527− <tbody>

528− <tr>

529− <td>OLS regression</td>

530− <td><span class="confinterval olstimelb4">xxx</span></td>

531− <td><span class="olstimept4">xxx</span></td>

532− <td><span class="confinterval olstimeub4">xxx</span></td>

533− </tr>

534− <tr>

535− <td>R² goodness-of-fit</td>

536− <td><span class="confinterval olsr2lb4">xxx</span></td>

537− <td><span class="olsr2pt4">xxx</span></td>

538− <td><span class="confinterval olsr2ub4">xxx</span></td>

539− </tr>

540− <tr>

541− <td>Mean execution time</td>

542− <td><span class="confinterval citime">0.9554592040840362</span></td>

543− <td><span class="time">0.9644028123334465</span></td>

544− <td><span class="confinterval citime">0.9742178227493241</span></td>

545− </tr>

546− <tr>

547− <td>Standard deviation</td>

548− <td><span class="confinterval citime">4.795288999957847e-3</span></td>

549− <td><span class="time">1.0832350628490363e-2</span></td>

550− <td><span class="confinterval citime">1.511725573135899e-2</span></td>

551− </tr>

552− </tbody>

553− </table>

554−

555− <span class="outliers">

556− <p>Outlying measurements have moderate

557− (<span class="percent">0.18749999999999994</span>%)

558− effect on estimated standard deviation.</p>

559− </span>

560−<h2><a name="b5">Matrix Composition/200x200/NF - 200x200</a></h2>

561− <table width="100%">

562− <tbody>

563− <tr>

564− <td><div id="kde5" class="kdechart"

565− style="width:450px;height:278px;"></div></td>

566− <td><div id="time5" class="timechart"

567− style="width:450px;height:278px;"></div></td>

568−<!--

569− <td><div id="cycle5" class="cyclechart"

570− style="width:300px;height:278px;"></div></td>

571−-->

572− </tr>

573− </tbody>

574− </table>

575−

576− <table>

577− <thead class="analysis">

578− <th></th>

579− <th class="cibound"

580− title="0.95 confidence level">lower bound</th>

581− <th>estimate</th>

582− <th class="cibound"

583− title="0.95 confidence level">upper bound</th>

584− </thead>

585− <tbody>

586− <tr>

587− <td>OLS regression</td>

588− <td><span class="confinterval olstimelb5">xxx</span></td>

589− <td><span class="olstimept5">xxx</span></td>

590− <td><span class="confinterval olstimeub5">xxx</span></td>

591− </tr>

592− <tr>

593− <td>R² goodness-of-fit</td>

594− <td><span class="confinterval olsr2lb5">xxx</span></td>

595− <td><span class="olsr2pt5">xxx</span></td>

596− <td><span class="confinterval olsr2ub5">xxx</span></td>

597− </tr>

598− <tr>

599− <td>Mean execution time</td>

600− <td><span class="confinterval citime">0.9654473458749635</span></td>

601− <td><span class="time">0.980254961521041</span></td>

602− <td><span class="confinterval citime">0.9994299113750458</span></td>

603− </tr>

604− <tr>

605− <td>Standard deviation</td>

606− <td><span class="confinterval citime">2.345238647867079e-4</span></td>

607− <td><span class="time">2.1335106928918192e-2</span></td>

608− <td><span class="confinterval citime">2.623796574056436e-2</span></td>

609− </tr>

610− </tbody>

611− </table>

612−

613− <span class="outliers">

614− <p>Outlying measurements have moderate

615− (<span class="percent">0.1875</span>%)

616− effect on estimated standard deviation.</p>

617− </span>

618−<h2><a name="b6">Matrix vs List - `select`/Distribution/List Distribution - Applicative version</a></h2>

619− <table width="100%">

620− <tbody>

621− <tr>

622− <td><div id="kde6" class="kdechart"

623− style="width:450px;height:278px;"></div></td>

624− <td><div id="time6" class="timechart"

625− style="width:450px;height:278px;"></div></td>

626−<!--

627− <td><div id="cycle6" class="cyclechart"

628− style="width:300px;height:278px;"></div></td>

629−-->

630− </tr>

631− </tbody>

632− </table>

633−

634− <table>

635− <thead class="analysis">

636− <th></th>

637− <th class="cibound"

638− title="0.95 confidence level">lower bound</th>

639− <th>estimate</th>

640− <th class="cibound"

641− title="0.95 confidence level">upper bound</th>

642− </thead>

643− <tbody>

644− <tr>

645− <td>OLS regression</td>

646− <td><span class="confinterval olstimelb6">xxx</span></td>

647− <td><span class="olstimept6">xxx</span></td>

648− <td><span class="confinterval olstimeub6">xxx</span></td>

649− </tr>

650− <tr>

651− <td>R² goodness-of-fit</td>

652− <td><span class="confinterval olsr2lb6">xxx</span></td>

653− <td><span class="olsr2pt6">xxx</span></td>

654− <td><span class="confinterval olsr2ub6">xxx</span></td>

655− </tr>

656− <tr>

657− <td>Mean execution time</td>

658− <td><span class="confinterval citime">0.43770282037348807</span></td>

659− <td><span class="time">0.4467186700617276</span></td>

660− <td><span class="confinterval citime">0.46234597762486374</span></td>

661− </tr>

662− <tr>

663− <td>Standard deviation</td>

664− <td><span class="confinterval citime">1.3631004574408759e-3</span></td>

665− <td><span class="time">1.5029085673470916e-2</span></td>

666− <td><span class="confinterval citime">1.8726410685291874e-2</span></td>

667− </tr>

668− </tbody>

669− </table>

670−

671− <span class="outliers">

672− <p>Outlying measurements have moderate

673− (<span class="percent">0.1875</span>%)

674− effect on estimated standard deviation.</p>

675− </span>

676−<h2><a name="b7">Matrix vs List - `select`/Distribution/Matrix Distribution - Applicative version</a></h2>

677− <table width="100%">

678− <tbody>

679− <tr>

680− <td><div id="kde7" class="kdechart"

681− style="width:450px;height:278px;"></div></td>

682− <td><div id="time7" class="timechart"

683− style="width:450px;height:278px;"></div></td>

684−<!--

685− <td><div id="cycle7" class="cyclechart"

686− style="width:300px;height:278px;"></div></td>

687−-->

688− </tr>

689− </tbody>

690− </table>

691−

692− <table>

693− <thead class="analysis">

694− <th></th>

695− <th class="cibound"

696− title="0.95 confidence level">lower bound</th>

697− <th>estimate</th>

698− <th class="cibound"

699− title="0.95 confidence level">upper bound</th>

700− </thead>

701− <tbody>

702− <tr>

703− <td>OLS regression</td>

704− <td><span class="confinterval olstimelb7">xxx</span></td>

705− <td><span class="olstimept7">xxx</span></td>

706− <td><span class="confinterval olstimeub7">xxx</span></td>

707− </tr>

708− <tr>

709− <td>R² goodness-of-fit</td>

710− <td><span class="confinterval olsr2lb7">xxx</span></td>

711− <td><span class="olsr2pt7">xxx</span></td>

712− <td><span class="confinterval olsr2ub7">xxx</span></td>

713− </tr>

714− <tr>

715− <td>Mean execution time</td>

716− <td><span class="confinterval citime">0.176601423499556</span></td>

717− <td><span class="time">0.1820171075632162</span></td>

718− <td><span class="confinterval citime">0.18602406437478045</span></td>

719− </tr>

720− <tr>

721− <td>Standard deviation</td>

722− <td><span class="confinterval citime">4.2476576439371165e-3</span></td>

723− <td><span class="time">6.298418696687028e-3</span></td>

724− <td><span class="confinterval citime">8.492295568985392e-3</span></td>

725− </tr>

726− </tbody>

727− </table>

728−

729− <span class="outliers">

730− <p>Outlying measurements have moderate

731− (<span class="percent">0.1388888888888888</span>%)

732− effect on estimated standard deviation.</p>

733− </span>

734−<h2><a name="b8">Matrix vs List - `select`/Distribution/Matrix Distribution - Selective version</a></h2>

735− <table width="100%">

736− <tbody>

737− <tr>

738− <td><div id="kde8" class="kdechart"

739− style="width:450px;height:278px;"></div></td>

740− <td><div id="time8" class="timechart"

741− style="width:450px;height:278px;"></div></td>

742−<!--

743− <td><div id="cycle8" class="cyclechart"

744− style="width:300px;height:278px;"></div></td>

745−-->

746− </tr>

747− </tbody>

748− </table>

749−

750− <table>

751− <thead class="analysis">

752− <th></th>

753− <th class="cibound"

754− title="0.95 confidence level">lower bound</th>

755− <th>estimate</th>

756− <th class="cibound"

757− title="0.95 confidence level">upper bound</th>

758− </thead>

759− <tbody>

760− <tr>

761− <td>OLS regression</td>

762− <td><span class="confinterval olstimelb8">xxx</span></td>

763− <td><span class="olstimept8">xxx</span></td>

764− <td><span class="confinterval olstimeub8">xxx</span></td>

765− </tr>

766− <tr>

767− <td>R² goodness-of-fit</td>

768− <td><span class="confinterval olsr2lb8">xxx</span></td>

769− <td><span class="olsr2pt8">xxx</span></td>

770− <td><span class="confinterval olsr2ub8">xxx</span></td>

771− </tr>

772− <tr>

773− <td>Mean execution time</td>

774− <td><span class="confinterval citime">8.933987364670819e-2</span></td>

775− <td><span class="time">9.136040249002575e-2</span></td>

776− <td><span class="confinterval citime">9.678633084819704e-2</span></td>

777− </tr>

778− <tr>

779− <td>Standard deviation</td>

780− <td><span class="confinterval citime">1.4344054494409147e-3</span></td>

781− <td><span class="time">5.3416383323926005e-3</span></td>

782− <td><span class="confinterval citime">9.301990711614968e-3</span></td>

783− </tr>

784− </tbody>

785− </table>

786−

787− <span class="outliers">

788− <p>Outlying measurements have moderate

789− (<span class="percent">0.18171701654949143</span>%)

790− effect on estimated standard deviation.</p>

791− </span>

792−<h2><a name="b9">Matrix vs List - `select`/Matrix `select`/Applicative version</a></h2>

793− <table width="100%">

794− <tbody>

795− <tr>

796− <td><div id="kde9" class="kdechart"

797− style="width:450px;height:278px;"></div></td>

798− <td><div id="time9" class="timechart"

799− style="width:450px;height:278px;"></div></td>

800−<!--

801− <td><div id="cycle9" class="cyclechart"

802− style="width:300px;height:278px;"></div></td>

803−-->

804− </tr>

805− </tbody>

806− </table>

807−

808− <table>

809− <thead class="analysis">

810− <th></th>

811− <th class="cibound"

812− title="0.95 confidence level">lower bound</th>

813− <th>estimate</th>

814− <th class="cibound"

815− title="0.95 confidence level">upper bound</th>

816− </thead>

817− <tbody>

818− <tr>

819− <td>OLS regression</td>

820− <td><span class="confinterval olstimelb9">xxx</span></td>

821− <td><span class="olstimept9">xxx</span></td>

822− <td><span class="confinterval olstimeub9">xxx</span></td>

823− </tr>

824− <tr>

825− <td>R² goodness-of-fit</td>

826− <td><span class="confinterval olsr2lb9">xxx</span></td>

827− <td><span class="olsr2pt9">xxx</span></td>

828− <td><span class="confinterval olsr2ub9">xxx</span></td>

829− </tr>

830− <tr>

831− <td>Mean execution time</td>

832− <td><span class="confinterval citime">0.7791034777492314</span></td>

833− <td><span class="time">0.840422457165308</span></td>

834− <td><span class="confinterval citime">0.8923891394988459</span></td>

835− </tr>

836− <tr>

837− <td>Standard deviation</td>

838− <td><span class="confinterval citime">2.769907342949611e-2</span></td>

839− <td><span class="time">6.898641320796234e-2</span></td>

840− <td><span class="confinterval citime">9.23128387071436e-2</span></td>

841− </tr>

842− </tbody>

843− </table>

844−

845− <span class="outliers">

846− <p>Outlying measurements have moderate

847− (<span class="percent">0.21376667254855866</span>%)

848− effect on estimated standard deviation.</p>

849− </span>

850−<h2><a name="b10">Matrix vs List - `select`/Matrix `select`/Selective version</a></h2>

851− <table width="100%">

852− <tbody>

853− <tr>

854− <td><div id="kde10" class="kdechart"

855− style="width:450px;height:278px;"></div></td>

856− <td><div id="time10" class="timechart"

857− style="width:450px;height:278px;"></div></td>

858−<!--

859− <td><div id="cycle10" class="cyclechart"

860− style="width:300px;height:278px;"></div></td>

861−-->

862− </tr>

863− </tbody>

864− </table>

865−

866− <table>

867− <thead class="analysis">

868− <th></th>

869− <th class="cibound"

870− title="0.95 confidence level">lower bound</th>

871− <th>estimate</th>

872− <th class="cibound"

873− title="0.95 confidence level">upper bound</th>

874− </thead>

875− <tbody>

876− <tr>

877− <td>OLS regression</td>

878− <td><span class="confinterval olstimelb10">xxx</span></td>

879− <td><span class="olstimept10">xxx</span></td>

880− <td><span class="confinterval olstimeub10">xxx</span></td>

881− </tr>

882− <tr>

883− <td>R² goodness-of-fit</td>

884− <td><span class="confinterval olsr2lb10">xxx</span></td>

885− <td><span class="olsr2pt10">xxx</span></td>

886− <td><span class="confinterval olsr2ub10">xxx</span></td>

887− </tr>

888− <tr>

889− <td>Mean execution time</td>

890− <td><span class="confinterval citime">0.41360511700137675</span></td>

891− <td><span class="time">0.42580509562564356</span></td>

892− <td><span class="confinterval citime">0.43034074300067005</span></td>

893− </tr>

894− <tr>

895− <td>Standard deviation</td>

896− <td><span class="confinterval citime">1.588963214600777e-4</span></td>

897− <td><span class="time">8.497480607539406e-3</span></td>

898− <td><span class="confinterval citime">1.0474628931683076e-2</span></td>

899− </tr>

900− </tbody>

901− </table>

902−

903− <span class="outliers">

904− <p>Outlying measurements have moderate

905− (<span class="percent">0.1875</span>%)

906− effect on estimated standard deviation.</p>

907− </span>

908−

909− <h2><a name="grokularation">understanding this report</a></h2>

910−

911− <p>In this report, each function benchmarked by criterion is assigned

912− a section of its own. The charts in each section are active; if

913− you hover your mouse over data points and annotations, you will see

914− more details.</p>

915−

916− <ul>

917− <li>The chart on the left is a

918− <a href="http://en.wikipedia.org/wiki/Kernel_density_estimation">kernel

919− density estimate</a> (also known as a KDE) of time

920− measurements. This graphs the probability of any given time

921− measurement occurring. A spike indicates that a measurement of a

922− particular time occurred; its height indicates how often that

923− measurement was repeated.</li>

924−

925− <li>The chart on the right is the raw data from which the kernel

926− density estimate is built. The <i>x</i> axis indicates the

927− number of loop iterations, while the <i>y</i> axis shows measured

928− execution time for the given number of loop iterations. The

929− line behind the values is the linear regression prediction of

930− execution time for a given number of iterations. Ideally, all

931− measurements will be on (or very near) this line.</li>

932− </ul>

933−

934− <p>Under the charts is a small table.

935− The first two rows are the results of a linear regression run

936− on the measurements displayed in the right-hand chart.</p>

937−

938− <ul>

939− <li><i>OLS regression</i> indicates the

940− time estimated for a single loop iteration using an ordinary

941− least-squares regression model. This number is more accurate

942− than the <i>mean</i> estimate below it, as it more effectively

943− eliminates measurement overhead and other constant factors.</li>

944− <li><i>R² goodness-of-fit</i> is a measure of how

945− accurately the linear regression model fits the observed

946− measurements. If the measurements are not too noisy, R²

947− should lie between 0.99 and 1, indicating an excellent fit. If

948− the number is below 0.99, something is confounding the accuracy

949− of the linear model.</li>

950− <li><i>Mean execution time</i> and <i>standard deviation</i> are

951− statistics calculated from execution time

952− divided by number of iterations.</li>

953− </ul>

954−

955− <p>We use a statistical technique called

956− the <a href="http://en.wikipedia.org/wiki/Bootstrapping_(statistics)">bootstrap</a>

957− to provide confidence intervals on our estimates. The

958− bootstrap-derived upper and lower bounds on estimates let you see

959− how accurate we believe those estimates to be. (Hover the mouse

960− over the table headers to see the confidence levels.)</p>

961−

962− <p>A noisy benchmarking environment can cause some or many

963− measurements to fall far from the mean. These outlying

964− measurements can have a significant inflationary effect on the

965− estimate of the standard deviation. We calculate and display an

966− estimate of the extent to which the standard deviation has been

967− inflated by outliers.</p>

968−

969−<script type="text/javascript">

970−$(function () {

971− function mangulate(rpt) {

972− var measured = function(key) {

973− var idx = rpt.reportKeys.indexOf(key);

974− return rpt.reportMeasured.map(function(r) { return r[idx]; });

975− };

976− var lowerBound = function(est) {

977− return est.estPoint - est.estError.confIntLDX;

978− };

979− var upperBound = function(est) {

980− return est.estPoint + est.estError.confIntUDX;

981− };

982− var number = rpt.reportNumber;

983− var name = rpt.reportName;

984− var mean = rpt.reportAnalysis.anMean.estPoint;

985− var iters = measured("iters");

986− var times = measured("time");

987− var kdetimes = rpt.reportKDEs[0].kdeValues;

988− var kdepdf = rpt.reportKDEs[0].kdePDF;

989−

990− var meanSecs = mean;

991− var units = $.timeUnits(mean);

992− var rgrs = rpt.reportAnalysis.anRegress[0];

993− var scale = units[0];

994− var olsTime = rgrs.regCoeffs.iters;

995− $(".olstimept" + number).text(function() {

996− return $.renderTime(olsTime.estPoint);

997− });

998− $(".olstimelb" + number).text(function() {

999− return $.renderTime(lowerBound(olsTime));

1000− });

1001− $(".olstimeub" + number).text(function() {

1002− return $.renderTime(upperBound(olsTime));

1003− });

1004− $(".olsr2pt" + number).text(function() {

1005− return rgrs.regRSquare.estPoint.toFixed(3);

1006− });

1007− $(".olsr2lb" + number).text(function() {

1008− return lowerBound(rgrs.regRSquare).toFixed(3);

1009− });

1010− $(".olsr2ub" + number).text(function() {

1011− return upperBound(rgrs.regRSquare).toFixed(3);

1012− });

1013− mean *= scale;

1014− kdetimes = $.scaleBy(scale, kdetimes);

1015− var kq = $("#kde" + number);

1016− var k = $.plot(kq,

1017− [{ label: name + " time densities",

1018− data: $.zip(kdetimes, kdepdf),

1019− }],

1020− { xaxis: { tickFormatter: $.unitFormatter(scale) },

1021− yaxis: { ticks: false },

1022− grid: { borderColor: "#777",

1023− hoverable: true, markings: [ { color: '#6fd3fb',

1024− lineWidth: 1.5, xaxis: { from: mean, to: mean } } ] },

1025− });

1026− var o = k.pointOffset({ x: mean, y: 0});

1027− kq.append('<div class="meanlegend" title="' + $.renderTime(meanSecs) +

1028− '" style="position:absolute;left:' + (o.left + 4) +

1029− 'px;bottom:139px;">mean</div>');

1030− $.addTooltip("#kde" + number,

1031− function(secs) { return $.renderTime(secs / scale); });

1032− var timepairs = new Array(times.length);

1033− var lastiter = iters[iters.length-1];

1034− var olspairs = [[0,0], [lastiter, lastiter * scale * olsTime.estPoint]];

1035− for (var i = 0; i < times.length; i++)

1036− timepairs[i] = [iters[i],times[i]*scale];

