├── .github ├── ISSUE_TEMPLATE │ └── 想法與相關內容.md └── workflows │ └── publish.yml ├── .gitignore ├── LICENSE ├── Makefile ├── README.md ├── assets ├── android-chrome-192x192.png ├── android-chrome-512x512.png ├── apple-touch-icon.png ├── favicon-16x16.png ├── favicon-32x32.png ├── favicon.ico ├── fonts │ ├── KaTeX_AMS-Regular.ttf │ ├── KaTeX_AMS-Regular.woff │ ├── KaTeX_AMS-Regular.woff2 │ ├── KaTeX_Caligraphic-Bold.ttf │ ├── KaTeX_Caligraphic-Bold.woff │ ├── KaTeX_Caligraphic-Bold.woff2 │ ├── KaTeX_Caligraphic-Regular.ttf │ ├── KaTeX_Caligraphic-Regular.woff │ ├── KaTeX_Caligraphic-Regular.woff2 │ ├── KaTeX_Fraktur-Bold.ttf │ ├── KaTeX_Fraktur-Bold.woff │ ├── KaTeX_Fraktur-Bold.woff2 │ ├── KaTeX_Fraktur-Regular.ttf │ ├── KaTeX_Fraktur-Regular.woff │ ├── KaTeX_Fraktur-Regular.woff2 │ ├── KaTeX_Main-Bold.ttf │ ├── KaTeX_Main-Bold.woff │ ├── KaTeX_Main-Bold.woff2 │ ├── KaTeX_Main-BoldItalic.ttf │ ├── KaTeX_Main-BoldItalic.woff │ ├── KaTeX_Main-BoldItalic.woff2 │ ├── KaTeX_Main-Italic.ttf │ ├── KaTeX_Main-Italic.woff │ ├── KaTeX_Main-Italic.woff2 │ ├── KaTeX_Main-Regular.ttf │ ├── KaTeX_Main-Regular.woff │ ├── KaTeX_Main-Regular.woff2 │ ├── KaTeX_Math-BoldItalic.ttf │ ├── KaTeX_Math-BoldItalic.woff │ ├── KaTeX_Math-BoldItalic.woff2 │ ├── KaTeX_Math-Italic.ttf │ ├── KaTeX_Math-Italic.woff │ ├── KaTeX_Math-Italic.woff2 │ ├── KaTeX_SansSerif-Bold.ttf │ ├── KaTeX_SansSerif-Bold.woff │ ├── KaTeX_SansSerif-Bold.woff2 │ ├── KaTeX_SansSerif-Italic.ttf │ ├── KaTeX_SansSerif-Italic.woff │ ├── KaTeX_SansSerif-Italic.woff2 │ ├── KaTeX_SansSerif-Regular.ttf │ ├── KaTeX_SansSerif-Regular.woff │ ├── KaTeX_SansSerif-Regular.woff2 │ ├── KaTeX_Script-Regular.ttf │ ├── KaTeX_Script-Regular.woff │ ├── KaTeX_Script-Regular.woff2 │ ├── KaTeX_Size1-Regular.ttf │ ├── KaTeX_Size1-Regular.woff │ ├── KaTeX_Size1-Regular.woff2 │ ├── KaTeX_Size2-Regular.ttf │ ├── KaTeX_Size2-Regular.woff │ ├── KaTeX_Size2-Regular.woff2 │ ├── KaTeX_Size3-Regular.ttf │ ├── KaTeX_Size3-Regular.woff │ ├── KaTeX_Size3-Regular.woff2 │ ├── KaTeX_Size4-Regular.ttf │ ├── KaTeX_Size4-Regular.woff │ ├── KaTeX_Size4-Regular.woff2 │ ├── KaTeX_Typewriter-Regular.ttf │ ├── KaTeX_Typewriter-Regular.woff │ ├── KaTeX_Typewriter-Regular.woff2 │ ├── inria-sans-v14-latin_latin-ext-300.woff2 │ ├── inria-sans-v14-latin_latin-ext-300italic.woff2 │ ├── inria-sans-v14-latin_latin-ext-700.woff2 │ ├── inria-sans-v14-latin_latin-ext-700italic.woff2 │ ├── inria-sans-v14-latin_latin-ext-italic.woff2 │ └── inria-sans-v14-latin_latin-ext-regular.woff2 ├── forest.xsl ├── forester.sty ├── images │ ├── fp-000L │ │ └── injection.png │ ├── fp-000M │ │ └── surjection.png │ └── fp-000N │ │ └── bijection.png ├── katex.min.css ├── preamble.sty └── style.css ├── javascript └── forester.js ├── package-lock.json ├── package.json └── trees ├── alg-0001.tree ├── alg-0002.tree ├── alg-0003.tree ├── alg-0004.tree ├── alg-0005.tree ├── base-macros.tree ├── cat-0001.tree ├── cat-0002.tree ├── cat-0003.tree ├── cat-0004.tree ├── cat-0005.tree ├── cat-0006.tree ├── cat-0007.tree ├── cat-0008.tree ├── cat-0009.tree ├── cat-000A.tree ├── cs-0001.tree ├── fp-0001.tree ├── fp-0002.tree ├── fp-0003.tree ├── fp-0004.tree ├── fp-0005.tree ├── fp-0006.tree ├── fp-0007.tree ├── fp-0008.tree ├── fp-0009.tree ├── fp-000A.tree ├── fp-000B.tree ├── fp-000C.tree ├── fp-000D.tree ├── fp-000E.tree ├── fp-000F.tree ├── fp-000G.tree ├── fp-000H.tree ├── fp-000I.tree ├── fp-000J.tree ├── fp-000K.tree ├── fp-000L.tree ├── fp-000M.tree ├── fp-000N.tree ├── fp-000O.tree ├── fp-000P.tree ├── people ├── dannypsnl.tree └── fizzyelt.tree ├── roadmap.tree └── set-0001.tree /.github/ISSUE_TEMPLATE/想法與相關內容.md: -------------------------------------------------------------------------------- 1 | --- 2 | name: 想法與相關內容 3 | about: 紀錄自己的想法與理解 4 | title: "[主題] 標題" 5 | labels: documentation 6 | assignees: FizzyElt 7 | 8 | --- 9 | 10 | # [相關主題] 11 | -------------------------------------------------------------------------------- /.github/workflows/publish.yml: -------------------------------------------------------------------------------- 1 | name: github pages 2 | on: 3 | push: 4 | branches: 5 | - main 6 | - forester 7 | paths: 8 | - trees/** 9 | - javascript/** 10 | - assets/** 11 | - Makefile 12 | - .github/workflows/publish.yml 13 | pull_request: 14 | branches: 15 | - main 16 | - forester 17 | 18 | jobs: 19 | deploy: 20 | runs-on: ubuntu-22.04 21 | steps: 22 | - uses: actions/checkout@v3 23 | with: 24 | fetch-depth: 1 25 | - name: "install emacs" 26 | run: sudo apt-get install -y emacs 27 | - name: Set-up OCaml 28 | uses: ocaml/setup-ocaml@v2 29 | with: 30 | ocaml-compiler: "5.1" 31 | 32 | - name: Install JS dependencies 33 | run: npm install 34 | 35 | - name: Install Forester 36 | run: | 37 | git clone https://git.sr.ht/~jonsterling/ocaml-forester 38 | cd ocaml-forester 39 | opam pin add -y . --locked 40 | 41 | - name: Setup TeX Live 42 | uses: teatimeguest/setup-texlive-action@v2 43 | with: 44 | packages: >- 45 | scheme-medium 46 | standalone 47 | mlmodern 48 | amsfonts 49 | amsmath 50 | zx-calculus 51 | tikz-cd 52 | l3packages 53 | dvisvgm 54 | pgf 55 | etoolbox 56 | mathtools 57 | stmaryrd 58 | newtx 59 | newpx 60 | xstring 61 | xpatch 62 | fontaxes 63 | kastrup 64 | spath3 65 | 66 | - name: Check `latex` version 67 | run: latex --version 68 | 69 | - name: "build forester" 70 | run: make release 71 | 72 | - name: "Deploy" 73 | uses: peaceiris/actions-gh-pages@v3 74 | with: 75 | github_token: ${{ secrets.GITHUB_TOKEN }} 76 | publish_dir: ./output 77 | publish_branch: gh-pages 78 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | build/ 2 | latex/ 3 | output/ 4 | node_modules/ 5 | assets/forester.js 6 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | MIT License 2 | 3 | Copyright (c) 2023 FizzyElt 4 | 5 | Permission is hereby granted, free of charge, to any person obtaining a copy 6 | of this software and associated documentation files (the "Software"), to deal 7 | in the Software without restriction, including without limitation the rights 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 | copies of the Software, and to permit persons to whom the Software is 10 | furnished to do so, subject to the following conditions: 11 | 12 | The above copyright notice and this permission notice shall be included in all 13 | copies or substantial portions of the Software. 14 | 15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 | SOFTWARE. 22 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | all: build 2 | .PHONY: all 3 | 4 | serve: 5 | @$(if $(shell which forest), echo "forest-server installed", cargo install forest-server) 6 | @forest watch 1313 -- "build --dev --base-url https://fizzyelt.github.io/functional-programming --root fp-0001 trees/ " 7 | .PHONY: serve 8 | 9 | release: assets/forester.js theme 10 | @echo "Build forester" 11 | @opam exec -- forester build --base-url https://fizzyelt.github.io/functional-programming --root fp-0001 trees/ 12 | .PHONY: release 13 | 14 | build: assets/forester.js theme 15 | @echo "Build forester" 16 | @opam exec -- forester build --dev --base-url https://fizzyelt.github.io/functional-programming --root fp-0001 trees/ 17 | .PHONY: build 18 | 19 | theme: 20 | mkdir theme 21 | 22 | assets/forester.js: node_modules assets 23 | @echo "Build JS" 24 | @./node_modules/.bin/esbuild --minify --bundle javascript/forester.js --outfile=assets/forester.js 25 | 26 | node_modules: 27 | npm install 28 | 29 | deps: 30 | cargo install forest-server 31 | brew install watch texlive 32 | opam install forester 33 | .PHONY: deps 34 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Functional Programming 知識大補帖 2 | 3 | 這是一個關於 Functional Programming 名詞解釋及知識點整理的地方，會包含一些情境及範例解說，目的在於消除學習 FP 過程中因不懂緣由所產生的困惑。 4 | 5 | ## Roadmap 6 | 7 | 1. lambda calculus 8 | 2. 用 recursion 解釋可計算性 9 | 3. 加上 simply type 10 | 4. 解釋型別規則 11 | 5. 建立 STLC categorical semantic 12 | 6. 加上 polymorphism 13 | 7. 解釋型別規則 14 | 8. 建立 \(E \to B\) categorical semantic 15 | 9. 解釋 \(\lambda 2\) 的 polymorphic property 16 | 10. 解釋 recursive type 17 | 11. 連結 F-algebra 18 | 12. 變成 dependent type 19 | 13. 解釋型別規則 20 | 14. 解釋模型（The natural model） 21 | - topos 22 | - topos fundamental theorem 23 | - presheaf 24 | - representable natural transformation 25 | 15. 進入真正的應用 26 | 16. side effect 27 | 17. concurrency 28 | 18. 