vault backup: 2025-02-04 10:17:22
This commit is contained in:
commit
a33da99c30
3
.obsidian/community-plugins.json
vendored
3
.obsidian/community-plugins.json
vendored
@ -14,5 +14,6 @@
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|||||||
"obsidian-plugin-toc",
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"obsidian-plugin-toc",
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||||||
"emoji-shortcodes",
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"emoji-shortcodes",
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||||||
"advanced-canvas",
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"advanced-canvas",
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||||||
"obsidian-tracker"
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"obsidian-tracker",
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||||||
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"better-export-pdf"
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||||||
]
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]
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||||||
29
.obsidian/plugins/better-export-pdf/data.json
vendored
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29
.obsidian/plugins/better-export-pdf/data.json
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@ -0,0 +1,29 @@
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|||||||
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{
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||||||
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"showTitle": true,
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||||||
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"maxLevel": "6",
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||||||
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"displayHeader": true,
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||||||
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"displayFooter": true,
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||||||
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"headerTemplate": "<div style=\"width: 100vw;font-size:10px;text-align:center;\"><span class=\"title\"></span></div>",
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||||||
|
"footerTemplate": "<div style=\"width: 100vw;font-size:10px;text-align:center;\"><span class=\"pageNumber\"></span> / <span class=\"totalPages\"></span></div>",
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||||||
|
"printBackground": false,
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||||||
|
"generateTaggedPDF": false,
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||||||
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"displayMetadata": false,
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||||||
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"debug": false,
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||||||
|
"isTimestamp": false,
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||||||
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"enabledCss": false,
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||||||
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"prevConfig": {
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||||||
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"pageSize": "A4",
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||||||
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"marginType": "1",
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||||||
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"showTitle": true,
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||||||
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"open": true,
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||||||
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"scale": 100,
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||||||
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"landscape": false,
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||||||
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"marginTop": "10",
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||||||
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"marginBottom": "10",