1037− iterFormatter = function() {

1038− var denom = 0;

1039− return function(iters) {

1040− if (iters == 0)

1041− return '';

1042− if (denom > 0)

1043− return (iters / denom).toFixed();

1044− var power;

1045− if (iters >= 1e9) {

1046− denom = '1e9'; power = '⁹';

1047− }

1048− if (iters >= 1e6) {

1049− denom = '1e6'; power = '⁶';

1050− }

1051− else if (iters >= 1e3) {

1052− denom = '1e3'; power = '³';

1053− }

1054− else denom = 1;

1055− if (denom > 1) {

1056− iters = (iters / denom).toFixed();

1057− iters += '×10' + power + ' iters';

1058− } else {

1059− iters += ' iters';

1060− }

1061− return iters;

1062− };

1063− };

1064− $.plot($("#time" + number),

1065− [{ label: "regression", data: olspairs,

1066− lines: { show: true } },

1067− { label: name + " times", data: timepairs,

1068− points: { show: true } }],

1069− { grid: { borderColor: "#777", hoverable: true },

1070− xaxis: { tickFormatter: iterFormatter() },

1071− yaxis: { tickFormatter: $.unitFormatter(scale) } });

1072− $.addTooltip("#time" + number,

1073− function(iters,secs) {

1074− return ($.renderTime(secs / scale) + ' / ' +

1075− iters.toLocaleString() + ' iters');

1076− });

1077− if (0) {

1078− var cyclepairs = new Array(cycles.length);

1079− for (var i = 0; i < cycles.length; i++)

1080− cyclepairs[i] = [cycles[i],i];

1081− $.plot($("#cycle" + number),

1082− [{ label: name + " cycles",

1083− data: cyclepairs }],

1084− { points: { show: true },

1085− grid: { borderColor: "#777", hoverable: true },

1086− xaxis: { tickFormatter:

1087− function(cycles,axis) { return cycles + ' cycles'; }},

1088− yaxis: { ticks: false },

1089− });

1090− $.addTooltip("#cycles" + number, function(x,y) { return x + ' cycles'; });

1091− }

1092− };

1093− var reports = 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1094− reports.map(mangulate);

1095−

1096− var benches = ["Matrix Composition/10x10/WHNF - 10x10","Matrix Composition/10x10/NF - 10x10","Matrix Composition/100x100/WHNF - 100x100","Matrix Composition/100x100/NF - 100x100","Matrix Composition/200x200/WHNF - 200x200","Matrix Composition/200x200/NF - 200x200","Matrix vs List - `select`/Distribution/List Distribution - Applicative version","Matrix vs List - `select`/Distribution/Matrix Distribution - Applicative version","Matrix vs List - `select`/Distribution/Matrix Distribution - Selective version","Matrix vs List - `select`/Matrix `select`/Applicative version","Matrix vs List - `select`/Matrix `select`/Selective version",];

1097− var ylabels = [[-0,"<a href=\"#b0\">Matrix Composition/10x10/WHNF - 10x10</a>"],[-1,"<a href=\"#b1\">Matrix Composition/10x10/NF - 10x10</a>"],[-2,"<a href=\"#b2\">Matrix Composition/100x100/WHNF - 100x100</a>"],[-3,"<a href=\"#b3\">Matrix Composition/100x100/NF - 100x100</a>"],[-4,"<a href=\"#b4\">Matrix Composition/200x200/WHNF - 200x200</a>"],[-5,"<a href=\"#b5\">Matrix Composition/200x200/NF - 200x200</a>"],[-6,"<a href=\"#b6\">Matrix vs List - `select`/Distribution/List Distribution - Applicative version</a>"],[-7,"<a href=\"#b7\">Matrix vs List - `select`/Distribution/Matrix Distribution - Applicative version</a>"],[-8,"<a href=\"#b8\">Matrix vs List - `select`/Distribution/Matrix Distribution - Selective version</a>"],[-9,"<a href=\"#b9\">Matrix vs List - `select`/Matrix `select`/Applicative version</a>"],[-10,"<a href=\"#b10\">Matrix vs List - `select`/Matrix `select`/Selective version</a>"],];

1098− var means = $.scaleTimes([1.2273010647022493e-4,1.2343493665093922e-4,0.11259088430757033,0.11065191628694417,0.9644028123334465,0.980254961521041,0.4467186700617276,0.1820171075632162,9.136040249002575e-2,0.840422457165308,0.42580509562564356,]);

1099− var stddevs = [1.0291105469156734e-5,7.497583796499309e-6,1.4513815882466386e-2,9.633081000847341e-3,1.0832350628490363e-2,2.1335106928918192e-2,1.5029085673470916e-2,6.298418696687028e-3,5.3416383323926005e-3,6.898641320796234e-2,8.497480607539406e-3,];

1100− stddevs = $.scaleBy(means[1], stddevs);

1101− var xs = [];

1102− var prev = null;

1103− var colors = ["#edc240","#afd8f8","#cb4b4b","#4da74d","#9440ed"];

1104− var max_time = 0;

1105− for (var i = 0; i < means[0].length; i++) {

1106− var name = benches[i].split(/\//);

1107− name.pop();

1108− name = name.join('/');

1109− if (name != prev) {

1110− var color = colors[xs.length / 2 % colors.length];

1111− xs.push({ data: [[means[0][i], -i]], label: name, color: color,

1112− bars: { show: true, horizontal: true, barWidth: 0.75,

1113− align: "center" } });

1114− xs.push({ data: [[means[0][i], -i, stddevs[i]]], color: color,

1115− lines: { show: false },

1116− points: { radius: 0, errorbars: "x",

1117− xerr: { show: true, radius: 5,

1118− lowerCap: "-", upperCap: "-" } } });

1119− prev = name;

1120− } else {

1121− xs[xs.length - 2].data.push([means[0][i], -i]);

1122− xs[xs.length - 1].data.push([means[0][i], -i, stddevs[i]]);

1123− }

1124− max_time = Math.max(max_time, means[0][i] + stddevs[i]);

1125− }

1126− var oq = $("#overview");

1127− var max_x = max_time * 1.02;

1128− o = $.plot(oq, xs, { grid: { borderColor: "#777", hoverable: true },

1129− legend: { show: xs.length > 1 },

1130− xaxis: { max: max_x, zoomRange: [false, max_x], panRange: [0, max_x] },

1131− yaxis: { ticks: ylabels, tickColor: '#ffffff', zoomRange: false, panRange: false },

1132− zoom: { interactive: true, amount: 1.05 },

1133− pan: { interactive: true } });

1134− if (benches.length > 3)

1135− o.getPlaceholder().height(28*benches.length);

1136− o.resize();

1137− o.setupGrid();

1138− o.draw();

1139− $.addTooltip("#overview", function(x,y) { return $.renderTime(x / means[1]); });

1140−});

1141−$(document).ready(function () {

1142− $(".time").text(function(_, text) {

1143− return $.renderTime(text);

1144− });

1145− $(".citime").text(function(_, text) {

1146− return $.renderTime(text);

1147− });

1148− $(".percent").text(function(_, text) {

1149− return (text*100).toFixed(1);

1150− });

1151− });

1152−</script>

1153−

1154− </div>

1155− </div>

1156− <div id="footer">

1157− <div class="body">

1158− <div class="footfirst">

1159− <h2>colophon</h2>

1160− <p>This report was created using the

1161− <a href="http://hackage.haskell.org/package/criterion">criterion</a>

1162− benchmark execution and performance analysis tool.</p>

1163− <p>Criterion is developed and maintained

1164− by <a href="http://www.serpentine.com/blog/">Bryan O'Sullivan</a>.</p>

1165− </div>

1166− </div>

1167− </div>

1168− </body>

1169−</html>

report.pdfdeleted

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src/LAoP/Dist.hs+4−2

@@ -39,8 +39,8 @@ module LAoP.Dist

3939 -- * Constraint type synonyms

4040 Countable,

4141 CountableN,

42− CountableDimensionsN,

43− FromListsN,

42+ CountableDimsN,

43+ FLN,

4444 Liftable,

4545 TrivialP,

4646

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

5353

5454 -- * Selective equivalent functions

5555 selectD,

56+ branchD,

57+ ifD,

5658

5759 -- * Monad equivalent functions

5860 returnD,

src/LAoP/Dist/Internal.hs+67−28

@@ -16,8 +16,8 @@ module LAoP.Dist.Internal

1616

1717 Countable,

1818 CountableN,

19− CountableDimensionsN,

20− FromListsN,

19+ CountableDimsN,

20+ FLN,

2121 Liftable,

2222 TrivialP,

2323

@@ -25,6 +25,8 @@ module LAoP.Dist.Internal

2525 unitD,

2626 multD,

2727 selectD,

28+ branchD,

29+ ifD,

2830 returnD,

2931 bindD,

3032 (??),

@@ -41,13 +43,15 @@ module LAoP.Dist.Internal

4143 )

4244 where

4345

44−import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDimensions, CountableN, CountableDimensionsN, Liftable, FromListsN)

46+import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDims, CountableN, CountableDimsN, Liftable, FLN)

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

4548import qualified LAoP.Matrix.Internal as I

4649import LAoP.Utils

4750import GHC.TypeLits

4851import Data.Proxy

4952import Data.List (sortBy)

5053import Control.DeepSeq

54+import Data.Bool

5155

5256-- | Type synonym for probability value

5357type Prob = Double

@@ -58,18 +62,18 @@ newtype Dist a = D (Matrix Prob () a)

5862 deriving (Show, Num, Eq, Ord, NFData) via (Matrix Prob () a)

5963

6064-- | Constraint type synonyms to keep the type signatures less convoluted

61−type Countable a = KnownNat (I.Count a)

62−type CountableN a = KnownNat (I.Count (I.Normalize a))

63−type CountableDimensionsN a b = (CountableN a, CountableN b)

64−type FromListsN a b = I.FromLists Prob (I.Normalize a) (I.Normalize b)

65−type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Prob, Ord Prob)

66−type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)

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)

6771

6872-- | Functor instance

6973fmapD ::

7074 ( Liftable a b,

71− CountableDimensionsN a b,

72− FromListsN b a

75+ CountableDimsN a b,

76+ FLN b a

7377 )

7478 =>

7579 (a -> b) -> Dist a -> Dist b

@@ -81,23 +85,58 @@ unitD = D (one 1)

8185

8286-- | Applicative/Monoidal instance 'mult' function

8387multD ::

84− ( CountableDimensionsN a b,

88+ ( CountableDimsN a b,

8589 CountableN (a, b),

86− FromListsN (a, b) a,

87− FromListsN (a, b) b,

90+ FLN (a, b) a,

91+ FLN (a, b) b,

8892 TrivialP a b

8993 ) => Dist a -> Dist b -> Dist (a, b)

90−multD (D a) (D b) = D (khatri a b)

94+multD (D a) (D b) = D (kr a b)

9195

9296-- | Selective instance function

9397selectD ::

94− ( FromListsN b b,

98+ ( FLN b b,

9599 CountableN b

96100 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b

97101selectD (D d) m = D (selectM d m)

98102

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

137+

99138-- | Monad instance 'return' function

100−returnD :: forall a . (Enum a, FromListsN () a, Countable a) => a -> Dist a

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

101140returnD a = D (col l)

102141 where

103142 i = fromInteger $ natVal (Proxy :: Proxy (Count a))

@@ -112,7 +151,7 @@ bindD (D d) m = D (m `comp` d)

112151(??) ::

113152 ( Enum a,

114153 Countable a,

115− FromListsN () a

154+ FLN () a

116155 ) => (a -> Bool) -> Dist a -> Prob

117156(??) p d =

118157 let l = toValues d

@@ -122,11 +161,11 @@ bindD (D d) m = D (m `comp` d)

122161-- Distribution Construction

123162

124163-- | Constructs a Bernoulli distribution

125−choose :: (FromListsN () a) => Prob -> Dist a

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

126165choose prob = D (col [prob, 1 - prob])

127166

128167-- | Creates a distribution given a shape function

129−shape :: (FromListsN () a) => (Prob -> Prob) -> [a] -> Dist a

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

130169shape _ [] = error "Probability.shape: empty list"

131170shape f xs =

132171 let incr = 1 / fromIntegral (length xs - 1)

@@ -134,23 +173,23 @@ shape f xs =

134173 in fromFreqs (zip xs ps)

135174

136175-- | Constructs a Linear distribution

137−linear :: (FromListsN () a) => [a] -> Dist a

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

138177linear = shape id

139178

140179-- | Constructs an Uniform distribution

141−uniform :: (FromListsN () a) => [a] -> Dist a

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

142181uniform = shape (const 1)

143182

144183-- | Constructs an Negative Exponential distribution

145−negExp :: (FromListsN () a) => [a] -> Dist a

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

146185negExp = shape (\x -> exp (-x))

147186

148187-- | Constructs an Normal distribution

149−normal :: (FromListsN () a) => [a] -> Dist a

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

150189normal = shape (normalCurve 0.5 0.5)

151190

152191-- | Transforms a 'Dist' into a list of pairs.

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

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

154193toValues (D d) =

155194 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))

156195 probs = toList d

@@ -158,7 +197,7 @@ toValues (D d) =

158197 in res

159198

160199-- | Pretty a distribution

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

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

162201prettyDist d =

163202 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)

164203 w = maximum (map (length . show . fst) values)

@@ -170,12 +209,12 @@ prettyDist d =

170209 showR n x = show x ++ " "

171210

172211-- | Pretty Print a distribution

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

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

174213prettyPrintDist = putStrLn . prettyDist @a

175214

176215-- Auxiliary functions

177216

178−fromFreqs :: (FromListsN () a) => [(a,Prob)] -> Dist a

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

179218fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))

180219 where q = sum $ map snd xs

181220

src/LAoP/Matrix/Internal.hs+396−396

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

3030-- motivation behind the library, the underlying theory, and implementation details.

3131--

3232-- This module offers many of the combinators mentioned in the work of

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

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

3434--

3535-- This is an Internal module and it is no supposed to be imported.

3636--

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

4343 --

4444 -- There exists two type families that make it easier to write

4545 -- matrix dimensions: 'FromNat' and 'Count'. This approach

46− -- leads to a very straightforward implementation

47− -- of LAoP combinators.

46+ -- leads to a very straightforward implementation

47+ -- of LAoP combinators.

4848

4949 -- * Type safe matrix representation

5050 Matrix (..),

5151

52+ -- * Constraint type aliases

53+ Countable,

54+ CountableDims,

55+ CountableN,

56+ CountableDimsN,

57+ FL,

58+ FLN,

59+ Liftable,

60+ Trivial,

61+

5262 -- * Primitives

53− empty,

5463 one,

55− junc,

56− split,

64+ join,

65+ fork,

5766

5867 -- * Auxiliary type families

5968 FromNat,

@@ -66,6 +75,7 @@ module LAoP.Matrix.Internal

6675 toLists,

6776 toList,

6877 matrixBuilder,

78+ matrixBuilder',

6979 row,

7080 col,

7181 zeros,

@@ -76,11 +86,17 @@ module LAoP.Matrix.Internal

7686 -- * Misc

7787 -- ** Get dimensions

7888 columns,

89+ columns',

7990 rows,

91+ rows',

8092

8193 -- ** Matrix Transposition

8294 tr,

8395

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

97+ (.|),

98+ (./),

99+

84100 -- ** Selective operator

85101 select,

86102 branch,

@@ -89,16 +105,19 @@ module LAoP.Matrix.Internal

89105 cond,

90106

91107 -- ** Matrix "abiding"

92− abideJS,

93− abideSJ,

108+ abideJF,

109+ abideFJ,

110+

111+ -- ** Zip matrices

112+ zipWithM,

94113

95114 -- * Biproduct approach

96− -- ** Split

115+ -- ** Fork

97116 (===),

98117 -- *** Projections

99118 p1,

100119 p2,

101− -- ** Junc

120+ -- ** Join

102121 (|||),

103122 -- *** Injections

104123 i1,

@@ -111,12 +130,13 @@ module LAoP.Matrix.Internal

111130

112131 -- | Note that given the restrictions imposed it is not possible to

113132 -- implement the standard type classes present in standard Haskell.

133+

114134 -- *** Matrix pairing projections

115− kp1,

116− kp2,

135+ fstM,

136+ sndM,

117137

118138 -- *** Matrix pairing

119− khatri,

139+ kr,

120140

121141 -- * Matrix composition and lifting

122142

@@ -124,10 +144,10 @@ module LAoP.Matrix.Internal

124144

125145 -- | Note that given the restrictions imposed it is not possible to

126146 -- implement the standard type classes present in standard Haskell.

127− identity,

147+ iden,

128148 comp,

129− fromF,

130149 fromF',

150+ fromF,

131151

132152 -- * Matrix printing

133153 pretty,

@@ -154,21 +174,22 @@ import LAoP.Utils.Internal

154174import Data.Bool

155175import Data.Kind

156176import Data.List

177+import Data.Maybe

157178import Data.Proxy

179+import Data.Typeable

158180import Data.Void

159181import GHC.TypeLits

160182import Data.Type.Equality

161183import GHC.Generics

162184import Control.DeepSeq

163−import Control.Category

164−import Prelude hiding ((.))

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

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

165187

166188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.

167189data Matrix e cols rows where

168− Empty :: Matrix e Void Void

169190 One :: e -> Matrix e () ()

170− Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

171− Split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

191+ Join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

192+ Fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

172193

173194deriving instance (Show e) => Show (Matrix e cols rows)

174195

@@ -195,122 +216,95 @@ type family Count (d :: Type) :: Nat where

195216--

196217-- Thanks to Li-Yao Xia this type family is super fast.

197218type family FromNat (n :: Nat) :: Type where

198− FromNat 0 = Void

199219 FromNat 1 = ()

200220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))

201221

202222type family FromNat' (b :: Bool) (m :: Type) :: Type where

203− FromNat' 'True m = Either m m

223+ FromNat' 'True m = Either m m

204224 FromNat' 'False m = Either () (Either m m)

205225

206226-- | Type family that normalizes the representation of a given data

207227-- structure

208228type family Normalize (d :: Type) :: Type where

209− Normalize (Either a b) = Either (Normalize a) (Normalize b)

210− Normalize d = FromNat (Count d)

229+ Normalize (Either a b) = Either (Normalize a) (Normalize b)

230+ Normalize d = FromNat (Count d)

211231

212232-- | Constraint type synonyms to keep the type signatures less convoluted

213233type Countable a = KnownNat (Count a)

214234type CountableN a = KnownNat (Count (Normalize a))

215−type CountableDimensions a b = (Countable a, Countable b)

216−type CountableDimensionsN a b = (CountableN a, CountableN b)

217−type FromListsN e a b = FromLists e (Normalize a) (Normalize b)

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)

218239type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

219240type Trivial a = FromNat (Count a) ~ a

220241

221−-- | It isn't possible to implement the 'id' function so it's

222−-- implementation is 'undefined'. However 'comp' can be and this partial

223−-- class implementation exists just to make the code more readable.

224−--

225−-- Please use 'identity' instead.

242+-- | It is possible to implement a constrained version of the category type

243+-- class.