安全性 29 | - 記憶體安全與子結構類型 30 | - (?) proof nets 31 | - 整體安全性不是只關乎記憶體 32 | 19. domain language 33 | 34 | ### ⚠️ 部份完成小主題內容，整理後移除 35 | 36 | - [pure function](./trees/fp-000J.tree) 37 | - immutable 38 | - curry 39 | - Lazy semantic 40 | - [data first, data last](./trees/fp-0002.tree) 41 | - [Magma](./trees/alg-0001.tree) 42 | - [Semigroup](./trees/alg-0003.tree) 43 | - [Monoid](./trees/alg-0005.tree) 44 | - [Functor](./trees/cat-0003.tree) 45 | - Applicative 46 | - Monad 47 | 48 | ## Development 49 | 50 | - [forester](https://www.jonmsterling.com/jms-005P.xml) 51 | - https://en.wikipedia.org/wiki/MacTeX 52 | 53 | Build command 54 | 55 | ```shell 56 | make build 57 | ``` 58 | 59 | Locally watch server 60 | 61 | ```shell 62 | ./server.sh 63 | ``` 64 | 65 | ## Contribution 66 | 67 | 建立新的 tree，prefix 根據你的內容的主題更換 68 | 69 | 目前現有的主題 70 | 71 | - fp (functional programming) 72 | - alg (algebra) 73 | - cat (category) 74 | - set 75 | 76 | ```bash 77 | forester new prefix=fp 78 | ``` 79 | 80 | 基本內容，如果是修改現有的 tree 就新增 author 上去 81 | 82 | ```text 83 | \title{XXX} 84 | \author{name} 85 | ``` 86 | 87 | 如果你願意貢獻你的一些資訊可以在 `trees/person` 資料夾開一個 tree 88 | 89 | trees/person/xxxx.tree 90 | 91 | ```text 92 | \title{XXXXX} 93 | \taxon{person} 94 | \meta{position}{Front-End Developer} 95 | \meta{external}{xxxxx.com} 96 | ``` 97 | 98 | 將你的內容關聯到某個 tree 內 99 | 100 | xx-XXXX.tree 101 | 102 | ```text 103 | \transclude{xx-XXXX} 104 | ``` 105 | 106 | 提及 xx-XXXX.tree 107 | 108 | ```text 109 | [text](xx-XXXX) 110 | ``` 111 | 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    470 |     471 |     472 | -------------------------------------------------------------------------------- /assets/forester.sty: -------------------------------------------------------------------------------- 1 | \RequirePackage{xparse} 2 | \RequirePackage{graphicx} 3 | \RequirePackage[dvipsnames]{xcolor} 4 | \RequirePackage{hyperref} 5 | \RequirePackage{xspace} 6 | \RequirePackage{mparhack} 7 | \RequirePackage{newpxtext,newpxmath} 8 | 9 | \definecolor{Matterhorn}{RGB}{77,77,77} 10 | \definecolor{RegalBlue}{RGB}{3,69,117} 11 | \definecolor{DarkBlue}{RGB}{3,69,117} 12 | \definecolor{FunGreen}{RGB}{14,87,54} 13 | \definecolor{VerdunGreen}{RGB}{54,87,14} 14 | \definecolor{RedDevil}{RGB}{134,1,17} 15 | \definecolor{DarkRed}{RGB}{134,1,17} 16 | 17 | \hypersetup{ 18 | colorlinks = true, 19 | linkcolor = Matterhorn, 20 | } 21 | 22 | \setcounter{secnumdepth}{5} 23 | 24 | \newcounter{forest} 25 | 26 | \ExplSyntaxOn 27 | 28 | \tl_new:N \l__forester_current_counter 29 | \clist_new:N \l__forester_counters 30 | \clist_set:Nn \l__forester_counters {section,subsection,subsubsection,paragraph,subparagraph} 31 | 32 | \keys_define:nn { forester } { 33 | treeCommands .clist_set:N = \l__forester_counters, 34 | forestSite .tl_set:N = \l__forester_site, 35 | } 36 | 37 | \keys_define:nn { forester_tree } { 38 | title .tl_set:N = \l__forester_tree_title, 39 | taxon .tl_set:N = \l__forester_tree_taxon, 40 | slug .tl_set:N = \l__forester_tree_slug, 41 | } 42 | 43 | \cs_new:Npn \forester_format_slug:n #1 { 44 | \textcolor{gray}{\texttt{#1}} 45 | } 46 | 47 | \cs_new:Npn \forester_slug_url:n #1 { 48 | \l__forester_site / #1.xml 49 | } 50 | 51 | \cs_new:Npn \forester_format_slug_link:n #1 { 52 | \tl_if_empty:NTF \l__forester_site { 53 | \hyperref[#1]{\forester_format_slug:n {#1}} 54 | } { 55 | \href{\forester_slug_url:n {#1}}{ 56 | \forester_format_slug:n {#1} 57 | } 58 | } 59 | } 60 | 61 | \cs_new:Npn \forester_format_result: { 62 | \medskip\par\noindent 63 | \marginpar{\forester_format_slug_link:n {\l__forester_tree_slug}} 64 | \textbf{\text_titlecase_first:n{\l__forester_tree_taxon}~\use:c{the\l__forester_current_counter}} 65 | \tl_if_empty:NTF \l__forester_tree_title {} {~(\l__forester_tree_title)} 66 | .\xspace 67 | } 68 | 69 | \cs_new:Npn \forester_format_section: { 70 | \use:c{\l__forester_current_counter}[ 71 | \text_titlecase_first:n{\l__forester_tree_title} 72 | ]{ 73 | \text_titlecase_first:n{\l__forester_tree_title} 74 | \marginpar{\forester_format_slug_link:n {\l__forester_tree_slug}} 75 | } 76 | } 77 | 78 | \NewDocumentEnvironment{tree}{m}{ 79 | \group_begin: 80 | \keys_set:nn { forester_tree } {#1} 81 | \clist_pop:NN \l__forester_counters \l__forester_current_counter 82 | \tl_if_empty:NTF \l__forester_tree_taxon { 83 | \forester_format_section: 84 | } { 85 | \refstepcounter{\l__forester_current_counter} 86 | \addcontentsline{toc}{\l__forester_current_counter}{ 87 | \protect{\numberline{\use:c{the\l__forester_current_counter}}} 88 | \textbf{\text_titlecase_first:n{\l__forester_tree_taxon}} 89 | \tl_if_empty:NTF \l__forester_tree_title {} {~(\l__forester_tree_title)} 90 | } 91 | \forester_format_result: 92 | } 93 | \bool_if_exist:cTF {g__forester_slug_defined_\l__forester_tree_slug} {} { 94 | \label{\l__forester_tree_slug} 95 | \bool_gset_true:c {g__forester_slug_defined_\l__forester_tree_slug} 96 | } 97 | }{ 98 | \medskip 99 | \group_end: 100 | } 101 | 102 | \cs_new:Npn \forester_ref:nn #1 #2 { 103 | \cs_if_exist:cTF {r@#1} { 104 | \hyperref[#1]{#2} 105 | } { 106 | #2~\cite{#1} 107 | } 108 | } 109 | 110 | \NewDocumentCommand\ForesterSetup{+m}{ 111 | \keys_set:nn { forester } { #1 } 112 | } 113 | 114 | \NewDocumentCommand\ForesterRef{mm}{ 115 | \forester_ref:nn {#1} {#2} 116 | } 117 | 118 | \ExplSyntaxOff 119 | 120 | 121 | -------------------------------------------------------------------------------- /assets/images/fp-000L/injection.png: 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-------------------------------------------------------------------------------- 1 | \usepackage{tikz, tikz-cd, mathtools, amssymb, stmaryrd} 2 | \usetikzlibrary{matrix,arrows} 3 | \usetikzlibrary{fit,positioning,calc,shapes} 4 | \usetikzlibrary{decorations.pathreplacing} 5 | \usetikzlibrary{decorations.pathmorphing} 6 | \usetikzlibrary{decorations.markings} 7 | 8 | \tikzset{ 9 | desc/.style={sloped, fill=white,inner sep=2pt}, 10 | upright desc/.style={fill=white,inner sep=2pt}, 11 | pullback/.style = { 12 | append after command={ 13 | \pgfextra{ 14 | \draw ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm); 15 | } 16 | } 17 | }, 18 | pullback 45/.style = { 19 | append after command={ 20 | \pgfextra{ 21 | \draw[rotate = 45] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm); 22 | } 23 | } 24 | }, 25 | ne pullback/.style = { 26 | append after command={ 27 | \pgfextra{ 28 | \draw ($(\tikzlastnode) + (-.2cm,-.5cm)$) -- ++(-0.3cm,0) -- ++(0,0.3cm); 29 | } 30 | } 31 | }, 32 | sw pullback/.style = { 33 | append after command={ 34 | \pgfextra{ 35 | \draw ($(\tikzlastnode) + (.2cm,.5cm)$) -- ++(0.3cm,0) -- ++(0,-0.3cm); 36 | } 37 | } 38 | }, 39 | dotted pullback/.style = { 40 | append after command={ 41 | \pgfextra{ 42 | \draw [densely dotted] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm); 43 | } 44 | } 45 | }, 46 | muted pullback/.style = { 47 | append after command={ 48 | \pgfextra{ 49 | \draw [gray] ($(\tikzlastnode) + (.2cm,-.5cm)$) -- ++(0.3cm,0) -- ++(0,0.3cm); 50 | } 51 | } 52 | }, 53 | pushout/.style = { 54 | append after command={ 55 | \pgfextra{ 56 | \draw ($(\tikzlastnode) + (-.2cm,.5cm)$) -- ++(-0.3cm,0) -- ++(0,-0.3cm); 57 | } 58 | } 59 | }, 60 | between/.style args={#1 and #2}{ 61 | at = ($(#1)!0.5!(#2)$) 62 | }, 63 | diagram/.style = { 64 | on grid, 65 | node distance=2cm, 66 | commutative diagrams/every diagram, 67 | line width = .5pt, 68 | every node/.append style = { 69 | commutative diagrams/every cell, 70 | } 71 | }, 72 | fibration/.style = { 73 | -{Triangle[open]} 74 | }, 75 | etale/.style = { 76 | -{Triangle[open]} 77 | }, 78 | etale cover/.style= { 79 | >={Triangle[open]},->.> 80 | }, 81 | opfibration/.style = { 82 | -{Triangle} 83 | }, 84 | lies over/.style = { 85 | |-{Triangle[open]} 86 | }, 87 | op lies over/.style = { 88 | |-{Triangle} 89 | }, 90 | embedding/.style = { 91 | {right hook}-> 92 | }, 93 | open immersion/.style = { 94 | {right hook}-{Triangle[open]} 95 | }, 96 | closed immersion/.style = { 97 | {right hook}-{Triangle} 98 | }, 99 | closed immersion*/.style = { 100 | {left hook}-{Triangle} 101 | }, 102 | embedding*/.style = { 103 | {left hook}-> 104 | }, 105 | open immersion*/.style = { 106 | {left hook}-{Triangle[open]} 107 | }, 108 | exists/.style = { 109 | densely dashed 110 | }, 111 | } 112 | 113 | \newlength{\dontworryaboutit} 114 | 115 | \tikzset{ 116 | inline diagram/.style = { 117 | commutative diagrams/every diagram, 118 | commutative diagrams/cramped, 119 | line width = .5pt, 120 | every node/.append style = { 121 | commutative diagrams/every cell, 122 | anchor = base, 123 | inner sep = 0pt 124 | }, 125 | every path/.append style = { 126 | outer xsep = 2pt 127 | } 128 | } 129 | } 130 | 131 | \tikzset{ 132 | square/nw/.style = {}, 133 | square/ne/.style = {}, 134 | square/se/.style = {}, 135 | square/sw/.style = {}, 136 | square/north/.style = {->}, 137 | square/south/.style = {->}, 138 | square/west/.style = {->}, 139 | square/east/.style = {->}, 140 | square/north/node/.style = {above}, 141 | square/south/node/.style = {below}, 142 | square/west/node/.style = {left}, 143 | square/east/node/.style = {right}, 144 | } 145 | 146 | \ExplSyntaxOn 147 | 148 | \bool_new:N \l_jon_glue_west 149 | 150 | \keys_define:nn { jon-tikz/diagram } { 151 | nw .tl_set:N = \l_jon_tikz_diagram_nw, 152 | sw .tl_set:N = \l_jon_tikz_diagram_sw, 153 | ne .tl_set:N = \l_jon_tikz_diagram_ne, 154 | se .tl_set:N = \l_jon_tikz_diagram_se, 155 | 156 | width .tl_set:N = \l_jon_tikz_diagram_width, 157 | height .tl_set:N = \l_jon_tikz_diagram_height, 158 | 159 | north .tl_set:N = \l_jon_tikz_diagram_north, 160 | south .tl_set:N = \l_jon_tikz_diagram_south, 161 | west .tl_set:N = \l_jon_tikz_diagram_west, 162 | east .tl_set:N = \l_jon_tikz_diagram_east, 163 | 164 | nw/style .code:n = {\tikzset{square/nw/.style = {#1}}}, 165 | sw/style .code:n = {\tikzset{square/sw/.style = {#1}}}, 166 | ne/style .code:n = {\tikzset{square/ne/.style = {#1}}}, 167 | se/style .code:n = {\tikzset{square/se/.style = {#1}}}, 168 | 169 | glue .choice:, 170 | glue / west .code:n = {\bool_set:Nn \l_jon_glue_west \c_true_bool}, 171 | 172 | glue~target .tl_set:N = \l_jon_tikz_glue_target, 173 | 174 | north/style .code:n = {\tikzset{square/north/.style = {#1}}}, 175 | north/node/style .code:n = {\tikzset{square/north/node/.style = {#1}}}, 176 | south/style .code:n = {\tikzset{square/south/.style = {#1}}}, 177 | south/node/style .code:n = {\tikzset{square/south/node/.style = {#1}}}, 178 | west/style .code:n = {\tikzset{square/west/.style = {#1}}}, 179 | west/node/style .code:n = {\tikzset{square/west/node/.style = {#1}}}, 180 | east/style .code:n = {\tikzset{square/east/.style = {#1}}}, 181 | east/node/style .code:n = {\tikzset{square/east/node/.style = {#1}}}, 182 | 183 | draft .meta:n = { 184 | nw = {\__jon_tikz_diagram_fmt_placeholder:n {nw}}, 185 | sw = {\__jon_tikz_diagram_fmt_placeholder:n {sw}}, 186 | se = {\__jon_tikz_diagram_fmt_placeholder:n {se}}, 187 | ne = {\__jon_tikz_diagram_fmt_placeholder:n {ne}}, 188 | north = {\__jon_tikz_diagram_fmt_placeholder:n {north}}, 189 | south = {\__jon_tikz_diagram_fmt_placeholder:n {south}}, 190 | west = {\__jon_tikz_diagram_fmt_placeholder:n {west}}, 191 | east = {\__jon_tikz_diagram_fmt_placeholder:n {east}}, 192 | } 193 | } 194 | 195 | \tl_set:Nn \l_jon_tikz_diagram_width { 2cm } 196 | \tl_set:Nn \l_jon_tikz_diagram_height { 2cm } 197 | 198 | 199 | \cs_new:Npn \__jon_tikz_diagram_fmt_placeholder:n #1 { 200 | \texttt{\textcolor{red}{#1}} 201 | } 202 | 203 | \keys_set:nn { jon-tikz/diagram } { 204 | glue~target = {}, 205 | } 206 | 207 | 208 | \cs_new:Nn \__jon_tikz_render_square:nn { 209 | \group_begin: 210 | \keys_set:nn {jon-tikz/diagram} {#2} 211 | \bool_if:nTF \l_jon_glue_west { 212 | \node (#1ne) [right = \l_jon_tikz_diagram_width~of~\l_jon_tikz_glue_target ne,square/ne] {$\l_jon_tikz_diagram_ne$}; 213 | \node (#1se) [below = \l_jon_tikz_diagram_height~of~#1ne,square/se] {$\l_jon_tikz_diagram_se$}; 214 | \draw[square/north] (\l_jon_tikz_glue_target ne) to node [square/north/node] {$\l_jon_tikz_diagram_north$} (#1ne); 215 | \draw[square/east] (#1ne) to node [square/east/node] {$\l_jon_tikz_diagram_east$} (#1se); 216 | \draw[square/south] (\l_jon_tikz_glue_target se) to node [square/south/node] {$\l_jon_tikz_diagram_south$} (#1se); 217 | } { 218 | \node (#1nw) [square/nw] {$\l_jon_tikz_diagram_nw$}; 219 | \node (#1sw) [below = \l_jon_tikz_diagram_height~of~#1nw,square/sw] {$\l_jon_tikz_diagram_sw$}; 220 | \draw[square/west] (#1nw) to node [square/west/node] {$\l_jon_tikz_diagram_west$} (#1sw); 221 | 222 | \node (#1ne) [right = \l_jon_tikz_diagram_width~of~#1nw,square/ne] {$\l_jon_tikz_diagram_ne$}; 223 | \node (#1se) [below = \l_jon_tikz_diagram_height~of~#1ne,square/se] {$\l_jon_tikz_diagram_se$}; 224 | \draw[square/north] (#1nw) to node [square/north/node] {$\l_jon_tikz_diagram_north$} (#1ne); 225 | \draw[square/east] (#1ne) to node [square/east/node] {$\l_jon_tikz_diagram_east$} (#1se); 226 | \draw[square/south] (#1sw) to node [square/south/node] {$\l_jon_tikz_diagram_south$} (#1se); 227 | } 228 | \group_end: 229 | } 230 | 231 | \NewDocumentCommand\SpliceDiagramSquare{D<>{}m}{ 232 | \__jon_tikz_render_square:nn {#1} {#2} 233 | } 234 | 235 | 236 | \NewDocumentCommand\DiagramSquare{D<>{}O{}m}{ 237 | \begin{tikzpicture}[diagram,#2,baseline=(#1sw.base)] 238 | \__jon_tikz_render_square:nn {#1} {#3} 239 | \end{tikzpicture} 240 | } 241 | 242 | \ExplSyntaxOff 243 | -------------------------------------------------------------------------------- /assets/style.css: -------------------------------------------------------------------------------- 1 | /* SPDX-License-Identifier: CC0-1.0 */ 2 | 3 | /* inria-sans-300 - latin_latin-ext */ 4 | @font-face { 5 | font-display: swap; 6 | /* Check https://developer.mozilla.org/en-US/docs/Web/CSS/@font-face/font-display for other options. */ 7 | font-family: 'Inria Sans'; 8 | font-style: normal; 9 | font-weight: 300; 10 | src: url('fonts/inria-sans-v14-latin_latin-ext-300.woff2') format('woff2'); 11 | /* Chrome 36+, Opera 23+, Firefox 39+, Safari 12+, iOS 10+ */ 12 | } 13 | 14 | /* inria-sans-300italic - latin_latin-ext */ 15 | @font-face { 16 | font-display: swap; 17 | /* Check https://developer.mozilla.org/en-US/docs/Web/CSS/@font-face/font-display for other options. */ 18 | font-family: 'Inria Sans'; 19 | font-style: italic; 20 | font-weight: 300; 21 | src: url('fonts/inria-sans-v14-latin_latin-ext-300italic.woff2') format('woff2'); 22 | /* Chrome 36+, Opera 23+, Firefox 39+, Safari 12+, iOS 10+ */ 23 | } 24 | 25 | /* inria-sans-regular - latin_latin-ext */ 26 | @font-face { 27 | font-display: swap; 28 | /* Check https://developer.mozilla.org/en-US/docs/Web/CSS/@font-face/font-display for other options. */ 29 | font-family: 'Inria Sans'; 30 | font-style: normal; 31 | font-weight: 400; 32 | src: url('fonts/inria-sans-v14-latin_latin-ext-regular.woff2') format('woff2'); 33 | /* Chrome 36+, Opera 23+, Firefox 39+, Safari 12+, iOS 10+ */ 34 | } 35 | 36 | /* inria-sans-italic - latin_latin-ext */ 37 | @font-face { 38 | font-display: swap; 39 | /* Check https://developer.mozilla.org/en-US/docs/Web/CSS/@font-face/font-display for other options. */ 40 | font-family: 'Inria Sans'; 41 | font-style: italic; 42 | font-weight: 400; 43 | src: url('fonts/inria-sans-v14-latin_latin-ext-italic.woff2') format('woff2'); 44 | /* Chrome 36+, Opera 23+, Firefox 39+, Safari 12+, iOS 10+ */ 45 | } 46 | 47 | /* inria-sans-700 - latin_latin-ext */ 48 | @font-face { 49 | font-display: swap; 50 | /* Check https://developer.mozilla.org/en-US/docs/Web/CSS/@font-face/font-display for other options. */ 51 | font-family: 'Inria Sans'; 52 | font-style: normal; 53 | font-weight: 700; 54 | src: url('fonts/inria-sans-v14-latin_latin-ext-700.woff2') format('woff2'); 55 | /* Chrome 36+, Opera 23+, Firefox 39+, Safari 12+, iOS 10+ */ 56 | } 57 | 58 | /* inria-sans-700italic - latin_latin-ext */ 59 | @font-face { 60 | font-display: swap; 61 | /* Check https://developer.mozilla.org/en-US/docs/Web/CSS/@font-face/font-display for other options. */ 62 | font-family: 'Inria Sans'; 63 | font-style: italic; 64 | font-weight: 700; 65 | src: url('fonts/inria-sans-v14-latin_latin-ext-700italic.woff2') format('woff2'); 66 | /* Chrome 36+, Opera 23+, Firefox 39+, Safari 12+, iOS 10+ */ 67 | } 68 | 69 | :root { 70 | --content-gap: 15px; 71 | --radius: 5px; 72 | } 73 | 74 | h1, 75 | h2, 76 | h3, 77 | h4, 78 | h5, 79 | h6 { 80 | line-height: 1.2; 81 | margin-bottom: 0; 82 | } 83 | 84 | h5, 85 | h6, 86 | p { 87 | margin-top: 0; 88 | } 89 | 90 | h1, 91 | h2, 92 | h3, 93 | h4 { 94 | margin-top: .5em; 95 | } 96 | 97 | pre, 98 | img, 99 | .katex-display, 100 | section, 101 | center { 102 | overflow-y: hidden; 103 | } 104 | 105 | pre { 106 | border-radius: var(--radius); 107 | background-color: rgba(0, 100, 100, 0.04); 108 | padding: .5em; 109 | font-size: 11pt; 110 | margin-top: 0em; 111 | overflow-x: auto; 112 | white-space: pre-wrap; 113 | white-space: -moz-pre-wrap; 114 | white-space: -pre-wrap; 115 | white-space: -o-pre-wrap; 116 | word-wrap: break-word; 117 | } 118 | 119 | code { 120 | border-radius: var(--radius); 121 | background-color: rgba(0, 100, 100, 0.04); 122 | padding: 0.2em; 123 | font-size: 11pt; 124 | } 125 | 126 | body { 127 | font-family: "Inria Sans"; 128 | font-size: 12pt; 129 | line-height: 1.5em; 130 | } 131 | 132 | .logo { 133 | font-weight: 1000; 134 | font-size: 24px; 135 | } 136 | 137 | .logo a { 138 | color: #666; 139 | text-decoration: none; 140 | } 141 | 142 | .logo a:hover { 143 | color: #aaa; 144 | } 145 | 146 | .block.hide-metadata>details>summary>header>.metadata { 147 | display: none; 148 | } 149 | 150 | article>section>details>summary>header>h1>.taxon { 151 | display: block; 152 | font-size: .9em; 153 | color: #888; 154 | padding-bottom: 5pt; 155 | } 156 | 157 | section section[data-taxon="Reference"]>details>summary>header>h1>.taxon, 158 | section section[data-taxon="Person"]>details>summary>header>h1>.taxon { 159 | display: none; 160 | } 161 | 162 | footer>section { 163 | margin-bottom: 1em; 164 | } 165 | 166 | footer h2 { 167 | font-size: 14pt; 168 | } 169 | 170 | .metadata>address { 171 | display: inline; 172 | } 173 | 174 | @media only screen and (max-width: 1000px) { 175 | body { 176 | margin-top: 1em; 177 | margin-left: .5em; 178 | margin-right: .5em; 179 | transition: ease all .2s; 180 | } 181 | 182 | #grid-wrapper>nav { 183 | display: none; 184 | transition: ease all .2s; 185 | } 186 | } 187 | 188 | @media only screen and (min-width: 1000px) { 189 | body { 190 | margin-top: 2em; 191 | margin-left: 2em; 192 | transition: ease all .2s; 193 | } 194 | 195 | #grid-wrapper { 196 | display: grid; 197 | grid-template-columns: 90ex; 198 | } 199 | } 200 | 201 | body>header { 202 | margin-bottom: 2em; 203 | } 204 | 205 | #grid-wrapper>article { 206 | max-width: 90ex; 207 | margin-right: auto; 208 | grid-column: 1; 209 | } 210 | 211 | #grid-wrapper>nav { 212 | grid-column: 2; 213 | } 214 | 215 | details>summary>header { 216 | display: inline; 217 | } 218 | 219 | a.heading-link { 220 | box-shadow: none; 221 | } 222 | 223 | details h1 { 224 | font-size: 13pt; 225 | display: inline; 226 | } 227 | 228 | section .block[data-taxon]>details>summary>header>h1 { 229 | font-size: 12pt; 230 | } 231 | 232 | span.taxon { 233 | color: #444; 234 | } 235 | 236 | 237 | .link-list>section>details>summary>header h1 { 238 | font-size: 12pt; 239 | } 240 | 241 | 242 | article>section>details>summary>header>h1 { 243 | font-size: 1.5em; 244 | } 245 | 246 | details>summary { 247 | list-style-type: none; 248 | } 249 | 250 | details>summary::marker, 251 | details>summary::-webkit-details-marker { 252 | display: none; 253 | } 254 | 255 | article>section>details>summary>header { 256 | display: block; 257 | margin-bottom: .5em; 258 | } 259 | 260 | section.block>details { 261 | margin-bottom: 0.4em; 262 | } 263 | 264 | 265 | section.block>details[open] { 266 | margin-bottom: 1em; 267 | } 268 | 269 | 270 | .link-list>section.block>details { 271 | margin-bottom: .25em; 272 | } 273 | 274 | nav#toc { 275 | margin-left: 1em; 276 | } 277 | 278 | nav#toc h1 { 279 | margin-top: 0; 280 | font-size: 16pt; 281 | } 282 | 283 | nav#toc, 284 | nav#toc a { 285 | color: #555; 286 | } 287 | 288 | nav#toc h2 { 289 | font-size: 1.1em; 290 | } 291 | 292 | nav#toc ul { 293 | padding-left: 1.5em; 294 | list-style-type: square; 295 | } 296 | 297 | .toc-item-label { 298 | font-weight: bolder; 299 | } 300 | 301 | .katex { 302 | font-size: 1.2em; 303 | } 304 | 305 | .block { 306 | border-radius: var(--radius) 307 | } 308 | 309 | .block.splice { 310 | padding-top: 1em; 311 | } 312 | 313 | .block:hover { 314 | background-color: rgba(0, 100, 255, 0.04); 315 | } 316 | 317 | .block.highlighted { 318 | border-style: solid; 319 | border-width: 1pt; 320 | } 321 | 322 | .highlighted { 323 | background-color: rgba(255, 255, 140, .3); 324 | border-color: #ccc; 325 | } 326 | 327 | .highlighted:hover { 328 | background-color: rgba(255, 255, 140, .6); 329 | border-color: #aaa; 330 | } 331 | 332 | .block>.block-uid { 333 | visibility: hidden; 334 | display: flex; 335 | justify-content: flex-end; 336 | } 337 | 338 | .block:hover>.block-uid { 339 | visibility: visible; 340 | } 341 | 342 | .slug, 343 | .doi, 344 | .orcid { 345 | color: gray; 346 | font-weight: 200; 347 | } 348 | 349 | .edit-button { 350 | color: rgb(180, 180, 180); 351 | font-weight: 200; 352 | } 353 | 354 | .error { 355 | color: rgb(255, 50, 50, .8); 356 | } 357 | 358 | .block { 359 | padding-left: 5px; 360 | padding-right: 10px; 361 | padding-bottom: 2px; 362 | border-radius: 5px; 363 | } 364 | 365 | .block>.toggle-children-button { 366 | padding-right: 3pt; 367 | visibility: hidden; 368 | float: right; 369 | } 370 | 371 | .block:hover>.toggle-children-button { 372 | visibility: visible; 373 | } 374 | 375 | a.external { 376 | text-decoration: underline; 377 | color:#1d4ed8; 378 | } 379 | 380 | a.toc { 381 | box-shadow: none; 382 | text-decoration: none; 383 | } 384 | 385 | a.local{ 386 | color:#047857; 387 | } 388 | 389 | a.local, 390 | a.slug { 391 | box-shadow: none; 392 | text-decoration: underline; 393 | text-decoration-style: dotted; 394 | } 395 | 396 | ninja-keys::part(ninja-action) { 397 | white-space: nowrap; 398 | } 399 | 400 | body { 401 | hyphens: auto; 402 | } 403 | 404 | table { 405 | margin-bottom: 1em; 406 | } 407 | 408 | table.macros { 409 | overflow-x: visible; 410 | overflow-y: visible; 411 | font-size: 0.9em; 412 | } 413 | 414 | table.macros td { 415 | padding-left: 5pt; 416 | padding-right: 15pt; 417 | vertical-align: baseline; 418 | } 419 | 420 | th { 421 | text-align: left; 422 | } 423 | 424 | th, 425 | td { 426 | padding: 0 15px; 427 | vertical-align: top; 428 | } 429 | 430 | td.macro-name, 431 | td.macro-body { 432 | white-space: nowrap; 433 | } 434 | 435 | td.macro-doc { 436 | font-size: .9em; 437 | } 438 | 439 | .enclosing.macro-scope>.enclosing { 440 | border-radius: 2px; 441 | } 442 | 443 | .enclosing.macro-scope>.enclosing:hover { 444 | background-color: rgba(0, 100, 255, 0.1); 445 | } 446 | 447 | [aria-label][role~="tooltip"]::after { 448 | font-family: "Inria Sans"; 449 | } 450 | 451 | .tooltip { 452 | position: relative; 453 | } 454 | 455 | .inline.tooltip { 456 | display: inline-block; 457 | } 458 | 459 | .display.tooltip { 460 | display: block; 461 | } 462 | 463 | 464 | /* The tooltip class is applied to the span element that is the tooltip */ 465 | 466 | .tooltip .tooltiptext { 467 | visibility: hidden; 468 | white-space: nowrap; 469 | min-width: fit-content; 470 | background-color: black; 471 | color: #fff; 472 | padding-left: 5px; 473 | padding-top: 5px; 474 | padding-right: 10px; 475 | border-radius: 6px; 476 | position: absolute; 477 | z-index: 1; 478 | top: 100%; 479 | left: 50%; 480 | margin-left: -60px; 481 | opacity: 0; 482 | transition: opacity 0.3s; 483 | } 484 | 485 | .tooltip .tooltiptext::after { 486 | content: ""; 487 | position: absolute; 488 | top: 100%; 489 | left: 50%; 490 | margin-left: -5px; 491 | border-width: 5px; 492 | } 493 | 494 | 495 | /* Show the tooltip */ 496 | 497 | .tooltip:hover .tooltiptext { 498 | visibility: visible; 499 | opacity: 1; 500 | } 501 | 502 | .tooltiptext a { 503 | color: white 504 | } 505 | 506 | .macro-doc { 507 | font-style: italic; 508 | } 509 | 510 | .macro-name { 511 | white-space: nowrap; 512 | } 513 | 514 | .macro-is-private { 515 | color: var(--secondary); 516 | } 517 | 518 | blockquote { 519 | border-inline-start: 1px solid var(--secondary); 520 | } 521 | 522 | a:hover { 523 | background-color: rgba(0, 100, 255, .1); 524 | } 525 | 526 | a { 527 | color: black; 528 | } 529 | 530 | .nowrap { 531 | white-space: nowrap; 532 | } 533 | 534 | .nocite { 535 | display: none 536 | } 537 | 538 | blockquote { 539 | font-style: italic; 540 | } 541 | 542 | 543 | 544 | address { 545 | display: inline; 546 | } 547 | 548 | 549 | .metadata ul { 550 | padding-left: 0; 551 | display: inline; 552 | } 553 | 554 | .metadata li::after { 555 | content: " · "; 556 | } 557 | 558 | .metadata li:last-child::after { 559 | content: ""; 560 | } 561 | 562 | .metadata ul li { 563 | display: inline 564 | } 565 | 566 | img { 567 | object-fit: cover; 568 | max-width: 100%; 569 | } 570 | 571 | figcaption { 572 | font-style: italic; 573 | padding: 3px; 574 | text-align: center; 575 | } 576 | 577 | mark { 578 | background-color: rgb(255, 255, 151); 579 | } 580 | 581 | hr { 582 | margin-top: 10px; 583 | margin-bottom: 20px; 584 | background-color: gainsboro; 585 | border: 0 none; 586 | width: 100%; 587 | height: 2px; 588 | } 589 | -------------------------------------------------------------------------------- /javascript/forester.js: -------------------------------------------------------------------------------- 1 | import 'ninja-keys'; 2 | import 'katex'; 3 | 4 | import autoRenderMath from 'katex/contrib/auto-render'; 5 | 6 | function partition(array, isValid) { 7 | return array.reduce(([pass, fail], elem) => { 8 | return isValid(elem) ? [[...pass, elem], fail] : [pass, [...fail, elem]]; 9 | }, [[], []]); 10 | } 11 | 12 | window.addEventListener("load", (event) => { 13 | autoRenderMath(document.body) 14 | 15 | const openAllDetailsAbove = elt => { 16 | while (elt != null) { 17 | if (elt.nodeName == 'DETAILS') { 18 | elt.open = true 19 | } 20 | 21 | elt = elt.parentNode; 22 | } 23 | } 24 | 25 | const openDetailsIfAnchorHidden = evt => { 26 | const link = evt.target.closest('a') 27 | const selector = link.getAttribute('href') 28 | const target = document.querySelector(selector) 29 | openAllDetailsAbove(target) 30 | } 31 | 32 | 33 | [...document.querySelectorAll("[href^='#']")].forEach( 34 | el => el.addEventListener("click", openDetailsIfAnchorHidden) 35 | ); 36 | }); 37 | 38 | const ninja = document.querySelector('ninja-keys'); 39 | 40 | fetch("./forest.json") 41 | .then((res) => res.json()) 42 | .then((data) => { 43 | const items = [] 44 | 45 | const editIcon = '' 46 | const bookmarkIcon = '' 47 | 48 | if (window.sourcePath) { 49 | items.push({ 50 | id: 'edit', 51 | title: 'Edit current tree in Visual Studio Code', 52 | section: 'Commands', 53 | hotkey: 'cmd+e', 54 | icon: editIcon, 55 | handler: () => { 56 | window.location.href = `vscode://file/${window.sourcePath}` 57 | } 58 | }) 59 | } 60 | 61 | const isTopTree = (addr) => { 62 | const item = data[addr] 63 | return item.tags ? item.tags.includes('top') : false 64 | } 65 | 66 | const addItemToSection = (addr, section, icon) => { 67 | const item = data[addr] 68 | const title = 69 | item.taxon 70 | ? (item.title ? `${item.taxon} (${item.title})` : item.taxon) 71 | : (item.title ? item.title : "Untitled") 72 | const fullTitle = `${title} [${addr}]` 73 | items.push({ 74 | id: addr, 75 | title: fullTitle, 76 | section: section, 77 | icon: icon, 78 | handler: () => { 79 | window.location.href = item.route 80 | } 81 | }) 82 | } 83 | 84 | const [top, rest] = partition(Object.keys(data), isTopTree) 85 | top.forEach((addr) => addItemToSection(addr, "Top Trees", bookmarkIcon)) 86 | rest.forEach((addr) => addItemToSection(addr, "All Trees", null)) 87 | 88 | ninja.data = items 89 | }); 90 | -------------------------------------------------------------------------------- /package-lock.json: -------------------------------------------------------------------------------- 1 | { 2 | "name": "functional-programming", 3 | "lockfileVersion": 3, 4 | "requires": true, 5 | "packages": { 6 | "": { 7 | "dependencies": { 8 | "esbuild": "^0.18.15", 9 | "katex": "^0.16.8", 10 | "ninja-keys": "^1.2.2" 11 | } 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\title{Magma} 2 | \taxon{definition} 3 | \date{2023-09-13} 4 | \author{fizzyelt} 5 | 6 | \p{[Magma](https://ncatlab.org/nlab/show/magma) 是代數結構的一種，由一個集合與運算子組成 #{(M,\ \bullet)}。