Mercurial > hg > Papers > 2024 > moririn-thesis
changeset 3:70346fe80c78
add slide
| author | mori |
|---|---|
| date | Thu, 08 Feb 2024 15:46:29 +0900 |
| parents | 04a44bb1bd7b |
| children | dd37dd73a8ea |
| files | Slide/Makefile Slide/README.md Slide/cr.css Slide/figs/RedBlackTree.png Slide/figs/stack.drawio Slide/figs/stack.png Slide/figs/stack.svg Slide/figs/stack1.drawio Slide/figs/stack1.png Slide/figs/stack1png.png Slide/figs/stack2.png Slide/figs/stack3.drawio Slide/figs/stack3.svg Slide/figs/stack4.drawio Slide/figs/stack4.svg Slide/figs/test.drawio Slide/figs/test.svg Slide/logo.svg Slide/slide3.html Slide/slide3.md Slide/slide3.pdf |
| diffstat | 21 files changed, 2773 insertions(+), 0 deletions(-) [+] |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Slide/Makefile Thu Feb 08 15:46:29 2024 +0900 @@ -0,0 +1,3 @@ +build: + marp template.md --theme cr.css + open template.html \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Slide/README.md Thu Feb 08 15:46:29 2024 +0900 @@ -0,0 +1,15 @@ +## 使用方法 + +基本的には`Marp for VSCode` Extensionを使用する + +Extensionをインストール後、次のVSCodeの設定を書き込む。 +テーマのパスは環境によって違うかもしれないので確認する。 +おそらく、VSCodeで開いているディレクトリからの相対パスを入れれば良い。 +```json + "markdown.marp.exportType": "html", + "markdown.marp.themes": ["template/cr.css"] +``` + +## CLIでの使用方法 + +Makefileを作成してある \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Slide/cr.css Thu Feb 08 15:46:29 2024 +0900 @@ -0,0 +1,30 @@ +/* @theme cr */ + +@import 'default'; + +section { + background-image: url('./logo.svg'); + background-repeat: no-repeat; + background-position: right 2rem bottom 0.6rem; + background-size: 12%; +} + +h1 { + color: #828db2; + font-size: 2.1rem; +} + +h2 { + background-color: #828db2; + color: #fff; + padding: 0.3rem; + font-size: 2.1rem; +} + +p { + font-size: 1.3rem; +} + +ul, ol { + font-size: 1.2rem; +}
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stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><path d="M 0.5 28 L 747.5 28" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 745px; height: 1px; padding-top: 17px; margin-left: 2px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; font-weight: bold; white-space: normal; overflow-wrap: normal;">tree0</div></div></div></foreignObject><text x="374" y="20" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle" font-weight="bold">tree0</text></switch></g><path d="M 387 163 L 387 140 L 711.5 140 L 711.5 163" fill="rgb(255, 255, 255)" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="all"/><path d="M 387 163 L 387 580 L 711.5 580 L 711.5 163" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><path d="M 387 163 L 711.5 163" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 323px; height: 1px; padding-top: 152px; margin-left: 388px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; 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visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 269px; height: 1px; padding-top: 279px; margin-left: 434px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; font-weight: bold; white-space: normal; overflow-wrap: normal;">tree2</div></div></div></foreignObject><text x="569" y="282" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle" font-weight="bold">tree2</text></switch></g><path d="M 438 406 L 438 383 L 605 383 L 605 406" fill="rgb(255, 255, 255)" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="all"/><path d="M 438 406 L 438 568 L 605 568 L 605 406" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><path d="M 438 406 L 605 406" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 165px; height: 1px; padding-top: 395px; margin-left: 439px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; font-weight: bold; white-space: normal; overflow-wrap: normal;">tree3</div></div></div></foreignObject><text x="522" y="398" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle" font-weight="bold">tree3</text></switch></g><rect x="466.25" y="350" width="115.5" height="30" fill="none" stroke="none" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 114px; height: 1px; padding-top: 365px; margin-left: 467px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">左に降りる<br />s-left</div></div></div></foreignObject><text x="524" y="369" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">左に降りる
s-left</text></switch></g><rect x="546.25" y="235" width="115.5" height="30" fill="none" stroke="none" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 114px; height: 1px; padding-top: 250px; margin-left: 547px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">右に降りる<br />s-right</div></div></div></foreignObject><text x="604" y="254" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">右に降りる
s-right</text></switch></g><rect x="483" y="110" width="115.5" height="30" fill="none" stroke="none" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 114px; height: 1px; padding-top: 125px; margin-left: 484px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">右に降りる<br />s-right</div></div></div></foreignObject><text x="541" y="129" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">右に降りる
s-right</text></switch></g><rect x="344" y="30" width="60" height="30" fill="none" stroke="none" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 58px; height: 1px; padding-top: 45px; margin-left: 345px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">s-nil</div></div></div></foreignObject><text x="374" y="49" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">s-nil</text></switch></g><path d="M 1103 30 L 1103 13.5 Q 1103 0 1089.5 0 L 796.5 0 Q 783 0 783 13.5 L 783 30" fill="rgb(255, 255, 255)" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="all"/><path d="M 783 30 L 783 132.5 Q 783 146 796.5 146 L 1089.5 146 Q 1103 146 1103 132.5 L 1103 30" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><path d="M 783 30 L 1103 30" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><path d="M 823 30 L 823 56 L 823 86 L 823 116 L 823 146" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><path d="M 963 30 L 963 56 L 963 86 L 963 116 L 963 146" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="none"/><g fill="rgb(0, 0, 0)" font-family="Helvetica" text-anchor="middle" font-size="16px"><text x="942.5" y="21.5"> stack | stackInvariant</text></g><rect x="783" y="30" width="40" height="26" fill="none" stroke="none" pointer-events="all"/><path d="M 783 30 M 823 30 M 823 56 M 783 56" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 38px; height: 1px; padding-top: 43px; margin-left: 784px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 22px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">Top</div></div></div></foreignObject><text x="803" y="48" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">Top</text></switch></g><rect x="823" y="30" width="140" height="26" fill="none" stroke="none" pointer-events="all"/><path d="M 823 30 M 963 30 M 963 56 M 823 56" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 132px; height: 1px; padding-top: 43px; margin-left: 831px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left; max-height: 22px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">tree3 (Key = 14)</div></div></div></foreignObject><text x="831" y="48" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px">tree3 (Key = 14)</text></switch></g><rect x="963" y="30" width="140" height="26" fill="none" stroke="none" pointer-events="all"/><path d="M 963 30 M 1103 30 M 1103 56 M 963 56" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 132px; height: 1px; padding-top: 43px; margin-left: 971px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left; max-height: 22px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">s-left (14 < 15)</div></div></div></foreignObject><text x="971" y="48" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px">s-left (14 < 15)</text></switch></g><rect x="783" y="56" width="40" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 783 56 M 823 56 M 823 86 M 783 86" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 38px; height: 1px; padding-top: 71px; margin-left: 784px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">:</div></div></div></foreignObject><text x="803" y="76" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">:</text></switch></g><rect x="823" y="56" width="140" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 823 56 M 963 56 M 963 86 M 823 86" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 132px; height: 1px; padding-top: 71px; margin-left: 831px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">tree2 (Key = 15)</div></div></div></foreignObject><text x="831" y="76" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px">tree2 (Key = 15)</text></switch></g><rect x="963" y="56" width="140" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 963 56 M 1103 56 M 1103 86 M 963 86" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 132px; height: 1px; padding-top: 71px; margin-left: 971px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">s-right (10 < 15)</div></div></div></foreignObject><text x="971" y="76" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px">s-right (10 < 15)</text></switch></g><rect x="783" y="86" width="40" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 783 86 M 823 86 M 823 116 M 783 116" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; 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text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 323px; height: 1px; padding-top: 417px; margin-left: 789px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">stackInvariantを見ることで、tree0からtree3に行くには、右、右、左の経路をたどっていることがわかる！