226244instance (Num e) => Category (Matrix e) where

227− id = undefined

228− (.) = comp

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

246+ id = iden

247+ (.) = comp

229248

230249instance NFData e => NFData (Matrix e cols rows) where

231− rnf Empty = ()

232− rnf (One e) = rnf e

233− rnf (Junc a b) = rnf a `seq` rnf b

234− rnf (Split a b) = rnf a `seq` rnf b

250+ rnf (One e) = rnf e

251+ rnf (Join a b) = rnf a `seq` rnf b

252+ rnf (Fork a b) = rnf a `seq` rnf b

235253

236254instance Eq e => Eq (Matrix e cols rows) where

237− Empty == Empty = True

238− (One a) == (One b) = a == b

239− (Junc a b) == (Junc c d) = a == c && b == d

240− (Split a b) == (Split c d) = a == c && b == d

241− x@(Split _ _) == y@(Junc _ _) = x == abideJS y

242− x@(Junc _ _) == y@(Split _ _) = abideJS x == y

255+ (One a) == (One b) = a == b

256+ (Join a b) == (Join c d) = a == c && b == d

257+ (Fork a b) == (Fork c d) = a == c && b == d

258+ x@(Fork _ _) == y@(Join _ _) = x == abideJF y

259+ x@(Join _ _) == y@(Fork _ _) = abideJF x == y

243260

244261instance 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

245267

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− Empty - Empty = Empty

254− (One a) - (One b) = One (a - b)

255− (Junc a b) - (Junc c d) = Junc (a - c) (b - d)

256− (Split a b) - (Split c d) = Split (a - c) (b - d)

257− x@(Split _ _) - y@(Junc _ _) = x - abideJS y

258− x@(Junc _ _) - y@(Split _ _) = abideJS x - y

259−

260− Empty * Empty = Empty

261− (One a) * (One b) = One (a * b)

262− (Junc a b) * (Junc c d) = Junc (a * c) (b * d)

263− (Split a b) * (Split c d) = Split (a * c) (b * d)

264− x@(Split _ _) * y@(Junc _ _) = x * abideJS y

265− x@(Junc _ _) * y@(Split _ _) = abideJS x * y

266−

267− abs Empty = Empty

268− abs (One a) = One (abs a)

269− abs (Junc a b) = Junc (abs a) (abs b)

270− abs (Split a b) = Split (abs a) (abs b)

271−

272− signum Empty = Empty

273− signum (One a) = One (signum a)

274− signum (Junc a b) = Junc (signum a) (signum b)

275− signum (Split a b) = Split (signum a) (signum b)

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)

271+

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)

276275

277276instance Ord e => Ord (Matrix e cols rows) where

278− Empty <= Empty = True

279− (One a) <= (One b) = a <= b

280− (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)

281− (Split a b) <= (Split c d) = (a <= c) && (b <= d)

282− x@(Split _ _) <= y@(Junc _ _) = x <= abideJS y

283− x@(Junc _ _) <= y@(Split _ _) = abideJS x <= y

277+ (One a) <= (One b) = a <= b

278+ (Join a b) <= (Join c d) = (a <= c) && (b <= d)

279+ (Fork a b) <= (Fork c d) = (a <= c) && (b <= d)

280+ x@(Fork _ _) <= y@(Join _ _) = x <= abideJF y

281+ x@(Join _ _) <= y@(Fork _ _) = abideJF x <= y

284282

285283-- Primitives

286284

287−-- | Empty matrix constructor

288−empty :: Matrix e Void Void

289−empty = Empty

290−

291285-- | Unit matrix constructor

292286one :: e -> Matrix e () ()

293287one = One

294288

295−-- | Matrix 'Junc' constructor

296−junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

297−junc = Junc

289+-- | Matrix 'Join' constructor

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

291+join = Join

298292

299293infixl 3 |||

300294

301−-- | Matrix 'Junc' constructor

295+-- | Matrix 'Join' constructor

302296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows

303−(|||) = Junc

297+(|||) = Join

304298

305−-- | Matrix 'Split' constructor

306−split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

307−split = Split

299+-- | Matrix 'Fork' constructor

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

301+fork = Fork

308302

309303infixl 2 ===

310304

311−-- | Matrix 'Split' constructor

305+-- | Matrix 'Fork' constructor

312306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)

313−(===) = Split

307+(===) = Fork

314308

315309-- Construction

316310

@@ -318,84 +312,101 @@ infixl 2 ===

318312--

319313-- Given that it is not possible to branch on types at the term level type

320314-- classes are needed very much like an inductive definition but on types.

321−class FromLists e cols rows where

315+class FromLists cols rows where

322316 -- | Build a matrix out of a list of list of elements. Throws a runtime

323317 -- error if the dimensions do not match.

324318 fromLists :: [[e]] -> Matrix e cols rows

325319

326−instance FromLists e Void Void where

327− fromLists [] = Empty

328− fromLists _ = error "Wrong dimensions"

329−

330−instance {-# OVERLAPPING #-} FromLists e () () where

320+instance FromLists () () where

331321 fromLists [[e]] = One e

332322 fromLists _ = error "Wrong dimensions"

333323

334−instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where

335− fromLists [h : t] = Junc (One h) (fromLists [t])

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

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

336326 fromLists _ = error "Wrong dimensions"

337327

338−instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), Countable a) => FromLists e (Either a b) () where

339− fromLists [l] =

328+instance {-# OVERLAPPABLE #-}

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

330+ fromLists [l] =

340331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

341− in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])

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

342333 fromLists _ = error "Wrong dimensions"

343334

344−instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where

345− fromLists ([h] : t) = Split (One h) (fromLists t)

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

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

346337 fromLists _ = error "Wrong dimensions"

347338

348−instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, Countable a) => FromLists e () (Either a b) where

349− fromLists l@([_] : _) =

339+instance {-# OVERLAPPABLE #-}

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

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

350342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))

351− in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))

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

352344 fromLists _ = error "Wrong dimensions"

353345

354−instance {-# OVERLAPPABLE #-} (FromLists e (Either a b) c, FromLists e (Either a b) d, Countable c) => FromLists e (Either a b) (Either c d) where

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

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

355348 fromLists l@(h : t) =

356349 let lh = length h

357350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))

358− condition = all (== lh) (map length t)

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

359352 in if lh > 0 && condition

360− then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))

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

361354 else error "Not all rows have the same length"

362355

363356-- | Matrix builder function. Constructs a matrix provided with

364−-- a construction function.

365−matrixBuilder ::

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

358+matrixBuilder' ::

366359 forall e cols rows.

367− ( FromLists e cols rows,

368− CountableDimensions cols rows

369− ) =>

370− ((Int, Int) -> e) ->

371− Matrix e cols rows

372−matrixBuilder f =

360+ ( FL cols rows,

361+ CountableDims cols rows

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

363+matrixBuilder' f =

373364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

374365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))

375366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]

376367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions

377368

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+

378389-- | Constructs a column vector matrix

379−col :: (FromLists e () rows) => [e] -> Matrix e () rows

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

380391col = fromLists . map (: [])

381392

382393-- | Constructs a row vector matrix

383−row :: (FromLists e cols ()) => [e] -> Matrix e cols ()

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

384395row = fromLists . (: [])

385396

386397-- | Lifts functions to matrices with arbitrary dimensions.

387398--

388399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

389400-- types @a@ or @b@ allows.

390−fromF ::

401+fromF' ::

391402 forall a b cols rows e.

392403 ( Liftable e a b,

393− CountableDimensions cols rows,

394− FromLists e rows cols

404+ CountableDims cols rows,

405+ FL rows cols

395406 ) =>

396407 (a -> b) ->

397408 Matrix e cols rows

398−fromF f =

409+fromF' f =

399410 let minA = minBound @a

400411 maxA = maxBound @a

401412 minB = minBound @b

@@ -405,8 +416,8 @@ fromF f =

405416 elementsA = take ccols [minA .. maxA]

406417 elementsB = take rrows [minB .. maxB]

407418 combinations = (,) <$> elementsA <*> elementsB

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

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

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

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

410421 else ((fromEnum a, fromEnum b), 0)) $ combinations

411422 mList = buildList combAp rrows

412423 in tr $ fromLists mList

@@ -416,41 +427,23 @@ fromF f =

416427

417428-- | Lifts functions to matrices with dimensions matching @a@ and @b@

418429-- cardinality's.

419−fromF' ::

430+fromF ::

420431 forall a b e.

421432 ( Liftable e a b,

422− CountableDimensionsN a b,

423− FromListsN e b a

433+ CountableDimsN a b,

434+ FLN b a

424435 ) =>

425436 (a -> b) ->

426437 Matrix e (Normalize a) (Normalize b)

427−fromF' f =

428− let minA = minBound @a

429− maxA = maxBound @a

430− minB = minBound @b

431− maxB = maxBound @b

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

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

434− elementsA = take ccols [minA .. maxA]

435− elementsB = take rrows [minB .. maxB]

436− combinations = (,) <$> elementsA <*> elementsB

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

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

439− else ((fromEnum a, fromEnum b), 0)) $ combinations

440− mList = buildList combAp rrows

441− in tr $ fromLists mList

442− where

443− buildList [] _ = []

444− buildList l r = take r l : buildList (drop r l) r

438+fromF = fromF'

445439

446440-- Conversion

447441

448442-- | Converts a matrix to a list of lists of elements.

449443toLists :: Matrix e cols rows -> [[e]]

450−toLists Empty = []

451444toLists (One e) = [[e]]

452−toLists (Split l r) = toLists l ++ toLists r

453−toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)

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

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

454447

455448-- | Converts a matrix to a list of elements.

456449toList :: Matrix e cols rows -> [e]

@@ -459,38 +452,38 @@ toList = concat . toLists

459452-- Zeros Matrix

460453

461454-- | The zero matrix. A matrix wholly filled with zeros.

462−zeros :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows

463−zeros = matrixBuilder (const 0)

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

456+zeros = matrixBuilder' (const 0)

464457

465458-- Ones Matrix

466459

467460-- | The ones matrix. A matrix wholly filled with ones.

468461--

469462-- Also known as T (Top) matrix.

470−ones :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows

471−ones = matrixBuilder (const 1)

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

464+ones = matrixBuilder' (const 1)

472465

473466-- Const Matrix

474467

475468-- | The constant matrix constructor. A matrix wholly filled with a given

476469-- value.

477−constant :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => e -> Matrix e cols rows

478−constant e = matrixBuilder (const e)

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

471+constant e = matrixBuilder' (const e)

479472

480473-- Bang Matrix

481474

482475-- | The T (Top) row vector matrix.

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

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

484477bang =

485478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))

486479 in fromLists [take c [1, 1 ..]]

487480

488−-- Identity Matrix

481+-- iden Matrix

489482

490−-- | Identity matrix.

491−identity :: (Num e, FromLists e cols cols, Countable cols) => Matrix e cols cols

492−identity = matrixBuilder (bool 0 1 . uncurry (==))

493−{-# NOINLINE identity #-}

483+-- | iden matrix.

484+iden :: forall e cols . (Num e, FL cols cols, Countable cols) => Matrix e cols cols

485+iden = matrixBuilder' (bool 0 1 . uncurry (==))

486+{-# NOINLINE iden #-}

494487

495488-- Matrix composition (MMM)

496489

@@ -499,65 +492,90 @@ identity = matrixBuilder (bool 0 1 . uncurry (==))

499492-- This definition takes advantage of divide-and-conquer and fusion laws

500493-- from LAoP.

501494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

502−comp Empty Empty = Empty

503−comp (One a) (One b) = One (a * b)

504−comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law

505−comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law

506−comp c (Junc a b) = Junc (comp c a) (comp c b) -- Junc fusion law

495+comp (One a) (One b) = One (a * b)

496+comp (Join a b) (Fork c d) = comp a c + comp b d -- Divide-and-conquer law

497+comp (Fork a b) c = Fork (comp a c) (comp b c) -- Fork fusion law

498+comp c (Join a b) = Join (comp c a) (comp c b) -- Join fusion law

507499{-# NOINLINE comp #-}

508−{-# RULES

509− "comp/identity1" forall m. comp m identity = m ;

510− "comp/identity2" forall m. comp identity m = m

500+{-# RULES

501+ "comp/iden1" forall m. comp m iden = m ;

502+ "comp/iden2" forall m. comp iden m = m

511503#-}

512504

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)

522+

513523-- Projections

514524

515525-- | Biproduct first component projection

516−p1 :: forall e m n. (Num e, CountableDimensions n m, FromLists e n m, FromLists e m m) => Matrix e (Either m n) m

517−p1 =

518− let iden = identity :: Matrix e m m

519− zero = zeros :: Matrix e n m

520− in junc iden zero

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

527+p1 = join id zeros

521528

522529-- | Biproduct second component projection

523−p2 :: forall e m n. (Num e, CountableDimensions n m, FromLists e m n, FromLists e n n) => Matrix e (Either m n) n

524−p2 =

525− let iden = identity :: Matrix e n n

526− zero = zeros :: Matrix e m n

527− in junc zero iden

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

531+p2 = join zeros id

528532

529533-- Injections

530534

531535-- | Biproduct first component injection

532−i1 :: (Num e, CountableDimensions n m, FromLists e n m, FromLists e m m) => Matrix e m (Either m n)

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

533537i1 = tr p1

534538

535539-- | Biproduct second component injection

536−i2 :: (Num e, CountableDimensions n m, FromLists e m n, FromLists e n n) => Matrix e n (Either m n)

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

537541i2 = tr p2

538542

539543-- Dimensions

540544

541545-- | Obtain the number of rows.

542546--

543−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

544−-- dimensions in constant time.

545−--

546−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

547−-- exchange for performance.

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''.

548550rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int

549551rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))

550552

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

554+-- constraints.

555+--

556+-- For a more efficient version see 'rows'.

557+rows' :: Matrix e cols rows -> Int

558+rows' (One _) = 1

559+rows' (Join lhs _) = rows' lhs

560+rows' (Fork top bottom) = rows' top + rows' bottom

561+

551562-- | Obtain the number of columns.

552−--

553−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

554−-- dimensions in constant time.

555563--

556−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

557−-- exchange for performance.

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''.

558567columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int

559568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))

560569

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+

561579-- Coproduct Bifunctor

562580

563581infixl 5 -|-

@@ -566,164 +584,161 @@ infixl 5 -|-

566584(-|-) ::

567585 forall e n k m j.

568586 ( Num e,

569− CountableDimensions j k,

570− FromLists e k k,

571− FromLists e j k,

572− FromLists e k j,

573− FromLists e j j

587+ CountableDims j k,

588+ FL k k,

589+ FL j k,

590+ FL k j,

591+ FL j j

574592 ) =>

575593 Matrix e n k ->

576594 Matrix e m j ->

577595 Matrix e (Either n m) (Either k j)

578−(-|-) a b = Junc (i1 . a) (i2 . b)

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

579597

580598-- Khatri Rao Product and projections

581599

582600-- | Khatri Rao product first component projection matrix.

583−kp1 ::

601+fstM ::

584602 forall e m k .

585603 ( Num e,

586− CountableDimensions k m,

587− FromLists e (Normalize (m, k)) m,

604+ CountableDims k m,

605+ FL (Normalize (m, k)) m,

588606 CountableN (m, k)

589607 ) => Matrix e (Normalize (m, k)) m

590−kp1 = matrixBuilder f

608+fstM = matrixBuilder' f

591609 where

592610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

593611 f (x, y)

594612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1

595− | otherwise = 0

613+ | otherwise = 0

596614

597615-- | Khatri Rao product second component projection matrix.

598−kp2 ::

616+sndM ::

599617 forall e m k .

600618 ( Num e,

601− CountableDimensions k m,

602− FromLists e (Normalize (m, k)) k,

619+ CountableDims k m,

620+ FL (Normalize (m, k)) k,

603621 CountableN (m, k)

604622 ) => Matrix e (Normalize (m, k)) k

605−kp2 = matrixBuilder f

623+sndM = matrixBuilder' f

606624 where

607625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))

608626 f (x, y)

609627 | x == y || mod (y - x) offset == 0 = 1

610− | otherwise = 0

628+ | otherwise = 0

611629

612630-- | Khatri Rao Matrix product also known as matrix pairing.

613631--

614632-- NOTE: That this is not a true categorical product, see for instance:

615−--

633+--

616634-- @

617−-- | kp1 . khatri a b == a

618−-- khatri a b ==> |

619−-- | kp2 . khatri a b == b

635+-- | fstM . kr a b == a

636+-- kr a b ==> |

637+-- | sndM . kr a b == b

620638-- @

621639--

622640-- __Emphasis__ on the implication symbol.

623−khatri ::

624− forall e cols a b.

641+kr ::

642+ forall e cols a b.

625643 ( Num e,

626− CountableDimensions a b,

644+ CountableDims a b,

627645 CountableN (a, b),

628− FromLists e (Normalize (a, b)) a,

629− FromLists e (Normalize (a, b)) b

646+ FL (Normalize (a, b)) a,

647+ FL (Normalize (a, b)) b

630648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))

631−khatri a b =

632− let kp1' = kp1 @e @a @b

633− kp2' = kp2 @e @a @b

634− in (tr kp1') . a * (tr kp2') . b

649+kr a b =

650+ let fstM' = fstM @e @a @b

651+ sndM' = sndM @e @a @b

652+ in (tr fstM' . a) * (tr sndM' . b)

635653

636654-- Product Bifunctor (Kronecker)

637655

638656infixl 4 ><

639657

640658-- | Matrix product functor also known as kronecker product

641−(><) ::

642− forall e m p n q.

659+(><) ::

660+ forall e m p n q.

643661 ( Num e,

644− CountableDimensions m n,

645− CountableDimensions p q,

646− CountableDimensionsN (m, n) (p, q),

647− FromLists e (Normalize (m, n)) m,

648− FromLists e (Normalize (m, n)) n,

649− FromLists e (Normalize (p, q)) p,

650− FromLists e (Normalize (p, q)) q

651− )

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+ )

652670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))

653671(><) a b =

654− let kp1' = kp1 @e @m @n

655− kp2' = kp2 @e @m @n

656− in khatri (a . kp1') (b . kp2')

672+ let fstM' = fstM @e @m @n

673+ sndM' = sndM @e @m @n

674+ in kr (a . fstM') (b . sndM')

657675

658−-- Matrix abide Junc Split

676+-- Matrix abide Join Fork

659677

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

661−--

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

679+--

662680-- Law:

663681--

664682-- @

665−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

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

666684-- @

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

668−abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law

669−abideJS Empty = Empty

670−abideJS (One e) = One e

671−abideJS (Junc a b) = Junc (abideJS a) (abideJS b)

672−abideJS (Split a b) = Split (abideJS a) (abideJS b)

685+abideJF :: Matrix e cols rows -> Matrix e cols rows

686+abideJF (Join (Fork a c) (Fork b d)) = Fork (Join (abideJF a) (abideJF b)) (Join (abideJF c) (abideJF d)) -- Join-Fork abide law

687+abideJF (One e) = One e

688+abideJF (Join a b) = Join (abideJF a) (abideJF b)

689+abideJF (Fork a b) = Fork (abideJF a) (abideJF b)

673690

674−-- Matrix abide Split Junc

691+-- Matrix abide Fork Join

675692

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

677−--

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

694+--

678695-- @

679−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

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

680697-- @

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

682−abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law

683−abideSJ Empty = Empty

684−abideSJ (One e) = One e

685−abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)

686−abideSJ (Split a b) = Split (abideSJ a) (abideSJ b)

698+abideFJ :: Matrix e cols rows -> Matrix e cols rows

699+abideFJ (Fork (Join a b) (Join c d)) = Join (Fork (abideFJ a) (abideFJ c)) (Fork (abideFJ b) (abideFJ d)) -- Fork-Join abide law

700+abideFJ (One e) = One e

701+abideFJ (Join a b) = Join (abideFJ a) (abideFJ b)

702+abideFJ (Fork a b) = Fork (abideFJ a) (abideFJ b)

687703

688704-- Matrix transposition

689705

690706-- | Matrix transposition.

691707tr :: Matrix e cols rows -> Matrix e rows cols

692−tr Empty = Empty

693−tr (One e) = One e

694−tr (Junc a b) = Split (tr a) (tr b)

695−tr (Split a b) = Junc (tr a) (tr b)

708+tr (One e) = One e

709+tr (Join a b) = Fork (tr a) (tr b)

710+tr (Fork a b) = Join (tr a) (tr b)

696711

697712-- Selective 'select' operator

698713

699714-- | Selective functors 'select' operator equivalent inspired by the

700715-- ArrowMonad solution presented in the paper.