定義為「對於在集合 #{M} 裡所有 #{a,\ b}，#{a\ \bullet \ b} 的結果也是集合 #{M} 裡的元素」} 7 | 8 | ##{ 9 | a,\ b \in M \Longrightarrow a\ \bullet\ b \in M 10 | } 11 | 12 | \transclude{alg-0002} 13 | -------------------------------------------------------------------------------- /trees/alg-0002.tree: -------------------------------------------------------------------------------- 1 | \title{不是 semigroup 的 magma} 2 | \taxon{example} 3 | \author{dannypsnl} 4 | \date{2023-09-13} 5 | 6 | \p{我們很容易造出一堆其實是 semigroup（滿足結合律）的 magma，然後騙自己說這樣就是 magma，但這無助於我們思考結合律的意義。刻意構造一個 magma 會發現這其實沒那麼容易，這裡給出一個是 magma 但不是 semigroup 的案例。} 7 | 8 | \ol{ 9 | \li{ 10 | #{M} 是整數有序對構成的集合 11 | } 12 | \li{ 13 | #{\bullet} 在輸入兩對的相接序對有序時直接回傳，在無序時回傳全部元素的總和為唯一元素的有序對 14 | } 15 | } 16 | 17 | \p{第二條彎彎繞繞的意思就是 #{[1] \bullet [2] = [1, 2]}，而 #{[2] \bullet [1] = [3]}。所以現在可以說結合律對運算 #{\bullet} 不成立，因為} 18 | 19 | \ul{ 20 | \li{#{[0] \bullet ([2] \bullet [1]) = [0,3]}} 21 | \li{#{([0] \bullet [2]) \bullet [1] = [3]}} 22 | } 23 | 24 | \p{也可以看到交換律不成立的例子。} 25 | -------------------------------------------------------------------------------- /trees/alg-0003.tree: -------------------------------------------------------------------------------- 1 | \title{Semigroup} 2 | \taxon{definition} 3 | \date{2023-09-13} 4 | \author{fizzyelt} 5 | 6 | \p{[Semigroup](https://ncatlab.org/nlab/show/semigroup) 是滿足結合律的 [Magma](alg-0001)。結合律是指對所有 semigroup 中的元素 #{x,y,z \in S}，皆滿足} 7 | 8 | ##{ 9 | x \bullet (y \bullet z) = (x \bullet y) \bullet z 10 | } 11 | 12 | \transclude{alg-0004} 13 | -------------------------------------------------------------------------------- /trees/alg-0004.tree: -------------------------------------------------------------------------------- 1 | \title{不滿足交換律的 semigroup} 2 | \taxon{example} 3 | \author{dannypsnl} 4 | \date{2023-09-13} 5 | 6 | \p{沿襲 [Magma 案例](alg-0002)的傳統，這裡不舉滿足交換律的 semigroup。在程式語言裡面已經有一個我們很常看到的案例了，也就是字串相接，我們可以著手檢查：} 7 | 8 | \ul{ 9 | \li{字串與字串相接還是字串} 10 | \li{字串先接 #{x, y} 或 #{y, z} 不影響結果} 11 | } 12 | -------------------------------------------------------------------------------- /trees/alg-0005.tree: -------------------------------------------------------------------------------- 1 | \title{Monoid} 2 | \taxon{definition} 3 | \date{2023-09-13} 4 | \author{fizzyelt} 5 | 6 | \p{[Monoid](https://en.wikipedia.org/wiki/Monoid) 是有 identity element 的 [semigroup](alg-0003)，其中 identity element #{e} 滿足} 7 | 8 | ##{ 9 | e \bullet a = a = a \bullet e 10 | } 11 | 12 | \p{如果把 id morphism 對應到 #{e}，而剩餘的 morphism 對應到每個 monoid 的元素上，那麼只有一個 object 的 [small category](cat-0001) 正好是一個 monoid。} 13 | 14 | \transclude{fp-000O} 15 | \transclude{cs-0001} 16 | -------------------------------------------------------------------------------- /trees/base-macros.tree: -------------------------------------------------------------------------------- 1 | \def\br{\xml{br}{}} 2 | 3 | \def\img[src]{\xml{img}[src]{\src}{}} 4 | 5 | \def\proof[body]{ 6 | \scope{ 7 | \put\transclude/toc{false} 8 | \subtree{ 9 | \taxon{proof} 10 | \body 11 | } 12 | } 13 | } 14 | -------------------------------------------------------------------------------- /trees/cat-0001.tree: -------------------------------------------------------------------------------- 1 | \title{直覺 Category} 2 | \date{2023-09-09} 3 | \taxon{definition} 4 | \author{dannypsnl} 5 | 6 | \def\CCat{#{\mathcal{C}}} 7 | 8 | \p{ 9 | 說直覺是因為我們這裡未有完善定義何謂 [class](set-0001) 跟 [set](set-0001)，但是作為入門或是簡要使用的定義「夠好」。 10 | 我們說 \CCat 是一個範疇的意思是： 11 | 12 | \ol{ 13 | \li{ 14 | 有 #{Ob(\CCat)} 物件 \strong{class}，表示範疇中的物件。作為方便的記號，當 #{a} 為 #{\CCat} 中一物件，我們記成 #{a \in Ob(\CCat)} 15 | } 16 | \li{ 17 | 當有 #{a, b \in Ob(\CCat)}，而 #{f} 為一 #{a} 到 #{b} 之態射(morphism) ，記為 #{f \in \CCat(a, b)} 或是 #{f : a \to b}。 18 | 態射可以組合，也就是說當有 #{f : a \to b} 且 #{g : b \to c}，存在一個兩個的組合態射 #{f \circ g}。 19 | 與上面相同，#{\CCat(a, b)} 是一個態射 \strong{class} 20 | } 21 | \li{ 22 | #{id_a\ id_b} 對 #{f:a\to b} 滿足 #{f\circ id_a = f} 跟 #{id_b\circ f = f} 23 | } 24 | \li{ 25 | #{(f\circ g)\circ h=f\circ (g\circ h)} 26 | } 27 | } 28 | 29 | 要是對每個 #{a,b} 之間的全部 morphism #{\CCat(a, b)} 是一個集合，就說範疇 locally small；要是整個範疇中所有的 morphism 是一個集合，那範疇就是 small。 30 | 深入技術細節會讓你更了解為什麼需要定義這麼多東西，但在最開始的時候，只要簡單的相信範疇給定的定義即可。 31 | } 32 | -------------------------------------------------------------------------------- /trees/cat-0002.tree: -------------------------------------------------------------------------------- 1 | \title{Functor} 2 | \taxon{definition} 3 | \date{2023-09-11} 4 | \author{dannypsnl} 5 | 6 | \def\cat1{#{\mathcal{C}}} 7 | \def\cat2{#{\mathcal{D}}} 8 | 9 | \p{ 10 | Let #{\cat1 \to \cat2} be [categories](cat-0001). To specify a \em{functor} #{F} from \cat1 to \cat2, denoted #{F : \cat1 \to \cat2} 11 | 12 | \ol{ 13 | \li{for every object #{c \in Ob(\cat1)}, one specifies an object #{F(c) \in Ob(\cat2)};} 14 | \li{for every morphism #{f : c_1 \to c_2} in \cat1, one specifies a morphism #{F(f) : F(c_1) \to F(c_2)} in \cat2.} 15 | } 16 | 17 | And they must satisfy two properties: 18 | 19 | \ol{ 20 | \li{ 21 | for every object #{c \in Ob(\cat1)}, we have #{F(id_c) = id_{F(c)}}; 22 | } 23 | \li{ 24 | for every three objects #{c_1, c_2, c_3 \in Ob(\cat1)} and two morphisms #{f \in \cat1(c_1, c_2)} and #{g \in \cat1(c_2, c_3)}, the equation #{F(f \circ g) = F(f) \circ F(g)} holds in \cat2. 25 | } 26 | } 27 | } 28 | -------------------------------------------------------------------------------- /trees/cat-0003.tree: -------------------------------------------------------------------------------- 1 | \title{Functor} 2 | \author{fizzyelt} 3 | \date{2023-09-11} 4 | 5 | \transclude{cat-0002} 6 | \transclude{fp-000H} 7 | \transclude{fp-000G} 8 | \transclude{fp-000I} 9 | -------------------------------------------------------------------------------- /trees/cat-0004.tree: -------------------------------------------------------------------------------- 1 | \title{cartesian closed category} 2 | \taxon{definition} 3 | \author{dannypsnl} 4 | \date{2023-10-24} 5 | 6 | \p{ 7 | 一個[範疇](cat-0001) #{C} 被稱為 cartesian closed 表示 8 | \ol{ 9 | \li{任意有限多個 objects #{\{a_i \mid i \in \mathbb{N}\}} 的 product #{a_0 \times ... \times a_n} 亦是範疇的 object} 10 | \li{對所有 object #{a \in C} 而言，functor #{a \times - : C \to C} 是 #{(-)^a : C \to C} 的[右伴隨](https://en.wikipedia.org/wiki/Adjoint_functors)} 11 | } 12 | } 13 | -------------------------------------------------------------------------------- /trees/cat-0005.tree: -------------------------------------------------------------------------------- 1 | \title{#{F}-algebra} 2 | \taxon{definition} 3 | \author{dannypsnl} 4 | \author{fizzyelt} 5 | 6 | \p{要理解 #{F}-algebra，我們需要一些觀察，第一個是代數可以表示成一組簽名。例如，Monoid 有一組簽名：} 7 | 8 | ##{\begin{cases} 9 | 1:1\to m \\ 10 | \bullet: m \times m \to m 11 | \end{cases}} 12 | 13 | \p{或者你可以將 環（Ring）寫成：} 14 | 15 | ##{\begin{cases} 16 | 0: 1 \to m \\ 17 | 1:1 \to m \\ 18 | +: m \times m \to m \\ 19 | \times: m \times m \to m \\ 20 | - : m \to m 21 | \end{cases}} 22 | 23 | \p{我們令 #{m} 為範疇 #{C} 中的一個 object，要選一個足以表示簽名的範疇，以上面的例子來說需要 [cartesian closed category](cat-0004)。} 24 | 25 | \ul{ 26 | \li{對 #{m} 來說，Monoid 是 #{C} 中的一個態射 #{1 + m \times m \to m}} 27 | \li{對 #{m} 來說，環是 #{C} 中的一個態射 #{1 + 1 + m \times m + m \times m + m \to m}} 28 | } 29 | 30 | \p{根據這些，我們把代數的定義推廣為：固定 #{F} 函子後，每個 #{F(m) \to m} 都是一個代數。} 31 | 32 | \transclude{cat-0006} 33 | \transclude{cat-0007} 34 | 35 | \p{這些是跟範疇中初始對象 (initial object) 有關的一個定理} 36 | 37 | \transclude{cat-0008} 38 | \transclude{cat-0009} 39 | \transclude{cat-000A} 40 | -------------------------------------------------------------------------------- /trees/cat-0006.tree: -------------------------------------------------------------------------------- 1 | \title{#{F}-algebra (algebra for an endofunctor)} 2 | \taxon{definition} 3 | \author{dannypsnl} 4 | \author{fizzyelt} 5 | 6 | 7 | \p{在一個適當的範疇 #{C} 中，我們把簽名改成一個 functor #{F}，這個函子投射出的 #{F(m) \to m} 是 #{F}-algebra，記為三元組：} 8 | 9 | ##{(F,x,\alpha)} 10 | 11 | \p{其中 #{\alpha : Fx \to x} 是一個 #{C} 中的態射。