</div></div></div></foreignObject><text x="951" y="420" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">stackInvariantを見ることで、tree0からtree3に行くには、右、右、左の経路をたどっていることがわかる！</text></switch></g><rect x="784" y="186" width="329" height="144" rx="21.6" ry="21.6" fill="rgb(255, 255, 255)" stroke="rgb(0, 0, 0)" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 327px; height: 1px; padding-top: 258px; margin-left: 786px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;"><font style="font-size: 14px;">・stackはRedBlackTreeのリストで構成される<br />・stackは辿った木を保存している（今回は8→14）<br />・stackInvariantは辿った木の正しい経路が保存されている<br /><br />stackInvariantは、stackが木を辿った正しい順番で構成されていることを示している</font></div></div></div></foreignObject><text x="786" y="262" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px">・stackはRedBlackTreeのリストで構成される...</text></switch></g></g></svg> \ No newline at end of file
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1px; padding-top: 395px; margin-left: 439px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; font-weight: bold; white-space: normal; overflow-wrap: normal;">tree3</div></div></div></foreignObject><text x="522" y="398" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle" font-weight="bold">tree3</text></switch></g><rect x="466.25" y="350" width="115.5" height="30" fill="none" stroke="none" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 114px; height: 1px; padding-top: 365px; margin-left: 467px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">左に降りる<br />s-left</div></div></div></foreignObject><text x="524" y="369" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">左に降りる
s-left</text></switch></g><rect x="546.25" y="235" width="115.5" height="30" fill="none" stroke="none" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 114px; height: 1px; padding-top: 250px; margin-left: 547px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">右に降りる<br />s-right</div></div></div></foreignObject><text x="604" y="254" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">右に降りる
s-right</text></switch></g><rect x="483" y="110" width="115.5" height="30" fill="none" stroke="none" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 114px; height: 1px; padding-top: 125px; margin-left: 484px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">右に降りる<br />s-right</div></div></div></foreignObject><text x="541" y="129" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">右に降りる
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text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 123px; height: 1px; padding-top: 43px; margin-left: 907px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 22px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">tree3</div></div></div></foreignObject><text x="969" y="48" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">tree3</text></switch></g><rect x="1031" y="30" width="434" height="26" fill="none" stroke="none" pointer-events="all"/><path d="M 1031 30 M 1465 30 M 1465 56 M 1031 56" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; 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text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 121px; height: 1px; padding-top: 71px; margin-left: 784px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">:</div></div></div></foreignObject><text x="845" y="76" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">:</text></switch></g><rect x="906" y="56" width="125" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 906 56 M 1031 56 M 1031 86 M 906 86" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 123px; height: 1px; padding-top: 71px; margin-left: 907px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">tree2</div></div></div></foreignObject><text x="969" y="76" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">tree2</text></switch></g><rect x="1031" y="56" width="434" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 1031 56 M 1465 56 M 1465 86 M 1031 86" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 426px; height: 1px; padding-top: 71px; margin-left: 1039px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">(node 15 ...) :: (node 10 ...) :: (node 8 ...) :: []</div></div></div></foreignObject><text x="1039" y="76" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px">(node 15 ...) :: (node 10 ...) :: (node 8 ...) :: []</text></switch></g><rect x="783" y="86" width="123" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 783 86 M 906 86 M 906 116 M 783 116" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 121px; height: 1px; padding-top: 101px; margin-left: 784px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">:</div></div></div></foreignObject><text x="845" y="106" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">:</text></switch></g><rect x="906" y="86" width="125" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 906 86 M 1031 86 M 1031 116 M 906 116" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 123px; height: 1px; padding-top: 101px; margin-left: 907px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">tree1</div></div></div></foreignObject><text x="969" y="106" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">tree1</text></switch></g><rect x="1031" y="86" width="434" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 1031 86 M 1465 86 M 1465 116 M 1031 116" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 426px; height: 1px; padding-top: 101px; margin-left: 1039px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">(node 10 ...) :: (node 8 ...) :: []</div></div></div></foreignObject><text x="1039" y="106" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px">(node 10 ...) :: (node 8 ...) :: []</text></switch></g><rect x="783" y="116" width="123" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 783 116 M 906 116 M 906 146 M 783 146" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 121px; height: 1px; padding-top: 131px; margin-left: 784px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">orig</div></div></div></foreignObject><text x="845" y="136" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">orig</text></switch></g><rect x="906" y="116" width="125" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 906 116 M 1031 116 M 1031 146 M 906 146" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 123px; height: 1px; padding-top: 131px; margin-left: 907px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">tree0</div></div></div></foreignObject><text x="969" y="136" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px" text-anchor="middle">tree0</text></switch></g><rect x="1031" y="116" width="434" height="30" fill="none" stroke="none" pointer-events="all"/><path d="M 1031 116 M 1465 116 M 1465 146 M 1031 146" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 426px; height: 1px; padding-top: 131px; margin-left: 1039px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left; max-height: 26px; overflow: hidden;"><div style="display: inline-block; font-size: 16px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">(node 8 ...) :: []</div></div></div></foreignObject><text x="1039" y="136" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="16px">(node 8 ...) :: []</text></switch></g><ellipse cx="817" cy="503.63" rx="13.5" ry="13.625" fill="rgb(255, 255, 255)" stroke="rgb(0, 0, 0)" pointer-events="all"/><path d="M 817 517.25 L 817 562.67 M 817 526.33 L 790 526.33 M 817 526.33 L 844 526.33 M 817 562.67 L 790 599 M 817 562.67 L 844 599" fill="none" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe flex-start; justify-content: unsafe center; width: 1px; height: 1px; padding-top: 606px; margin-left: 817px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: nowrap;">Actor</div></div></div></foreignObject><text x="817" y="618" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">Actor</text></switch></g><path d="M 788 361 L 1113 361 L 1113 472 L 970.5 472 L 1044.75 502 L 950.5 472 L 788 472 Z" fill="rgb(255, 255, 255)" stroke="rgb(0, 0, 0)" stroke-miterlimit="10" transform="translate(950.5,0)scale(-1,1)translate(-950.5,0)" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe center; width: 323px; height: 1px; padding-top: 417px; margin-left: 789px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: center;"><div style="display: inline-block; font-size: 12px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;">stackInvariantを見ることで、tree0からtree3に行くには、右、右、左の経路をたどっていることがわかる！</div></div></div></foreignObject><text x="951" y="420" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="12px" text-anchor="middle">stackInvariantを見ることで、tree0からtree3に行くには、右、右、左の経路をたどっていることがわかる！