701−select :: (Num e, FromLists e b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

702−select (Split a b) y = y . a + b -- Divide-and-conquer law

703−select (Junc (Split a c) (Split b d)) y = junc (y . a + c) (y . b + d) -- Pattern matching + DnC law

704−select m y = junc y identity . m

716+select :: (Num e, FL b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

717+select (Fork a b) y = y . a + b -- Divide-and-conquer law

718+select (Join (Fork a c) (Fork b d)) y = join (y . a + c) (y . b + d) -- Pattern matching + DnC law

719+select m y = join y id . m

705720

706−branch ::

721+branch ::

707722 ( Num e,

708− CountableDimensions a b,

709− CountableDimensions c (Either b c),

710− FromLists e c b,

711− FromLists e a b,

712− FromLists e a a,

713− FromLists e b b,

714− FromLists e c c,

715− FromLists e b a,

716− FromLists e b c,

717− FromLists e (Either b c) b,

718− FromLists e (Either b c) c

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

719734 )

720735 => Matrix e cols (Either a b) -> Matrix e a c -> Matrix e b c -> Matrix e cols c

721736branch x l r = f x `select` g l `select` r

722737 where

723− f :: (Num e, Countable a, CountableDimensions b c, FromLists e a b, FromLists e c b, FromLists e b b, FromLists e b a, FromLists e a a)

738+ f :: (Num e, Countable a, CountableDims b c, FL a b, FL c b, FL b b, FL b a, FL a a)

724739 => Matrix e cols (Either a b) -> Matrix e cols (Either a (Either b c))

725− f m = split (tr i1) (i1 . tr i2) . m

726− g :: (Num e, CountableDimensions b c, FromLists e b c, FromLists e c c) => Matrix e a c -> Matrix e 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)

727742 g m = i2 . m

728743

729744-- McCarthy's Conditional

@@ -732,9 +747,9 @@ branch x l r = f x `select` g l `select` r

732747cond ::

733748 ( Trivial cols,

734749 Countable cols,

735− FromLists e () cols,

736− FromLists e cols (),

737− FromLists e cols cols,

750+ FL () cols,

751+ FL cols (),

752+ FL cols cols,

738753 Bounded a,

739754 Enum a,

740755 Num e,

@@ -742,14 +757,14 @@ cond ::

742757 )

743758 =>

744759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows

745−cond p f g = junc f g . grd p

760+cond p f g = join f g . grd p

746761

747−grd ::

762+grd ::

748763 ( Trivial q,

749764 Countable q,

750− FromLists e () q,

751− FromLists e q (),

752− FromLists e q q,

765+ FL () q,

766+ FL q (),

767+ FL q q,

753768 Bounded a,

754769 Enum a,

755770 Num e,

@@ -757,60 +772,86 @@ grd ::

757772 )

758773 =>

759774 (a -> Bool) -> Matrix e q (Either q q)

760−grd f = split (corr f) (corr (not . f))

775+grd f = fork (corr f) (corr (not . f))

761776

762−corr ::

763− forall e a q .

777+corr ::

778+ forall e a q .

764779 ( Trivial q,

765780 Countable q,

766− FromLists e () q,

767− FromLists e q (),

768− FromLists e q q,

781+ FL () q,

782+ FL q (),

783+ FL q q,

769784 Liftable e a Bool

770− )

785+ )

771786 => (a -> Bool) -> Matrix e q q

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

773− in khatri f (identity :: Matrix e q q)

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

788+ in kr f (id :: Matrix e q q)

774789

775790-- Pretty print

776791

777792prettyAux :: Show e => [[e]] -> [[e]] -> String

778793prettyAux [] _ = ""

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

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

780795 where

781796 v = fmap show m

782797 widest = maximum $ fmap length v

783798 fill str = replicate (widest - length str - 2) ' ' ++ str

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

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

785800 where

786− v = fmap show m

787− widest = maximum $ fmap length v

801+ v = fmap show m

802+ widest = maximum $ fmap length v

788803 fill str = replicate (widest - length str - 2) ' ' ++ str

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

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

790805 prettyAux t l

791806 where

792− v = fmap show l

793− widest = maximum $ fmap length v

807+ v = fmap show l

808+ widest = maximum $ fmap length v

794809 fill str = replicate (widest - length str - 2) ' ' ++ str

795810

796811-- | Matrix pretty printer

797−pretty :: (Countable cols, Show e) => Matrix e cols rows -> String

798−pretty m = "┌ " ++ unwords (replicate (columns m) blank) ++ " ┐\n" ++

799− prettyAux (toLists m) (toLists m) ++

800− "└ " ++ unwords (replicate (columns m) blank) ++ " ┘"

801− where

802− v = fmap show (toList m)

803− widest = maximum $ fmap length v

812+pretty :: (CountableDims cols rows, Show e) => Matrix e cols rows -> String

813+pretty m = concat

814+ [ "┌ ", unwords (replicate (columns m) blank), " ┐\n"

815+ , unlines

816+ [ "│ " ++ unwords (fmap (\j -> fill $ show $ getElem i j m) [1..columns m]) ++ " │" | i <- [1..rows m] ]

817+ , "└ ", unwords (replicate (columns m) blank), " ┘"

818+ ]

819+ where

820+ strings = map show (toList m)

821+ widest = maximum $ map length strings

804822 fill str = replicate (widest - length str) ' ' ++ str

805− blank = fill ""

823+ blank = fill ""

824+ safeGet i j m

825+ | i > rows m || j > columns m || i < 1 || j < 1 = Nothing

826+ | otherwise = Just $ unsafeGet i j m (toList m)

827+ unsafeGet i j m l = l !! encode (columns m) (i,j)

828+ encode m (i,j) = (i-1)*m + j - 1

829+ getElem i j m =

830+ fromMaybe

831+ (error $

832+ "getElem: Trying to get the "

833+ ++ show (i, j)

834+ ++ " element from a "

835+ ++ show (rows m) ++ "x" ++ show (columns m)

836+ ++ " matrix."

837+ )

838+ (safeGet i j m)

806839

807840-- | Matrix pretty printer

808−prettyPrint :: (Countable cols, Show e) => Matrix e cols rows -> IO ()

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

809842prettyPrint = putStrLn . pretty

810843

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+

811852-- Relational operators functions

812853

813−type Boolean = Natural 0 1

854+type Boolean = Natural 0 1

814855type Relation a b = Matrix Boolean a b

815856

816857-- | Helper conversion function

@@ -826,54 +867,47 @@ fromBool False = nat 0

826867

827868-- | Relational negation

828869negateM :: Relation cols rows -> Relation cols rows

829−negateM Empty = Empty

830−negateM (One (Nat 0)) = One (Nat 1)

831−negateM (One (Nat 1)) = One (Nat 0)

832−negateM (Junc a b) = Junc (negateM a) (negateM b)

833−negateM (Split a b) = Split (negateM a) (negateM b)

870+negateM (One (Nat p)) = One (Nat (negate p))

871+negateM (Join a b) = Join (negateM a) (negateM b)

872+negateM (Fork a b) = Fork (negateM a) (negateM b)

834873

835874-- | Relational addition

836875orM :: Relation cols rows -> Relation cols rows -> Relation cols rows

837−orM Empty Empty = Empty

838−orM (One a) (One b) = One (fromBool (toBool a || toBool b))

839−orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)

840−orM (Split a b) (Split c d) = Split (orM a c) (orM b d)

841−orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)

842−orM x@(Junc _ _) y@(Split _ _) = orM (abideJS x) y

876+orM (One a) (One b) = One (fromBool (toBool a || toBool b))

877+orM (Join a b) (Join c d) = Join (orM a c) (orM b d)

878+orM (Fork a b) (Fork c d) = Fork (orM a c) (orM b d)

879+orM x@(Fork _ _) y@(Join _ _) = orM x (abideJF y)

880+orM x@(Join _ _) y@(Fork _ _) = orM (abideJF x) y

843881

844882-- | Relational multiplication

845883andM :: Relation cols rows -> Relation cols rows -> Relation cols rows

846−andM Empty Empty = Empty

847−andM (One a) (One b) = One (fromBool (toBool a && toBool b))

848−andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)

849−andM (Split a b) (Split c d) = Split (andM a c) (andM b d)

850−andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)

851−andM x@(Junc _ _) y@(Split _ _) = andM (abideJS x) y

884+andM (One a) (One b) = One (fromBool (toBool a && toBool b))

885+andM (Join a b) (Join c d) = Join (andM a c) (andM b d)

886+andM (Fork a b) (Fork c d) = Fork (andM a c) (andM b d)

887+andM x@(Fork _ _) y@(Join _ _) = andM x (abideJF y)

888+andM x@(Join _ _) y@(Fork _ _) = andM (abideJF x) y

852889

853890-- | Relational subtraction

854891subM :: Relation cols rows -> Relation cols rows -> Relation cols rows

855−subM Empty Empty = Empty

856−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)

857−subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)

858−subM (Split a b) (Split c d) = Split (subM a c) (subM b d)

859−subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)

860−subM x@(Junc _ _) y@(Split _ _) = subM (abideJS x) y

892+subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)

893+subM (Join a b) (Join c d) = Join (subM a c) (subM b d)

894+subM (Fork a b) (Fork c d) = Fork (subM a c) (subM b d)

895+subM x@(Fork _ _) y@(Join _ _) = subM x (abideJF y)

896+subM x@(Join _ _) y@(Fork _ _) = subM (abideJF x) y

861897

862898-- | Matrix relational composition.

863899compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows

864−compRel Empty Empty = Empty

865−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))

866−compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law

867−compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law

868−compRel c (Junc a b) = Junc (compRel c a) (compRel c b) -- Junc fusion law

900+compRel (One a) (One b) = One (fromBool (toBool a && toBool b))

901+compRel (Join a b) (Fork c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law

902+compRel (Fork a b) c = Fork (compRel a c) (compRel b c) -- Fork fusion law

903+compRel c (Join a b) = Join (compRel c a) (compRel c b) -- Join fusion law

869904

870905-- | Matrix relational right division

871906divR :: Relation b c -> Relation b a -> Relation a c

872−divR Empty Empty = Empty

873907divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a

874−divR (Junc a b) (Junc c d) = andM (divR a c) (divR b d)

875−divR (Split a b) c = Split (divR a c) (divR b c)

876−divR c (Split a b) = Junc (divR c a) (divR c b)

908+divR (Join a b) (Join c d) = andM (divR a c) (divR b d)

909+divR (Fork a b) c = Fork (divR a c) (divR b c)

910+divR c (Fork a b) = Join (divR c a) (divR c b)

877911

878912-- | Matrix relational left division

879913divL :: Relation c b -> Relation a b -> Relation a c

@@ -889,86 +923,52 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)

889923--

890924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of

891925-- types @a@ or @b@ allows.

892−fromFRel ::

926+fromFRel' ::

893927 forall a b cols rows.

894928 ( Liftable Boolean a b,

895− CountableDimensions cols rows,

896− FromLists Boolean rows cols

929+ CountableDims cols rows,

930+ FL rows cols

897931 ) =>

898932 (a -> b) ->

899933 Relation cols rows

900−fromFRel f =

901− let minA = minBound @a

902− maxA = maxBound @a

903− minB = minBound @b

904− maxB = maxBound @b

905− ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))

906− rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))

907− elementsA = take ccols [minA .. maxA]

908− elementsB = take rrows [minB .. maxB]

909− combinations = (,) <$> elementsA <*> elementsB

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

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

912− else ((fromEnum a, fromEnum b), nat 0)) $ combinations

913− mList = buildList combAp rrows

914− in tr $ fromLists mList

915− where

916− buildList [] _ = []

917− buildList l r = take r l : buildList (drop r l) r

934+fromFRel' = fromF'

918935

919936-- | Lifts functions to relations with dimensions matching @a@ and @b@

920937-- cardinality's.

921−fromFRel' ::

938+fromFRel ::

922939 forall a b.

923940 ( Liftable Boolean a b,

924− CountableDimensionsN a b,

925− FromLists Boolean (Normalize b) (Normalize a)

941+ CountableDimsN a b,

942+ FLN b a

926943 ) =>

927944 (a -> b) ->

928945 Relation (Normalize a) (Normalize b)

929−fromFRel' f =

930− let minA = minBound @a

931− maxA = maxBound @a

932− minB = minBound @b

933− maxB = maxBound @b

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

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

936− elementsA = take ccols [minA .. maxA]

937− elementsB = take rrows [minB .. maxB]

938− combinations = (,) <$> elementsA <*> elementsB

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

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

941− else ((fromEnum a, fromEnum b), nat 0)) $ combinations

942− mList = buildList combAp rrows

943− in tr $ fromLists mList

944− where

945− buildList [] _ = []

946− buildList l r = take r l : buildList (drop r l) r

946+fromFRel = fromFRel'

947947

948948-- | Lifts a relation function to a Boolean Matrix

949−toRel ::

949+toRel ::

950950 forall a b.

951951 ( Bounded a,

952952 Bounded b,

953953 Enum a,

954954 Enum b,

955955 Eq b,

956− CountableDimensionsN a b,

957− FromListsN Boolean b a

958− )

956+ CountableDims a b,

957+ FLN b a

958+ )

959959 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)

960960toRel f =

961961 let minA = minBound @a

962962 maxA = maxBound @a

963963 minB = minBound @b

964964 maxB = maxBound @b

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

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

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

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

967967 elementsA = take ccols [minA .. maxA]

968968 elementsB = take rrows [minB .. maxB]

969969 combinations = (,) <$> elementsA <*> elementsB

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

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

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

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

972972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations

973973 mList = buildList combAp rrows

974974 in tr $ fromLists mList

src/LAoP/Matrix/Nat.hs+139−124

@@ -56,21 +56,20 @@ module LAoP.Matrix.Nat

5656

5757 -- * Constraint type synonyms

5858 Countable,

59− CountableDimensions,

59+ CountableDims,

6060 CountableN,

6161 CountableNz,

62− CountableDimensionsN,

63− FromListsN,

64− FromListsNz,

62+ CountableDimsN,

63+ FLN,

64+ FLNz,

6565 Liftable,

6666 TrivialE,

6767 TrivialP,

6868

6969 -- * Primitives

70− empty,

7170 one,

72− junc,

73− split,

71+ join,

72+ fork,

7473

7574 -- * Auxiliary type families

7675 I.FromNat,

@@ -78,11 +77,11 @@ module LAoP.Matrix.Nat

7877 I.Normalize,

7978

8079 -- * Matrix construction and conversion

81− I.FromLists,

80+ I.FL,

8281 fromLists,

8382 toLists,

8483 toList,

85− matrixBuilder,

84+ matrixBuilder',

8685 row,

8786 col,

8887 zeros,

@@ -98,6 +97,10 @@ module LAoP.Matrix.Nat

9897 -- ** Matrix Transposition

9998 tr,

10099

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

101+ (.|),

102+ (./),

103+

101104 -- ** Selective operator

102105 select,

103106

@@ -105,16 +108,19 @@ module LAoP.Matrix.Nat

105108 cond,

106109

107110 -- ** Matrix "abiding"

108− abideJS,

109− abideSJ,

111+ abideJF,

112+ abideFJ,

113+

114+ -- ** Zip Matrices

115+ zipWithM,

110116

111117 -- * Biproduct approach

112− -- ** Split

118+ -- ** Fork

113119 (===),

114120 -- *** Projections

115121 p1,

116122 p2,

117− -- ** Junc

123+ -- ** Join

118124 (|||),

119125 -- *** Injections

120126 i1,

@@ -128,11 +134,11 @@ module LAoP.Matrix.Nat

128134 -- | Note that given the restrictions imposed it is not possible to

129135 -- implement the standard type classes present in standard Haskell.

130136 -- *** Matrix pairing projections

131− kp1,

132− kp2,

137+ fstM,

138+ sndM,

133139

134140 -- *** Matrix pairing

135− khatri,

141+ kr,

136142

137143 -- * Matrix composition and lifting

138144

@@ -140,10 +146,10 @@ module LAoP.Matrix.Nat

140146

141147 -- | Note that given the restrictions imposed it is not possible to

142148 -- implement the standard type classes present in standard Haskell.

143− identity,

149+ iden,

144150 comp,

145− fromF,

146151 fromF',

152+ fromF,

147153

148154 -- * Matrix printing

149155 pretty,

@@ -154,47 +160,34 @@ where

154160import Data.Proxy

155161import GHC.TypeLits

156162import Control.DeepSeq

157−import qualified Control.Category as C

158163import qualified LAoP.Matrix.Internal as I

159164

160165newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))

161166 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))

162167

163168-- | Constraint type synonyms to keep the type signatures less convoluted

164−type Countable a = KnownNat (I.Count a)

165−type CountableDimensions a b = (Countable a, Countable b)

166−type CountableN a = KnownNat (I.Count (I.FromNat a))

167−type CountableNz a = KnownNat (I.Count (I.Normalize a))

168−type CountableDimensionsN a b = (CountableN a, CountableN b)

169−type FromListsN e a b = I.FromLists e (I.FromNat a) (I.FromNat b)

170−type FromListsNz e a b = I.FromLists e (I.Normalize a) (I.Normalize b)

171−type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

172−type TrivialE a b = I.FromNat (a + b) ~ Either (I.FromNat a) (I.FromNat b)

173−type TrivialP a b = I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))

174−

175−-- | It isn't possible to implement the 'id' function so it's

176−-- implementation is 'undefined'. However 'comp' can be and this partial

177−-- class implementation exists just to make the code more readable.

178−--

179−-- Please use 'identity' instead.

180−instance (Num e) => C.Category (Matrix e) where

181− id = undefined

182− (.) = comp

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))

183179

184180-- Primitives

185181

186−empty :: Matrix e 0 0

187−empty = M I.Empty

188−

189182one :: e -> Matrix e 1 1

190183one = M . I.One

191184

192−junc ::

185+join ::

193186 (TrivialE a b) =>

194187 Matrix e a rows ->

195188 Matrix e b rows ->

196189 Matrix e (a + b) rows

197−junc (M a) (M b) = M (I.Junc a b)

190+join (M a) (M b) = M (I.Join a b)

198191

199192infixl 3 |||

200193(|||) ::

@@ -202,14 +195,14 @@ infixl 3 |||

202195 Matrix e a rows ->

203196 Matrix e b rows ->

204197 Matrix e (a + b) rows

205−(|||) = junc

198+(|||) = join

206199

207−split ::

200+fork ::

208201 (TrivialE a b) =>

209202 Matrix e cols a ->

210203 Matrix e cols b ->

211204 Matrix e cols (a + b)

212−split (M a) (M b) = M (I.Split a b)

205+fork (M a) (M b) = M (I.Fork a b)

213206

214207infixl 2 ===

215208(===) ::

@@ -217,43 +210,45 @@ infixl 2 ===

217210 Matrix e cols a ->

218211 Matrix e cols b ->

219212 Matrix e cols (a + b)

220−(===) = split

213+(===) = fork

221214

222215-- Construction

223216

224−fromLists :: (FromListsN e cols rows) => [[e]] -> Matrix e cols rows

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

225218fromLists = M . I.fromLists

226219

227−matrixBuilder ::

228− (FromListsN e cols rows, CountableN cols, CountableN rows)

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)

229224 => ((Int, Int) -> e) -> Matrix e cols rows

230−matrixBuilder = M . I.matrixBuilder

225+matrixBuilder' = M . I.matrixBuilder'

231226

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

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

233228col = M . I.col

234229

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

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

236231row = M . I.row

237232

238−fromF ::

233+fromF' ::

239234 ( Liftable e a b,

240235 CountableN cols,

241236 CountableN rows,

242− FromListsN e rows cols

237+ FLN e rows cols

243238 ) =>

244239 (a -> b) ->

245240 Matrix e cols rows

246−fromF = M . I.fromF

241+fromF' = M . I.fromF'

247242

248−fromF' ::

243+fromF ::

249244 ( Liftable e a b,

250245 CountableNz a,

251246 CountableNz b,

252− FromListsNz e b a

247+ FLNz e b a

253248 ) =>

254249 (a -> b) ->

255250 Matrix e (I.Count a) (I.Count b)

256−fromF' = undefined -- M . I.fromF'

251+fromF = undefined -- M . I.fromF

257252

258253-- Conversion

259254

@@ -266,21 +261,21 @@ toList (M m) = I.toList m

266261-- Zeros Matrix

267262

268263zeros ::

269− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>

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

270265 Matrix e cols rows

271266zeros = M I.zeros

272267

273268-- Ones Matrix

274269

275270ones ::

276− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>

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

277272 Matrix e cols rows

278273ones = M I.ones

279274

280275-- Const Matrix

281276

282277constant ::

283− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>

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

284279 e ->

285280 Matrix e cols rows

286281constant = M . I.constant

@@ -289,27 +284,41 @@ constant = M . I.constant

289284

290285bang ::

291286 forall e cols.

