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  <title>Thunks &amp; Thoughts</title>
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  <link href="https://thunk.blog"/>
  <updated>2026-06-17T06:44:29Z</updated>
  <id>https://thunk.blog</id><entry>
    <title>Yes - Cabal custom scripts do have uses</title>
    <updated>2026-05-21T11:45:00.000Z</updated>
    <id>https://thunk.blog/posts/yes-cabal-custom-scripts-do-have-uses/</id>
    <content type="html">&lt;p&gt;This will be a quick one. I&#39;ve been working on implementing replication in my &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/tree/master&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Redis server&lt;/a&gt;. The first step for this, per the &lt;a href=&quot;https://codecrafters.io/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;codecrafters&lt;/a&gt; challenge that I&#39;m following, is to implement the &lt;a href=&quot;https://redis.io/docs/latest/commands/info/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;INFO&lt;/code&gt;&lt;/a&gt; command, which I&#39;ve done in this &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/pull/1&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;PR&lt;/a&gt; and may be the focus of a future blog post. The framework for the &lt;code&gt;INFO&lt;/code&gt; command was pretty easy to implement (types ftw). What I found myself spending the most time on was figuring out where the information in the &lt;code&gt;INFO&lt;/code&gt; response comes from.&lt;/p&gt;
&lt;p&gt;For the challenge, the goal was to implement the &lt;code&gt;Replication&lt;/code&gt; section with just the &lt;code&gt;role&lt;/code&gt; field, but I decided to also take a stab at implementing the &lt;code&gt;Server&lt;/code&gt; section. It was a bit of a tangent. It took a while to hunt down what each field of the section meant and how Redis acquired the field&#39;s value. GitHub Copilot was very helpful in navigating and tracing things around in the codebase. I&#39;ll spare you all the mundane details. Suffice it to say, I ended up translating the &lt;a href=&quot;https://github.com/redis/redis/blob/47c51369eeffd55e1baf20df7955a3dfbe842fc4/src/server.c#L6325&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Redis code&lt;/a&gt; for the server section into this &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/9408501f6f9483f79c25ec69e97d4ebe0a448ae7/codecrafters-redis/src/Redis/Commands/Info/Server.hs&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Haskell module&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;I&#39;m pretty proud of it overall, and through it I found a decent use for &lt;a href=&quot;https://cabal.readthedocs.io/en/stable/cabal-package-description-file.html#more-complex-packages&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Cabal&#39;s custom script mechanism&lt;/a&gt;. I use the script to collect and generate static build/release information like the current git commit hash, current git dirty status, and server build ID for the &lt;code&gt;redis_git_sha1&lt;/code&gt;, &lt;code&gt;redis_git_dirty&lt;/code&gt;, and &lt;code&gt;redis_build_id&lt;/code&gt; fields under the &amp;quot;Server&amp;quot; INFO section. Redis does something similar with its &lt;a href=&quot;https://github.com/redis/redis/blob/47c51369eeffd55e1baf20df7955a3dfbe842fc4/src/mkreleasehdr.sh&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;mkreleasehdr.sh script&lt;/a&gt;, which generates a header file with release info variables that it then uses to, among other things, seed those &amp;quot;Server&amp;quot; section fields I mentioned in &lt;a href=&quot;https://github.com/redis/redis/blob/47c51369eeffd55e1baf20df7955a3dfbe842fc4/src/server.c#L6291&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;its &lt;code&gt;INFO&lt;/code&gt; command implementation&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;I used Cabal&#39;s custom script mechanism to mimic this since I didn&#39;t want to have to bend over backwards to try to collect this information at runtime. The setup script creates a &lt;code&gt;BuildInfo.hs&lt;/code&gt; module that is seeded with constants, which I think is more efficient than whatever runtime attempt at fetching the same data I&#39;d have needed to concoct, which would have required shelling out to git or potentially generating a build ID on each INFO request. I also made it such that the &lt;code&gt;BuildInfo.hs&lt;/code&gt; module is only regenerated when certain data points change, similar to what the Redis script does. The full setup script can be perused &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/9408501f6f9483f79c25ec69e97d4ebe0a448ae7/codecrafters-redis/Setup.hs&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;Though Cabal custom scripts are &lt;a href=&quot;https://cabal.readthedocs.io/en/stable/cabal-package-description-file.html#custom-setup-scripts&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;deprecated&lt;/a&gt;, it was much easier for me to get set up with them than the &lt;a href=&quot;https://cabal.readthedocs.io/en/stable/cabal-package-description-file.html#setup-hooks&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;newer recommended setup hooks API&lt;/a&gt;. There were also more examples on the former to draw inspiration from than the latter. I think this is because of how new the hooks API seems to be. I &lt;em&gt;will&lt;/em&gt; add a TODO to migrate it over; who knows, I might even get to it someday. But since it is non-blocking, I plan to just move on for now.&lt;/p&gt;
&lt;p&gt;An alternative approach would have been to use Template Haskell. Template Haskell could do compile-time generation too, but would push metaprogramming into the app code for a small amount of static metadata. It seemed a little overkill for this particular scenario.&lt;/p&gt;
&lt;p&gt;In any case, that&#39;s all for this one folks. If you need to generate a module at compile time, Cabal custom scripts (or Hooks API) got your back. Till next time!&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/yes-cabal-custom-scripts-do-have-uses/"/>