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||||||
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"marginLeft": "10",
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||||||
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"marginRight": "10",
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||||||
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"displayHeader": true,
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||||||
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"displayFooter": true,
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||||||
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"cssSnippet": "0"
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||||||
|
}
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||||||
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}
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||||||
21276
.obsidian/plugins/better-export-pdf/main.js
vendored
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21276
.obsidian/plugins/better-export-pdf/main.js
vendored
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File diff suppressed because one or more lines are too long
11
.obsidian/plugins/better-export-pdf/manifest.json
vendored
Normal file
11
.obsidian/plugins/better-export-pdf/manifest.json
vendored
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@ -0,0 +1,11 @@
|
|||||||
|
{
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||||||
|
"id": "better-export-pdf",
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||||||
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"name": "Better Export PDF",
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||||||
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"version": "1.10.2",
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||||||
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"minAppVersion": "0.15.0",
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||||||
|
"description": "Export your notes to PDF, support export preview, add bookmarks outline and header/footer.",
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||||||
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"author": "l1xnan",
|
||||||
|
"authorUrl": "https://github.com/l1xnan",
|
||||||
|
"fundingUrl": "https://www.buymeacoffee.com/l1xnan",
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||||||
|
"isDesktopOnly": true
|
||||||
|
}
|
||||||
61
.obsidian/plugins/better-export-pdf/styles.css
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61
.obsidian/plugins/better-export-pdf/styles.css
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@ -0,0 +1,61 @@
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|||||||
|
#better-export-pdf {
|
||||||
|
display: flex;
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||||||
|
flex-direction: row;
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||||||
|
height: 75vh;
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||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf .pdf-preview {
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||||||
|
flex: auto;
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||||||
|
position: relative;
|
||||||
|
display: flex;
|
||||||
|
flex-direction: column;
|
||||||
|
overflow-x: hidden;
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||||||
|
overflow-y: scroll;
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||||||
|
align-content: flex-start;
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||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf .pdf-preview .webview-wrapper {
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||||||
|
position: relative;
|
||||||
|
height: 100%;
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||||||
|
width: 100%;
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||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf .pdf-preview .print-size {
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||||||
|
position: absolute;
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||||||
|
right: 8px;
|
||||||
|
top: 8px;
|
||||||
|
z-index: 99;
|
||||||
|
font-size: 0.6rem;
|
||||||
|
white-space: pre-wrap;
|
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|
text-align: right;
|
||||||
|
visibility: hidden;