292− (Num e, Enum e, I.FromLists e (I.FromNat cols) (), CountableN cols) =>

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

293288 Matrix e cols 1

294289bang = M I.bang

295290

296−-- Identity Matrix

291+-- iden Matrix

297292

298−identity ::

299− (Num e, FromListsN e cols cols, CountableN cols) =>

293+iden ::

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

300295 Matrix e cols cols

301−identity = M I.identity

296+iden = M I.iden

302297

303298-- Matrix composition (MMM)

304299

305300comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

306301comp (M a) (M b) = M (I.comp a b)

307302

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+

308317p1 ::

309318 ( Num e,

310− CountableDimensionsN n m,

311− FromListsN e n m,

312− FromListsN e m m,

319+ CountableDimsN n m,

320+ FLN e n m,

321+ FLN e m m,

313322 TrivialE m n

314323 ) =>

315324 Matrix e (m + n) m

@@ -317,9 +326,9 @@ p1 = M I.p1

317326

318327p2 ::

319328 ( Num e,

320− CountableDimensionsN n m,

321− FromListsN e m n,

322− FromListsN e n n,

329+ CountableDimsN n m,

330+ FLN e m n,

331+ FLN e n n,

323332 TrivialE m n

324333 ) =>

325334 Matrix e (m + n) n

@@ -329,9 +338,9 @@ p2 = M I.p2

329338

330339i1 ::

331340 ( Num e,

332− CountableDimensionsN n rows,

333− FromListsN e n rows,

334− FromListsN e rows rows,

341+ CountableDimsN n rows,

342+ FLN e n rows,

343+ FLN e rows rows,

335344 TrivialE rows n

336345 ) =>

337346 Matrix e rows (rows + n)

@@ -339,9 +348,9 @@ i1 = tr p1

339348

340349i2 ::

341350 ( Num e,

342− CountableDimensionsN rows m,

343− FromListsN e m rows,

344− FromListsN e rows rows,

351+ CountableDimsN rows m,

352+ FLN e m rows,

353+ FLN e rows rows,

345354 TrivialE m rows

346355 ) =>

347356 Matrix e rows (m + rows)

@@ -355,16 +364,16 @@ rows (M m) = I.rows m

355364columns :: (CountableN cols) => Matrix e cols rows -> Int

356365columns (M m) = I.columns m

357366

358−-- | Coproduct Bifunctor

359367infixl 5 -|-

360368

369+-- | Coproduct Bifunctor (Direct sum)

361370(-|-) ::

362371 ( Num e,

363− CountableDimensionsN j k,

364− FromListsN e k k,

365− FromListsN e j k,

366− FromListsN e k j,

367− FromListsN e j j,

372+ CountableDimsN j k,

373+ FLN e k k,

374+ FLN e j k,

375+ FLN e k j,

376+ FLN e j j,

368377 TrivialE n m,

369378 TrivialE k j

370379 ) =>

@@ -373,40 +382,42 @@ infixl 5 -|-

373382 Matrix e (n + m) (k + j)

374383(-|-) (M a) (M b) = M ((I.-|-) a b)

375384

376−-- | Khatri Rao Product and projections

377−kp1 ::

385+-- | Khatri Rao Product first projection

386+fstM ::

378387 forall e m k .

379388 ( Num e,

380− CountableDimensionsN m k,

389+ CountableDimsN m k,

381390 CountableN (m * k),

382− FromListsN e (m * k) m,

391+ FLN e (m * k) m,

383392 TrivialP m k

384393 ) => Matrix e (m * k) m

385−kp1 = M (I.kp1 @e @(I.FromNat m) @(I.FromNat k))

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

386395

387−kp2 ::

396+-- | Khatri Rao Product second projection

397+sndM ::

388398 forall e m k.

389399 ( Num e,

390− CountableDimensionsN k m,

391− FromListsN e (m * k) k,

400+ CountableDimsN k m,

401+ FLN e (m * k) k,

392402 CountableN (m * k),

393403 TrivialP m k

394404 ) => Matrix e (m * k) k

395−kp2 = M (I.kp2 @e @(I.FromNat m) @(I.FromNat k))

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

396406

397−khatri ::

407+-- | Khatri Rao Product

408+kr ::

398409 forall e cols a b.

399410 ( Num e,

400− CountableDimensionsN a b,

411+ CountableDimsN a b,

401412 CountableN (a * b),

402− FromListsN e (a * b) a,

403− FromListsN e (a * b) b,

413+ FLN e (a * b) a,

414+ FLN e (a * b) b,

404415 TrivialP a b

405416 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)

406−khatri a b =

407− let kp1' = kp1 @e @a @b

408− kp2' = kp2 @e @a @b

409− in comp (tr kp1') a * comp (tr kp2') 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

410421

411422-- | Product Bifunctor (Kronecker)

412423infixl 4 ><

@@ -414,28 +425,28 @@ infixl 4 ><

414425(><) ::

415426 forall e m p n q.

416427 ( Num e,

417− CountableDimensionsN m n,

418− CountableDimensionsN p q,

419− CountableDimensionsN (m * n) (p * q),

420− FromListsN e (m * n) m,

421− FromListsN e (m * n) n,

422− FromListsN e (p * q) p,

423− FromListsN e (p * q) q,

428+ CountableDimsN m n,

429+ CountableDimsN p q,

430+ CountableDimsN (m * n) (p * q),

431+ FLN e (m * n) m,

432+ FLN e (m * n) n,

433+ FLN e (p * q) p,

434+ FLN e (p * q) q,

424435 TrivialP m n,

425436 TrivialP p q

426437 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)

427438(><) a b =

428− let kp1' = kp1 @e @m @n

429− kp2' = kp2 @e @m @n

430− in khatri (comp a kp1') (comp b kp2')

439+ let fstM' = fstM @e @m @n

440+ sndM' = sndM @e @m @n

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

431442

432−-- | Matrix abide Junc Split

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

434−abideJS (M m) = M (I.abideJS m)

443+-- | Matrix abide Join Fork

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

445+abideJF (M m) = M (I.abideJF m)

435446

436−-- | Matrix abide Split Junc

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

438−abideSJ (M m) = M (I.abideSJ m)

447+-- | Matrix abide Fork Join

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

449+abideFJ (M m) = M (I.abideFJ m)

439450

440451-- | Matrix transposition

441452tr :: Matrix e cols rows -> Matrix e rows cols

@@ -444,7 +455,7 @@ tr (M m) = M (I.tr m)

444455-- Selective 'select' operator

445456select ::

446457 ( Num e,

447− FromListsN e rows1 rows1,

458+ FLN e rows1 rows1,

448459 CountableN rows1,

449460 I.FromNat rows2 ~ I.FromNat rows1,

450461 I.FromNat cols1 ~ I.FromNat cols2,

@@ -457,9 +468,9 @@ select (M m) (M y) = M (I.select m y)

457468cond ::

458469 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,

459470 CountableN cols,

460− I.FromLists e () (I.FromNat cols),

461− I.FromLists e (I.FromNat cols) (),

462− FromListsN e cols cols,

471+ I.FL () (I.FromNat cols),

472+ I.FL (I.FromNat cols) (),

473+ FLN e cols cols,

463474 Liftable e a Bool

464475 )

465476 =>

@@ -468,8 +479,12 @@ cond p (M a) (M b) = M (I.cond p a b)

468479

469480-- Pretty print

470481

471−pretty :: (CountableN cols, Show e) => Matrix e cols rows -> String

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

472483pretty (M m) = I.pretty m

473484

474−prettyPrint :: (CountableN cols, Show e) => Matrix e cols rows -> IO ()

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

475486prettyPrint (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+231−175

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

11{-# LANGUAGE ConstraintKinds #-}

2+{-# LANGUAGE TypeFamilies #-}

23{-# LANGUAGE AllowAmbiguousTypes #-}

34{-# LANGUAGE DataKinds #-}

45{-# LANGUAGE DerivingVia #-}

56{-# LANGUAGE FlexibleContexts #-}

67{-# LANGUAGE FlexibleInstances #-}

78{-# LANGUAGE GADTs #-}

8−{-# LANGUAGE KindSignatures #-}

99{-# LANGUAGE MultiParamTypeClasses #-}

1010{-# LANGUAGE NoStarIsType #-}

1111{-# LANGUAGE RankNTypes #-}

@@ -58,10 +58,10 @@ module LAoP.Matrix.Type

5858

5959 -- * Constraint type aliases

6060 Countable,

61− CountableDimensions,

61+ CountableDims,

6262 CountableN,

63− CountableDimensionsN,

64− FromListsN,

63+ CountableDimsN,

64+ FLN,

6565 Liftable,

6666 Trivial,

6767 TrivialP,

@@ -71,10 +71,9 @@ module LAoP.Matrix.Type

7171 One,

7272

7373 -- * Primitives

74− empty,

7574 one,

76− junc,

77− split,

75+ join,

76+ fork,

7877

7978 -- * Auxiliary type families

8079 I.FromNat,

@@ -86,6 +85,7 @@ module LAoP.Matrix.Type

8685 fromLists,

8786 toLists,

8887 toList,

88+ matrixBuilder',

8989 matrixBuilder,

9090 row,

9191 col,

@@ -97,14 +97,15 @@ module LAoP.Matrix.Type

9797

9898 -- * Functor instance equivalent function

9999 fmapM,

100+ bimapM,

100101

101102 -- * Applicative/Monoidal instance equivalent functions

102103 unitM,

103104 multM,

104105

105106 -- * Selective equivalent instance function

106− selectM,

107−

107+ selectM,

108+

108109 -- * Monad equivalent instance function

109110 returnM,

110111 bindM,

@@ -112,25 +113,34 @@ module LAoP.Matrix.Type

112113 -- * Misc

113114 -- ** Get dimensions

114115 columns,

116+ columns',

115117 rows,

118+ rows',

116119

117120 -- ** Matrix Transposition

118121 tr,

119122

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

124+ (.|),

125+ (./),

126+

120127 -- ** McCarthy's Conditional

121128 cond,

122129

123130 -- ** Matrix "abiding"

124− abideJS,

125− abideSJ,

131+ abideJF,

132+ abideFJ,

133+

134+ -- ** Zip Matrices

135+ zipWithM,

126136

127137 -- * Biproduct approach

128− -- ** Split

138+ -- ** Fork

129139 (===),

130140 -- *** Projections

131141 p1,

132142 p2,

133− -- ** Junc

143+ -- ** Join

134144 (|||),

135145 -- *** Injections

136146 i1,

@@ -144,11 +154,11 @@ module LAoP.Matrix.Type

144154 -- | Note that given the restrictions imposed it is not possible to

145155 -- implement the standard type classes present in standard Haskell.

146156 -- *** Matrix pairing projections

147− kp1,

148− kp2,

157+ fstM,

158+ sndM,

149159

150160 -- *** Matrix pairing

151− khatri,

161+ kr,

152162

153163 -- * Matrix composition and lifting

154164

@@ -156,10 +166,10 @@ module LAoP.Matrix.Type

156166

157167 -- | Note that given the restrictions imposed it is not possible to

158168 -- implement the standard type classes present in standard Haskell.

159− identity,

169+ iden,

160170 comp,

161− fromF,

162171 fromF',

172+ fromF,

163173

164174 -- * Relation

165175 toRel,

@@ -173,35 +183,44 @@ where

173183import Data.Void

174184import Data.Proxy

175185import Data.Kind

176−import GHC.TypeLits

186+import GHC.TypeLits

177187import Control.DeepSeq

178188import LAoP.Utils

179−import qualified Control.Category as C

180189import qualified LAoP.Matrix.Internal as I

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

181191

182192newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))

183193 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))

184194

185195-- | Constraint type synonyms to keep the type signatures less convoluted

186−type Countable a = KnownNat (I.Count a)

187−type CountableDimensions a b = (Countable a, Countable b)

188−type CountableN a = KnownNat (I.Count (I.Normalize a))

189−type CountableDimensionsN a b = (CountableN a, CountableN b)

190−type FromListsN e a b = I.FromLists e (I.Normalize a) (I.Normalize b)

191−type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)

192−type Trivial a = I.Normalize (I.Normalize a) ~ I.Normalize (I.Normalize (I.Normalize a))

193−type Trivial2 a = I.Normalize a ~ I.Normalize (I.Normalize a)

194−type Trivial3 a = I.FromNat (I.Count (I.Normalize (I.Normalize a))) ~ I.Normalize (I.Normalize a)

195−type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

196−

197−-- | It isn't possible to implement the 'id' function so it's

198−-- implementation is 'undefined'. However 'comp' can be and this partial

199−-- class implementation exists just to make the code more readable.

200−--

201−-- Please use 'identity' instead.

202−instance (Num e) => C.Category (Matrix e) where

203− id = undefined

204− (.) = comp

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)

205224

206225-- | Zero type alias

207226type Zero = Void

@@ -211,78 +230,74 @@ type One = ()

211230

212231-- Primitives

213232

214−-- | Empty matrix constructor

215−empty :: Matrix e Zero Zero

216−empty = M I.Empty

217−

218233-- | Unit matrix constructor

219234one :: e -> Matrix e One One

220235one = M . I.One

221236

222−-- | Matrix 'Junc' constructor

223−junc ::

237+-- | Matrix 'Join' constructor

238+join ::

224239 Matrix e a rows ->

225240 Matrix e b rows ->

226241 Matrix e (Either a b) rows

227−junc (M a) (M b) = M (I.Junc a b)

242+join (M a) (M b) = M (I.Join a b)

228243

229244infixl 3 |||

230−-- | Matrix 'Junc' constructor

245+-- | Matrix 'Join' constructor

231246(|||) ::

232247 Matrix e a rows ->

233248 Matrix e b rows ->

234249 Matrix e (Either a b) rows

235−(|||) = junc

250+(|||) = join

236251

237−-- | Matrix 'Split' constructor

238−split ::

252+-- | Matrix 'Fork' constructor

253+fork ::

239254 Matrix e cols a ->

240255 Matrix e cols b ->

241256 Matrix e cols (Either a b)

242−split (M a) (M b) = M (I.Split a b)

257+fork (M a) (M b) = M (I.Fork a b)

243258

244259infixl 2 ===

245−-- | Matrix 'Split' constructor

260+-- | Matrix 'Fork' constructor

246261(===) ::

247262 Matrix e cols a ->

248263 Matrix e cols b ->

249264 Matrix e cols (Either a b)

250−(===) = split

265+(===) = fork

251266

252267-- Functor hierarchy

253268

254269-- | Functor instance equivalent function

255−fmapM ::

270+fmapM ::

256271 ( Liftable e a b,

257− CountableDimensionsN a b,

258− FromListsN e b a

272+ CountableDimsN a b,

273+ FLN b a

259274 )

260275 =>

261276 (a -> b) -> Matrix e c a -> Matrix e c b

262−fmapM f m = fromF' f `comp` m

277+fmapM f m = fromF f . m

263278

264279-- | Applicative instance equivalent 'unit' function,

265280unitM :: (Num e) => Matrix e () ()

266281unitM = one 1

267282

268283-- | Applicative instance equivalent 'unit' function,

269−multM ::

270− ( CountableDimensionsN a b,

284+multM ::

285+ ( CountableDimsN a b,

271286 CountableN (a, b),

272287 Num e,

273− FromListsN e (a, b) a,

274− FromListsN e (a, b) b,

288+ FLN (a, b) a,

289+ FLN (a, b) b,

275290 TrivialP a b

276291 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)

277−multM = khatri

292+multM = kr

278293

279294-- | Monad instance equivalent 'return' function,

280−returnM ::

281− forall e a .

295+returnM ::

296+ forall e a .

282297 ( Num e,

283298 Enum e,

284299 Enum a,

285− FromListsN e () a,

300+ FLN () a,

286301 Countable a

287302 ) => a -> Matrix e One a

288303returnM a = col l

@@ -299,51 +314,64 @@ bindM = flip comp

299314

300315-- | Build a matrix out of a list of list of elements. Throws a runtime

301316-- error if the dimensions do not match.

302−fromLists :: (FromListsN e cols rows) => [[e]] -> Matrix e cols rows

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

303318fromLists = M . I.fromLists

304319

305320-- | Matrix builder function. Constructs a matrix provided with

306−-- a construction function.

307−matrixBuilder ::

308− (FromListsN e cols rows, CountableDimensionsN cols rows )

321+-- a construction function that operates with indices.

322+matrixBuilder' ::

323+ (FLN cols rows, CountableDimsN cols rows )

309324 => ((Int, Int) -> e) -> Matrix e cols rows

310−matrixBuilder = M . I.matrixBuilder

325+matrixBuilder' = M . I.matrixBuilder'

326+

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

328+-- a construction function that operates with arbitrary types.

329+matrixBuilder ::

330+ ( FLN a b,

331+ Enum a,

332+ Enum b,

333+ Bounded a,

334+ Bounded b,

335+ Eq a,

336+ Countable b

337+ ) => ((a, b) -> e) -> Matrix e a b

338+matrixBuilder f = M (I.matrixBuilder f)

311339

312340-- | Constructs a column vector matrix

313−col :: (FromListsN e () rows) => [e] -> Matrix e One rows

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

314342col = M . I.col

315343

316344-- | Constructs a row vector matrix

317−row :: (FromListsN e cols ()) => [e] -> Matrix e cols One

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

318346row = M . I.row

319347

320348-- | Lifts functions to matrices with arbitrary dimensions.

321349--

322350-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

323351-- types @a@ or @b@ allows.

324−fromF ::

352+fromF' ::

325353 ( Liftable e a b,

326− CountableDimensionsN cols rows,

327− FromListsN e rows cols

354+ CountableDimsN cols rows,

355+ FLN rows cols

328356 ) =>

329357 (a -> b) -> Matrix e cols rows

330−fromF = M . I.fromF

358+fromF' = M . I.fromF'

331359

332360-- | Lifts functions to matrices with dimensions matching @a@ and @b@

333361-- cardinality's.

334−fromF' ::

362+fromF ::

335363 ( Liftable e a b,

336− CountableDimensionsN a b,

337− FromListsN e b a

364+ CountableDimsN a b,

365+ FLN b a

338366 ) =>

339367 (a -> b) -> Matrix e a b

340−fromF' = M . I.fromF'

368+fromF = M . I.fromF

341369

342370-- | Lifts relation functions to Boolean Matrix

343371toRel ::

344372 ( Liftable (Natural 0 1) a b,

345− CountableDimensionsN a b,

346− FromListsN (Natural 0 1) b a

373+ CountableDims a b,

374+ FLN b a

347375 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b

348376toRel = M . I.toRel

349377

@@ -361,7 +389,7 @@ toList (M m) = I.toList m

361389

362390-- | The zero matrix. A matrix wholly filled with zeros.

363391zeros ::

364− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)

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

365393 => Matrix e cols rows

366394zeros = M I.zeros

367395

@@ -371,7 +399,7 @@ zeros = M I.zeros

371399--

372400-- Also known as T (Top) matrix.

373401ones ::

374− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)

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

375403 => Matrix e cols rows

376404ones = M I.ones

377405

@@ -380,7 +408,7 @@ ones = M I.ones

380408-- | The constant matrix constructor. A matrix wholly filled with a given

381409-- value.

382410constant ::

383− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)

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

384412 => e -> Matrix e cols rows

385413constant = M . I.constant

386414

@@ -389,30 +417,30 @@ constant = M . I.constant

389417-- | The T (Top) row vector matrix.

390418bang ::

391419 forall e cols.