</text></switch></g><rect x="1131" y="351" width="329" height="144" rx="21.6" ry="21.6" fill="rgb(255, 255, 255)" stroke="rgb(0, 0, 0)" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; 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text-align: left;"><div xmlns="http://www.w3.org/1999/xhtml" style="display: flex; align-items: unsafe center; justify-content: unsafe flex-start; width: 533px; height: 1px; padding-top: 465px; margin-left: 718px;"><div data-drawio-colors="color: rgb(0, 0, 0); " style="box-sizing: border-box; font-size: 0px; text-align: left;"><div style="display: inline-block; font-size: 19px; font-family: Helvetica; color: rgb(0, 0, 0); line-height: 1.2; pointer-events: all; white-space: normal; overflow-wrap: normal;"><font style="font-size: 19px;">・stackはRedBlackTreeのリストで構成される<br style="font-size: 19px;" />・stackは辿った木を保存している（今回は8→14を辿る）<br style="font-size: 19px;" />・stackInvariantは辿った木の正しい経路が保存されている<br style="font-size: 19px;" />・8→14は右、右、左の順番に降りていけば辿り着くことが、stackInvariantを見るとわかる<br /><br style="font-size: 19px;" />stackとstackInvariantは対応していており、stackが木を辿った順番が正しい順番で構成されていることを示している</font></div></div></div></foreignObject><text x="718" y="471" fill="rgb(0, 0, 0)" font-family="Helvetica" font-size="19px">・stackはRedBlackTreeのリストで構成される...</text></switch></g><rect x="704" y="33" width="62" height="26" fill="none" stroke="none" pointer-events="all"/><path d="M 704 33 M 766 33 M 766 59 M 704 59" fill="none" stroke="rgb(0, 0, 0)" stroke-linecap="square" stroke-miterlimit="10" pointer-events="all"/><g transform="translate(-0.5 -0.5)"><switch><foreignObject pointer-events="none" width="100%" height="100%" requiredFeatures="http://www.w3.org/TR/SVG11/feature#Extensibility" style="overflow: visible; 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.pl-mi2{background-color:var(--color-prettylights-syntax-markup-ignored-bg);color:var(--color-prettylights-syntax-markup-ignored-text)}div#\:\$p>svg>foreignObject>section .pl-mdr{color:var(--color-prettylights-syntax-meta-diff-range);font-weight:700}div#\:\$p>svg>foreignObject>section .pl-ba{color:var(--color-prettylights-syntax-brackethighlighter-angle)}div#\:\$p>svg>foreignObject>section .pl-sg{color:var(--color-prettylights-syntax-sublimelinter-gutter-mark)}div#\:\$p>svg>foreignObject>section .pl-corl{color:var(--color-prettylights-syntax-constant-other-reference-link);text-decoration:underline}div#\:\$p>svg>foreignObject>section g-emoji{display:inline-block;font-family:Apple Color Emoji,Segoe UI Emoji,Segoe UI Symbol;font-size:1em;font-style:normal!important;font-weight:var(--base-text-weight-normal,400);line-height:1;min-width:1ch;vertical-align:-.075em}div#\:\$p>svg>foreignObject>section g-emoji img{height:1em;width:1em}div#\:\$p>svg>foreignObject>section .task-list-item{list-style-type:none}div#\:\$p>svg>foreignObject>section .task-list-item label{font-weight:var(--base-text-weight-normal,400)}div#\:\$p>svg>foreignObject>section .task-list-item.enabled label{cursor:pointer}div#\:\$p>svg>foreignObject>section .task-list-item+.task-list-item{margin-top:4px}div#\:\$p>svg>foreignObject>section .task-list-item .handle{display:none}div#\:\$p>svg>foreignObject>section .task-list-item-checkbox{margin:0 .2em .25em -1.4em;vertical-align:middle}div#\:\$p>svg>foreignObject>section .contains-task-list:dir(rtl) .task-list-item-checkbox{margin:0 -1.6em .25em .2em}div#\:\$p>svg>foreignObject>section .contains-task-list{position:relative}div#\:\$p>svg>foreignObject>section .contains-task-list:focus-within .task-list-item-convert-container,div#\:\$p>svg>foreignObject>section .contains-task-list:hover .task-list-item-convert-container{clip:auto;display:block;height:24px;overflow:visible;width:auto}div#\:\$p>svg>foreignObject>section ::-webkit-calendar-picker-indicator{filter:invert(50%)}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1){color:var(--h1-color);font-size:1.6em}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1),div#\:\$p>svg>foreignObject>section :is(h2,marp-h2){border-bottom:none}div#\:\$p>svg>foreignObject>section :is(h2,marp-h2){font-size:1.3em}div#\:\$p>svg>foreignObject>section :is(h3,marp-h3){font-size:1.1em}div#\:\$p>svg>foreignObject>section :is(h4,marp-h4){font-size:1.05em}div#\:\$p>svg>foreignObject>section :is(h5,marp-h5){font-size:1em}div#\:\$p>svg>foreignObject>section :is(h6,marp-h6){font-size:.9em}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1) strong,div#\:\$p>svg>foreignObject>section :is(h2,marp-h2) strong,div#\:\$p>svg>foreignObject>section :is(h3,marp-h3) strong,div#\:\$p>svg>foreignObject>section :is(h4,marp-h4) strong,div#\:\$p>svg>foreignObject>section :is(h5,marp-h5) strong,div#\:\$p>svg>foreignObject>section :is(h6,marp-h6) strong{color:var(--heading-strong-color);font-weight:inherit}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h2,marp-h2)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h3,marp-h3)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h4,marp-h4)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h5,marp-h5)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h6,marp-h6)::part(auto-scaling){max-height:563px}div#\:\$p>svg>foreignObject>section hr{height:0;padding-top:.25em}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre){border:1px solid var(--color-border-default);line-height:1.15;overflow:visible}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre)::part(auto-scaling){max-height:529px}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs){color:var(--color-prettylights-syntax-storage-modifier-import)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-doctag),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-keyword),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-meta .hljs-keyword),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-template-tag),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-template-variable),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-type),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-variable.language_){color:var(--color-prettylights-syntax-keyword)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title.class_),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title.class_.inherited__),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title.function_){color:var(--color-prettylights-syntax-entity)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-attr),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-attribute),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-literal),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-meta),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-number),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-operator),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-attr),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-class),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-id),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-variable){color:var(--color-prettylights-syntax-constant)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-meta .hljs-string),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-regexp),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-string){color:var(--color-prettylights-syntax-string)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-built_in),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-symbol){color:var(--color-prettylights-syntax-variable)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-code),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-comment),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-formula){color:var(--color-prettylights-syntax-comment)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-name),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-quote),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-pseudo),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-tag){color:var(--color-prettylights-syntax-entity-tag)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-subst){color:var(--color-prettylights-syntax-storage-modifier-import)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-section){color:var(--color-prettylights-syntax-markup-heading);font-weight:700}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-bullet){color:var(--color-prettylights-syntax-markup-list)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-emphasis){color:var(--color-prettylights-syntax-markup-italic);font-style:italic}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-strong){color:var(--color-prettylights-syntax-markup-bold);font-weight:700}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-addition){background-color:var(--color-prettylights-syntax-markup-inserted-bg);color:var(--color-prettylights-syntax-markup-inserted-text)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-deletion){background-color:var(--color-prettylights-syntax-markup-deleted-bg);color:var(--color-prettylights-syntax-markup-deleted-text)}div#\:\$p>svg>foreignObject>section footer,div#\:\$p>svg>foreignObject>section header{color:var(--header-footer-color);font-size:18px;left:30px;margin:0;position:absolute}div#\:\$p>svg>foreignObject>section