|
||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf .pdf-preview > div {
|
||||||
|
flex: 1;
|
||||||
|
height: 100%;
|
||||||
|
width: 100%;
|
||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf .pdf-preview .filename {
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||||||
|
font-size: 0.75rem;
|
||||||
|
color: var(--color-base-60);
|
||||||
|
}
|
||||||
|
#better-export-pdf .pdf-preview .filename:not(:first-child) {
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||||||
|
padding-top: calc(var(--p-spacing));
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||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf webview {
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||||||
|
flex: 1;
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||||||
|
height: 100%;
|
||||||
|
width: 100%;
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||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf .setting-wrapper {
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||||||
|
width: 320px;
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||||||
|
margin-left: 16px;
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||||||
|
}
|
||||||
|
|
||||||
|
#better-export-pdf .setting-wrapper .setting-item[hidden] {
|
||||||
|
display: none;
|
||||||
|
}
|
||||||
60
.obsidian/plugins/obsidian-enhancing-export/data.json
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60
.obsidian/plugins/obsidian-enhancing-export/data.json
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@ -0,0 +1,60 @@
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|||||||
|
{
|
||||||
|
"items": [
|
||||||
|
{
|
||||||
|
"name": "Markdown"
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||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Markdown (Hugo)"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Html"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "TextBundle"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Typst"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "PDF"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Word (.docx)"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "OpenOffice"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "RTF"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Epub"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Latex"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Media Wiki"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "reStructuredText"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Textile"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "OPML"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"name": "Bibliography"
|
||||||
|
}
|
||||||
|
],
|
||||||
|
"defaultExportDirectoryMode": "Auto",
|
||||||
|
"openExportedFile": true,
|
||||||
|
"env": {},
|
||||||
|
"showExportProgressBar": true,
|
||||||
|
"lastExportDirectory": {
|
||||||
|
"linux": "/home/boyan/Documents"
|
||||||
|
},
|
||||||
|
"lastExportType": "PDF"
|
||||||
|
}
|
||||||
28
.obsidian/workspace.json
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28
.obsidian/workspace.json
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@ -34,9 +34,23 @@
|
|||||||
"icon": "lucide-file",
|
"icon": "lucide-file",
|
||||||
"title": "Introductory Lecture"
|
"title": "Introductory Lecture"
|
||||||
}
|
}
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"id": "1ac4561fed3e4c16",
|
||||||
|
"type": "leaf",
|
||||||
|
"state": {
|
||||||
|
"type": "markdown",
|
||||||
|
"state": {
|
||||||
|
"file": "conflict-files-obsidian-git.md",
|
||||||
|
"mode": "source",
|
||||||
|
"source": false
|
||||||
|
},
|
||||||
|
"icon": "lucide-file",
|
||||||
|
"title": "conflict-files-obsidian-git"
|
||||||
|
}
|
||||||
}
|
}
|
||||||
],
|
],
|
||||||
"currentTab": 1
|
"currentTab": 2
|
||||||
}
|
}
|
||||||
],
|
],
|
||||||
"direction": "vertical"
|
"direction": "vertical"
|
||||||
@ -213,11 +227,15 @@
|
|||||||
"omnisearch:Omnisearch": false
|
"omnisearch:Omnisearch": false
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"active": "56bef0584922cba9",
|
"active": "1ac4561fed3e4c16",
|
||||||
"lastOpenFiles": [
|
"lastOpenFiles": [
|
||||||
|
"Software Engineering/Introductory Lecture.md",
|
||||||
|
"conflict-files-obsidian-git.md",
|
||||||
|