392− (Num e, Enum e, FromListsN e cols (), CountableN cols) =>

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

393421 Matrix e cols One

394422bang = M I.bang

395423

396424-- | Point constant relation

397−point ::

425+point ::

398426 ( Bounded a,

399427 Enum a,

400428 Eq a,

401429 Num e,

402430 Ord e,

403431 CountableN a,

404− FromListsN e a One

432+ FLN a One

405433 ) => a -> Matrix e One a

406−point = fromF' . const

434+point = fromF . const

407435

408−-- Identity Matrix

436+-- iden Matrix

409437

410−-- | Identity matrix

411−identity ::

412− (Num e, FromListsN e a a, CountableN a) =>

438+-- | iden matrix

439+iden ::

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

413441 Matrix e a a

414−identity = M I.identity

415−{-# NOINLINE identity #-}

442+iden = M I.iden

443+{-# NOINLINE iden #-}

416444

417445-- Matrix composition (MMM)

418446

@@ -423,17 +451,31 @@ identity = M I.identity

423451comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows

424452comp (M a) (M b) = M (I.comp a b)

425453{-# NOINLINE comp #-}

426−{-# RULES

427− "comp/identity1" forall m. comp m identity = m ;

428− "comp/identity2" forall m. comp identity m = m

454+{-# RULES

455+ "comp/iden1" forall m. comp m iden = m ;

456+ "comp/iden2" forall m. comp iden m = m

429457#-}

430458

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)

472+

431473-- | Biproduct first component projection

432474p1 ::

433475 ( Num e,

434− CountableDimensionsN n m,

435− FromListsN e n m,

436− FromListsN e m m

476+ CountableDimsN n m,

477+ FLN n m,

478+ FLN m m

437479 ) =>

438480 Matrix e (Either m n) m

439481p1 = M I.p1

@@ -441,9 +483,9 @@ p1 = M I.p1

441483-- | Biproduct second component projection

442484p2 ::

443485 ( Num e,

444− CountableDimensionsN n m,

445− FromListsN e m n,

446− FromListsN e n n

486+ CountableDimsN n m,

487+ FLN m n,

488+ FLN n n

447489 ) =>

448490 Matrix e (Either m n) n

449491p2 = M I.p2

@@ -453,9 +495,9 @@ p2 = M I.p2

453495-- | Biproduct first component injection

454496i1 ::

455497 ( Num e,

456− CountableDimensionsN n m,

457− FromListsN e n m,

458− FromListsN e m m

498+ CountableDimsN n m,

499+ FLN n m,

500+ FLN m m

459501 ) =>

460502 Matrix e m (Either m n)

461503i1 = tr p1

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

463505-- | Biproduct second component injection

464506i2 ::

465507 ( Num e,

466− CountableDimensionsN n m,

467− FromListsN e m n,

468− FromListsN e n n

508+ CountableDimsN n m,

509+ FLN m n,

510+ FLN n n

469511 ) =>

470512 Matrix e n (Either m n)

471513i2 = tr p2

@@ -474,24 +516,34 @@ i2 = tr p2

474516

475517-- | Obtain the number of rows.

476518--

477−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

478−-- dimensions in constant time.

479−--

480−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in

481−-- exchange for performance.

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''.

482522rows :: (CountableN rows) => Matrix e cols rows -> Int

483523rows (M m) = I.rows m

484524

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

526+-- constraints.

527+--

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

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

530+rows' (M m) = I.rows' m

531+

485532-- | Obtain the number of columns.

486−--

487−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the

488−-- dimensions in constant time.

489533--

490−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in

491−-- exchange for performance.

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''.

492537columns :: (CountableN cols) => Matrix e cols rows -> Int

493538columns (M m) = I.columns m

494539

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+

495547-- Coproduct Bifunctor

496548

497549infixl 5 -|-

@@ -499,11 +551,11 @@ infixl 5 -|-

499551-- | Matrix coproduct functor also known as matrix direct sum.

500552(-|-) ::

501553 ( Num e,

502− CountableDimensionsN j k,

503− FromListsN e k k,

504− FromListsN e j k,

505− FromListsN e k j,

506− FromListsN e j j

554+ CountableDimsN j k,

555+ FLN k k,

556+ FLN j k,

557+ FLN k j,

558+ FLN j j

507559 ) =>

508560 Matrix e n k ->

509561 Matrix e m j ->

@@ -513,98 +565,98 @@ infixl 5 -|-

513565-- Khatri Rao Product and projections

514566

515567-- | Khatri Rao product first component projection matrix.

516−kp1 ::

568+fstM ::

517569 forall e m k .

518570 ( Num e,

519− CountableDimensionsN m k,

571+ CountableDimsN m k,

520572 CountableN (m, k),

521− FromListsN e (m, k) m,

573+ FLN (m, k) m,

522574 TrivialP m k

523575 ) => Matrix e (m, k) m

524−kp1 = M (I.kp1 @e @(I.Normalize m) @(I.Normalize k))

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

525577

526578-- | Khatri Rao product second component projection matrix.

527−kp2 ::

579+sndM ::

528580 forall e m k.

529581 ( Num e,

530− CountableDimensionsN k m,

582+ CountableDimsN k m,

531583 CountableN (m, k),

532− FromListsN e (m, k) k,

584+ FLN (m, k) k,

533585 TrivialP m k

534586 ) => Matrix e (m, k) k

535−kp2 = M (I.kp2 @e @(I.Normalize m) @(I.Normalize k))

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

536588

537589-- | Khatri Rao Matrix product also known as matrix pairing.

538590--

539591-- NOTE: That this is not a true categorical product, see for instance:

540−--

592+--

541593-- @

542−-- | kp1 `comp` khatri a b == a

543−-- khatri a b ==> |

544−-- | kp2 `comp` khatri a b == b

594+-- | fstM . kr a b == a

595+-- kr a b ==> |

596+-- | sndM . kr a b == b

545597-- @

546598--

547599-- __Emphasis__ on the implication symbol.

548−khatri ::

600+kr ::

549601 forall e cols a b.

550602 ( Num e,

551− CountableDimensionsN a b,

603+ CountableDimsN a b,

552604 CountableN (a, b),

553− FromListsN e (a, b) a,

554− FromListsN e (a, b) b,

605+ FLN (a, b) a,

606+ FLN (a, b) b,

555607 TrivialP a b

556608 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)

557−khatri a b =

558− let kp1' = kp1 @e @a @b

559− kp2' = kp2 @e @a @b

560− in comp (tr kp1') a * comp (tr kp2') b

609+kr a b =

610+ let fstM' = fstM @e @a @b

611+ sndM' = sndM @e @a @b

612+ in comp (tr fstM') a * comp (tr sndM') b

561613

562614-- Product Bifunctor (Kronecker)

563615

564616infixl 4 ><

565617

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

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

567619(><) ::

568620 forall e m p n q.

569621 ( Num e,

570− CountableDimensionsN m n,

571− CountableDimensionsN p q,

572− CountableDimensionsN (m, n) (p, q),

573− FromListsN e (m, n) m,

574− FromListsN e (m, n) n,

575− FromListsN e (p, q) p,

576− FromListsN e (p, q) q,

622+ CountableDimsN m n,

623+ CountableDimsN p q,

624+ CountableDimsN (m, n) (p, q),

625+ FLN (m, n) m,

626+ FLN (m, n) n,

627+ FLN (p, q) p,

628+ FLN (p, q) q,

577629 TrivialP m n,

578630 TrivialP p q

579631 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)

580632(><) a b =

581− let kp1' = kp1 @e @m @n

582− kp2' = kp2 @e @m @n

583− in khatri (comp a kp1') (comp b kp2')

633+ let fstM' = fstM @e @m @n

634+ sndM' = sndM @e @m @n

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

584636

585−-- Matrix abide Junc Split

637+-- Matrix abide Join Fork

586638

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

588−--

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

640+--

589641-- Law:

590642--

591643-- @

592−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)

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

593645-- @

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

595−abideJS (M m) = M (I.abideJS m)

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

647+abideJF (M m) = M (I.abideJF m)

596648

597−-- Matrix abide Split Junc

649+-- Matrix abide Fork Join

598650

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

600−--

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

652+--

601653-- Law:

602654--

603655-- @

604−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)

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

605657-- @

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

607−abideSJ (M m) = M (I.abideSJ m)

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

659+abideFJ (M m) = M (I.abideFJ m)

608660

609661-- Matrix transposition

610662

@@ -616,9 +668,9 @@ tr (M m) = M (I.tr m)

616668

617669-- | Selective functors 'select' operator equivalent inspired by the

618670-- ArrowMonad solution presented in the paper.

619−selectM ::

671+selectM ::

620672 ( Num e,

621− FromListsN e b b,

673+ FLN b b,

622674 CountableN b

623675 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b

624676selectM (M m) (M y) = M (I.select m y)

@@ -631,9 +683,9 @@ cond ::

631683 Trivial2 a,

632684 Trivial3 a,

633685 CountableN a,

634− FromListsN e () a,

635− FromListsN e a (),

636− FromListsN e a a,

686+ FLN () a,

687+ FLN a (),

688+ FLN a a,

637689 Liftable e a Bool

638690 )

639691 =>

@@ -643,9 +695,13 @@ cond p (M a) (M b) = M (I.cond p a b)

643695-- Pretty print

644696

645697-- | Matrix pretty printer

646−pretty :: (CountableN cols, Show e) => Matrix e cols rows -> String

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

647699pretty (M m) = I.pretty m

648700

649701-- | Matrix pretty printer

650−prettyPrint :: (CountableN cols, Show e) => Matrix e cols rows -> IO ()

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

651703prettyPrint (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+8−11

@@ -32,22 +32,19 @@ module LAoP.Relation

3232

3333 -- * Constraint type synonyms

3434 Countable,

35− CountableDimensions,

35+ CountableDims,

3636 CountableN,

37− CountableDimensionsN,

38− FromListsN,

37+ CountableDimsN,

38+ FLN,

3939 Liftable,

4040 Trivial,

41− TrivialE,

4241 TrivialP,

43− TrivialP2,

4442

4543 -- * Primitives

46− empty,

4744 one,

48− junc,

45+ join,

4946 (|||),

50− split,

47+ fork,

5148 (===),

5249

5350 -- * Auxiliary type families

@@ -110,8 +107,8 @@ module LAoP.Relation

110107 splitR,

111108

112109 -- ** Projections

113− p1,

114− p2,

110+ fstR,

111+ sndR,

115112 -- ** Bifunctor

116113 (><),

117114 -- * Relational coproduct

@@ -150,7 +147,7 @@ module LAoP.Relation

150147 cond,

151148

152149 -- * Relational composition and lifting

153− identity,

150+ iden,

154151 comp,

155152 fromF,

156153 fromF',

src/LAoP/Relation/Internal.hs+250−254

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

1+{-# LANGUAGE TypeFamilies #-}

12{-# LANGUAGE ConstraintKinds #-}

23{-# LANGUAGE GADTs #-}

34{-# LANGUAGE UndecidableInstances #-}

@@ -42,22 +43,19 @@ module LAoP.Relation.Internal

4243

4344 -- * Constraint type synonyms

4445 Countable,

45− CountableDimensions,

46+ CountableDims,

4647 CountableN,

47− CountableDimensionsN,

48− FromListsN,

48+ CountableDimsN,

49+ FLN,

4950 Liftable,

5051 Trivial,

51− TrivialE,

5252 TrivialP,

53− TrivialP2,

5453

5554 -- * Primitives

56− empty,

5755 one,

58− junc,

56+ join,

5957 (|||),

60− split,

58+ fork,

6159 (===),

6260

6361 -- * Auxiliary type families

@@ -68,14 +66,15 @@ module LAoP.Relation.Internal

6866 -- * Matrix construction and conversion

6967 I.FromLists,

7068 fromLists,

71− fromF,

7269 fromF',

70+ fromF,

7371 toRel,

7472 toLists,

7573 toList,

7674 toBool,

7775 pt,

7876 belongs,

77+ relationBuilder',

7978 relationBuilder,

8079 zeros,

8180 ones,

@@ -120,8 +119,8 @@ module LAoP.Relation.Internal

120119 splitR,

121120

122121 -- ** Projections

123− p1,

124− p2,

122+ fstR,

123+ sndR,

125124 -- ** Bifunctor

126125 (><),

127126 -- * Relational coproduct

@@ -160,10 +159,10 @@ module LAoP.Relation.Internal

160159 cond,

161160

162161 -- * Relational composition and lifting

163− identity,

162+ iden,

164163 comp,

165− fromF,

166164 fromF',

165+ fromF,

167166

168167 -- ** Relational application

169168 pointAp,

@@ -179,9 +178,9 @@ import Data.Void

179178import qualified LAoP.Matrix.Internal as I

180179import LAoP.Utils.Internal

181180import Control.DeepSeq

182−import qualified Control.Category as C

183181import Data.Bool

184182import GHC.TypeLits

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

185184

186185-- | Boolean type synonym for working with boolean matrices

187186type Boolean = Natural 0 1

@@ -194,34 +193,30 @@ newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))

194193deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)

195194

196195-- | Constraint type synonyms to keep the type signatures less convoluted

197−type Countable a = KnownNat (I.Count a)

198−type CountableDimensions a b = (Countable a, Countable b)

199−type CountableN a = KnownNat (I.Count (I.Normalize a))

200−type CountableDimensionsN a b = (CountableN a, CountableN b)

201−type FromListsN a b = I.FromLists Boolean (I.Normalize a) (I.Normalize b)

202−type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Boolean, Ord Boolean)

203−type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)

204−type TrivialE a b = I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)

205−type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)

206−type TrivialP2 a b = I.Normalize (I.Normalize a, I.Normalize b) ~ I.Normalize (I.Normalize (a, b))

207−

208−-- | It isn't possible to implement the 'id' function so it's

209−-- implementation is 'undefined'. However 'comp' can be and this partial

210−-- class implementation exists just to make the code more readable.

211−--

212−-- Please use 'identity' instead.

213−instance C.Category Relation where

214− id = undefined

215− (.) = comp

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

216211

217212instance Num (Relation a b) where

218− -- | Matrix addition becomes Boolean matrix disjunction

213+ -- | Matrix addition becomes Boolean matrix disjointion

219214 (R a) + (R b) = R (I.orM a b)

220215

221216 -- | Matrix subtraction becomes Relational subtraction

222217 (R a) - (R b) = R (I.subM a b)

223218

224− -- | Matrix multiplication becomes Boolean matrix conjunction

219+ -- | Matrix multiplication becomes Boolean matrix conjointion

225220 (R a) * (R b) = R (I.andM a b)

226221

227222 -- | Matrix negation becomes Boolean matrix negation

@@ -233,87 +228,92 @@ type One = ()

233228

234229-- Primitives

235230

236−-- | Empty matrix constructor

237−empty :: Relation Zero Zero

238−empty = R I.Empty

239−

240231-- | Unit matrix constructor

241232one :: Boolean -> Relation One One

242233one = R . I.One

243234

244−-- | Boolean Matrix 'Junc' constructor, also known as relational coproduct.

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

245236--

246237-- See 'eitherR'.

247−junc :: (TrivialE a b)

248− => Relation a c -> Relation b c -> Relation (Either a b) c

249−junc (R a) (R b) = R (I.Junc a b)

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

239+join (R a) (R b) = R (I.Join a b)

250240

251241infixl 3 |||

252−-- | Boolean Matrix 'Junc' constructor

242+-- | Boolean Matrix 'Join' constructor

253243--

254244-- See 'eitherR'.

255245(|||) ::

256− (TrivialE a b) =>

257246 Relation a c ->

258247 Relation b c ->

259248 Relation (Either a b) c

260−(|||) = junc

249+(|||) = join

261250

262−-- | Boolean Matrix 'Split' constructor, also known as relational product.

263−split :: (TrivialE a b)

264− => Relation c a -> Relation c b -> Relation c (Either a b)

265−split (R a) (R b) = R (I.Split a b)

251+-- | Boolean Matrix 'Fork' constructor, also known as relational product.

252+fork :: Relation c a -> Relation c b -> Relation c (Either a b)

253+fork (R a) (R b) = R (I.Fork a b)

266254

267255infixl 2 ===

268−-- | Boolean Matrix 'Split' constructor

256+-- | Boolean Matrix 'Fork' constructor

269257(===) ::

270− (TrivialE a b) =>

271258 Relation c a ->

272259 Relation c b ->

273260 Relation c (Either a b)

274−(===) = split

261+(===) = fork

275262

276263-- Construction

277264

278265-- | Build a matrix out of a list of list of elements. Throws a runtime

279266-- error if the dimensions do not match.

280−fromLists :: (FromListsN a b) => [[Boolean]] -> Relation a b

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

281268fromLists = R . I.fromLists

282269

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

284−-- a construction function.

285−relationBuilder ::

286− (FromListsN a b, CountableDimensionsN a b) =>

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) =>

287274 ((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.

279+relationBuilder ::

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

288288relationBuilder = R . I.matrixBuilder

289289

290290-- | Lifts functions to matrices with arbitrary dimensions.

291291--

292292-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of

293293-- types @a@ or @b@ allows.

294−fromF ::

294+fromF' ::

295295 ( Liftable a b,

296− CountableDimensionsN c d,

297− FromListsN d c

296+ CountableDimsN c d,

297+ FLN d c

298298 )

299299 => (a -> b) -> Relation c d

300−fromF f = R (I.fromFRel f)

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

301301

302302-- | Lifts functions to matrices with dimensions matching @a@ and @b@

303303-- cardinality's.

304−fromF' ::

304+fromF ::

305305 ( Liftable a b,

306− CountableDimensionsN a b,

307− FromListsN b a

306+ CountableDimsN a b,

307+ FLN b a

308308 )

309309 => (a -> b) -> Relation a b

310−fromF' f = R (I.fromFRel' f)

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

311311

312312-- | Lifts relation functions to 'Relation'

313313toRel ::

314314 ( Liftable a b,

315− CountableDimensionsN a b,

316− FromListsN b a

315+ CountableDims a b,

316+ FLN b a

317317 )

318318 => (a -> b -> Bool) -> Relation a b

319319toRel = R . I.toRel

@@ -322,9 +322,9 @@ toRel = R . I.toRel

322322fromRel ::

323323 ( Liftable a b,

324324 Eq a,

325− CountableDimensionsN a b,

326− FromListsN a One,

327− FromListsN b One

325+ CountableDimsN a b,

326+ FLN a One,

327+ FLN b One

328328 )

329329 => Relation a b -> (a -> b -> Bool)

330330fromRel r a b = pointApBool a b r

@@ -348,12 +348,12 @@ toBool r = case toList r of

348348-- | Power transpose.

349349--

350350-- Maps a relation to a set valued function.

351−pt ::

351+pt ::

352352 ( Liftable a b,

353353 Eq a,

354− CountableDimensionsN a b,

355− FromListsN a One,

356− FromListsN b One

354+ CountableDimsN a b,

355+ FLN a One,

356+ FLN b One

357357 )

358358 => Relation a b -> (a -> List b)

359359pt r a =

@@ -365,8 +365,8 @@ belongs ::

365365 ( Bounded a,

366366 Enum a,

367367 Eq a,

368− CountableDimensionsN (List a) a,

369− FromListsN a (List a)

368+ CountableDims (List a) a,

369+ FLN a (List a)

370370 )

371371 => Relation (List a) a

372372belongs = toRel elemR

@@ -378,16 +378,16 @@ belongs = toRel elemR

378378-- | The zero relation. A relation where no element of type @a@ relates

379379-- with elements of type @b@.

380380--

381−-- Also known as ⊥ (Bottom) Relation or empty Relation.

381+-- Also known as ⊥ (Bottom) Relation.

382382--

383383-- @

384−-- r ``comp`` ⊥ == ⊥ ``comp`` r == ⊥

384+-- r `.` ⊥ == ⊥ `.` r == ⊥

385385-- ⊥ ``sse`` R && R ``sse`` T == True

386386-- @

387387zeros ::

388− (FromListsN a b, CountableDimensionsN a b) =>

388+ (FLN a b, CountableDimsN a b) =>

389389 Relation a b

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

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

391391

392392-- Ones Matrix

393393

@@ -400,43 +400,43 @@ zeros = relationBuilder (const (nat 0))

400400-- ⊥ ``sse`` R && R ``sse`` T == True

401401-- @

402402ones ::

403− (FromListsN a b, CountableDimensionsN a b) =>

403+ (FLN a b, CountableDimsN a b) =>

404404 Relation a b

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

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

406406

407407-- Bang Matrix

408408

409409-- | The T (Top) row vector relation.