  </entry><entry>
    <title>On maths and engineering</title>
    <updated>2026-05-21T11:44:00.000Z</updated>
    <id>https://thunk.blog/posts/on-maths-and-engineering/</id>
    <content type="html">&lt;p&gt;I want to be dangerous with maths, at least, with regard to software engineering. I know math isn&#39;t &lt;em&gt;necessarily&lt;/em&gt; a requirement for the field, but being competent never hurts. Moreover, time and time again I have found myself trying to read through some CS literature like &lt;em&gt;Intro to Algorithms&lt;/em&gt;, or SICP, or (most recently) &lt;em&gt;The Algorithm Design Manual&lt;/em&gt; only to find myself waylaid by the math weaved into the essence of the prose. Some of it, I am able to overcome, but often it ends up being a point of frustration that causes me to lose interest or take too long to make any meaningful progress.&lt;/p&gt;
&lt;p&gt;When I was just starting out, I didn&#39;t really think much of this gap. I was more occupied with just learning programming, but now I think I want to do more to study computer science: do a little more theory rather than just practicals. I want to get good at the relevant math which, from my research, seems to just be mostly discrete math (plus linear algebra and statistics for ML). Maybe some Calculus, but that seems more useful in specific niches, if anything.&lt;/p&gt;
&lt;p&gt;Oddly enough, I &lt;em&gt;have&lt;/em&gt; taken a Calculus class, Calculus I specifically, back in community college (CC). However, since I wasn&#39;t able to complete a Bachelor&#39;s degree immediately after CC, I haven&#39;t taken a discrete math class. Well, that&#39;s not entirely true. I did do some study in the semester before my hiatus at WGU. Needless to say, for all intents and purposes, I haven&#39;t &lt;em&gt;really&lt;/em&gt; taken a discrete math class, so I&#39;m just gonna do that.&lt;/p&gt;
&lt;p&gt;I&#39;ve found the textbook, &lt;em&gt;Discrete Mathematics with Applications&lt;/em&gt; by Susanna S. Epp, to be fantastic so far. Super accessible, and I hope to get through it cover-to-cover or close enough to achieve my ends. In parallel, I am also reading &lt;em&gt;The Algorithm Design Manual&lt;/em&gt; for DSA practice. Yes, sure I &lt;em&gt;could&lt;/em&gt; just take one of the many DSA/Leetcode courses geared specifically towards getting a person interview-ready, and I did start with that, but I guess I wanted something a little more comprehensive? I also wanted to make progress on this &lt;a href=&quot;https://teachyourselfcs.com/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;curriculum&lt;/a&gt;, so I figured reading TADM would be killing two birds with one stone.&lt;/p&gt;
&lt;p&gt;I&#39;m in a pretty good spot to dive into this, I think. Let&#39;s just hope it stays that way long enough for me to get to where I want to go.&lt;/p&gt;
&lt;p&gt;Anyways, while I go through TADM, particularly the exercises, I find myself needing to do things like come up with counterexamples or prove algorithms. This stuff was never interesting to me, but I find it more and more fascinating as the days go by. I&#39;m even looking to learn Lean to get better at formally expressing the proofs I write. I haven&#39;t gotten to the proof section of &lt;em&gt;Discrete Mathematics with Applications&lt;/em&gt; yet, so the proof exercises in TADM are still a bit of a slog. I have found Claude to be particularly helpful, but I don&#39;t let it just give me the answers. I have system prompts that specifically disable this behavior. No, I have configured Claude to be a kind of tutor, and it&#39;s been working wonders in helping me get through the TADM exercises while I shore up my math.&lt;/p&gt;
&lt;p&gt;A couple of days ago, Claude and I worked through a proof for Horner&#39;s method using loop invariants. Below is what I was able to come up with following Claude&#39;s assistance (all of this below is in my own words):&lt;/p&gt;
&lt;div data-callout-metadata=&quot;&quot; class=&quot;callout &quot; data-callout=&quot;note&quot;&gt;&lt;div class=&quot;callout-title&quot;&gt;&lt;div class=&quot;callout-title-inner&quot;&gt; Horner&#39;s Method&lt;/div&gt;&lt;/div&gt;
&lt;div class=&quot;callout-content&quot;&gt;&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;text&quot;&gt;&lt;code class=&quot;language-text&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span&gt;Horner(a, x)&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span&gt;	p = a_n&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span&gt;	for i from n − 1 to 0&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;span&gt;		p = p · x + a_i&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;5&quot;&gt;&lt;span&gt;	return p&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;What is/are the invariant(s) that directly assert on the expected end result of this algorithm? Horner&#39;s method is an algorithm for evaluating [[Polynomials]]. So it stands to reason that our invariant ought to be something related to the evaluation of polynomials. The algorithm returns a value &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; which is probably the evaluated polynomial. Ok, then what is true about &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; at every step (initialization, after each iteration, and after termination)?&lt;/p&gt;