"Untitled.md",
|
||||||
|
"Statistics and Probability/Mock exam run 1.md",
|
||||||
|
"Functional Programming/Proofs.md",
|
||||||
"Operating Systems",
|
"Operating Systems",
|
||||||
"Statistics and Probability/Support Lecture.md",
|
"Statistics and Probability/Support Lecture.md",
|
||||||
"Software Engineering/Introductory Lecture.md",
|
|
||||||
"Software Engineering",
|
"Software Engineering",
|
||||||
"Pasted image 20250113151159.png",
|
"Pasted image 20250113151159.png",
|
||||||
"Extracurricular/Misc/Ideas.md",
|
"Extracurricular/Misc/Ideas.md",
|
||||||
@ -234,7 +252,6 @@
|
|||||||
"Discrete Structures/Midterm",
|
"Discrete Structures/Midterm",
|
||||||
"Extracurricular/satQuest/img/Pasted image 20241206134156.png",
|
"Extracurricular/satQuest/img/Pasted image 20241206134156.png",
|
||||||
"Extracurricular/satQuest/Parts Proposal.md",
|
"Extracurricular/satQuest/Parts Proposal.md",
|
||||||
"Untitled.md",
|
|
||||||
"Untitled.canvas",
|
"Untitled.canvas",
|
||||||
"Discrete Structures/Mathematical Data Structures.md",
|
"Discrete Structures/Mathematical Data Structures.md",
|
||||||
"Discrete Structures/Relations and Digraphs.md",
|
"Discrete Structures/Relations and Digraphs.md",
|
||||||
@ -251,8 +268,6 @@
|
|||||||
"Advanced Algorithms/Recurrence relations.md",
|
"Advanced Algorithms/Recurrence relations.md",
|
||||||
"Advanced Algorithms/Graph Algorithms.md",
|
"Advanced Algorithms/Graph Algorithms.md",
|
||||||
"Linear Algebra/Matrices.md",
|
"Linear Algebra/Matrices.md",
|
||||||
"Advanced Programming/Intro.md",
|
|
||||||
"Advanced Programming/Annotation Repository.md",
|
|
||||||
"Advanced Algorithms/assets/pnp/Pasted image 20241203234013.png",
|
"Advanced Algorithms/assets/pnp/Pasted image 20241203234013.png",
|
||||||
"Advanced Algorithms/assets/pnp/Pasted image 20241203234032.png",
|
"Advanced Algorithms/assets/pnp/Pasted image 20241203234032.png",
|
||||||
"Advanced Algorithms/assets/graph/1_WR4AtjT_nhwSOtAW99Yd5g.gif",
|
"Advanced Algorithms/assets/graph/1_WR4AtjT_nhwSOtAW99Yd5g.gif",
|
||||||
@ -261,7 +276,6 @@
|
|||||||
"Advanced Algorithms/practicals",
|
"Advanced Algorithms/practicals",
|
||||||
"Statistics and Probability/R",
|
"Statistics and Probability/R",
|
||||||
"Discrete Structures/img",
|
"Discrete Structures/img",
|
||||||
"Algorithmic Programming Contests",
|
|
||||||
"Web Engineering/canvae/server_client.canvas",
|
"Web Engineering/canvae/server_client.canvas",
|
||||||
"Advanced Programming/projects/second/Refactoring.canvas",
|
"Advanced Programming/projects/second/Refactoring.canvas",
|
||||||
"Advanced Programming/assets/assignment/assignment_organization.canvas",
|
"Advanced Programming/assets/assignment/assignment_organization.canvas",
|
||||||
|
|||||||
@ -89,6 +89,8 @@ To prove $P$:
|
|||||||
|
|
||||||
## Mathematical Induction
|
## Mathematical Induction
|
||||||
[^2]
|
[^2]
|
||||||
|
Generalized in [Proofs (FP)](Proofs.md)
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|||||||
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Load Diff
@ -1,595 +0,0 @@
|
|||||||
---
|
|
||||||
|
|
||||||
excalidraw-plugin: parsed
|
|
||||||
tags: [excalidraw]
|
|
||||||
|
|
||||||
---
|
|
||||||
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
|
|
||||||
|
|
||||||
|
|
||||||
# Excalidraw Data
|
|
||||||
## Text Elements
|
|
||||||
%%
|
|
||||||
## Drawing
|
|
||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
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|
||||||
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||||||
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|
||||||
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||||||
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|
||||||
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||||||
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|
||||||
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|
|
||||||
|
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||||||
OFAsGXTUJ1uK5/Rsst4xEkM+IkffGrjADMgNXJfUmsmvHOVbduV1EsiUUgsm2zKG/FPZaW8fZstv5QsiYEfsHt/e+HM/GpykMfl74FezZR6wrTXBNnqw5Kp5l/IgLN3Ah4Iv2y5RjjPQkUSpWtrDd84wwmOMO8kfFCIt6ldvck4cd8Oun8i/4lN/tE2+o9j90TCt4x+XatYsJL0rpxE7sw8EXUSCrjuuGELvYlkcqOG0jQqlFquvbPaKGZ8k/HDM
|
|
||||||
|
|
||||||