410410bang ::

411− (FromListsN a One, CountableN a) =>

411+ (FLN a One, CountableN a) =>

412412 Relation a One

413413bang = ones

414414

415415-- | Point constant relation

416−point ::

416+point ::

417417 ( Bounded a,

418418 Enum a,

419419 Eq a,

420420 CountableN a,

421− FromListsN a One

421+ FLN a One

422422 ) => a -> Relation One a

423−point = fromF' . const

423+point = fromF . const

424424

425−-- Identity Matrix

425+-- iden Matrix

426426

427−-- | Identity matrix

427+-- | iden matrix

428428--

429429-- @

430−-- 'identity' ``comp`` r == r == r ``comp`` 'identity'

430+-- 'iden' `.` r == r == r `.` 'iden'

431431-- @

432−identity ::

433− (FromListsN a a, CountableN a) => Relation a a

434−identity = relationBuilder (bool (nat 0) (nat 1) . uncurry (==))

432+iden ::

433+ (FLN a a, CountableN a) => Relation a a

434+iden = relationBuilder' (bool (nat 0) (nat 1) . uncurry (==))

435435

436436-- | Relational composition

437437--

438438-- @

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

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

440440-- @

441441comp :: Relation b c -> Relation a b -> Relation a c

442442comp (R a) (R b) = R (I.compRel a b)

@@ -484,7 +484,7 @@ shrunkBy r s = r `intersection` divR s (conv r)

484484-- r ``overriddenBy`` r == r

485485-- @

486486overriddenBy ::

487− ( FromListsN b b,

487+ ( FLN b b,

488488 CountableN b

489489 ) => Relation a b -> Relation a b -> Relation a b

490490overriddenBy r s = s `union` r `intersection` divR zeros (conv s)

@@ -496,11 +496,11 @@ overriddenBy r s = s `union` r `intersection` divR zeros (conv s)

496496pointAp ::

497497 ( Liftable a b,

498498 Eq a,

499− CountableDimensionsN a b,

500− FromListsN a One,

501− FromListsN b One

499+ CountableDimsN a b,

500+ FLN a One,

501+ FLN b One

502502 ) => a -> b -> Relation a b -> Relation One One

503−pointAp a b r = conv (point b) `comp` r `comp` point a

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

504504

505505-- | Relational application

506506--

@@ -508,11 +508,11 @@ pointAp a b r = conv (point b) `comp` r `comp` point a

508508pointApBool ::

509509 ( Liftable a b,

510510 Eq a,

511− CountableDimensionsN a b,

512− FromListsN a One,

513− FromListsN b One

511+ CountableDimsN a b,

512+ FLN a One,

513+ FLN b One

514514 ) => a -> b -> Relation a b -> Bool

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

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

516516

517517-- | Relational converse

518518--

@@ -538,7 +538,7 @@ iff r s = r == s

538538

539539-- | Relational intersection

540540--

541−-- Lifts pointwise conjunction.

541+-- Lifts pointwise conjointion.

542542--

543543-- @

544544-- (r ``intersection`` s) ``intersection`` t == r ``intersection`` (s ``intersection`` t)

@@ -549,13 +549,13 @@ intersection a b = a * b

549549

550550-- | Relational union

551551--

552−-- Lifts pointwise disjunction.

552+-- Lifts pointwise disjointion.

553553--

554554-- @

555555-- (r ``union`` s) ``union`` t == r `'union' (s ``union`` t)

556556-- r ``union`` s ``sse`` x == r ``sse`` x && s ``sse`` x

557−-- r ``comp`` (s ``union`` t) == (r ``comp`` s) ``union`` (r ``comp`` t)

558−-- (s ``union`` t) ``comp`` r == (s ``comp`` r) ``union`` (t ``comp`` r)

557+-- r `.` (s ``union`` t) == (r `.` s) ``union`` (r `.` t)

558+-- (s ``union`` t) `.` r == (s `.` r) ``union`` (t `.` r)

559559-- @

560560union :: Relation a b -> Relation a b -> Relation a b

561561union a b = a + b

@@ -563,46 +563,46 @@ union a b = a + b

563563-- | Relation Kernel

564564--

565565-- @

566−-- 'ker' r == 'conv' r ``comp`` r

566+-- 'ker' r == 'conv' r `.` r

567567-- 'ker' r == 'img' ('conv' r)

568568-- @

569569ker :: Relation a b -> Relation a a

570−ker r = conv r `comp` r

570+ker r = conv r . r

571571

572572-- | Relation Image

573573--

574574-- @

575−-- 'img' r == r ``comp`` conv r

575+-- 'img' r == r `.` conv r

576576-- 'img' r == 'ker' ('conv' r)

577577-- @

578578img :: Relation a b -> Relation b b

579−img r = r `comp` conv r

579+img r = r . conv r

580580

581581-- | Function division. Special case of 'divS'.

582582--

583583-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and

584584-- 'entire'.

585585--

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

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

587587divisionF :: Relation a c -> Relation b c -> Relation a b

588−divisionF f g = conv g `comp` f

588+divisionF f g = conv g . f

589589

590590-- Taxonomy of binary relations

591591

592592-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@

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

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

594594simple = coreflexive . img

595595

596596-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@

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

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

598598injective = coreflexive . ker

599599

600600-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@

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

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

602602entire = reflexive . ker

603603

604604-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@

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

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

606606surjective = reflexive . img

607607

608608-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@

@@ -611,13 +611,13 @@ surjective = reflexive . img

611611-- relations:

612612--

613613-- @

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

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

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

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

616616-- @

617617function ::

618− ( CountableDimensionsN a b,

619− FromListsN a a,

620− FromListsN b b

618+ ( CountableDimsN a b,

619+ FLN a a,

620+ FLN b b

621621 )

622622 => Relation a b -> Bool

623623function r = simple r && entire r

@@ -625,7 +625,7 @@ function r = simple r && entire r

625625-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@

626626representation ::

627627 ( CountableN a,

628− FromListsN a a

628+ FLN a a

629629 )

630630 => Relation a b -> Bool

631631representation r = injective r && entire r

@@ -633,91 +633,91 @@ representation r = injective r && entire r

633633-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@

634634abstraction ::

635635 ( CountableN b,

636− FromListsN b b

636+ FLN b b

637637 )

638638 => Relation a b -> Bool

639639abstraction r = surjective r && simple r

640640

641641-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@

642642surjection ::

643− ( CountableDimensionsN a b,

644− FromListsN a a,

645− FromListsN b b

643+ ( CountableDimsN a b,

644+ FLN a a,

645+ FLN b b

646646 )

647647 => Relation a b -> Bool

648648surjection r = function r && abstraction r

649649

650650-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@

651651injection ::

652− ( CountableDimensionsN a b,

653− FromListsN a a,

654− FromListsN b b

652+ ( CountableDimsN a b,

653+ FLN a a,

654+ FLN b b

655655 )

656656 => Relation a b -> Bool

657657injection r = function r && representation r

658658

659659-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@

660660bijection ::

661− ( CountableDimensionsN a b,

662− FromListsN b b,

663− FromListsN a a

661+ ( CountableDimsN a b,

662+ FLN b b,

663+ FLN a a

664664 ) => Relation a b -> Bool

665665bijection r = injection r && surjection r

666666

667667-- Properties of relations

668668

669−-- | A 'Relation' @r@ is 'reflexive' 'iff' @'identity' ``sse`` r@

670−reflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool

671−reflexive r = identity <= r

669+-- | A 'Relation' @r@ is 'reflexive' 'iff' @'id' ``sse`` r@

670+reflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

671+reflexive r = id <= r

672672

673−-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'identity'@

674−coreflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool

675−coreflexive r = r <= identity

673+-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'id'@

674+coreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

675+coreflexive r = r <= id

676676

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

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

678678transitive :: Relation a a -> Bool

679−transitive r = (r `comp` r) `sse` r

679+transitive r = (r . r) `sse` r

680680

681681-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@

682682symmetric :: Relation a a -> Bool

683683symmetric r = r == conv r

684684

685−-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'identity'@

686−antiSymmetric :: (CountableN a, FromListsN a a) => Relation a a -> Bool

687−antiSymmetric r = (r `intersection` conv r) `sse` identity

685+-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'id'@

686+antiSymmetric :: (CountableN a, FLN a a) => Relation a a -> Bool

687+antiSymmetric r = (r `intersection` conv r) `sse` id

688688

689−-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'identity') == 'zeros'@

690−irreflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool

691−irreflexive r = (r `intersection` identity) == zeros

689+-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'id') == 'zeros'@

690+irreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool

691+irreflexive r = (r `intersection` id) == zeros

692692

693693-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@

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

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

695695connected r = (r `union` conv r) == ones

696696

697697-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@

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

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

699699preorder r = reflexive r && transitive r

700700

701701-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@

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

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

703703partialOrder r = antiSymmetric r && preorder r

704704

705705-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@

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

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

707707linearOrder r = connected r && partialOrder r

708708

709709-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@

710−equivalence :: (CountableN a, FromListsN a a) => Relation a a -> Bool

710+equivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

711711equivalence r = symmetric r && preorder r

712712

713713-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@

714−partialEquivalence :: (CountableN a, FromListsN a a) => Relation a a -> Bool

714+partialEquivalence :: (CountableN a, FLN a a) => Relation a a -> Bool

715715partialEquivalence r = partialOrder r && equivalence r

716716

717717-- | A 'Relation' @r@ is 'difunctional' or regular wherever

718−-- @r ``comp`` 'conv' r ``comp`` r == r@

718+-- @r `.` 'conv' r `.` r == r@

719719difunctional :: Relation a b -> Bool

720−difunctional r = r `comp` conv r `comp` r == r

720+difunctional r = r . conv r . r == r

721721

722722-- Relational pairing

723723

@@ -726,61 +726,61 @@ difunctional r = r `comp` conv r `comp` r == r

726726-- NOTE: That this is not a true categorical product, see for instance:

727727--

728728-- @

729−-- | 'p1' ``comp`` 'splitR' a b ``sse`` a

729+-- | 'fstR' `.` 'splitR' a b ``sse`` a

730730-- 'splitR' a b <=> |

731−-- | 'p2' ``comp`` 'splitR' a b ``sse`` b

731+-- | 'sndR' `.` 'splitR' a b ``sse`` b

732732-- @

733733--

734734-- __Emphasis__ on the 'sse'.

735735--

736736-- @

737−-- 'splitR' r s ``comp`` f == 'splitR' (r ``comp`` f) (s ``comp`` f)

738−-- (R '><' S) ``comp`` 'splitR' p q == 'splitR' (r ``comp`` p) (s ``comp`` q)

739−-- 'conv' ('splitR' r s) ``comp`` 'splitR' x y == ('conv' r ``comp`` x) ``intersection`` ('conv' s ``comp`` y)

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)

740740-- @

741741--

742742-- @

743743-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)

744744-- @

745745splitR ::

746− ( CountableDimensionsN a b,

746+ ( CountableDimsN a b,

747747 CountableN (a, b),

748− FromListsN (a, b) a,

749− FromListsN (a, b) b,

748+ FLN (a, b) a,

749+ FLN (a, b) b,

750750 TrivialP a b

751751 )

752752 => Relation c a -> Relation c b -> Relation c (a, b)

753−splitR (R f) (R g) = R (I.khatri f g)

753+splitR (R f) (R g) = R (I.kr f g)

754754

755755-- | Relational pairing first component projection

756756--

757757-- @

758−-- 'p1' ``comp`` 'splitR' r s ``sse`` r

758+-- 'fstR' `.` 'splitR' r s ``sse`` r

759759-- @

760−p1 ::

760+fstR ::

761761 forall a b .

762− ( CountableDimensionsN a b,

762+ ( CountableDimsN a b,

763763 CountableN (a, b),

764− FromListsN (a, b) a,

764+ FLN (a, b) a,

765765 TrivialP a b

766766 )

767767 => Relation (a, b) a

768−p1 = R (I.kp1 @Boolean @(I.Normalize a) @(I.Normalize b))

768+fstR = R (I.fstM @Boolean @(I.Normalize a) @(I.Normalize b))

769769

770770-- | Relational pairing second component projection

771771--

772772-- @

773−-- 'p2' ``comp`` 'splitR' r s ``sse`` s

773+-- 'sndR' `.` 'splitR' r s ``sse`` s

774774-- @

775−p2 ::

775+sndR ::

776776 forall a b .

777− ( CountableDimensionsN a b,

777+ ( CountableDimsN a b,

778778 CountableN (a, b),

779− FromListsN (a, b) b,

779+ FLN (a, b) b,

780780 TrivialP a b

781781 )

782782 => Relation (a, b) b

783−p2 = R (I.kp2 @Boolean @(I.Normalize a) @(I.Normalize b))

783+sndR = R (I.sndM @Boolean @(I.Normalize a) @(I.Normalize b))

784784

785785-- Relational pairing functor

786786

@@ -788,17 +788,17 @@ infixl 4 ><

788788-- | Relational pairing functor

789789--

790790-- @

791−-- r '><' s == 'splitR' (r ``comp`` p1) (s ``comp`` p2)

792−-- (r '><' s) ``comp`` (p '><' q) == (r ``comp`` p) '><' (s ``comp`` q)

791+-- r '><' s == 'splitR' (r `.` fstR) (s `.` sndR)

792+-- (r '><' s) `.` (p '><' q) == (r `.` p) '><' (s `.` q)

793793-- @

794794(><) ::

795− ( CountableDimensionsN a b,

796− CountableDimensionsN c d,

797− CountableDimensionsN (a, c) (b, d),

798− FromListsN (a, c) a,

799− FromListsN (a, c) c,

800− FromListsN (b, d) b,

801− FromListsN (b, d) d,

795+ ( CountableDimsN a b,

796+ CountableDimsN c d,

797+ CountableDimsN (a, c) (b, d),

798+ FLN (a, c) a,

799+ FLN (a, c) c,

800+ FLN (b, d) b,

801+ FLN (b, d) d,

802802 TrivialP a c,

803803 TrivialP b d

804804 )

@@ -810,33 +810,31 @@ infixl 4 ><

810810-- | Relational coproduct.

811811--

812812-- @

813−-- | 'eitherR' a b ``comp`` 'i1' == a

813+-- | 'eitherR' a b `.` 'i1' == a

814814-- 'eitherR' a b <=> |

815−-- | 'eitherR' a b ``comp`` 'i2' == b

815+-- | 'eitherR' a b `.` 'i2' == b

816816-- @

817817--

818818-- @

819−-- 'eitherR' r s ``comp`` 'conv' ('eitherR' t u) == (r ``comp`` 'conv' t) ``union`` (s ``comp`` 'conv' u)

819+-- 'eitherR' r s `.` 'conv' ('eitherR' t u) == (r `.` 'conv' t) ``union`` (s `.` 'conv' u)

820820-- @

821821--

822822-- @

823823-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)

824824-- @

825−eitherR :: (TrivialE a b)

826− => Relation a c -> Relation b c -> Relation (Either a b) c

827−eitherR = junc

825+eitherR :: Relation a c -> Relation b c -> Relation (Either a b) c

826+eitherR = join

828827

829828-- | Relational coproduct first component injection

830829--

831830-- @

832−-- 'img' 'i1' ``union`` 'img' 'i2' == 'identity'

833−-- 'i1' ``comp`` 'i2' = 'zeros'

831+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

832+-- 'i1' `.` 'i2' = 'zeros'

834833-- @

835834i1 ::

836− ( CountableDimensionsN a b,

837− FromListsN b a,

838− FromListsN a a,

839− TrivialE a b

835+ ( CountableDimsN a b,

836+ FLN b a,

837+ FLN a a

840838 )

841839 => Relation a (Either a b)

842840i1 = R I.i1

@@ -844,14 +842,13 @@ i1 = R I.i1

844842-- | Relational coproduct second component injection

845843--

846844-- @

847−-- 'img' 'i1' ``union`` 'img' 'i2' == 'identity'

848−-- 'i1' ``comp`` 'i2' = 'zeros'

845+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'

846+-- 'i1' `.` 'i2' = 'zeros'

849847-- @

850848i2 ::

851− ( CountableDimensionsN a b,

852− FromListsN a b,

853− FromListsN b b,

854− TrivialE a b

849+ ( CountableDimsN a b,

850+ FLN a b,

851+ FLN b b

855852 )

856853 => Relation b (Either a b)

857854i2 = R I.i2

@@ -861,16 +858,14 @@ infixl 5 -|-

861858-- | Relational coproduct functor.

862859--

863860-- @

864−-- r '-|-' s == 'eitherR' ('i1' ``comp`` r) ('i2' ``comp`` s)

861+-- r '-|-' s == 'eitherR' ('i1' `.` r) ('i2' `.` s)

865862-- @

866863(-|-) ::

867− ( CountableDimensionsN b d,

868− FromListsN b b,

869− FromListsN d b,

870− FromListsN b d,

871− FromListsN d d,

872− TrivialE a c,

873− TrivialE b d

864+ ( CountableDimsN b d,

865+ FLN b b,

866+ FLN d b,

867+ FLN b d,

868+ FLN d d

874869 )

875870 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)

876871(-|-) (R a) (R b) = R ((I.-|-) a b)

@@ -883,20 +878,20 @@ infixl 5 -|-

883878-- 'trans'/'untrans';

884879-- more-over, where each particular attribute is placed (input/output) is irrelevant.

885880trans ::

886− ( CountableDimensionsN a b,

881+ ( CountableDimsN a b,

887882 CountableN c,

888− CountableDimensionsN (a, b) (c, b),

889− FromListsN (c, b) c,

890− FromListsN (c, b) b,

891− FromListsN (a, b) a,

892− FromListsN (a, b) b,

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,

893888 Trivial (a, b),

894889 Trivial (c, b),

895− TrivialP2 a b,

896− TrivialP2 c b

890+ TrivialP a b,

891+ TrivialP c b

897892 )

898893 => Relation (a, b) c -> Relation a (c, b)

899−trans r = splitR r p2 `comp` conv p1

894+trans r = splitR r sndR . conv fstR

900895

901896-- | Relational 'untrans'

902897--

@@ -904,109 +899,110 @@ trans r = splitR r p2 `comp` conv p1

904899-- 'trans'/'untrans';

905900-- more-over, where each particular attribute is placed (input/output) is irrelevant.

906901untrans ::

907− ( CountableDimensionsN a b,

902+ ( CountableDimsN a b,

908903 CountableN c,

909− CountableDimensionsN (a, b) (c, b),

910− FromListsN (c, b) c,

911− FromListsN (c, b) b,

912− FromListsN (a, b) b,

913− FromListsN (a, b) a,

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,

914909 Trivial (a, b),

915910 Trivial (c, b),

916− TrivialP2 a b,

917− TrivialP2 c b

911+ TrivialP a b,

912+ TrivialP c b

918913 )

919914 => Relation a (c, b) -> Relation (a, b) c

920−untrans s = p1 `comp` conv (splitR (conv s) p2)

915+untrans s = fstR . conv (splitR (conv s) sndR)

921916

922917-- | Transforms predicate @p@ into a correflexive relation.

923918--

924919-- @

925−-- 'predR' ('const' True) == 'identity'

920+-- 'predR' ('const' True) == 'id'

926921-- 'predR' ('const' False) == 'zeros'

927922-- @

928923--

929924-- @

930−-- 'predR' q ``comp`` 'predR' p == 'predR' q ``union`` 'predR' p

925+-- 'predR' q `.` 'predR' p == 'predR' q ``union`` 'predR' p

931926-- @

932927predR ::

933928 ( Bounded a,

934929 Enum a,

930+ Countable a,

935931 CountableN a,

936− FromListsN a a,

937− FromListsN Bool a

932+ FLN a a,

933+ FLN Bool a

938934 )

939935 => Relation a Bool -> Relation a a

940−predR p = identity `intersection` divisionF (fromF' (const True)) p

936+predR p = id `intersection` divisionF (fromF (const True)) p

941937

942938-- | Equalizes functions @f@ and @g@.