} 12 | -------------------------------------------------------------------------------- /trees/cat-0007.tree: -------------------------------------------------------------------------------- 1 | \title{Category of #{F}-algebra} 2 | \taxon{definition} 3 | \author{dannypsnl} 4 | \author{fizzyelt} 5 | 6 | \p{functor 與 natural transformation 構成 functor category} 7 | 8 | \p{對於在一個適當的範疇 #{C} 中固定的自函子 #{F} (以下省略 #{F} 符號)} 9 | 10 | \ol{ 11 | \li{object 是全部的 #{F}-algebra。} 12 | \li{態射皆為 #{(x,\alpha)\xrightarrow{Fm} (y,\beta)} 對象的同態，組合由函子定律給出。} 13 | } 14 | 15 | \p{同態是令下圖交換的態射 #{m}} 16 | 17 | \tex{\usepackage{preamble}}{ 18 | % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGeCJdLFsyLDAsIkZ5Il0sWzAsMiwieCJdLFsyLDIsInkiXSxbMCwxLCJGbSJdLFsyLDMsIm0iLDJdLFswLDIsIlxcYWxwaGEiLDJdLFsxLDMsIlxcYmV0YSJdXQ== 19 | \begin{tikzcd} 20 | Fx && Fy \\ 21 | \\ 22 | x && y 23 | \arrow["Fm", from=1-1, to=1-3] 24 | \arrow["m"', from=3-1, to=3-3] 25 | \arrow["\alpha"', from=1-1, to=3-1] 26 | \arrow["\beta", from=1-3, to=3-3] 27 | \end{tikzcd} 28 | } 29 | 30 | \p{額外檢查我們可以發現 identity 也如預期運作} 31 | 32 | \tex{\usepackage{preamble}}{ 33 | % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGeCJdLFsyLDAsIkZ4Il0sWzAsMiwieCJdLFsyLDIsIngiXSxbMCwxLCIxX3tGeH0iXSxbMiwzLCIxX3t4fSIsMl0sWzAsMiwiXFxhbHBoYSIsMl0sWzEsMywiXFxhbHBoYSJdXQ== 34 | \begin{tikzcd} 35 | Fx && Fx \\ 36 | \\ 37 | x && x 38 | \arrow["{1_{Fx}}", from=1-1, to=1-3] 39 | \arrow["{1_{x}}"', from=3-1, to=3-3] 40 | \arrow["\alpha"', from=1-1, to=3-1] 41 | \arrow["\alpha", from=1-3, to=3-3] 42 | \end{tikzcd} 43 | } 44 | 45 | -------------------------------------------------------------------------------- /trees/cat-0008.tree: -------------------------------------------------------------------------------- 1 | \title{Lambek's} 2 | \taxon{theorem} 3 | \author{dannypsnl} 4 | \author{fizzyelt} 5 | \import{base-macros} 6 | 7 | \p{初始代數是同構態射} 8 | 9 | \proof{ 10 | 讓 #{Fi} 成為 #{F}-algebra 範疇中的初始對象，對於 #{C} 中所有對象 #{a} 產生以下交換圖 11 | } 12 | 13 | \tex{\usepackage{preamble}}{ 14 | % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGaSJdLFsyLDAsIkZhIl0sWzAsMiwiaSJdLFsyLDIsImEiXSxbMCwxLCJGbSJdLFsyLDMsIm0iLDJdLFswLDIsImoiLDJdLFsxLDMsImYiXV0= 15 | \begin{tikzcd} 16 | Fi && Fa \\ 17 | \\ 18 | i && a 19 | \arrow["Fm", from=1-1, to=1-3] 20 | \arrow["m"', from=3-1, to=3-3] 21 | \arrow["j"', from=1-1, to=3-1] 22 | \arrow["f", from=1-3, to=3-3] 23 | \end{tikzcd} 24 | } 25 | 26 | \p{現在我們將 #{a} 替換成 #{Fi}} 27 | 28 | \tex{\usepackage{preamble}}{ 29 | % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGaSJdLFsyLDAsIkZcXCBGaSJdLFswLDIsImkiXSxbMiwyLCJGaSJdLFswLDEsIkZtIl0sWzIsMywibSIsMl0sWzAsMiwiaiIsMl0sWzEsMywiRmoiXV0= 30 | \begin{tikzcd} 31 | Fi && {F\ Fi} \\ 32 | \\ 33 | i && Fi 34 | \arrow["Fm", from=1-1, to=1-3] 35 | \arrow["m"', from=3-1, to=3-3] 36 | \arrow["j"', from=1-1, to=3-1] 37 | \arrow["Fj", from=1-3, to=3-3] 38 | \end{tikzcd} 39 | } 40 | 41 | \p{再來透過複製路徑 #{Fi \to i} 得出以下交換圖} 42 | 43 | \tex{\usepackage{preamble}}{ 44 | % https://q.uiver.app/#q=WzAsNSxbMCwwLCJGXFwgRmkiXSxbMiwwLCJGaSJdLFswLDIsIkZpIl0sWzIsMiwiaSJdLFswLDFdLFswLDEsIkZqIl0sWzIsMywiaiIsMl0sWzAsMiwiRmoiLDJdLFsxLDMsImoiXV0= 45 | \begin{tikzcd} 46 | {F\ Fi} && Fi \\ 47 | {} \\ 48 | Fi && i 49 | \arrow["Fj", from=1-1, to=1-3] 50 | \arrow["j"', from=3-1, to=3-3] 51 | \arrow["Fj"', from=1-1, to=3-1] 52 | \arrow["j", from=1-3, to=3-3] 53 | \end{tikzcd} 54 | } 55 | 56 | \p{將兩張交換圖合併} 57 | 58 | \tex{\usepackage{preamble}}{ 59 | % https://q.uiver.app/#q=WzAsNixbMiwwLCJGXFwgRmkiXSxbNCwwLCJGaSJdLFsyLDIsIkZpIl0sWzQsMiwiaSJdLFswLDAsIkZpIl0sWzAsMiwiaSJdLFswLDEsIkZqIl0sWzIsMywiaiIsMl0sWzAsMiwiRmoiLDJdLFsxLDMsImoiXSxbNCwwLCJGbSJdLFs1LDIsIm0iLDJdLFs0LDUsImoiLDJdXQ== 60 | \begin{tikzcd} 61 | Fi && {F\ Fi} && Fi \\ 62 | \\ 63 | i && Fi && i 64 | \arrow["Fj", from=1-3, to=1-5] 65 | \arrow["j"', from=3-3, to=3-5] 66 | \arrow["Fj"', from=1-3, to=3-3] 67 | \arrow["j", from=1-5, to=3-5] 68 | \arrow["Fm", from=1-1, to=1-3] 69 | \arrow["m"', from=3-1, to=3-3] 70 | \arrow["j"', from=1-1, to=3-1] 71 | \end{tikzcd} 72 | } 73 | 74 | \p{透過定義，我們現在知道 #{j \circ m} 是同態(homomorphism)，並且因為 #{Fi} 為初始對象我們得到} 75 | 76 | ##{Fj \circ Fm = 1_{Fi}} 77 | 78 | \p{所以，我們也得出} 79 | 80 | ##{j \circ m = 1_{i}} 81 | 82 | \p{因此 #{m} 是 #{j} 的 inverse，證明 #{j} 是同構。} 83 | -------------------------------------------------------------------------------- /trees/cat-0009.tree: -------------------------------------------------------------------------------- 1 | \title{Catamorphism} 2 | \author{dannypsnl} 3 | \author{fizzyelt} 4 | 5 | \p{根據 Lambek's 理論可以畫出交換圖} 6 | 7 | \tex{\usepackage{preamble}}{ 8 | % https://q.uiver.app/#q=WzAsNCxbMiwwLCJGYSJdLFsyLDIsImEiXSxbMCwwLCJGaSJdLFswLDIsImkiXSxbMCwxLCJhbGciXSxbMiwwLCJGbSJdLFszLDEsIm0iLDJdLFsyLDMsImoiLDJdXQ== 9 | \begin{tikzcd} 10 | Fi && Fa \\ 11 | \\ 12 | i && a 13 | \arrow["alg", from=1-3, to=3-3] 14 | \arrow["Fm", from=1-1, to=1-3] 15 | \arrow["m"', from=3-1, to=3-3] 16 | \arrow["{j^{-1}}"', from=1-1, to=3-1] 17 | \end{tikzcd} 18 | } 19 | 20 | \p{在 Haskell 中，我們可以這樣定義初始代數} 21 | 22 | \subtree{ 23 | \pre{\startverb% 24 | newtype Fix f = Fix (f (Fix f)) 25 | 26 | unFix :: Fix f -> f (Fix f) 27 | unFix (Fix x) = x 28 | \stopverb}} 29 | 30 | \p{將 #{j} 視為 \code{Fix} 建構子，#{j^{-1}} 為 \code{unFix}，再來我們可以定義 \code{m = alg . fmap m . unFix}。因為 \code{m :: Fix f -> a}，我們有了 Catamorphism 的程式定義} 31 | 32 | \subtree{ 33 | \pre{\startverb% 34 | cata :: Functor f => (f a -> a) -> Fix f -> a 35 | cata alg = alg . fmap (cata alg) . unFix 36 | \stopverb}} 37 | 38 | \p{\code{foldr} 是一個常見的例子，它是 Catamorphism 的一種便利版本。} 39 | -------------------------------------------------------------------------------- /trees/cat-000A.tree: -------------------------------------------------------------------------------- 1 | \title{Anamorphism} 2 | \author{dannypsnl} 3 | \author{fizzyelt} 4 | 5 | \p{作為一個對偶概念，可以畫出一個 coalgebra 的圖，我們稱這種關係為 anamorphism。} 6 | 7 | \tex{\usepackage{preamble}}{ 8 | % https://q.uiver.app/#q=WzAsNCxbMiwwLCJGXFwgYSJdLFsyLDIsImEiXSxbMCwwLCJGXFwgdCJdLFswLDIsInQiXSxbMSwzLCJtIl0sWzEsMCwiZiIsMl0sWzAsMiwiRm0iLDJdLFszLDIsInUiXV0= 9 | \begin{tikzcd} 10 | {F\ t} && {F\ a} \\ 11 | \\ 12 | t && a 13 | \arrow["m", from=3-3, to=3-1] 14 | \arrow["f"', from=3-3, to=1-3] 15 | \arrow["Fm"', from=1-3, to=1-1] 16 | \arrow["u", from=3-1, to=1-1] 17 | \end{tikzcd} 18 | } 19 | -------------------------------------------------------------------------------- /trees/cs-0001.tree: -------------------------------------------------------------------------------- 1 | \title{半自動機} 2 | \taxon{example} 3 | \author{fizzyelt} 4 | \date{2023-09-13} 5 | 6 | \p{對給定的半自動機可以討論其 [characteristic monoid](https://planetmath.org/CharacteristicMonoid)。一個半自動機是個三元組 #{(Q, \Sigma, T)}} 7 | 8 | \ul{ 9 | \li{#{Q} 是一個非空的「狀態集合」} 10 | \li{#{\Sigma} 是一個「輸入字母表」的非空集合} 11 | \li{#{T} 是轉移函數} 12 | } 13 | 14 | \p{假設一個 #{M(Q, \Sigma, T)} 集合} 15 | 16 | ##{ 17 | M(Q,\ \Sigma,\ T) = \{ T_w\ |\ w\in \Sigma^* \} 18 | } 19 | 20 | \p{集合 #{M(Q,\ \Sigma,\ T)} 在\strong{函數複合}下閉合；就是說，對於所有 #{v,w \in \Sigma^*} ，有 #{T_w \circ T_v = T_{vw}} 。它還包含 #{T_\epsilon} ，而這個 #{T_\epsilon} 是個\strong{恆等函數(即 identity function)}。 } 21 | -------------------------------------------------------------------------------- /trees/fp-0001.tree: -------------------------------------------------------------------------------- 1 | \title{帶你入門 functional programming} 2 | \meta{author}{false} 3 | 4 | \p{什麼是函數式程式設計是一個大哉問，但作為一個入門導引，我們簡單的定義成受 lambda calculus 啟發，並且格外注重可組合性的語言。在其上有兩種變體} 5 | 6 | \ol{ 7 | \li{一種現在相當著重在對形式計算時機的控制，也就是 macro 上，也就是 Lisp 與其延伸語言} 8 | \li{一種則注重對數學概念的引進，乃自裡面有一種叫做代數類型的型別定義方法，其基本出發自 F-algebra，但其內涵相當深遠，並不止於此，例如定理證明器往往只使用其中的 inductives 這一類變體。} 9 | } 10 | 11 | \p{作為入門網站，這裡著重在第二類語言，因為 macro 與這類語言並沒有顯著衝突，並且也非入門概念，引進理論討論 macro 的運算控制也沒有問題，因此沒有理由特別用入門時的觀點分類。那麼，第二類語言裡面，同樣有許多著重不同的變體} 12 | 13 | \ol{ 14 | \li{OCaml 著重類型正確，卻不在意 side effect} 15 | \li{Haskell 注重類型也注重 side effect 控制，同時對 type level 有一定的操作能力} 16 | \li{Agda, Coq, Lean 等語言作為寫程式的工具可能有點失格，但主要是因為目標完全不同，分類作語言的子類別：定理證明器 比較合適} 17 | } 18 | 19 | \p{這個網站的準則，是教導抽象的數學概念，讓學習者最終對實際 functional programming 有一定的了解，並大致理解現代語義學研究，或許能夠應用其成果。那麼，我們開始回顧 lambda calculus 與可計算性的沿革，開始學習 functional programming 吧} 20 | 21 | \transclude{roadmap} 22 | -------------------------------------------------------------------------------- /trees/fp-0002.tree: -------------------------------------------------------------------------------- 1 | \title{Data first or Data last?} 2 | \author{fizzyelt} 3 | \date{2023-09-09} 4 | 5 | \p{ 6 | data first 跟 data last 區別在於主要被操作的資料是擺在最前面或最後面，而 functional programming 習慣 data last 的方式設計函數，目的是提高函數的可組合性。 7 | } 8 | \transclude{fp-0003} 9 | \transclude{fp-0004} 10 | \transclude{fp-0005} 11 | \transclude{fp-0006} 12 | \transclude{fp-0008} 13 | \transclude{fp-000B} 14 | -------------------------------------------------------------------------------- /trees/fp-0003.tree: -------------------------------------------------------------------------------- 1 | \title{一般函數的設計} 2 | 3 | \p{ 4 | 在一般情況下我們撰寫一個函數習慣將資料擺在最前面，其他參數擺後面。假設今天我們設計一個 Array \code{map} 函數： 5 | } 6 | 7 | \pre{\startverb% 8 | function map(data, callback){ 9 | let res = []; 10 | 11 | for(const item of data){ 12 | res.push(callback(item)) 13 | } 14 | 15 | return res 16 | } 17 | \stopverb} 18 | 19 | \p{我們可以利用 \code{map} 它再做出新的函數 \code{double}：} 20 | 21 | \pre{\startverb% 22 | function double(data) { 23 | return map(data, (num) => num * 2); 24 | } 25 | \stopverb} 26 | 27 | \p{但你慢慢就會發現你每利用 \code{map} 做一個新的函數出來，你都要重複做一次傳遞 \code{data} 的動作。