nrnzUvuJmyWjibkH/CKIruOb2fZo6kF9GlcYeLmEyq5eiG0UWPZIZNKZ8lqkMqD9uPan9vevbrurtoHBLxTnmqCnaEuddVmztdSrt3lml8N6fkKs3JUrzSwEgBdY0HqH+kzNuPF6zFTuntgOFadzu2ruN3c7Vl7b+lpC6lmS5LnN8yE+h4SrlISvbS+V2ZHzEzrAAhj98AhnGc901lqaPnvB/cLooy3luEqigjV5z9yVdxjswfQpBi+d3OZKo2mr
|
|
||||||
|
|
||||||
uMiTd/T8chJZMpOuYai32CdU+wkWOnAbtEG0ZccFBTluIwrfF695jzOK9xAlevvl9UyYVLSZlCcAlnYlAvQCk6uSKWtf8tHUoXv+usXu9vff4oNT4ESrBXC6y9CdRaMfvPNkQIz97dwtxZfviltfucFP9kpIoRN4F6ptz9x9NkWq/ve0DgptaqYkT1vP7pw7/uawhLyOEoAfOeQ07o2tgMfzlBrLxxv4hkvsrMfEuFrdr8XieXSZFYkmE1FIE0HA
|
|
||||||
|
|
||||||
u/uDYiQFqgiqFdkfJ3jEqlz4naYFVKCy5maraxbojJEVZy/FyKRPoP8yeYuErcQZ5Uo0CEiWK/crLoXCacAa0t84snTGDB4uIksxXkOelatF/xFgl8fH8pIAhFuicne3WrXaML5HfFBumCJ3zGGFaFo900d6xZ/xCYEvQtCYxRuDuLYgFFySi0tIUJ1yw20nETqi18DUHkGCFxGMD1XSZgIlqd4k846v6qahKIvmEULTDXKOEuFUAoSGdkIOWfzk
|
|
||||||
|
|
||||||
TUBA/lFGq5RxPS1WAKzgvobYvFMwwm75/D9Ee3ZrEfBosc0BuOKyDUA/ZtyutFmOpzs8LOGE8Gpkf38okefDzfyhg+EpJGnYWbp1VU7p5fJZtygEdZtUe95AfPXvP5EGjyY7stAZQE3pTafSsUsVjhkebYOUecj/yg+uiHUpSdikda6S1vV4eKy7fxuTUQOlme8f89kJRwl97QsUmpjYs9u9wBCqoLhRKLQsCNYf5HfinG1g4eHZA+vc5zC0ItpC
|
|
||||||
|
|
||||||
hIPvSmZyvCGINM5LU/DN4jm1OpIUA9QAAqZnee+ApkBjGkBws+FQ1/ui4tEn5TAlVz8IsPPO53friar3OvKKJKh4rRFVSgmJsrlqkAAaidRLoNMS3CVImY3plD6ntPnQkBkB+HOVi8291aQlL64cAQYX/EkEhVx/48LBEr+9u2pIWyuFAxOZ27KuxtSHv7MGJqpsWghEL3vDyGLHSQVwhi0xdd5Cso1j75Ul0H6hZ/WZMMFNFp9VooE4oI89Mh0E
|
|
||||||
|
|
||||||
m89WFBx5L2KwAvuSiVz7zug6cBRYVQ+uoqF8oidFOFJPOqrSesvfKMqZ9AaeWXG4FmQiTDvZQRw/lImklWIGFi9XUV+sud4vZdgj3KA+EEqrlJuzNGtoPCguzBWguhvMJ5Jw22t8l6xtLzlGo2aN/kwz52S9I7zLXwmkukTLGeQzwmfTgOGfet+1FNytTqlGkJ5MF54l21P6HfIBTzmV1v51qmUn4WAyv7c/ITlUr5BKIg4Eihxm63EbssxV6FNc
|
|
||||||
|
|
||||||
pKVJWRumvrumYEERQqg9V7jE3l/dF+2J57raAhk4ycGaU0qMfWqxjrj1ebMJ9AFFl110uPApxFemyzRGuWEK6k89FRlhEF2fD0v8FzCsxJRGIsIgyjTl7MccgiVYgRU4kjRRlN+OGqegcBqe5vTAg7RsrkYTz3PmTE+HxV2hJ/wreOYEKMG79pPpCfsvKuKHs90Crj25nRto85Q6E8nIRNlHZyYxe7zciiaLCONBquuxjgEb7QLWn7YTXOFFIG/K
|
|
||||||
|
|
||||||
A1EZ1w+Vm6uSRSsmJmgxQOFdKE1LeNwwlkrjEf+9g+Qt/ACFL1JMfk+g0Pky3hqq9v4dAGVbpb1Cbu4cGbuTC5xWQBVQeF/VZF4plUegcDUeAhEnVjbctqV98FXeZmPq5T8UxOuyuGleqQRMsrPP2t6+w5eeVah2oRE3nQjVIAtN6t16agZd1wM5S/sryz/PlavVWfSkZMom+i3OJ5gol6EimWdZgqNaEobOhwyqgHEp+Ea0lV7gB9O06FjieBoX
|
|
||||||
|
|
||||||
ieOhzgQdaqpefZoc96SBr2CEm5tj6N+pt4kiC0Vwmbx+7KGAwgXKCL9wD3Zacs/1lGnZSO/aezzC6BoiApovPryiqSWvG5ypXoT91liaip3f2pkvqUyRiclyR5qYe3DW1NBFkDNbOmTlGeQ6gAYaL6NuM3WqvFTE4vR1lCvK9SKhg4D6lWZmWMN/GsU8guaEs875Uhc4RJjxsfEIlUMls7FSHN9mOE9kD3GnT+DUDnqe39oqOSkrsUsewsk5SCGJ
|
|
||||||
|
|
||||||
4jCsXYsEoe4vu50b5EpqlgOeWdG14SrVvUVSnSukDAJGpD3V+9eEhsE8H7POVnEKrvW96bgfAoDe38h9fgnpTHxT8Q4fKw9Q6CvJVEBSdU/SwjfT6BKf+V8HMRleZEyN5Bgf8cOOna6Te4uyOPyb2TeqBxQOdGxTfqb+CPKb4zf6b3Tfab1TfWb0zf2b8zfOTiZO4MxOOoR1GQknNSHsb0jeGSCjfPcoTeMb3mxwAL+B1iN552QC1BuACtBoAJ8B
|
|
||||||
|
|
||||||
MgBUAiACUQ1gAwBCAAgAKACog3Vl5J5gIbejb9iAIANgARAPTALQI2B9AOyA0mx9OTb2bfQaUnBLbxkA9b0hWSoSPJTb+benb1bfHrZr77b17fsgM7frb/DLPb47fA71bebbxmnQEaHeLb1bfn6VmmY797eMgMpJ7B4nfw7xkAjoJwBrpc6BmQFDJNbw7fY7xnes76yBC+oiZPwGneoAEHeY6JgBd7erfziENQEAPMAcdBXeg7/LeK0BERQaWwAK
|
|
||||||
|
|
||||||
AJ8BcAJEUW71beNoF99O793eQgGIg6YFCBLqP3eMgB3eJ75HolFGTATb+B0oQCyBUbAvgGJiwlUYkuiHyMDg+QNgBl73OayA+Lrl0wsXgb5rejAGwADAIrecwAQARgqCBdOdVVuFFPeQvOjngMDiESQCbfEKcQAS70YAy75rfP7+yAEAKs04vH/fSACQBdoGwA3sBtAao8EBktJE4SAONgPCLQJ8AGIgUQEMBWgGg+0H7yB7OIoVmoF5IE6Pg+ZW
|
|
||||||
|
|
||||||
I/eC70nBI77CBlJFAAMwCvxzktg+fQKA/AkFfeJQFkBoH/uBrihKAxgKnzuAKw/SgBwAHsJw/XstqAhAFAB1wFcVXso/e7AOHxsADkBWQDw+4AOA/IHzw++meyp1iJI/CAIwAzoBffBSmgAPCJnyMcmbeIQAYA5709AyCw9TtqBQ+1Hxo/3kItBwAG4QG9PoRgAAtAQAAtAgAA==
|
|
||||||
```
|
|
||||||
%%
|
|
||||||
80
Functional Programming/Proofs.md
Normal file
80
Functional Programming/Proofs.md
Normal file
@ -0,0 +1,80 @@
|
|||||||
|
---
|
||||||
|
title: Understanding Induction
|
||||||
|
abstract: I am fucking stupid and I can't understand this
|
||||||
|
author:
|
||||||
|
---
|
||||||
|
|
||||||
|
## Understanding Induction
|
||||||
|
|
||||||
|
There are a few assumptions to be made when performing induction proofs (in general):
|
||||||
|
- There is *some* non-decreasing property about the category we're observing (i.e. difference between elements in a sequence, length of a list, height of a tree)
|
||||||
|
- The structure needs to be *well-defined*[^1]
|
||||||
|
|
||||||
|
And then we move on to the mechanism for proofs itself:
|
||||||
|
1. Induction Base Case: The base case(s) must be **sufficient** to cover all minimal or simplest cases of the structure.
|
||||||
|
For instance, when proving properties of lists, we often start by showing the property holds for the empty list.
|
||||||
|
|
||||||
|
2. Inductive Hypothesis: We assume that the property we're trying to prove holds for an arbitrary case. It must explicitly account for all necessary assumptions about the intermediate state of the structure.
|
||||||
|
|
||||||
|
3. Inductive Step: To establish the general inductive step, we should demonstrate that if the property holds for a given category (the induction hypothesis), it also holds for the next element (based on the non-decreasing property mentioned earlier).
|
||||||
|
|
||||||
|
We should be able to guarantee termination and avoid ambiguity in the process.
|
||||||
|
|
||||||
|
## Induction in Haskell
|
||||||
|
Given Haskell's [purity](Introduction%20to%20Functional%20Programming.md), we can perform induction proofs on most of its structure.
|
||||||
|
|
||||||
|
The non-decreasing property is defined by the data structure (`data` keyword) and/or the logic we have implemented (i.e. natural numbers - n > n-1 > 0, lists -> length, tree -> height).
|
||||||
|
|
||||||
|
We then proceed with the process mentioned above, although slightly altered as to fit the data structure category:
|
||||||
|
- **BC**: We must first establish a base case, which is a minimal instance of the structure that satisfies the property being proven. For example, this could be an empty list, a single element, or a tree with only one node. The base case must be simple enough to verify directly.
|
||||||
|
|
||||||
|
- **IH**: Assume that the property holds for some arbitrary but fixed instance of the structure. This is often referred to as the **induction step assumption**, where you posit[^2] that the property is valid for a smaller or simpler instance of the structure.
|
||||||
|
|
||||||
|
- **IS**: Using the inductive hypothesis, prove that the property holds for a larger or more complex instance of the structure.
|
||||||
|
|
||||||
|
For example, if the property holds for a list of length $n$, you need to show it holds for a list of length $n+1$. Similarly, for a tree of height $h$, show it holds for a tree of height $h+1$. This step bridges the smaller case to the larger one.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
For example, when proving that reversing twice returns the original list, we can use mathematical induction:
|
||||||
|
|
||||||
|
1. **Base Case**: We need to show the property holds for the empty list (`[]`). Here, it's clear that `reverseList [] = []` and `reverseList (reverseList [])` is also `[]` since `reverseList` appends elements to the left `(++)`, and an empty list cannot be appended to anything.
|
||||||
|
|
||||||
|
2. **Inductive Step**: Suppose the property holds for a given non-empty list `x:xs` (induction hypothesis). To prove it holds for the longer list xs (`x:xs`), we apply `reverseList` and use the defined function properties as follows:
|
||||||
|
```haskell
|
||||||
|
-- Induction Hypothesis (property holds for xs)
|
||||||
|
prop_reverseTwice xs =>
|
||||||
|
|
||||||
|
-- Show that property also holds for x:xs (inductive step)
|
||||||
|
prop_reverseTwice (reverseList x:xs) =
|
||||||
|
-- Break down the list using reverseList and pattern matching
|
||||||
|
prop_reverseTwice ((reverseList xs) ++ [x]) =
|
||||||
|
-- Use the definition of concatenation, reverse, and the induction hypothesis
|
||||||
|
prop_reverseTwice (xs ++ reverseList [] ++ [x]) =
|
||||||
|
-- Use the fact that concatenating an empty list with any other list returns the original list.
|
||||||
|
prop_reverseTwice (xs ++ [x]) =
|
||||||
|
-- The property holds for xs by our induction hypothesis
|
||||||
|
True
|
||||||
|
```
|
||||||
|
By showing the base case and inductive step, we have successfully proven that `reverseList` reverses lists twice, returning the original list.