943939-- That is, @'equalizer' f g@ is the largest coreflexive

944940-- that restricts @g@ so that @f@ and @g@ yield the same outputs.

945941--

946942-- @

947−-- 'equalizer' r r == 'identity'

943+-- 'equalizer' r r == 'id'

948944-- 'equalizer' ('point' True) ('point' False) == 'zeros'

949945-- @

950946equalizer ::

951947 ( CountableN a,

952− FromListsN a a

948+ FLN a a

953949 )

954950 => Relation a b -> Relation a b -> Relation a a

955−equalizer f g = identity `intersection` divisionF f g

951+equalizer f g = id `intersection` divisionF f g

956952

957953-- | Relational conditional guard.

958954--

959955-- @

960−-- 'guard' p = 'i2' ``overriddenBy`` 'i1' ``comp`` 'predR' p

956+-- 'guard' p = 'i2' ``overriddenBy`` 'i1' `.` 'predR' p

961957-- @

962−guard ::

958+guard ::

963959 ( Bounded b,

964960 Enum b,

965961 CountableN b,

966− FromListsN b b,

967− FromListsN Bool b,

968− TrivialE b b

962+ Countable b,

963+ FLN b b,

964+ FLN Bool b

969965 ) => Relation b Bool -> Relation b (Either b b)

970966guard p = conv (eitherR (predR p) (predR (negate p)))

971967

972968-- | Relational McCarthy's conditional.

973−cond ::

969+cond ::

974970 ( Bounded b,

975971 Enum b,

972+ Countable b,

976973 CountableN b,

977− FromListsN b b,

978− FromListsN Bool b,

979− TrivialE b b

980− )

974+ FLN b b,

975+ FLN Bool b

976+ )

981977 => Relation b Bool -> Relation b c -> Relation b c -> Relation b c

982−cond p r s = eitherR r s `comp` guard p

978+cond p r s = eitherR r s . guard p

983979

984980-- | Relational domain.

985981--

986982-- For injective relations, 'domain' and 'ker'nel coincide,

987−-- since @'ker' r ``sse`` 'identity'@ in such situations.

983+-- since @'ker' r ``sse`` 'id'@ in such situations.

988984domain ::

989985 ( CountableN a,

990− FromListsN a a

986+ FLN a a

991987 ) => Relation a b -> Relation a a

992−domain r = ker r `intersection` identity

988+domain r = ker r `intersection` id

993989

994990-- | Relational range.

995991--

996992-- For functions, 'range' and 'img' (image) coincide,

997−-- since @'img' f ``sse`` 'identity'@ for any @f@.

993+-- since @'img' f ``sse`` id@ for any @f@.

998994range ::

999995 ( CountableN b,

1000− FromListsN b b

996+ FLN b b

1001997 ) => Relation a b -> Relation b b

1002−range r = img r `intersection` identity

998+range r = img r `intersection` id

1003999

10041000-- Relation pretty print

10051001

10061002-- | Relation pretty printing

1007−pretty :: (CountableN a) => Relation a b -> String

1003+pretty :: (CountableDimsN a b) => Relation a b -> String

10081004pretty (R a) = I.pretty a

10091005

10101006-- | Relation pretty printing

1011−prettyPrint :: (CountableN a) => Relation a b -> IO ()

1007+prettyPrint :: (CountableDimsN a b) => Relation a b -> IO ()

10121008prettyPrint (R a) = I.prettyPrint a

src/LAoP/Utils.hs+5−2

@@ -95,8 +95,11 @@ module LAoP.Utils

9595 coerceNat2,

9696 coerceNat3,

9797

98− -- * Powerset data type

99− List (..)

98+ -- * List data type

99+ List (..),

100+

101+ -- * Category type-class

102+ Category(..)

100103 )

101104where

102105

src/LAoP/Utils/Internal.hs+30−74

@@ -1,83 +1,13 @@

1+{-# LANGUAGE TypeFamilies #-}

12{-# LANGUAGE DeriveGeneric #-}

23{-# LANGUAGE TypeApplications #-}

34{-# LANGUAGE DataKinds #-}

45{-# LANGUAGE FlexibleInstances #-}

56{-# LANGUAGE GeneralizedNewtypeDeriving #-}

6−{-# LANGUAGE KindSignatures #-}

77{-# LANGUAGE ScopedTypeVariables #-}

99module LAoP.Utils.Internal

10− ( -- | This is an internal module and it is not meant to be imported.

11− --

12− -- Utility module that provides the 'Natural' data type.

13− -- The semantic associated with this data type is that

14− -- it's meant to be a restricted 'Int' value. For example

15− -- the type @Natural 6@ can only be instanciated with @Nat n@

16− -- where @0 <= n <= 6@. Why, You might ask, because with normal

17− -- 'Int's it is not possible to have a decent @Enum (Int, Int)@

18− -- instance. See the following probabilistic programming model as and

19− -- example:

20− --

21− -- We want to calculate the probability of the sum of two dice throws.

22− -- To do this we start by defining the sample space:

23− --

24− -- @

25− -- type SampleSpace = Int -- We think 'Int' are enough

26− --

27− -- die :: Dist Int 6

28− -- die = unifrom [1..6]

29− --

30− -- -- Promote 'Int' addition to a matrix

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

32− -- @

33− --

34− -- The last line is impossible because @(Int, Int)@ does not have

35− -- a good 'Enum' instance: @[(0, 1), (0, 2), .. (0, maxBound), (1, 0),

36− -- ..]@. And we'd like the addition matrix to be of 36 columns by 12

37− -- rows but limited to integers up to @6@!

38− --

39− -- One way to solve this issue is by defining and auxilary data type to

40− -- represent the sample space:

41− --

42− -- @

43− -- data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6

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

45− -- important

46− -- @

47− --

48− -- And write the sample space addition function:

49− --

50− -- @

51− -- ssAdd :: SampleSpace -> SampleSpace -> Int

52− -- ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)

53− -- @

54− --

55− -- And then promote that function to matrix and everything is alright:

56− --

57− -- @

58− -- ssAddM = fromF' (uncurry ssAdd)

59− --

60− -- dieSumProb = ssAddM `comp` (khatri die die)

61− -- @

62− --

63− -- This is a nice solution for small sample spaces. But for larger ones

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

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

66− -- To mitigate this limitation the 'Natural' type comes a long way and

67− -- allows one to model the sample in an easier way. See for instance:

68− --

69− -- @

70− -- ssAdd :: Natural 6 -> Natural 6 -> Natural 12

71− -- ssAdd = coerceNat (+)

72− --

73− -- ssAddM = fromF' (uncurry sumSS)

74− --

75− -- die :: Dist (Natural 6) 6

76− -- die = uniform [nat @6 1 .. nat 6]

77− --

78− -- dieSumProb = ssAddM `comp` (khatri die die)

79− -- @

80− --

10+ (

8111 -- * 'Natural' data type

8212 Natural(..),

8313 nat,

@@ -88,8 +18,11 @@ module LAoP.Utils.Internal

8818 coerceNat2,

8919 coerceNat3,

9020

91− -- * Powerset data type

92− List (..)

21+ -- * 'List' data type

22+ List (..),

23+

24+ -- * Category type class

25+ Category(..)

9326 )

9427where

9528

@@ -100,6 +33,9 @@ import Data.Maybe

10033import GHC.TypeLits

10134import Control.DeepSeq

10235import GHC.Generics

36+import Data.Kind

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

38+import qualified Prelude

10339

10440-- | Wrapper around 'Int's that have a restrictive semantic associated.

10541-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'

@@ -238,3 +174,23 @@ instance

238174 fromEnum (L x) =

239175 let as = [minBound .. maxBound]

240176 in fromMaybe (error "Does not exist") $ elemIndex x (powerset as)

177+

178+-- | Constrained category instance

179+class Category k where

180+ type Object k o :: Constraint

181+ type Object k o = ()

182+ id :: Object k a => k a a

183+ (.) :: k b c -> k a b -> k a c

184+

185+{-# RULES

186+"identity/left" forall p .

187+ id . p = p

188+"identity/right" forall p .

189+ p . id = p

190+"association" forall p q r .

191+ (p . q) . r = p . (q . r)

192+ #-}

193+

194+instance Category (->) where

195+ id = Prelude.id

196+ (.) = (Prelude..)

stack-8.8.3.yamlnew file+1

@@ -0,0 +1 @@

1+resolver: lts-15.5

stack.yaml+1−66

@@ -1,66 +1 @@

1−# This file was automatically generated by 'stack init'

2−#

3−# Some commonly used options have been documented as comments in this file.

4−# For advanced use and comprehensive documentation of the format, please see:

5−# https://docs.haskellstack.org/en/stable/yaml_configuration/

6−

7−# Resolver to choose a 'specific' stackage snapshot or a compiler version.

8−# A snapshot resolver dictates the compiler version and the set of packages

9−# to be used for project dependencies. For example:

10−#

11−# resolver: lts-3.5

12−# resolver: nightly-2015-09-21

13−# resolver: ghc-7.10.2

14−#

15−# The location of a snapshot can be provided as a file or url. Stack assumes

16−# a snapshot provided as a file might change, whereas a url resource does not.

17−#

18−# resolver: ./custom-snapshot.yaml

19−# resolver: https://example.com/snapshots/2018-01-01.yaml

20−resolver: lts-14.18

21−

22−# User packages to be built.

23−# Various formats can be used as shown in the example below.

24−#

25−# packages:

26−# - some-directory

27−# - https://example.com/foo/bar/baz-0.0.2.tar.gz

28−# subdirs:

29−# - auto-update

30−# - wai

31−packages:

32−- .

33−# Dependency packages to be pulled from upstream that are not in the resolver.

34−# These entries can reference officially published versions as well as

35−# forks / in-progress versions pinned to a git hash. For example:

36−#

37−# extra-deps:

38−# - acme-missiles-0.3

39−# - git: https://github.com/commercialhaskell/stack.git

40−# commit: e7b331f14bcffb8367cd58fbfc8b40ec7642100a

41−#

42−# extra-deps: []

43−

44−# Override default flag values for local packages and extra-deps

45−# flags: {}

46−

47−# Extra package databases containing global packages

48−# extra-package-dbs: []

49−

50−# Control whether we use the GHC we find on the path

51−# system-ghc: true

52−#

53−# Require a specific version of stack, using version ranges

54−# require-stack-version: -any # Default

55−# require-stack-version: ">=2.1"

56−#

57−# Override the architecture used by stack, especially useful on Windows

58−# arch: i386

59−# arch: x86_64

60−#

61−# Extra directories used by stack for building

62−# extra-include-dirs: [/path/to/dir]

63−# extra-lib-dirs: [/path/to/dir]

64−#

65−# Allow a newer minor version of GHC than the snapshot specifies

66−# compiler-check: newer-minor

1+resolver: lts-14.3

stack.yaml.lock+4−4

@@ -6,7 +6,7 @@

66packages: []

77snapshots:

88- completed:

9− size: 524789

10− url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/18.yaml

11− sha256: 646be71223e08234131c6989912e6011e01b9767bc447b6d466a35e14360bdf2

12− original: lts-14.18

9+ size: 523878

10+ url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/3.yaml

11+ sha256: 470c46c27746a48c7c50f829efc0cf00112787a7804ee4ac7a27754658f6d92c

12+ original: lts-14.3

test/Examples/Quantum.hsnew file+31

@@ -0,0 +1,31 @@

1+{-# LANGUAGE StandaloneDeriving #-}

2+module Examples.Quantum where

4+import LAoP.Utils

5+import LAoP.Matrix.Type

6+import Data.Complex

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

9+deriving instance Ord a => Ord (Complex a)

10+

11+xor :: (Bool, Bool) -> Bool

12+xor (False, b) = b

13+xor (True, b) = not b

14+

15+ker :: Num e => Matrix e a b -> Matrix e a a

16+ker m = tr m . m

17+

18+cnot :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)

19+cnot = kr fstM (fromF xor)

20+

21+ccnot :: (Num e, Ord e) => Matrix e ((Bool, Bool), Bool) ((Bool, Bool), Bool)

22+ccnot = kr fstM (fromF f)

23+ where

24+ f = xor . tp (uncurry (&&)) id

25+ tp f g (a,b) = (f a, g b)

26+

27+had :: Matrix (Complex Double) Bool Bool

28+had = (1/sqrt 2) .| fromLists [[1, 1], [1, -1]]

29+

30+bell :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)

31+bell = cnot . (had >< iden)

app/Examples.hs → test/Examples/Readme.hsrenamed+56−31

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

1+{-# LANGUAGE GADTs #-}

12{-# LANGUAGE DeriveGeneric #-}

23{-# LANGUAGE FlexibleContexts #-}

3−{-# LANGUAGE TypeOperators #-}

44{-# LANGUAGE TypeApplications #-}

55{-# LANGUAGE DataKinds #-}

7−module Examples

7+module Examples.Readme

88 ( exec

99 )

1010 where

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

1515import LAoP.Dist

1616import GHC.TypeLits

1717import Data.Coerce

18−import GHC.Generics

19−import Control.Category hiding (id)

20−import Prelude hiding ((.))

18+import qualified GHC.Generics as G

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

2120

2221-- Monty Hall Problem

2322data Outcome = Win | Lose

24− deriving (Bounded, Enum, Eq, Show, Generic)

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

2524

2625switch :: Outcome -> Outcome

2726switch Win = Lose

@@ -31,7 +30,7 @@ firstChoice :: Matrix Double () Outcome

3130firstChoice = col [1/3, 2/3]

3231

3332secondChoice :: Matrix Double Outcome Outcome

34−secondChoice = fromF' switch

33+secondChoice = fromF switch

3534

3635-- Dice sum

3736

@@ -40,7 +39,7 @@ type SS = Natural 1 6 -- Sample Space

4039sumSS :: SS -> SS -> Natural 2 12

4140sumSS = coerceNat (+)

4241

43−sumSSM = fromF' (uncurry sumSS)

42+sumSSM = fromF (uncurry sumSS)

4443

4544condition :: (Int, Int) -> Int -> Int

4645condition (fst, snd) thrd = if fst == snd

@@ -50,38 +49,64 @@ condition (fst, snd) thrd = if fst == snd

5049conditionSS :: (SS, SS) -> SS -> Natural 3 18

5150conditionSS = coerceNat2 condition

5251

53−conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die

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

5453

5554die :: Matrix Double () SS

5655die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]

5756

5857-- Sprinkler

58+data G = Dry | Wet

59+ deriving (Bounded, Enum, Eq, Show, G.Generic)

5960

60−rain :: Matrix Double () Bool

61−rain = col [0.8, 0.2]

61+data S = Off | On

62+ deriving (Bounded, Enum, Eq, Show, G.Generic)

6263

63−sprinkler :: Matrix Double Bool Bool

64−sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]

64+data R = No | Yes

65+ deriving (Bounded, Enum, Eq, Show, G.Generic)

6566

66−grass :: Matrix Double (Bool, Bool) Bool

67−grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]

67+rain :: Matrix Double () R

68+rain = matrixBuilder gen

69+ where

70+ gen (_, No) = 0.8

71+ gen (_, Yes) = 0.2

72+

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

92+

93+tag f = kr f id

6894

69−state :: Matrix Double () (Bool, (Bool, Bool))

70−state = khatri grass identity . khatri sprinkler identity . rain

95+state g s r = tag g . tag s . r

7196

72−grass_wet :: Matrix Double (Bool, (Bool, Bool)) One

73−grass_wet = row [0,1] . kp1

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

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

7499

75−rainning :: Matrix Double (Bool, (Bool, Bool)) One

76−rainning = row [0,1] . kp2 . kp2

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

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

77102

78103-- Alcuin Puzzle

79104

80105data Being = Farmer | Fox | Goose | Beans

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

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

82107

83108data Bank = LeftB | RightB

84− deriving (Bounded, Enum, Eq, Show, Generic)

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

85110

86111eats :: Being -> Being -> Bool

87112eats Fox Goose = True

@@ -96,21 +121,21 @@ cross LeftB = RightB

96121cross RightB = LeftB

97122

98123crossR :: R.Relation Bank Bank

99−crossR = R.fromF' cross

124+crossR = R.fromF cross

100125

101126-- | Initial state, everyone in the left bank

102127locationLeft :: Being -> Bank

103128locationLeft _ = LeftB

104129

105130locationLeftR :: R.Relation Being Bank

106−locationLeftR = R.fromF' locationLeft

131+locationLeftR = R.fromF locationLeft

107132

108133-- | Initial state, everyone in the right bank

109134locationRight :: Being -> Bank

110135locationRight _ = RightB

111136

112137locationRightR :: R.Relation Being Bank

113−locationRightR = R.fromF' locationRight

138+locationRightR = R.fromF locationRight

114139

115140-- Properties

116141

@@ -127,7 +152,7 @@ inv :: R.Relation Being Bank -> Bool

127152inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)

128153 where

129154 farmer :: R.Relation Being Being

130− farmer = R.fromF' (const Farmer)

155+ farmer = R.fromF (const Farmer)

131156

132157-- Arbitrary state

133158bankState :: Being -> Bank -> Bool

@@ -146,14 +171,14 @@ exec = do

146171 putStrLn "Monty Hall Problem solution:"

147172 prettyPrint (secondChoice . firstChoice)

148173 putStrLn "\n Sum of dices probability:"

149− prettyPrint (sumSSM `comp` khatri die die)

174+ prettyPrint (sumSSM . kr die die)

150175 putStrLn "\n Conditional dice throw:"

151176 prettyPrint conditionalThrows

152177 putStrLn "\n Checking that the last result is indeed a distribution: "

153− prettyPrint (bang . sumSSM . khatri die die)

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

154179 putStrLn "\n Probability of grass being wet:"

155− prettyPrint (grass_wet . state)

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

156181 putStrLn "\n Probability of rain:"

157− prettyPrint (rainning . state)

182+ prettyPrint (rainning . state grass sprinkler rain)

158183 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"

159184 print (inv bankStateR)

test/Examples/Spreadsheets.hsnew file+47

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

1+{-# LANGUAGE DeriveGeneric #-}

3+module Examples.Spreadsheets

5+import LAoP.Utils

6+import LAoP.Matrix.Type

7+import GHC.Generics

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

10+data Student = Student1 | Student2 | Student3 | Student4

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

12+

13+data Question = Question1 | Question2 | Question3 | Question4

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

15+

16+data Results = Exam | Test | Final

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

18+

19+test :: Matrix Float One Results

20+test = point Test

21+

22+exam :: Matrix Float One Results

23+exam = point Exam

24+

25+final :: Matrix Float One Results

26+final = point Final

27+

28+m :: Matrix Float Question Student

29+m = fromLists [[95, 90, 100, 40], [20, 90, 90, 0], [30, 20, 95, 0], [50, 80, 100, 30]]

30+

31+w :: Matrix Float Question One

32+w = fromLists [[0.2, 0.3, 0.2, 0.3]]

33+

34+xls :: Matrix Float Question One

35+ -> Matrix Float Question Student

36+ -> Matrix Float Student One

37+ -> Matrix Float (Either Question Results) (Either One Student)

38+xls w m t = join (fork w m) (fork zeros r)

39+ where

40+ rExam = m . tr w

41+ rTest = tr t

42+ rFinal = rTest `maxPW` rExam

43+ r = (rExam . tr exam) + (rTest . tr test) + (rFinal . tr final)

44+

45+-- | Overloaded, point-wise 'max' function

46+maxPW :: Ord e => Matrix e a b -> Matrix e a b -> Matrix e a b

47+maxPW = zipWithM max

test/Spec.hs+7−1

@@ -1,2 +1,8 @@

1+module Main (main) where

3+import qualified Examples.Readme as RD

15main :: IO ()

2−main = putStrLn "Test suite not yet implemented"

6+main = do

7+ putStrLn ("Testing if README works" :: String)

8+ RD.exec