header{top:21px}div#\:\$p>svg>foreignObject>section footer{bottom:21px}div#\:\$p>svg>foreignObject>section{--h1-color:#246;--header-footer-color:hsla(0,0%,40%,.75);--heading-strong-color:#48c;--paginate-color:#777;align-items:stretch;display:flex;flex-flow:column nowrap;font-size:29px;height:720px;justify-content:center;padding:78.5px;width:1280px}div#\:\$p>svg>foreignObject>section{--marpit-root-font-size:29px}div#\:\$p>svg>foreignObject>section:where(.invert){--h1-color:#cee7ff;--header-footer-color:hsla(0,0%,60%,.75);--heading-strong-color:#7bf;--paginate-color:#999}div#\:\$p>svg>foreignObject>section>:last-child,div#\:\$p>svg>foreignObject>section[data-footer]>:nth-last-child(2){margin-bottom:0}div#\:\$p>svg>foreignObject>section>:first-child,div#\:\$p>svg>foreignObject>section>header:first-child+*{margin-top:0}div#\:\$p>svg>foreignObject>section:after{bottom:21px;color:var(--paginate-color);font-size:24px;padding:0;position:absolute;right:30px}div#\:\$p>svg>foreignObject>section:after{--marpit-root-font-size:24px}div#\:\$p>svg>foreignObject>section[data-color] :is(h1,marp-h1),div#\:\$p>svg>foreignObject>section[data-color] :is(h2,marp-h2),div#\:\$p>svg>foreignObject>section[data-color] :is(h3,marp-h3),div#\:\$p>svg>foreignObject>section[data-color] :is(h4,marp-h4),div#\:\$p>svg>foreignObject>section[data-color] :is(h5,marp-h5),div#\:\$p>svg>foreignObject>section[data-color] :is(h6,marp-h6){color:currentcolor}div#\:\$p>svg>foreignObject>:where(section){container-type:size}div#\:\$p>svg>foreignObject>section img[data-marp-twemoji]{background:transparent;height:1em;margin:0 .05em 0 .1em;vertical-align:-.1em;width:1em} + +/* @theme cr */div#\:\$p>svg>foreignObject>section{background-image:url('./logo.svg');background-repeat:no-repeat;background-position:right calc(var(--marpit-root-font-size, 1rem) * 2) bottom calc(var(--marpit-root-font-size, 1rem) * 0.6);background-size:12%}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1){color:#828db2;font-size:calc(var(--marpit-root-font-size, 1rem) * 2.1)}div#\:\$p>svg>foreignObject>section :is(h2,marp-h2){background-color:#828db2;color:#fff;padding:calc(var(--marpit-root-font-size, 1rem) * 0.3);font-size:calc(var(--marpit-root-font-size, 1rem) * 2.1)}div#\:\$p>svg>foreignObject>section p{font-size:calc(var(--marpit-root-font-size, 1rem) * 1.3)}div#\:\$p>svg>foreignObject>section ol,div#\:\$p>svg>foreignObject>section ul{font-size:calc(var(--marpit-root-font-size, 1rem) * 1.2)}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]{columns:initial!important;display:block!important;padding:0!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]:after,div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]:before,div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=content]:after,div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=content]:before{display:none!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container]{all:initial;display:flex;flex-direction:row;height:100%;overflow:hidden;width:100%}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container][data-marpit-advanced-background-direction=vertical]{flex-direction:column}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background][data-marpit-advanced-background-split]>div[data-marpit-advanced-background-container]{width:var(--marpit-advanced-background-split,50%)}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background][data-marpit-advanced-background-split=right]>div[data-marpit-advanced-background-container]{margin-left:calc(100% - var(--marpit-advanced-background-split, 50%))}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container]>figure{all:initial;background-position:center;background-repeat:no-repeat;background-size:cover;flex:auto;margin:0}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container]>figure>figcaption{position:absolute;border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;white-space:nowrap;width:1px}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=content],div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=pseudo]{background:transparent!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=pseudo],div#\:\$p>svg[data-marpit-svg]>foreignObject[data-marpit-advanced-background=pseudo]{pointer-events:none!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background-split]{width:100%;height:100%}</style></head><body><div class="bespoke-marp-osc"><button data-bespoke-marp-osc="prev" tabindex="-1" title="Previous slide">Previous slide</button><span data-bespoke-marp-osc="page"></span><button data-bespoke-marp-osc="next" tabindex="-1" title="Next slide">Next slide</button><button data-bespoke-marp-osc="fullscreen" tabindex="-1" title="Toggle fullscreen (f)">Toggle fullscreen</button><button data-bespoke-marp-osc="presenter" tabindex="-1" title="Open presenter view (p)">Open presenter view</button></div><div id=":$p"><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="1" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="1" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h1 id="gearsagda-%E3%81%AB%E3%82%88%E3%82%8B-red-black-tree-%E3%81%AE%E8%A8%BC%E6%98%8E%E4%BB%98%E3%81%8D%E5%AE%9F%E8%A3%85%E3%81%AB%E9%96%A2%E3%81%99%E3%82%8B%E7%A0%94%E7%A9%B6">GearsAgda による Red Black Tree の証明付き実装に関する研究</h1> + +<p>琉球大学 工学部工学科 知能情報コース<br /> +並列信頼研究室</p> +<p>森 逸汰</p> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="2" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="2" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="gearsagda%E3%82%92%E4%BD%BF%E3%81%A3%E3%81%9F%E3%83%97%E3%83%AD%E3%82%B0%E3%83%A9%E3%83%A0%E3%81%AE%E8%A8%BC%E6%98%8E">GearsAgdaを使ったプログラムの証明</h2> +<ul> +<li>GearsAgdaは、gotoからgotoまでのコードの一部を指す +<ul> +<li>例えば,赤黒木を一段下がるコード</li> +</ul> +</li> +<li>関数呼び出しを含まない特徴を持つ</li> +<li>GearsAgdaは、入力のInvariantから出力のInvariantを生成する</li> +<li>LoopConnectorでcodeGearを接続する +<ul> +<li>ループの停止性とInvariantの接続を示す</li> +</ul> +</li> +<li>GearsAgdaのコードはGearOSの実行単位であるcodeGearに直接対応する</li> +</ul> +<p>これにより、codeGearの正しさを証明できる</p> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="3" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="3" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="binarytree%E3%81%AE%E5%AE%9A%E7%BE%A9">BinaryTreeの定義</h2> +<ul> +<li>bt型を定義している</li> +<li>Aは任意の型になっている</li> +<li>leafとnodeはコンストラクタになっている</li> +<li>nodeは左の子と右の子を再帰的に持つ</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">data bt {n : Level} (A : Set n) : Set n where + leaf : bt A + node : (key : N) → (value : A) → + (left : bt A ) → (right : bt A ) → bt A +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="4" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="4" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="color%E3%81%AE%E5%AE%9A%E7%BE%A9">Colorの定義</h2> +<ul> +<li>Color型を定義している</li> +<li>Colorは赤と黒の場合に分けられる</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">data Color : Set where + Red : Color + Black : Color +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="5" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="5" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="agda%E3%81%A7redblacktree%E3%81%AE%E6%A7%8B%E9%80%A0%E3%82%92%E4%BD%9C%E3%82%8B">AgdaでRedBlackTreeの構造を作る</h2> +<ul> +<li>valueにColorを組として渡すように設計した。</li> +<li>これにより、BinaryTreeのコードを応用することが可能になる。</li> +<li>BinaryTreeのコードを応用してRedBlackTreeを構成している。</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">RBTreeTest : bt (Color ∧ N) +RBTreeTest = node <span class="hljs-number">8</span> ⟪ Black , <span class="hljs-number">200</span> ⟫ (node <span class="hljs-number">5</span> ⟪ Red , <span class="hljs-number">100</span> ⟫ (leaf) (leaf)) (node <span class="hljs-number">10</span> ⟪ Red , <span class="hljs-number">300</span> ⟫ (leaf) (leaf)) +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="6" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="6" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="insert%E3%81%A8find%E3%81%AE%E9%96%A2%E4%BF%82">Insertとfindの関係</h2> +<p>木の操作としてInsertなどを実装したい</p> +<ul> +<li>Insert操作では、挿入する場所を探す必要がある。</li> +<li>そのため、指定したKeyと一致するノードを探し出すfind操作が必要になる。</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="7" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="7" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="findrbt%E3%81%AE%E5%8B%95%E4%BD%9C%E3%81%A8%E7%9B%AE%E7%9A%84">findRBTの動作と目的</h2> +<ul> +<li>動作 +<ul> +<li>RedBlackTreeを辿っていき、指定されたKeyを探す</li> +</ul> +</li> +<li>目的 +<ul> +<li>指定されたKeyを見つけるまでに辿った正しい経路をstackに保存して返したい</li> +</ul> +</li> +</ul> +<p>木を保存したstackが正しい順番で構成されているかを証明する必要がある</p> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="8" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="8" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="invariant%E3%82%92%E7%94%A8%E3%81%84%E3%81%9F%E8%A8%BC%E6%98%8E">Invariantを用いた証明</h2> +<ul> +<li>Invariantとはループや再帰で不変である条件や命題<br /> +例えば... +<ul> +<li>「stackが辿った木の順番通りに保存されている」ことを示すInvariantがある</li> +<li>木の操作ではループや再帰などの処理が行われる</li> +<li>処理の後にInvariantが正しく保持されていることを示すことで、stackが木の順番通りに構成されていることが証明できる。