|
||||||
|
|
||||||
|
|
||||||
|
### TL;DR
|
||||||
|
1. Base case - empty list or tree
|
||||||
|
2. Inductive hypothesis - assume it works for an arbitrary list `xs` or tree (i.e.)
|
||||||
|
```haskell
|
||||||
|
data Tree a = Empty -- A tree can be empty
|
||||||
|
| Node a (Tree a) (Tree a) -- Or a node with a value and two
|
||||||
|
deriving (Show, Eq)
|
||||||
|
|
||||||
|
-- when a Tree a is Empty
|
||||||
|
|
||||||
|
```
|
||||||
|
|
||||||
|
3. Inductive Step - show that it works for a larger list `x:xs` or a non-empty tree
|
||||||
|
|
||||||
|
This process involves knowing *some* implementation details of higher-order functions (or at least a vague idea as to how they work).
|
||||||
|
|
||||||
|
|
||||||
|
[^1]: Clear rules as to how it's constructed
|
||||||
|
[^2]: to propose as an explanation **:** [suggest](https://www.merriam-webster.com/dictionary/suggest)
|
||||||
21
Statistics and Probability/Mock exam run 1.md
Normal file
21
Statistics and Probability/Mock exam run 1.md
Normal file
@ -0,0 +1,21 @@
|
|||||||
|
|
||||||
|
### Null hyposthesis
|
||||||
|
|
||||||
|
### Confidence interval
|
||||||
|
Interval which is expected to contain the parameter
|
||||||
|
$$
|
||||||
|
CI = \bar{x} \pm z \frac{s}{\sqrt{n}}
|
||||||
|
$$
|
||||||
|
Where $z\frac{s}{\sqrt{n}}$ is the variation in our estimate.
|
||||||
|
|
||||||
|
|
||||||
|
### T-Test
|
||||||
|
Used to determine if there is a significant difference between the means of two groups.
|
||||||
|
|
||||||
|
We have to assume:
|
||||||
|
- Data follows a normal distribution
|
||||||
|
- each observation is independent
|
||||||
|
|
||||||
|
$$
|
||||||
|
t = \frac{\text{mean} - \text{theoretical value}}{s\sqrt{n}}
|
||||||
|
$$
|
||||||
50
Untitled.md
Normal file
50
Untitled.md
Normal file
@ -0,0 +1,50 @@
|
|||||||
|
```haskell
|
||||||
|
add :: (Int -> Int) -> Int
|
||||||
|
add x y = x+y
|
||||||
|
|
||||||
|
instance Eq Bool where
|
||||||
|
True == True = True
|
||||||
|
False == False = True
|
||||||
|
_ == _ = False
|
||||||
|
|
||||||
|
instance (Eq a, Eq b) => Eq (a, b) where
|
||||||
|
(x1, y1) == (x2, y2) = x1 == x2 && y1 == y2
|
||||||
|
|
||||||
|
-- A value that exists
|
||||||
|
myValue :: Maybe Int
|
||||||
|
myValue = Just 42
|
||||||
|
|
||||||
|
-- A value that doesn't exist
|
||||||
|
noValue :: Maybe Int
|
||||||
|
noValue = Nothing
|
||||||
|
|
||||||
|
|
||||||
|
```
|
||||||
|
|
||||||
|
|
||||||
|
```python
|
||||||
|
|
||||||
|
class Number:
|
||||||
|
# Some implementation
|
||||||
|
|
||||||
|
class Float(Number):
|
||||||
|
# Some implentation
|
||||||
|
|
||||||
|
class Integer(Number):
|
||||||
|
# Some implementation
|
||||||
|
|
||||||
|
|
||||||
|
def add(numA:Number, numB:Number) -> Number:
|
||||||
|
|
||||||
|
```
|
||||||
|
|
||||||
|
|
||||||
|
```python
|
||||||
|
|
||||||
|
class MyClass:
|
||||||
|
def __eq__(self):
|
||||||
|
# MyClass :: Eq
|
||||||
|
def __str__(self)
|
||||||
|
# MyClass :: Show
|
||||||
|
def __
|
||||||
|
```
|
||||||
Loading…
x
Reference in New Issue
Block a user