} 28 | -------------------------------------------------------------------------------- /trees/fp-0004.tree: -------------------------------------------------------------------------------- 1 | \title{data last} 2 | 3 | \p{今天我們轉成 data last 形式來設計 \code{map}：} 4 | 5 | \pre{\startverb% 6 | function map(callback, data) { 7 | let res = []; 8 | 9 | for (const item of data) { 10 | res.push(callback(item)); 11 | } 12 | 13 | return res; 14 | } 15 | \stopverb} 16 | 17 | \p{我們再嘗試做一個 \code{double}：} 18 | 19 | \pre{\startverb% 20 | function double(data) { 21 | return map((num) => num * 2, data); 22 | } 23 | \stopverb} 24 | 25 | \p{你會發現你還是要一直傳遞 \code{data} ，這到底有什麼差，但我們忘了一件事情是 FP 通常會將函數做\strong{柯里化(curry)}。} 26 | -------------------------------------------------------------------------------- /trees/fp-0005.tree: -------------------------------------------------------------------------------- 1 | \title{data last 搭配柯里化} 2 | 3 | \p{ 4 | 我們將 \code{map} 函數做柯里化 5 | } 6 | 7 | \pre{\startverb% 8 | function map(callback) { 9 | return function (data) { 10 | let res = []; 11 | 12 | for (const item of data) { 13 | res.push(callback(item)); 14 | } 15 | 16 | return res; 17 | }; 18 | } 19 | \stopverb} 20 | 21 | \p{現在做一個 double 函數只需要傳 callback 即可} 22 | 23 | \pre{\startverb% 24 | map((num) => num * 2); 25 | \stopverb} 26 | -------------------------------------------------------------------------------- /trees/fp-0006.tree: -------------------------------------------------------------------------------- 1 | \title{利於函式組合} 2 | 3 | \p{一般我們做函式組合通常是 #{f} 函數的輸出對應到 #{g} 函數的輸入。} 4 | 5 | ##{ 6 | f = a \to b,\ g = b \to c 7 | } 8 | 9 | \p{ 10 | 但現實是並不是每個函數的輸入參數都只有一個，我們必須先給一些參數後才能再跟其他函數接在一起。 11 | 而一般情況下組合會是長這樣： 12 | } 13 | 14 | \pre{\startverb% 15 | function fn(data) { 16 | return map( 17 | map(data, (num) => num * 2), 18 | (num) => num + 1 19 | ); 20 | } 21 | \stopverb} 22 | 23 | \p{你每組一個新的函數出來你都得做一次傳遞資料的動作。} 24 | 25 | \transclude{fp-0007} 26 | -------------------------------------------------------------------------------- /trees/fp-0007.tree: -------------------------------------------------------------------------------- 1 | \title{利用 compose} 2 | 3 | \p{在函數 \strong{data last 搭配柯里化} 的情況下用 \code{compose} 來組合函式會乾淨許多。} 4 | 5 | \pre{\startverb% 6 | const fn = compose( 7 | map((num) => num + 1), 8 | map((num) => num * 2) 9 | ); 10 | \stopverb} 11 | -------------------------------------------------------------------------------- /trees/fp-0008.tree: -------------------------------------------------------------------------------- 1 | \title{data first 較優雅的作法} 2 | 3 | \p{如果不想像一開始那樣一直傳遞 data，但還是想走 data first 形式，有兩種} 4 | 5 | \transclude{fp-0009} 6 | \transclude{fp-000A} 7 | -------------------------------------------------------------------------------- /trees/fp-0009.tree: -------------------------------------------------------------------------------- 1 | \title{chaining} 2 | 3 | \p{一種是就是用我們一般 Array methods 的形式串接下去} 4 | 5 | \pre{\startverb% 6 | data.map((num) => num * 2).map((num) => num + 1); 7 | \stopverb} 8 | 9 | \p{可以從 [lodash](https://lodash.com/docs/#chain) 找到這種設計方式，但伴隨的缺點是跟其他函式庫沒辦法配合的這麼好。} 10 | -------------------------------------------------------------------------------- /trees/fp-000A.tree: -------------------------------------------------------------------------------- 1 | \title{略過參數的語法糖} 2 | 3 | \p{在 Scala 中可以利用 \code{_} 來暫時略過參數：} 4 | 5 | \pre{\startverb% 6 | List(1, 2, 3, 4).map(_ * 2) 7 | \stopverb} 8 | 9 | \pre{\startverb% 10 | def multiplier(a: Int, b: Int): Int = a * b 11 | 12 | val fourTimes = multiplier(_: Int, 4) 13 | 14 | fourTimes(2) // 8 15 | \stopverb} 16 | 17 | \p{ 18 | 我們在 \code{ramda} 中會看到一個叫做 [placeholder function](https://ramdajs.com/docs/#__) 來模擬類似 Scala 的效果，它可以保留位置來安插後續進來的參數。 19 | 20 | 假設我們利用 data first 方式設計 \code{map} 並做柯里化，使用 \code{__} 函數就會是以下這樣： 21 | } 22 | 23 | \pre{\startverb% 24 | const fn = compose( 25 | map(R.__, (num) => num + 1), 26 | map(R.__, (num) => num * 2) 27 | ); 28 | \stopverb} 29 | -------------------------------------------------------------------------------- /trees/fp-000B.tree: -------------------------------------------------------------------------------- 1 | \title{參數的順序會影響結果的函數} 2 | 3 | \p{有些函數的參數資料格式相同，並且擺放順序不同會影響執行結果。} 4 | 5 | \ul{ 6 | \li{\code{<} lessThan} 7 | \li{\code{<=} lessThanEqual} 8 | \li{\code{>} greaterThan} 9 | \li{\code{>=} greaterThanEqual} 10 | \li{\code{concat}} 11 | } 12 | 13 | \p{ 14 | 如果是一般的比大小，我們反轉一下邏輯就可以解決(\code{<} 改成 \code{>=})，但 \code{concat} 就不行 15 | 假設我們的需求是將主流程的資訊接在 \code{arr} 的前面，但一般作法似乎會有錯誤： 16 | } 17 | 18 | \pre{\startverb% 19 | flow( 20 | // ... 21 | concat(arr) // 前一個的結果會被接在 arr 後面 22 | ); 23 | \stopverb} 24 | 25 | \transclude{fp-000C} 26 | \transclude{fp-000D} 27 | \transclude{fp-000E} 28 | -------------------------------------------------------------------------------- /trees/fp-000C.tree: -------------------------------------------------------------------------------- 1 | \title{重新包裝} 2 | 3 | \p{一種辦法是我們再包裝一個函數，重新將 data 擺在正確位置} 4 | 5 | \pre{\startverb% 6 | flow( 7 | // ... 8 | (data) => concat(data, arr) // 多包一層來擺放正確位置 9 | ); 10 | \stopverb} 11 | -------------------------------------------------------------------------------- /trees/fp-000D.tree: -------------------------------------------------------------------------------- 1 | \title{flip} 2 | 3 | \p{flip 函數可以將你的函數的前兩個參數對調} 4 | 5 | \pre{\startverb% 6 | flow( 7 | // ... 8 | flip(concat)(arr) // 將參數對調 9 | ); 10 | \stopverb} 11 | 12 | [Ramda \code{flip}](https://ramdajs.com/docs/#flip) 13 | -------------------------------------------------------------------------------- /trees/fp-000E.tree: -------------------------------------------------------------------------------- 1 | \title{placeholder} 2 | 3 | \p{也可以利用 placeholder function 來消除不必要的包裝跟邏輯反轉。} 4 | 5 | \pre{\startverb% 6 | flow( 7 | // ... 8 | R.concat(R.__, arr) // 保留第一個位置給後續進來的資料 9 | ); 10 | \stopverb} 11 | -------------------------------------------------------------------------------- /trees/fp-000F.tree: -------------------------------------------------------------------------------- 1 | \title{透過單元測試難易度辨別函式純度} 2 | \author{fizzyelt} 3 | \date{2023-10-15} 4 | \import{base-macros} 5 | 6 | \p{身為一個初學者該如何知道自己的函數有多純？其實在進行單元測試時可以很容易觀察到，以下會有兩點可以注意：} 7 | 8 | \ul{ 9 | \li{是否依賴外部資源及變數導致你需要使用 mock data？ 10 | \br你必須創造一些虛假的上下文才能使你的函數正常運作，代表這函數過度依賴環境，在遷移上相對困難許多。} 11 | 12 | \li{你的函數執行完是否會導致外部變數被修改？ 13 | \br函數應該盡可能減少對外部影響，否則在追蹤錯誤時會變得越來越困難。} 14 | } 15 | 16 | \p{除了函數本身的行為必須正確之外，一個函數無論遷移到何種新環境時都不會改變其行為，也不會對該環境產生任何影響，這個函數才稱得上足夠純。然而，在現實程式中，很難實現所有函數都是純函數的理想情況。我們能做的是盡量減少非純函數，以降低維護成本。} 17 | 18 | \p{透過單元測試，不僅可以檢驗你的函數是否如預期執行，還可以讓你重新審視該函數的定位，並且培養撰寫純函數的直覺。} 19 | -------------------------------------------------------------------------------- /trees/fp-000G.tree: -------------------------------------------------------------------------------- 1 | \title{實際上的用途} 2 | \author{fizzyelt} 3 | \author{dannypsnl} 4 | 5 | \p{ 6 | 可以將一個 pure function 應用到容器型別之中。 7 | 單純從容器的角度來看，就像是隱藏了「取出 → 運算 → 放回去」的過程，而根據每個容器用途不同在過程中執行不同操作及判斷。 8 | } 9 | 10 | \tex{\usepackage{preamble}}{ 11 | % http://quiver.localhost/?q=WzAsNCxbMCwxLCJGXFw7YSJdLFszLDEsIkZcXDtiIl0sWzEsMCwiYSJdLFsyLDAsImIiXSxbMCwxXSxbMiwzXSxbNSw0LCJcXGJvbGR7bGlmdH0iLDAseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9fV1d 12 | \begin{tikzcd} 13 | & a & b \\ 14 | {F\;a} &&& {F\;b} 15 | \arrow[""{name=0, anchor=center, inner sep=0}, from=2-1, to=2-4] 16 | \arrow[""{name=1, anchor=center, inner sep=0}, from=1-2, to=1-3] 17 | \arrow["{\bold{lift}}", shorten <=4pt, shorten >=4pt, Rightarrow, from=1, to=0] 18 | \end{tikzcd} 19 | } 20 | 21 | \p{ 22 | 雖然 polymorphism 的容器解釋非常方便，卻是眾多誤解的根源之一。實際上不能做這麼簡單的解釋，舉例來說我可以定義 #{F(-) = (c \to -)}，於是 #{F(a) \to F(b)} 視同 #{(c \to a) \to (c \to b)}，令 #{g : c \to a, f : a \to b}，#{\lambda c. \lambda g. f(g(c))} 正是一個 functor。 23 | } 24 | -------------------------------------------------------------------------------- /trees/fp-000H.tree: -------------------------------------------------------------------------------- 1 | \title{FP 的 Functor 與範疇論的 Functor 之間的關聯} 2 | \author{fizzyelt} 3 | \author{dannypsnl} 4 | 5 | \p{ 6 | 即便我們看了[範疇論](cat-0002)跟 FP 語言中的 Functor 定義，但在概念的銜接上仍然有困難，如果你看了 [Functor(functional programming) wiki](https://en.wikipedia.org/wiki/Functor_(functional_programming)) 會發現他是受範疇論啟發的 design pattern。而到底引用了哪些概念，以及與語言能做到什麼程度，並沒有很好的說明。 7 | } 8 | 9 | \p{ 10 | 如果以「Functor 是兩個範疇之間的映射」這個觀點來看程式中的 Functor 似乎有些奇怪，通常我們看到的範例都是舉兩個不同範疇 #{C,D}，如果某一程式語言是一個範疇 #{C} 的話，那範疇 #{D} 就會是另一個語言，雖然我們常常這樣思考，但是我們定義程式裡面的 Functor 時不想弄得太複雜。所以這裡指的都是「 #{C} 與 #{C} 之間的映射」，也就是 \strong{Endofunctor} 。你可以在 [Haskell wiki](https://wiki.haskell.org/Typeclassopedia) 以及 fp-ts 作者所寫的 [Introduction to Functional Programming](https://github.com/enricopolanski/functional-programming#a-boundary-that-leads-to-functors) 中找到相關說明。 