&lt;p&gt;Some scrutiny suggests that the invariant we care about here is that &amp;quot;&lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; always contains the evaluated polynomial up to &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.357ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.937ex&quot; height=&quot;1.355ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -441 856 598.8&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;msub&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;&amp;quot;.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;At initialization, &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;6.5ex&quot; height=&quot;1.758ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -583 2872.8 777&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(780.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(1836.6,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; which adheres to our invariant since, at this point, &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;0.781ex&quot; height=&quot;1.52ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -661 345 672&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; is not initialized, there is no &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;0.781ex&quot; height=&quot;1.52ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -661 345 672&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; but &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; holds the first term of the polynomial.&lt;/li&gt;
&lt;li&gt;After each iteration, we can see that our invariant holds true as with every iteration, &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; still contains an evaluated polynomial up to &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.357ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.937ex&quot; height=&quot;1.355ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -441 856 598.8&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;msub&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;.
&lt;ul&gt;
&lt;li&gt;For iteration 1, we are 1 away from &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.357ex&quot; height=&quot;1.025ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 600 453&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, so &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;5.254ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 2322.4 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(822.2,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(1822.4,0)&quot;&gt;&lt;path data-c=&quot;31&quot; d=&quot;M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. 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style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;TeXAtom&quot; transform=&quot;translate(562,-150) scale(0.707)&quot; data-mjx-texclass=&quot;ORD&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(600,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(1378,0)&quot;&gt;&lt;path data-c=&quot;31&quot; d=&quot;M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(8607.4,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(9663.2,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; 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style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;TeXAtom&quot; transform=&quot;translate(562,-150) scale(0.707)&quot; data-mjx-texclass=&quot;ORD&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(600,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(1378,0)&quot;&gt;&lt;path data-c=&quot;31&quot; d=&quot;M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. Since at this iteration &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;9.052ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 4001 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(622.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1678.6,0)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2500.8,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(3501,0)&quot;&gt;&lt;path data-c=&quot;31&quot; d=&quot;M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; is also equal to &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.357ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;10.478ex&quot; height=&quot;1.676ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -583 4631.1 740.8&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;msub&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(1258.5,0)&quot;&gt;&lt;path data-c=&quot;2217&quot; d=&quot;M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1980.7,0)&quot;&gt;&lt;path data-c=&quot;1D465&quot; d=&quot;M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2774.9,0)&quot;&gt;&lt;path data-c=&quot;2B&quot; d=&quot;M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(3775.2,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, matching our invariant&lt;/li&gt;
&lt;li&gt;For iteration 2, we are 2 away from &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.357ex&quot; height=&quot;1.025ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 600 453&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, so &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;5.254ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 2322.4 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(822.2,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(1822.4,0)&quot;&gt;&lt;path data-c=&quot;32&quot; d=&quot;M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. Thus &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.566ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;60.472ex&quot; height=&quot;2.452ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -833.9 26728.5 1083.9&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(780.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(1836.6,0)&quot;&gt;&lt;path data-c=&quot;28&quot; d=&quot;M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2225.6,0)&quot;&gt;&lt;path data-c=&quot;28&quot; d=&quot;M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z&quot; 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d=&quot;M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(6167.5,0)&quot;&gt;&lt;path data-c=&quot;2B&quot; d=&quot;M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(7167.7,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; 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display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. 