</li> +</ul> +</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="9" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="9" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="stack%E3%81%A8stackinvariant%E3%81%AE%E8%AA%AC%E6%98%8E">stackとstackInvariantの説明</h2> +<p><img src="figs/stack4.svg" alt="Width 50" /></p> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="10" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="10" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="stackinvariant%E5%9E%8B%E3%81%AE%E5%AE%9A%E7%BE%A9">stackInvariant型の定義</h2> +<ul> +<li>s-nil : 初期化状態で、stackに元の木以外は何も入っていない</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (<span class="hljs-built_in">stack</span> : List (bt A)) → Set n where +s-nil : {tree0 : bt A} → stackInvariant key tree0 <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree0 ∷ [])</span> +</code></pre> +<ul> +<li>s-right : 右側を辿った木がstackに入っている</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">s-right : (tree tree0 tree₁ : bt A) → {st : List (bt A)} + → key₁ < key → stackInvariant <span class="hljs-title function_">key</span> <span class="hljs-params">(node key₁ v1 tree₁ tree)</span> tree0 st → stackInvariant key tree <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree ∷ st)</span> +</code></pre> +<ul> +<li>s-left : 左側を辿った木がstackに入っている</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">s-left : (tree₁ tree0 tree : bt A) → {st : List (bt A)} + → key < key₁ → stackInvariant <span class="hljs-title function_">key</span> <span class="hljs-params">(node key₁ v1 tree₁ tree)</span> tree0 st → stackInvariant key tree₁ <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree₁ ∷ st)</span> +</code></pre> +<p>stackがtreeを辿っていることを示す命題であることがわかる</p> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="11" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="11" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="stackinvariant%E3%81%AE%E5%80%A4%E3%81%AE%E4%BE%8B">stackInvariantの値の例</h2> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c"> s-left (<span class="hljs-number">14</span> < <span class="hljs-number">15</span>)(s-right (<span class="hljs-number">10</span> < <span class="hljs-number">15</span>)(s-right (<span class="hljs-number">8</span> < <span class="hljs-number">10</span>)) s-nil) +</code></pre> +<ul> +<li>stackが辿ったtreeを保存していることを示す証明になっている</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="12" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="12" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="findrbt%E3%81%AE%E5%AE%9F%E8%A3%85%E5%AE%9A%E7%BE%A9%E9%83%A8%E5%88%86">findRBTの実装:定義部分</h2> +<ul> +<li>findRBTは、treeとstackとInvariantを受け取って、Loopであるnext又は、exitに渡す</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c"> findRBT : {n m : Level} {A : Set n} {t : Set m} + → (key : N) + → (tree tree0 : bt (Color ∧ A) ) + → (<span class="hljs-built_in">stack</span> : List (bt (Color ∧ A))) + → RBtreeInvariant tree ∧ stackInvariant key tree tree0 <span class="hljs-built_in">stack</span> + → (next : ... → t) + → (<span class="hljs-built_in">exit</span> : ... → t) + → t +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="13" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="13" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="findrbt%E3%81%AE%E5%AE%9F%E8%A3%85%E5%AE%9A%E7%BE%A9%E9%83%A8%E5%88%86next">findRBTの実装:定義部分next</h2> +<ul> +<li>次のLoopに接続する部分</li> +<li>新しいstackとstackInvariantが作られている</li> +<li>nextには、次の木の高さは元の木の高さより低くなることが条件として含まれている</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c"> → (next : (tree1 : bt (Color ∧ A) ) + → (<span class="hljs-built_in">stack</span> : List (bt (Color ∧ A))) + → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 <span class="hljs-built_in">stack</span> + → bt-depth tree1 < bt-depth tree + → t ) +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="14" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="14" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="findrbt%E3%81%AE%E5%AE%9F%E8%A3%85%E5%AE%9A%E7%BE%A9%E9%83%A8%E5%88%86exit">findRBTの実装:定義部分exit</h2> +<ul> +<li>leafに到達するか指定したKeyのノードが見つかってLoopから脱出する部分</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c"> → (<span class="hljs-built_in">exit</span> : (tree1 : bt (Color ∧ A)) + → (<span class="hljs-built_in">stack</span> : List (bt (Color ∧ A))) + → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 <span class="hljs-built_in">stack</span> + → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) + → t ) + → t +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="15" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="15" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="loopconnector%E3%81%AFfindrbt%E3%81%AEnext%E3%82%92%E6%8E%A5%E7%B6%9A%E3%81%99%E3%82%8B">LoopConnectorはfindRBTのnextを接続する</h2> +<ul> +<li>LoopConnectorは引数として<br /> +(loop : (r : Index) → Invraiant r<br /> +→ (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t)をとる</li> +<li>ここにcodeGearが入り、codeGearにあるtreeやInvariantを接続できる</li> +<li>自然数reduceが減少することによりLoopの停止性を証明できる</li> +<li>LoopConnectorで接続することによりInvariantの接続と停止性が証明される</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="16" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="16" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="terminatingloops%E3%81%AE%E5%AE%9F%E9%9A%9B%E3%81%AE%E5%9E%8B">TerminatingLoopSの実際の型</h2> +<ul> +<li>Indexにはtreeとstackが入る</li> +<li>Invariantが(Set m)の形になるため任意の命題を入れることができる</li> +<li>reduceはIndexから減少する自然数を得る関数が入る</li> +<li>TermnatingLoopSの証明は実際に記述されている</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) + → (r : Index) → (p : Invraiant r) + → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="17" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="17" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="terminatingloops%E3%82%92%E4%BD%BF%E3%81%A3%E3%81%9F%E6%8E%A5%E7%B6%9A">TerminatingLoopSを使った接続</h2> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">findTest {n} {m} {A} {t} k tr0 rb0 <span class="hljs-built_in">exit</span> = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) + Index + {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant k (proj1 p) tr0 (proj2 p) } + Invariant + (λ p → bt-depth (proj1 p)) ⟪ tr0 , tr0 ∷ [] ⟫ ⟪ rb0 , s-nil ⟫ + reduce 初期値 + $ λ p RBP loop → findRBT k (proj1 p) tr0 (proj2 p) RBP (λ t1 s1 P2 lt1 → loop ⟪ t1 , s1 ⟫ P2 lt1 ) + GearsAgdaの呼び出し + $ λ tr1 st P2 O → <span class="hljs-built_in">exit</span> tr1 st P2 O + GearsAgdaからの脱出 +</code></pre> +<ul> +<li>LoopConnectorでcodeGearが接続されていることが確認できる。</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="18" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="18" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="findrbt%E3%81%AE%E5%AE%9F%E8%A1%8C%E7%B5%90%E6%9E%9C">findRBTの実行結果</h2> +<ul> +<li>stackとstackInvariantが出力されている</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">--指定したKeyの値 <span class="hljs-number">14</span> (木の中に存在するKey) +record + { tree = node <span class="hljs-number">14</span> ⟪ Red , <span class="hljs-number">1400</span> ⟫ leaf leaf + ; <span class="hljs-built_in">stack</span> = + node <span class="hljs-number">14</span> ⟪ Red , <span class="hljs-number">1400</span> ⟫ leaf leaf + :: node <span class="hljs-number">15</span> ⟪ Black , <span class="hljs-number">1500</span> ⟫ (node <span class="hljs-number">14</span> ⟪ Red , <span class="hljs-number">1400</span> ⟫ leaf leaf)leaf + :: node <span class="hljs-number">10</span> ⟪ Red , <span class="hljs-number">1000</span> ⟫ leaf (node <span class="hljs-number">15</span> ⟪ Black , <span class="hljs-number">1500</span> ⟫ (node <span class="hljs-number">14</span> ⟪ Red , <span class="hljs-number">1400</span> ⟫ leaf leaf) leaf) + :: node <span class="hljs-number">8</span> ⟪ Black , <span class="hljs-number">800</span> ⟫ + (node <span class="hljs-number">5</span> ⟪ Red , <span class="hljs-number">500</span> ⟫ (node <span class="hljs-number">2</span> ⟪ Black , <span class="hljs-number">200</span> ⟫ leaf leaf) + (node <span class="hljs-number">6</span> ⟪ Black , <span class="hljs-number">600</span> ⟫ leaf leaf)) + (node <span class="hljs-number">10</span> ⟪ Red , <span class="hljs-number">1000</span> ⟫ leaf + (node <span class="hljs-number">15</span> ⟪ Black , <span class="hljs-number">1500</span> ⟫ (node <span class="hljs-number">14</span> ⟪ Red , <span class="hljs-number">1400</span> ⟫ leaf leaf) leaf)) + :: [] + ; ti = rb-single <span class="hljs-number">14</span> <span class="hljs-number">1400</span> + ; si = + s-left + (s≤s(s≤s (s≤s(s≤s(s≤s (s≤s(s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s z≤n))))))))))))))) + (s-right (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s z≤n))))))))))) + (s-right (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s z≤n))))))))) s-nil)) + ; <span class="hljs-built_in">exit</span> = case2 refl + } +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="19" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="19" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="findrbt%E3%81%AE%E5%AE%9F%E8%A1%8C%E7%B5%90%E6%9E%9C%E3%81%8B%E3%82%89">findRBTの実行結果から</h2> +<ul> +<li>stackInvariantを導出できたことにより、stackが正しい順序を保っていることが証明できる</li> +</ul> +<p>このことから、「目的の木への正しい経路を保存したstack」を返したことになるため、findRBTの動作が正しいことが証明できたと言える。