11 | } 12 | 13 | 14 | \blockquote{ 15 | \p{Technically, these laws make f and fmap together an endofunctor on Hask, the category of Haskell types (ignoring [⊥](https://wiki.haskell.org/Bottom), which is a party pooper). See [Wikibook: Category theory](https://en.wikibooks.org/wiki/Haskell/Category_theory).} 16 | } 17 | 18 | \blockquote{ 19 | \p{ 20 | Even though a map between two different programming languages is a fascinating idea, we're more interested in a map where C and D are the same (the TS category). In that case we're talking about \strong{endofunctors} (from the greek "endo" meaning "inside", "internal"). 21 | } 22 | } 23 | 24 | \p{ 25 | 範疇論跟程式語言定義的 Functor 不是完全一樣的概念，因為大部分的語言並不可能寫出[命題](https://ncatlab.org/nlab/show/type+of+propositions)，所以範疇論的限制語言裡不一定有加上，這之間的一些漏洞就會產生對使用者來說可能很奇怪的結果，我們能做的就是將兩者之間重疊的部份做連結。而 Haskell 是透過 [Typeclass](https://wiki.haskell.org/Typeclassopedia) 來做約束，所以在程式語言的語境下所說的 Functor 是指 type class 的定義。另外，我們常說「Maybe 是一個 Functor」 指的是 Maybe 是 Functor class 的一個實例 (Maybe 可替換任何其他 Functor class 實例)。 26 | } 27 | 28 | \p{ 29 | 其他 FP 語言雖然不一定都使用 Typeclass 這個詞，但大多 Functor 的定義都差不多。 30 | } 31 | 32 | \subtree{ 33 | \title{haskell} 34 | \pre{\startverb% 35 | class Functor f where 36 | fmap :: (a -> b) -> f a -> f b 37 | \stopverb} 38 | } 39 | 40 | \subtree{ 41 | \title{haskell} 42 | \pre{\startverb% 43 | trait Functor[F[_]] { 44 | def map[A,B](a: F[A])(f: A => B): F[B] 45 | } 46 | \stopverb} 47 | } 48 | 49 | \subtree{ 50 | \title{lean} 51 | \pre{\startverb% 52 | class Functor (f : Type u → Type v) : Type (max (u+1) v) where 53 | map : {α β : Type u} → (α → β) → f α → f β 54 | \stopverb} 55 | } 56 | 57 | \transclude{fp-000P} 58 | -------------------------------------------------------------------------------- /trees/fp-000I.tree: -------------------------------------------------------------------------------- 1 | \title{常見的 Functor Type} 2 | \author{fizzyelt} 3 | 4 | \ul{ 5 | \li{ 6 | Maybe / Option 7 | \ul{ 8 | \li{Haskell (Maybe)} 9 | \li{Lean (Option)} 10 | \li{Rust (Option)} 11 | } 12 | } 13 | \li{ 14 | Either / Result / Except 15 | \ul{ 16 | \li{Haskell (Either)} 17 | \li{Result (Rust)} 18 | \li{Lean (Except)} 19 | } 20 | } 21 | \li{ 22 | IO 23 | \ul{ 24 | \li{Haskell (IO)} 25 | \li{Lean (IO)} 26 | } 27 | } 28 | 29 | \li{ 30 | Reader 31 | 待補 32 | } 33 | \li{ 34 | Writer 35 | 待補 36 | } 37 | \li{ 38 | State 39 | 待補 40 | } 41 | 42 | } 43 | -------------------------------------------------------------------------------- /trees/fp-000J.tree: -------------------------------------------------------------------------------- 1 | \title{純函數 (Pure function)} 2 | \taxon{definition} 3 | \author{fizzyelt} 4 | \author{dannypsnl} 5 | 6 | \p{純函數的意義是我們想讓編譯器可以做公共子常式提取跟幾種循環等最佳化而試圖去探尋的某一類函數，進而是一種有效減輕開發上的心智負擔，是能夠被靈活組合的一類函數。在程式中你可以利用函數做許多事情，不只是單純的運算，還包含了各種值的修改及各種非同步操作，雖然方便但也造成了閱讀及維護上的困難。} 7 | 8 | \p{由於上述的最佳化都涉及到複製或是共用函數呼叫，因此可以導出這些函數在系統下的任何情形中對相同輸入都能給出相同的輸出。這個部分讓我們推導出一些性質，比如這類函數絕對不跟其他函數共用任何儲存機制，而是只用參數跟回傳處理讀寫的概念。再來函數不應該有副作用，否則增加或是減少函數呼叫都會導致不同的副作用結果。} 9 | 10 | \p{因為程式語言裡面必須面對無法判定任意計算是否停機的關係，所以建模型的時候都會留下一個對應到 \strong{undefined} (#{\bot}) 的位置，這樣的概念叫做 partial function，跟數學上的[全函數](fp-000K)有所差異。} 11 | 12 | \transclude{fp-000F} 13 | -------------------------------------------------------------------------------- /trees/fp-000K.tree: -------------------------------------------------------------------------------- 1 | \title{函數} 2 | \taxon{definition} 3 | \author{fizzyelt} 4 | \author{dannypsnl} 5 | 6 | \p{在數學中，函數是一個集合 (稱為\strong{定義域 domain}) 到另一個集合 (稱為\strong{值域 codomain}) 的映射，假設我們有個函數 #{f:A \to B}，則 #{A} (domain) 中每個元素都必須對應到 #{B}(codomain) 中的一個元素，能達成 #{A \to B} 條件的函數總共有 #{|B|^{|A|}} 種，我們能實現的是有限的語法能實現的一小部分，而不是所有都實現。} 7 | 8 | \p{Typescript} 9 | 10 | \pre{ 11 | // isEven : number -> boolean 12 | function isEven(num: number): boolean { 13 | return num \% 2 === 0 14 | } 15 | } 16 | 17 | \p{Haskell} 18 | 19 | \pre{\startverb% 20 | is_even :: Int -> Bool 21 | is_even n = n `mod` 2 == 0 22 | \stopverb} 23 | 24 | \p{函數分三種類型，包含 \strong{單射 (injection)}、\strong{滿射 (surjection)}、\strong{對射 (bijection)}。} 25 | 26 | \transclude{fp-000L} 27 | \transclude{fp-000M} 28 | \transclude{fp-000N} 29 | -------------------------------------------------------------------------------- /trees/fp-000L.tree: -------------------------------------------------------------------------------- 1 | \title{單射 (injection)} 2 | 3 | \p{假設我們稱一個函數 #{f:A \to B} 是單射，則 #{A} 與 #{B} 的元素關係都是一對一的，即} 4 | 5 | ##{\forall a,b\in A,\ a\neq b \Longrightarrow f(a)\neq f(b)} 6 | 7 | \xml{figure}{ 8 | \xml{img}[src]{images/fp-000L/injection.png}{} 9 | \xml{figcaption}{injection} 10 | } 11 | 12 | \p{我們可以從中得知另一個訊息是 #{A} 中的元素數量會小於等於 #{B} 元素數量，即 #{|A|\le |B|}} 13 | -------------------------------------------------------------------------------- /trees/fp-000M.tree: -------------------------------------------------------------------------------- 1 | \title{滿射 (surjection)} 2 | 3 | \p{我們稱一個函數 #{f : A \to B} 是滿射，則對於 #{B} 中所有的元素，都存在至少一個 #{A} 元素與其對應，即} 4 | 5 | ##{\forall b \in B,\ \exist a \in A,\ f(a) = b} 6 | 7 | \xml{figure}{ 8 | \xml{img}[src]{images/fp-000M/surjection.png}{} 9 | \xml{figcaption}{surjection} 10 | } 11 | 12 | \p{我們可以從中得知另一個訊息是 #{A} 中的元素數量會大於等於 #{B} 元素數量，即 #{|A| \ge |B|}} 13 | -------------------------------------------------------------------------------- /trees/fp-000N.tree: -------------------------------------------------------------------------------- 1 | \title{對射 (bijection)} 2 | 3 | \p{我們稱一個函數 #{f : A \to B} 是對射，則該函數同時達成單射與滿射條件，因為同時滿足 #{|A|\le |B|} 跟 #{|A|\ge |B|} 則 #{|A| = |B|}} 4 | 5 | \xml{figure}{ 6 | \xml{img}[src]{images/fp-000N/bijection.png}{} 7 | \xml{figcaption}{bijection} 8 | } 9 | -------------------------------------------------------------------------------- /trees/fp-000O.tree: -------------------------------------------------------------------------------- 1 | \title{函數組合} 2 | \taxon{example} 3 | \author{fizzyelt} 4 | \author{dannypsnl} 5 | \date{2023-09-13} 6 | 7 | \p{在禁止 free variable 的 lambda calculus 中，可以將 monoid 用作模型} 8 | 9 | \ul{ 10 | \li{#{e} 就是 #{\lambda x. x}} 11 | \li{函數 #{f, g} 的組合 #{f \circ g} 是 #{\lambda x. f (g \; x)}} 12 | } 13 | 14 | \p{identity law 的檢查比較容易，這裡我們就來看結合律如何運作：} 15 | 16 | ##{\begin{aligned} 17 | &f \circ (g \circ h) 18 | \\ 19 | &= \lambda x. f ((g \circ h) \; x) 20 | \\ 21 | &= \lambda x. f ((\lambda x. g (h \; x)) \; x) 22 | \\ 23 | &= \lambda x. f (g (h \; x)) 24 | \\ 25 | &(f \circ g) \circ h 26 | \\ 27 | &= \lambda x. (f \circ g)(h \; x) 28 | \\ 29 | &= \lambda x. (\lambda x. f (g \; x)) (h \; x) 30 | \\ 31 | &= \lambda x. f (g (h \; x)) 32 | \end{aligned}} 33 | 34 | \p{確實滿足要求。} 35 | -------------------------------------------------------------------------------- /trees/fp-000P.tree: -------------------------------------------------------------------------------- 1 | \title{Functor Laws} 2 | \author{fizzyelt} 3 | 4 | \p{ 5 | Functor class 的實例不能視為範疇論中的 Functor，必須檢查 Functor 的約束才會是真正的 Functor。 6 | } 7 | 8 | ##{ 9 | fmap\ id = id 10 | \\ 11 | fmap\ (g\circ f) = fmap\ g \circ fmap\ f 12 | } 13 | -------------------------------------------------------------------------------- /trees/people/dannypsnl.tree: -------------------------------------------------------------------------------- 1 | \title{Lîm Tsú-thuàn} 2 | \taxon{person} 3 | \meta{position}{Compiler Developer} 4 | \meta{external}{https://dannypsnl.me} 5 | -------------------------------------------------------------------------------- /trees/people/fizzyelt.tree: -------------------------------------------------------------------------------- 1 | \title{FizzyElt} 2 | \taxon{person} 3 | \meta{position}{Front-End Developer} 4 | \meta{external}{https://fizzyelt.github.io} 5 | -------------------------------------------------------------------------------- /trees/roadmap.tree: -------------------------------------------------------------------------------- 1 | \title{地圖} 2 | \meta{author}{false} 3 | 4 | \p{此地圖為一大方向，每個主體內會有延伸，包含實例、名詞解釋等等，類似於心智圖，而未來也有可能異動} 5 | 6 | \ol{ 7 | \li{lambda calculus} 8 | \li{用 recursion 解釋可計算性} 9 | \li{加上 simply type} 10 | \li{解釋型別規則} 11 | \li{建立 STLC categorical semantic} 12 | \li{加上 polymorphism} 13 | \li{解釋型別規則} 14 | \li{建立 #{E \to B} categorical semantic} 15 | \li{解釋 #{\lambda 2} 的 polymorphic property} 16 | \li{解釋 recursive type} 17 | \li{連結 F-algebra} 18 | \li{變成 dependent type} 19 | \li{解釋型別規則} 20 | \li{解釋模型（The natural model） 21 | \ul{ 22 | \li{topos} 23 | \li{topos fundamental theorem} 24 | \li{presheaf} 25 | \li{representable natural transformation} 26 | } 27 | } 28 | \li{進入真正的應用} 29 | \li{side effect} 30 | \li{concurrency} 31 | \li{安全性 32 | 33 | \ul{ 34 | \li{記憶體安全與子結構類型} 35 | \ul{ 36 | \li{(?) proof nets} 37 | } 38 | \li{整體安全性不是只關乎記憶體} 39 | } 40 | } 41 | \li{domain language} 42 | } 43 | -------------------------------------------------------------------------------- /trees/set-0001.tree: -------------------------------------------------------------------------------- 1 | \date{2023-09-09} 2 | \title{本性類與良性類} 3 | \author{dannypsnl} 4 | 5 | \p{ 6 | 類區分為兩種：一種是可以順利進行類運算的「良性類」，我們把這種「良性類」稱為集合；另一種是要限制運算的「本性類」，對於本性類，類運算並不是都能進行的。 7 | 更多資料可以參照數學導論 8 | } --------------------------------------------------------------------------------