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&lt;li&gt;For iteration 3, we are 3 away from &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.357ex&quot; height=&quot;1.025ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 600 453&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, so &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;5.254ex&quot; height=&quot;1.69ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -665 2322.4 747&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(822.2,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(1822.4,0)&quot;&gt;&lt;path data-c=&quot;33&quot; d=&quot;M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. 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role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 11101.9 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(622.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; 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d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(1378,0)&quot;&gt;&lt;path data-c=&quot;32&quot; d=&quot;M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(13325.8,0)&quot;&gt;&lt;path data-c=&quot;2217&quot; d=&quot;M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(14048,0)&quot;&gt;&lt;path data-c=&quot;1D465&quot; d=&quot;M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(14842.3,0)&quot;&gt;&lt;path data-c=&quot;2B&quot; d=&quot;M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(15842.5,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, matching our invariant&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;Notice how with each iteration, the polynomial held by &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; increases by a term and all the exponents increase by 1? Also note that the exponent for the first time is equal to the number of iterations. At our first iteration the first term had an exponent of 1. At the second iteration, the exponent was 2, and in our third, it was 3. Likewise, at our first iteration &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;9.052ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 4001 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(622.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1678.6,0)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2500.8,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(3501,0)&quot;&gt;&lt;path data-c=&quot;31&quot; d=&quot;M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, at the second &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;9.052ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 4001 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(622.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1678.6,0)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2500.8,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(3501,0)&quot;&gt;&lt;path data-c=&quot;32&quot; d=&quot;M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;, and at the third iteration &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;9.052ex&quot; height=&quot;1.69ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -665 4001 747&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(622.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1678.6,0)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2500.8,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(3501,0)&quot;&gt;&lt;path data-c=&quot;33&quot; d=&quot;M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. Doing some quick algebra shows us that we could generalize our exponent as &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;4.904ex&quot; height=&quot;1.681ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -661 2167.4 743&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(822.2,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1822.4,0)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; (&lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;9.052ex&quot; height=&quot;1.69ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -665 4001 747&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(622.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1678.6,0)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2500.8,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(3501,0)&quot;&gt;&lt;path data-c=&quot;33&quot; d=&quot;M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; -&amp;gt; &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;12.949ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 5723.4 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mn&quot;&gt;&lt;path data-c=&quot;30&quot; d=&quot;M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(777.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1833.6,0)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2655.8,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(3656,0)&quot;&gt;&lt;path data-c=&quot;33&quot; d=&quot;M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(4378.2,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(5378.4,0)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; -&amp;gt; &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;19.299ex&quot; height=&quot;2.009ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; 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display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mtext&gt;3, the index&lt;/mtext&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;). And because each subsequent term has an exponent that is 1 less than the exponent of the previous term, the second and third terms will have an exponent of &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;8.801ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 3889.9 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; 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padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; and &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;8.801ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 3889.9 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; 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style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(600,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1378,0)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(1723,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(2501,0)&quot;&gt;&lt;path data-c=&quot;33&quot; d=&quot;M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; 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style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(29959.1,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;TeXAtom&quot; transform=&quot;translate(562,-150) scale(0.707)&quot; data-mjx-texclass=&quot;ORD&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. Notice how this is more or less equivalent to what we got as the value of &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.439ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.138ex&quot; height=&quot;1.439ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 503 636&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; after each iteration.&lt;/li&gt;