</p> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="20" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="20" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="%E3%81%BE%E3%81%A8%E3%82%81">まとめ</h2> +<ul> +<li>stackInvariantを用いることで、findRBTの正しさを証明することができた。</li> +<li>LoopConnectorを用いることで、GearsOSで複雑なアルゴリズムを実装することが可能になった。</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="21" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="21" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="%E4%BB%8A%E5%BE%8C%E3%81%AE%E8%AA%B2%E9%A1%8C">今後の課題</h2> +<ul> +<li>Insertなどの証明を完成させる +<ul> +<li>Insert操作の証明は未完成だが、大枠を掴むことができている。</li> +<li>Insertは木がバランスするところまで置き換えながら上昇する</li> +<li>必要ならばParent GrandParent Uncleで木の回転を行う</li> +<li>ルートまで木を置き換えながら上昇する</li> +<li>これらの部分は途中までかけている</li> +</ul> +</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="22" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="22" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="%E3%81%8A%E3%81%BE%E3%81%91">おまけ</h2> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="23" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="background" data-marpit-advanced-background-split="right"><div data-marpit-advanced-background-container="true" data-marpit-advanced-background-direction="horizontal"><figure style="background-image:url("figs/RedBlackTree.png");background-size:80%;"><figcaption>:</figcaption></figure></div></section></foreignObject><foreignObject width="50%" height="720"><section id="23" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="23" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="content" data-marpit-advanced-background-split="right"> +<h2 id="red-black-tree">Red Black Tree</h2> + +<ul> +<li>バランスした平衡二分木の一種</li> +<li>木はnodeとleafからなる</li> +<li>nodeにはkeyという数値が与えられる</li> +</ul> +</section> +</foreignObject><foreignObject width="1280" height="720" data-marpit-advanced-background="pseudo"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="23" style="" data-marpit-pagination-total="44" data-marpit-advanced-background="pseudo" data-marpit-advanced-background-split="right"></section></foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="24" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="background" data-marpit-advanced-background-split="right"><div data-marpit-advanced-background-container="true" data-marpit-advanced-background-direction="horizontal"><figure style="background-image:url("figs/RedBlackTree.png");background-size:80%;"><figcaption>:</figcaption></figure></div></section></foreignObject><foreignObject width="50%" height="720"><section id="24" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="24" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="content" data-marpit-advanced-background-split="right"> +<h2 id="red-black-tree-1">Red Black Tree</h2> + +<ul> +<li>nodeの配置はKeyの大小関係によって決定する</li> +<li>左の子のKey < 親のKey < 右の子のKey という大小関係を持つ</li> +</ul> +<p>これらは、BinaryTreeと同じ性質となっている。</p> +</section> +</foreignObject><foreignObject width="1280" height="720" data-marpit-advanced-background="pseudo"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="24" style="" data-marpit-pagination-total="44" data-marpit-advanced-background="pseudo" data-marpit-advanced-background-split="right"></section></foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="25" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="background" data-marpit-advanced-background-split="right"><div data-marpit-advanced-background-container="true" data-marpit-advanced-background-direction="horizontal"><figure style="background-image:url("figs/RedBlackTree.png");background-size:70%;"><figcaption>:</figcaption></figure></div></section></foreignObject><foreignObject width="50%" height="720"><section id="25" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="25" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="content" data-marpit-advanced-background-split="right"> +<h2 id="redblacktree%E3%81%AE%E7%89%B9%E5%BE%B4">RedBlackTreeの特徴</h2> + +<ul> +<li>各ノードは赤か黒の色を持つ</li> +<li>赤のノードは赤のノードを子に持たない</li> +<li>リーフノードはすべて黒である</li> +</ul> +</section> +</foreignObject><foreignObject width="1280" height="720" data-marpit-advanced-background="pseudo"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="25" style="" data-marpit-pagination-total="44" data-marpit-advanced-background="pseudo" data-marpit-advanced-background-split="right"></section></foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="26" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="background" data-marpit-advanced-background-split="right"><div data-marpit-advanced-background-container="true" data-marpit-advanced-background-direction="horizontal"><figure style="background-image:url("figs/RedBlackTree.png");background-size:80%;"><figcaption>:</figcaption></figure></div></section></foreignObject><foreignObject width="50%" height="720"><section id="26" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="26" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="content" data-marpit-advanced-background-split="right"> +<h2 id="redblacktree%E3%81%AE%E7%89%B9%E5%BE%B4-1">RedBlackTreeの特徴</h2> + +<ul> +<li>任意のノードについて、そのノードから子孫の葉までの道に含まれる黒いノードの数は、選んだ葉によらず一定である。</li> +</ul> +</section> +</foreignObject><foreignObject width="1280" height="720" data-marpit-advanced-background="pseudo"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="26" style="" data-marpit-pagination-total="44" data-marpit-advanced-background="pseudo" data-marpit-advanced-background-split="right"></section></foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="27" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="background" data-marpit-advanced-background-split="right"><div data-marpit-advanced-background-container="true" data-marpit-advanced-background-direction="horizontal"><figure style="background-image:url("figs/RedBlackTree.png");background-size:80%;"><figcaption>:</figcaption></figure></div></section></foreignObject><foreignObject width="50%" height="720"><section id="27" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="27" style="--paginate:true;--theme:cr;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="44" data-marpit-advanced-background="content" data-marpit-advanced-background-split="right"> +<h2 id="redblacktree%E3%81%AE%E7%89%B9%E5%BE%B4-2">RedBlackTreeの特徴</h2> + +<ul> +<li>根から葉まで道で最長のものの長さは、根から葉までの道で最短のものの長さの二倍を超えない。</li> +</ul> +<p>このことから、RedBlackTreeがある程度の平衡性を持つことがわかる。</p> +</section> +</foreignObject><foreignObject width="1280" height="720" data-marpit-advanced-background="pseudo"><section data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="27" style="" data-marpit-pagination-total="44" data-marpit-advanced-background="pseudo" data-marpit-advanced-background-split="right"></section></foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="28" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="28" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="%E5%AE%9A%E7%90%86%E8%A8%BC%E6%98%8E%E6%94%AF%E6%8F%B4%E7%B3%BB%E8%A8%80%E8%AA%9Eagda%E3%81%A8%E3%81%AF">定理証明支援系言語Agdaとは</h2> +<ul> +<li>Curry-Howard同型対応により、型付きラムダ計算と命題が１対１で対応するため、Agdaで書かれたプログラムは証明を書くことができる</li> +<li>Curry-Howard同型対応とは、型と命題が一致するという論理学の性質</li> +<li>型が正しいことを示すことで、命題が正しいと言える</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="29" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="29" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="stackinvariant%E3%81%AE%E8%A9%B3%E7%B4%B0">stackInvariantの詳細</h2> +<ul> +<li>s-nil : stackの初期化状態であり、stackに元の木以外は何も入っていないことを示す</li> +<li>s-right,s-left : １つ前の木からみて、どちら側の木をstackに積んだかを示している</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (<span class="hljs-built_in">stack</span> : List (bt A)) → Set n where + s-nil : {tree0 : bt A} → stackInvariant key tree0 <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree0 ∷ [])</span> + s-right : <span class="hljs-params">(tree tree0 tree₁ : bt A)</span> → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key₁ < key → stackInvariant <span class="hljs-title function_">key</span> <span class="hljs-params">(node key₁ v1 tree₁ tree)</span> tree0 st → stackInvariant key tree <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree ∷ st)</span> + s-left : <span class="hljs-params">(tree₁ tree0 tree : bt A)</span> → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key < key₁ → stackInvariant <span class="hljs-title function_">key</span> <span class="hljs-params">(node key₁ v1 tree₁ tree)</span> tree0 st → stackInvariant key tree₁ <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree₁ ∷ st)</span> +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="30" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="30" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="stackinvariant%E3%81%AE%E8%A9%B3%E7%B4%B0-1">stackInvariantの詳細</h2> +<p>s-right,s-leftでは、</p> +<ul> +<li>keyの大小関係を引数として渡す必要がある</li> +<li>1つ前の木を積んだ時のstackInvariantを引数として渡す必要がある</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (<span class="hljs-built_in">stack</span> : List (bt A)) → Set n where + s-nil : {tree0 : bt A} → stackInvariant