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style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow data-mjx-texclass=&quot;ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. This is equivalent to our invariant, so the invariant holds all through.&lt;/li&gt;
&lt;/ul&gt;
&lt;div data-callout-metadata=&quot;&quot; class=&quot;callout &quot; data-callout=&quot;aside&quot;&gt;&lt;div class=&quot;callout-title&quot;&gt;&lt;div class=&quot;callout-title-inner&quot;&gt;Aside&lt;/div&gt;&lt;/div&gt;
&lt;div class=&quot;callout-content&quot;&gt;&lt;p&gt;Note that &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.357ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.937ex&quot; height=&quot;1.355ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -441 856 598.8&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;msub&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; is the last term since only after the last iteration is &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;0.781ex&quot; height=&quot;1.52ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -661 345 672&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; fixed at 0. Using it as the index in any other term wouldn&#39;t be correct once we start adding more terms. So, on the first iteration, &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.357ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;10.478ex&quot; height=&quot;1.676ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -583 4631.1 740.8&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;msub&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(1258.5,0)&quot;&gt;&lt;path data-c=&quot;2217&quot; d=&quot;M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1980.7,0)&quot;&gt;&lt;path data-c=&quot;1D465&quot; d=&quot;M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2774.9,0)&quot;&gt;&lt;path data-c=&quot;2B&quot; d=&quot;M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(3775.2,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(562,-150) scale(0.707)&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; is correct because on the first iteration &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.186ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;9.052ex&quot; height=&quot;1.692ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -666 4001 748&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D456&quot; d=&quot;M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(622.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mi&quot; transform=&quot;translate(1678.6,0)&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(2500.8,0)&quot;&gt;&lt;path data-c=&quot;2212&quot; d=&quot;M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mn&quot; transform=&quot;translate(3501,0)&quot;&gt;&lt;path data-c=&quot;31&quot; d=&quot;M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;. However, it would be incorrect to say &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.471ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;26.786ex&quot; height=&quot;2.357ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -833.9 11839.3 1041.9&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45D&quot; d=&quot;M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;mo&quot; transform=&quot;translate(780.8,0)&quot;&gt;&lt;path data-c=&quot;3D&quot; d=&quot;M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;g data-mml-node=&quot;msub&quot; transform=&quot;translate(1836.6,0)&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D44E&quot; d=&quot;M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z&quot; 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Instead we want to reference &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.357ex&quot; height=&quot;1.025ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 600 453&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt; directly somehow, since &lt;mjx-container class=&quot;MathJax&quot; jax=&quot;SVG&quot; style=&quot;direction: ltr; position: relative;&quot;&gt;&lt;svg style=&quot;overflow: visible; min-height: 1px; min-width: 1px; vertical-align: -0.025ex;&quot; xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;1.357ex&quot; height=&quot;1.025ex&quot; role=&quot;img&quot; focusable=&quot;false&quot; viewBox=&quot;0 -442 600 453&quot; aria-hidden=&quot;true&quot;&gt;&lt;g stroke=&quot;currentColor&quot; fill=&quot;currentColor&quot; stroke-width=&quot;0&quot; transform=&quot;scale(1,-1)&quot;&gt;&lt;g data-mml-node=&quot;math&quot;&gt;&lt;g data-mml-node=&quot;mi&quot;&gt;&lt;path data-c=&quot;1D45B&quot; d=&quot;M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z&quot; style=&quot;stroke-width: 3;&quot;&gt;&lt;/path&gt;&lt;/g&gt;&lt;/g&gt;&lt;/g&gt;&lt;/svg&gt;&lt;mjx-assistive-mml unselectable=&quot;on&quot; display=&quot;inline&quot; style=&quot;top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; position: absolute; padding: 1px 0px 0px 0px; border: 0px; display: block; width: auto; overflow: hidden;&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;&#39;s value never changes with each iteration.&lt;/p&gt;