key tree0 <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree0 ∷ [])</span> + s-right : <span class="hljs-params">(tree tree0 tree₁ : bt A)</span> → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key₁ < key → stackInvariant <span class="hljs-title function_">key</span> <span class="hljs-params">(node key₁ v1 tree₁ tree)</span> tree0 st → stackInvariant key tree <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree ∷ st)</span> + s-left : <span class="hljs-params">(tree₁ tree0 tree : bt A)</span> → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key < key₁ → stackInvariant <span class="hljs-title function_">key</span> <span class="hljs-params">(node key₁ v1 tree₁ tree)</span> tree0 st → stackInvariant key tree₁ <span class="hljs-title function_">tree0</span> <span class="hljs-params">(tree₁ ∷ st)</span> +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="31" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="31" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="%E2%89%A4-%E3%81%AE%E8%AA%AC%E6%98%8E"><em>≤</em> の説明</h2> +<ul> +<li>自然数の大小関係を表す</li> +<li>以下の２通りのパターンですべてを記述することができる</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c"> data _≤_ : (a b : ℕ) → Set Where + z≤n : {n : ℕ } → zero ≤ n + s≤s : {a b : ℕ } → a ≤ b → suc a ≤ suc b +</code></pre> +<p>例えば 3 ≤ 6 を表したいときは、s≤s (s≤s (s≤s z≤n)) と書ける<br /> +これは、Keyの大小関係を証明する際などに使用する。</p> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="32" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="32" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="terminatingloops">TerminatingLoopS</h2> +<ul> +<li>t : 不定の型。codeGearとの接続のために使用する</li> +<li>Index : 任意の型</li> +<li>Invariant : ループで不変な条件</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) + → (r : Index) → (p : Invraiant r) + → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="33" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="33" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="terminatingloops-1">TerminatingLoopS</h2> +<ul> +<li>reduce : ループで減少していく数値(ループの停止性を示すことができる)</li> +<li>r , p : IndexとInvariantの初期値</li> +<li>loop : codeGearに渡すnextの部分。reduceが減っていることの証明を引数として受け取る。tを返す</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) + → (r : Index) → (p : Invraiant r) + → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="34" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="34" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="terminatingloops%E3%82%92%E4%BD%BF%E3%81%A3%E3%81%9F%E6%8E%A5%E7%B6%9A-1">TerminatingLoopSを使った接続</h2> +<ul> +<li>Index部分 : RedBlackTreeとstackを渡している。</li> +<li>Invariant部分 : RBtreeInvariantとstackInvariantを渡している。</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">findTest {n} {m} {A} {t} k tr0 rb0 <span class="hljs-built_in">exit</span> = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) + {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant k (proj1 p) tr0 (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tr0 , tr0 ∷ [] ⟫ ⟪ rb0 , s-nil ⟫ + $ λ p RBP loop → findRBT k (proj1 p) tr0 (proj2 p) RBP (λ t1 s1 P2 lt1 → loop ⟪ t1 , s1 ⟫ P2 lt1 ) + $ λ tr1 st P2 O → <span class="hljs-built_in">exit</span> tr1 st P2 O +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="35" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="35" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="terminatingloops%E3%82%92%E4%BD%BF%E3%81%A3%E3%81%9F%E6%8E%A5%E7%B6%9A-2">TerminatingLoopSを使った接続</h2> +<ul> +<li>reduce部分 : 木の深さを数値として渡している。</li> +<li>初期値部分 : 元の木と元の木のみが入っているstackを渡している。また、それぞれのInvariantを渡している。</li> +<li>loop部分 : findRBTを呼び出すことでnextとexitを接続している</li> +</ul> +<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c">findTest {n} {m} {A} {t} k tr0 rb0 <span class="hljs-built_in">exit</span> = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) + {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant k (proj1 p) tr0 (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tr0 , tr0 ∷ [] ⟫ ⟪ rb0 , s-nil ⟫ + $ λ p RBP loop → findRBT k (proj1 p) tr0 (proj2 p) RBP (λ t1 s1 P2 lt1 → loop ⟪ t1 , s1 ⟫ P2 lt1 ) + $ λ tr1 st P2 O → <span class="hljs-built_in">exit</span> tr1 st P2 O +</code></pre> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="36" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="36" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="redblacktree%E3%81%AE%E3%81%BB%E3%81%8B%E3%81%AEinvariant">RedBlackTreeのほかのInvariant</h2> +<ul> +<li>RBtreeInvariant<br /> +木がRedBlackTreeであることを示すInvariant</li> +<li>treeInvariant<br /> +木がBinaryTreeであることを示すInvariant</li> +<li>replacedTree<br /> +木が置き換えられたということを示すInvariant</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="37" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="37" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="redblacktree%E3%81%AB%E3%81%8A%E3%81%91%E3%82%8Binsert">RedBlackTreeにおけるInsert</h2> +<ul> +<li>RedBlackTreeのInsert操作では、黒の深さでバランスをとる必要がある。</li> +<li>バランスを行うためには、replaceやrotateなどの操作を行う。</li> +<li>対象のnodeから見た親、祖父、おじの色で6つの場合分けを行う必要がある。</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="38" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="38" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="insert%E3%81%AE%E3%82%B1%E3%83%BC%E3%82%B91">Insertのケース1</h2> +<ul> +<li>親が黒の場合<br /> +そのまま赤のnodeを挿入することができる</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="39" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="39" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="insert%E3%81%AE%E3%82%B1%E3%83%BC%E3%82%B92">Insertのケース2</h2> +<ul> +<li>親が赤 祖父が黒 おじが赤の場合</li> +<li>親が赤のためそのまま挿入できない</li> +<li>親とおじの色を黒に変更し、祖父の色を赤にすることでバランスする。</li> +<li>祖父の親が赤の場合、赤が連続してしまう</li> +<li>注目するノードを祖父に変更し1つ上の黒のノードまで戻る必要がある</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="40" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="40" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="insert%E3%81%AE%E3%82%B1%E3%83%BC%E3%82%B93">Insertのケース3</h2> +<ul> +<li>ノードが存在しない場合<br /> +そのまま挿入することができる</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="41" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="41" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="insert%E3%81%AE%E3%82%B1%E3%83%BC%E3%82%B94">Insertのケース4</h2> +<ul> +<li>親が赤 祖父、おじがいない場合</li> +<li>親が赤のためそのまま挿入できない<br /> +親の色を黒に変更し、子ノードとして挿入する</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="42" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="42" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="insert%E3%81%AE%E3%82%B1%E3%83%BC%E3%82%B95">Insertのケース5</h2> +<ul> +<li>親が赤 祖父が黒 おじが黒の場合且つ木が内側に曲がっている場合</li> +<li>親が赤なのでそのまま挿入できない</li> +<li>内側に曲がっていると回転がうまくできない</li> +<li>親と子の間で回転を行い、木を外側向きに修正する必要がある</li> +<li>修正した後の状態はケース６と同じであるため、ケース６に遷移する</li> +</ul> +</section> +</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="43" data-paginate="true" data-theme="cr" lang="ja-JP" data-marpit-pagination="43" style="--paginate:true;--theme:cr;" data-marpit-pagination-total="44"> +<h2 id="insert%E3%81%AE%E3%82%B1%E3%83%BC%E3%82%B96">Insertのケース6</h2> +<ul> +<li>親が赤 祖父が黒 おじが黒の場合且つ木が外側に曲がっている場合</li> +<li>親が赤なのでそのまま挿入できない</li> +<li>祖父ノードと親ノードの間で回転を行う</li> +<li>親ノードを黒、祖父ノードを赤に変更することで挿入する</li> +</ul> +</section> 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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Slide/slide3.md Thu Feb 08 15:46:29 2024 +0900 @@ -0,0 +1,422 @@ +--- +marp: false +theme: cr +paginate: true +--- + +# GearsAgda による Red Black Tree の証明付き実装に関する研究 + +<!-- +スピーカーノート +--> + +琉球大学 工学部工学科 知能情報コース +並列信頼研究室 + +森 逸汰 + +--- +## GearsAgdaを使ったプログラムの証明 +- GearsAgdaは、gotoからgotoまでのコードの一部を指す + - 例えば,赤黒木を一段下がるコード +- 関数呼び出しを含まない特徴を持つ +- GearsAgdaは、入力のInvariantから出力のInvariantを生成する +- LoopConnectorでcodeGearを接続する + - ループの停止性とInvariantの接続を示す +- GearsAgdaのコードはGearOSの実行単位であるcodeGearに直接対応する + +これにより、codeGearの正しさを証明できる + +--- + +## BinaryTreeの定義 +- bt型を定義している +- Aは任意の型になっている +- leafとnodeはコンストラクタになっている +- nodeは左の子と右の子を再帰的に持つ +```c +data bt {n : Level} (A : Set n) : Set n where + leaf : bt A + node : (key : N) → (value : A) → + (left : bt A ) → (right : bt A ) → bt A +``` + +--- +## Colorの定義 +- Color型を定義している +- Colorは赤と黒の場合に分けられる +```c +data Color : Set where + Red : Color + Black : Color +``` +--- + +## AgdaでRedBlackTreeの構造を作る +- valueにColorを組として渡すように設計した。 +- これにより、BinaryTreeのコードを応用することが可能になる。 +- BinaryTreeのコードを応用してRedBlackTreeを構成している。 +```c +RBTreeTest : bt (Color ∧ N) +RBTreeTest = node 8 ⟪ Black , 200 ⟫ (node 5 ⟪ Red , 100 ⟫ (leaf) (leaf)) (node 10 ⟪ Red , 300 ⟫ (leaf) (leaf)) +``` +--- + +## Insertとfindの関係 +木の操作としてInsertなどを実装したい +- Insert操作では、挿入する場所を探す必要がある。 +- そのため、指定したKeyと一致するノードを探し出すfind操作が必要になる。 +--- +## findRBTの動作と目的 +- 動作 + - RedBlackTreeを辿っていき、指定されたKeyを探す +- 目的 + - 指定されたKeyを見つけるまでに辿った正しい経路をstackに保存して返したい + +木を保存したstackが正しい順番で構成されているかを証明する必要がある + +--- +## Invariantを用いた証明 +- Invariantとはループや再帰で不変である条件や命題 +例えば... + - 「stackが辿った木の順番通りに保存されている」ことを示すInvariantがある + - 木の操作ではループや再帰などの処理が行われる + - 処理の後にInvariantが正しく保持されていることを示すことで、stackが木の順番通りに構成されていることが証明できる。 + +--- + +## stackとstackInvariantの説明 + + +--- +## stackInvariant型の定義 +- s-nil : 初期化状態で、stackに元の木以外は何も入っていない +```c +data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where +s-nil : {tree0 : bt A} → stackInvariant key tree0 tree0 (tree0 ∷ []) +``` +- s-right : 右側を辿った木がstackに入っている +```c +s-right : (tree tree0 tree₁ : bt A) → {st : List (bt A)} + → key₁ < key → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree tree0 (tree ∷ st) +``` +- s-left : 左側を辿った木がstackに入っている +```c +s-left : (tree₁ tree0 tree : bt A) → {st : List (bt A)} + → key < key₁ → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree₁ tree0 (tree₁ ∷ st) +``` + +stackがtreeを辿っていることを示す命題であることがわかる + +--- + +## stackInvariantの値の例 +```c + s-left (14 < 15)(s-right (10 < 15)(s-right (8 < 10)) s-nil) +``` +- stackが辿ったtreeを保存していることを示す証明になっている + +--- +## findRBTの実装:定義部分 +- findRBTは、treeとstackとInvariantを受け取って、Loopであるnext又は、exitに渡す + +```c + findRBT : {n m : Level} {A : Set n} {t : Set m} + → (key : N) + → (tree tree0 : bt (Color ∧ A) ) + → (stack : List (bt (Color ∧ A))) + → RBtreeInvariant tree ∧ stackInvariant key tree tree0 stack + → (next : ... → t) + → (exit : ... → t) + → t +``` +--- + +## findRBTの実装:定義部分next +- 次のLoopに接続する部分 +- 新しいstackとstackInvariantが作られている +- nextには、次の木の高さは元の木の高さより低くなることが条件として含まれている +```c + → (next : (tree1 : bt (Color ∧ A) ) + → (stack : List (bt (Color ∧ A))) + → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack + → bt-depth tree1 < bt-depth tree + → t ) +``` + +--- +## findRBTの実装:定義部分exit +- leafに到達するか指定したKeyのノードが見つかってLoopから脱出する部分 +```c + → (exit : (tree1 : bt (Color ∧ A)) + → (stack : List (bt (Color ∧ A))) + → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack + → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) + → t ) + → t +``` +--- + +## LoopConnectorはfindRBTのnextを接続する +- LoopConnectorは引数として +(loop : (r : Index) → Invraiant r + → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t)をとる +- ここにcodeGearが入り、codeGearにあるtreeやInvariantを接続できる +- 自然数reduceが減少することによりLoopの停止性を証明できる +- LoopConnectorで接続することによりInvariantの接続と停止性が証明される +--- + +## TerminatingLoopSの実際の型 +- Indexにはtreeとstackが入る +- Invariantが(Set m)の形になるため任意の命題を入れることができる +- reduceはIndexから減少する自然数を得る関数が入る +- TermnatingLoopSの証明は実際に記述されている +```c +TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) + → (r : Index) → (p : Invraiant r) + → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t +``` +--- +## TerminatingLoopSを使った接続 +```c +findTest {n} {m} {A} {t} k tr0 rb0 exit = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) + Index + {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant k (proj1 p) tr0 (proj2 p) } + Invariant + (λ p → bt-depth (proj1 p)) ⟪ tr0 , tr0 ∷ [] ⟫ ⟪ rb0 , s-nil ⟫ + reduce 初期値 + $ λ p RBP loop → findRBT k (proj1 p) tr0 (proj2 p) RBP (λ t1 s1 P2 lt1 → loop ⟪ t1 , s1 ⟫ P2 lt1 ) + GearsAgdaの呼び出し + $ λ tr1 st P2 O → exit tr1 st P2 O + GearsAgdaからの脱出 +``` +- LoopConnectorでcodeGearが接続されていることが確認できる。 +--- +## findRBTの実行結果 +- stackとstackInvariantが出力されている +```c +--指定したKeyの値 14 (木の中に存在するKey) +record + { tree = node 14 ⟪ Red , 1400 ⟫ leaf leaf + ; stack = + node 14 ⟪ Red , 1400 ⟫ leaf leaf + :: node 15 ⟪ Black , 1500 ⟫ (node 14 ⟪ Red , 1400 ⟫ leaf leaf)leaf + :: node 10 ⟪ Red , 1000 ⟫ leaf (node 15 ⟪ Black , 1500 ⟫ (node 14 ⟪ Red , 1400 ⟫ leaf leaf) leaf) + :: node 8 ⟪ Black , 800 ⟫ + (node 5 ⟪ Red , 500 ⟫ (node 2 ⟪ Black , 200 ⟫ leaf leaf) + (node 6 ⟪ Black , 600 ⟫ leaf leaf)) + (node 10 ⟪ Red , 1000 ⟫ leaf + (node 15 ⟪ Black , 1500 ⟫ (node 14 ⟪ Red , 1400 ⟫ leaf leaf) leaf)) + :: [] + ; ti = rb-single 14 1400 + ; si = + s-left + (s≤s(s≤s (s≤s(s≤s(s≤s (s≤s(s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s z≤n))))))))))))))) + (s-right (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s z≤n))))))))))) + (s-right (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s (s≤s z≤n))))))))) s-nil)) + ; exit = case2 refl + } +``` +--- +## findRBTの実行結果から +- stackInvariantを導出できたことにより、stackが正しい順序を保っていることが証明できる + +このことから、「目的の木への正しい経路を保存したstack」を返したことになるため、findRBTの動作が正しいことが証明できたと言える。 + +--- + +## まとめ + +- stackInvariantを用いることで、findRBTの正しさを証明することができた。 +- LoopConnectorを用いることで、GearsOSで複雑なアルゴリズムを実装することが可能になった。 +--- + +## 今後の課題 +- Insertなどの証明を完成させる + - Insert操作の証明は未完成だが、大枠を掴むことができている。 + - Insertは木がバランスするところまで置き換えながら上昇する + - 必要ならばParent GrandParent Uncleで木の回転を行う + - ルートまで木を置き換えながら上昇する + - これらの部分は途中までかけている +--- + +## おまけ +--- + +## Red Black Tree + +- バランスした平衡二分木の一種 +- 木はnodeとleafからなる +- nodeにはkeyという数値が与えられる + +--- + +## Red Black Tree + +- nodeの配置はKeyの大小関係によって決定する +- 左の子のKey < 親のKey < 右の子のKey という大小関係を持つ + +これらは、BinaryTreeと同じ性質となっている。 + +--- + +## RedBlackTreeの特徴 + +- 各ノードは赤か黒の色を持つ +- 赤のノードは赤のノードを子に持たない +- リーフノードはすべて黒である +--- +## RedBlackTreeの特徴 + +- 任意のノードについて、そのノードから子孫の葉までの道に含まれる黒いノードの数は、選んだ葉によらず一定である。 + +--- +## RedBlackTreeの特徴 + +- 根から葉まで道で最長のものの長さは、根から葉までの道で最短のものの長さの二倍を超えない。 + +このことから、RedBlackTreeがある程度の平衡性を持つことがわかる。 + +--- + +## 定理証明支援系言語Agdaとは +- Curry-Howard同型対応により、型付きラムダ計算と命題が１対１で対応するため、Agdaで書かれたプログラムは証明を書くことができる +- Curry-Howard同型対応とは、型と命題が一致するという論理学の性質 +- 型が正しいことを示すことで、命題が正しいと言える + + +--- +## stackInvariantの詳細 +- s-nil : stackの初期化状態であり、stackに元の木以外は何も入っていないことを示す +- s-right,s-left : １つ前の木からみて、どちら側の木をstackに積んだかを示している + +```c +data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where + s-nil : {tree0 : bt A} → stackInvariant key tree0 tree0 (tree0 ∷ []) + s-right : (tree tree0 tree₁ : bt A) → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key₁ < key → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree tree0 (tree ∷ st) + s-left : (tree₁ tree0 tree : bt A) → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key < key₁ → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree₁ tree0 (tree₁ ∷ st) +``` +--- + +## stackInvariantの詳細 +s-right,s-leftでは、 +- keyの大小関係を引数として渡す必要がある +- 1つ前の木を積んだ時のstackInvariantを引数として渡す必要がある + +```c +data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where + s-nil : {tree0 : bt A} → stackInvariant key tree0 tree0 (tree0 ∷ []) + s-right : (tree tree0 tree₁ : bt A) → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key₁ < key → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree tree0 (tree ∷ st) + s-left : (tree₁ tree0 tree : bt A) → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key < key₁ → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree₁ tree0 (tree₁ ∷ st) +``` +--- + +## _≤_ の説明 +- 自然数の大小関係を表す +- 以下の２通りのパターンですべてを記述することができる +```c + data _≤_ : (a b : ℕ) → Set Where + z≤n : {n : ℕ } → zero ≤ n + s≤s : {a b : ℕ } → a ≤ b → suc a ≤ suc b +``` +例えば 3 ≤ 6 を表したいときは、s≤s (s≤s (s≤s z≤n)) と書ける +これは、Keyの大小関係を証明する際などに使用する。 + +--- + +## TerminatingLoopS +- t : 不定の型。codeGearとの接続のために使用する +- Index : 任意の型 +- Invariant : ループで不変な条件 +```c +TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) + → (r : Index) → (p : Invraiant r) + → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t +``` +--- +## TerminatingLoopS +- reduce : ループで減少していく数値(ループの停止性を示すことができる) +- r , p : IndexとInvariantの初期値 +- loop : codeGearに渡すnextの部分。reduceが減っていることの証明を引数として受け取る。tを返す +```c +TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) + → (r : Index) → (p : Invraiant r) + → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t +``` +--- + +## TerminatingLoopSを使った接続 +- Index部分 : RedBlackTreeとstackを渡している。 +- Invariant部分 : RBtreeInvariantとstackInvariantを渡している。 +```c +findTest {n} {m} {A} {t} k tr0 rb0 exit = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) + {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant k (proj1 p) tr0 (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tr0 , tr0 ∷ [] ⟫ ⟪ rb0 , s-nil ⟫ + $ λ p RBP loop → findRBT k (proj1 p) tr0 (proj2 p) RBP (λ t1 s1 P2 lt1 → loop ⟪ t1 , s1 ⟫ P2 lt1 ) + $ λ tr1 st P2 O → exit tr1 st P2 O +``` +--- +## TerminatingLoopSを使った接続 +- reduce部分 : 木の深さを数値として渡している。 +- 初期値部分 : 元の木と元の木のみが入っているstackを渡している。また、それぞれのInvariantを渡している。 +- loop部分 : findRBTを呼び出すことでnextとexitを接続している +```c +findTest {n} {m} {A} {t} k tr0 rb0 exit = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) + {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant k (proj1 p) tr0 (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tr0 , tr0 ∷ [] ⟫ ⟪ rb0 , s-nil ⟫ + $ λ p RBP loop → findRBT k (proj1 p) tr0 (proj2 p) RBP (λ t1 s1 P2 lt1 → loop ⟪ t1 , s1 ⟫ P2 lt1 ) + $ λ tr1 st P2 O → exit tr1 st P2 O +``` +--- + +## RedBlackTreeのほかのInvariant +- RBtreeInvariant + 木がRedBlackTreeであることを示すInvariant +- treeInvariant + 木がBinaryTreeであることを示すInvariant +- replacedTree + 木が置き換えられたということを示すInvariant + +--- + +## RedBlackTreeにおけるInsert +- RedBlackTreeのInsert操作では、黒の深さでバランスをとる必要がある。 +- バランスを行うためには、replaceやrotateなどの操作を行う。 +- 対象のnodeから見た親、祖父、おじの色で6つの場合分けを行う必要がある。 + +--- +## Insertのケース1 +- 親が黒の場合 + そのまま赤のnodeを挿入することができる +--- +## Insertのケース2 +- 親が赤 祖父が黒 おじが赤の場合 +- 親が赤のためそのまま挿入できない +- 親とおじの色を黒に変更し、祖父の色を赤にすることでバランスする。 +- 祖父の親が赤の場合、赤が連続してしまう +- 注目するノードを祖父に変更し1つ上の黒のノードまで戻る必要がある +--- +## Insertのケース3 +- ノードが存在しない場合 + そのまま挿入することができる +--- +## Insertのケース4 +- 親が赤 祖父、おじがいない場合 +- 親が赤のためそのまま挿入できない + 親の色を黒に変更し、子ノードとして挿入する +--- +## Insertのケース5 +- 親が赤 祖父が黒 おじが黒の場合且つ木が内側に曲がっている場合 +- 親が赤なのでそのまま挿入できない +- 内側に曲がっていると回転がうまくできない +- 親と子の間で回転を行い、木を外側向きに修正する必要がある +- 修正した後の状態はケース６と同じであるため、ケース６に遷移する +--- +## Insertのケース6 +- 親が赤 祖父が黒 おじが黒の場合且つ木が外側に曲がっている場合 +- 親が赤なのでそのまま挿入できない +- 祖父ノードと親ノードの間で回転を行う +- 親ノードを黒、祖父ノードを赤に変更することで挿入する +--- \ No newline at end of file