&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;p&gt;Look at how beautiful that is. It&#39;s enough to make a grown man cry! Though I think what I have here is accurate, I&#39;m always open to correction.&lt;/p&gt;
&lt;p&gt;Learning about loop invariants and using them for this proof took me a while, and there was a lot of stumbling and fumbling, but I managed it. My goal now is to make it easier for me to do these kinds of things, and by extension open myself up to more CS literature. Not all of it, mind, but enough to help broaden my engineering capabilities. Let&#39;s see how it goes.&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/on-maths-and-engineering/"/>
  </entry><entry>
    <title>Numbers? Numerals? Oh Boy</title>
    <updated>2026-05-21T11:44:00.000Z</updated>
    <id>https://thunk.blog/posts/numbers-numerals-oh-boy/</id>
    <content type="html">&lt;p&gt;So a while back I was attempting to do some data marshalling for the persistence of the store in my &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;redis server implementation&lt;/a&gt; per the &lt;a href=&quot;https://rdb.fnordig.de/file_format.html#length-encoding&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;RDB file specification&lt;/a&gt;. It turns out that when you attempt to get the char code for a character in Haskell, it&#39;ll give you a decimal number. This stumped me for a bit since in the example I was testing out (getting the char code for the character &#39;R&#39;) I was expecting to get 52 per what I saw in the docs, but I was getting 82 instead?&lt;/p&gt;
&lt;p&gt;Then while I was having a friendly chat with some of the gods at the functional programming discord, I was told that those numbers are the same, wtf? I was expecting 52, which, it turns out, is the char code for &#39;R&#39; in &lt;em&gt;hexadecimal&lt;/em&gt;, while 82 is the char code in &lt;em&gt;decimal&lt;/em&gt;. For some reason, my brain somehow didn&#39;t even consider that those two symbols (52 and 82) could be the same in any way. If you too are a little confused, read on.&lt;/p&gt;
&lt;p&gt;You see, it turns out that we all conflate the idea of numbers and numerals because the distinction tends not to have any practical significance for the most part. Numbers are in reality not a sequence of symbols, but are abstract mathematical objects. They exist independently of any base system, they just are, like the concept of &#39;distance&#39; or &#39;quantity&#39;. They are a pure mathematical concept that transcends notation (like a soul), independent of the symbols we typically associate them with/use to represent them. These symbols are actually numerals.&lt;/p&gt;
&lt;p&gt;Think about it, when you have three apples, the &amp;quot;threeness&amp;quot; exists whether you write it as 3, III, 三, or 11 (in binary). The quantity itself is the number; how we write it down is the numeral.&lt;/p&gt;
&lt;p&gt;A helpful analogy here might be the game &lt;a href=&quot;https://store.steampowered.com/app/1110910/Mortal_Shell/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Mortal Shell&lt;/a&gt;. Great game btw, though a little too tedious for me. Anyway, this game makes a great analogy because it follows a sort of spirit (number) that can be taken into combat with different shells (numerals). Just like in the game where the same spirit can inhabit different shells, the same number can be expressed through different numeral systems.&lt;/p&gt;
&lt;p&gt;Numerals are what we mean when we typically refer to a &amp;quot;number&amp;quot; in everyday speech. If numbers are the soul, numerals are the physical shells with which we interact with said soul in everyday life. Numerals are a representation of numbers in the form of some base (10 for decimal, 16 for hex, 2 for binary, etc.) using an available sequence of digits. They&#39;re the physical manifestation of the abstract mathematical idea for practical use.&lt;/p&gt;
&lt;p&gt;I mean, how else can a notion of equality between completely disparate symbols like &amp;quot;42&amp;quot;, &amp;quot;101010&amp;quot;, and &amp;quot;2A&amp;quot; even exist without some sort of common denominator to refer back to? In this case, that common denominator is a number in the mathematical sense.&lt;/p&gt;
&lt;p&gt;Although computers represent numbers using binary numerals for practical reasons (hardware limitations, to speed up bitwise operations, etc.), what we deal with as engineers are numbers in the mathematical sense because that&#39;s the level of abstraction we think in. The fact that it&#39;s binary beneath the covers is irrelevant to us, so for all intents and purposes (excluding maybe systems engineering), numeric types are a digital representation of numbers not numerals.&lt;/p&gt;
&lt;p&gt;So in essence, numbers are abstract mathematical objects that represent quantity. Numerals are the symbols we use to represent/concretize said objects. Crazy right? Who would&#39;ve thought numbers to be like spirits? What if they&#39;re the ghost of some vengeful mathematicians? Anyway, understanding this helped clarify my confusion while giving me a cool concept to explain when I want to sound smart, so it&#39;s a win win.&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/numbers-numerals-oh-boy/"/>
  </entry><entry>
    <title>NixOS, the start of something new (part 1)</title>
    <updated>2026-05-21T11:44:00.000Z</updated>
    <id>https://thunk.blog/posts/nixos-the-start-of-something-new-part-1/</id>
    <content type="html">&lt;p&gt;I am really digging this Nix thing. I&#39;ve known about Nix for a while now. It didn&#39;t really seem like much, especially since I already had my &lt;a href=&quot;https://github.com/OlaoluwaM/distro-setup&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;distro-setup&lt;/a&gt; repository for reproducing my Fedora system. I considered Nix to be needless complexity until I started using it at work and my eyes were opened.&lt;/p&gt;
&lt;p&gt;I realized that most, if not all, of the things I have scripted in my distro-setup repository could easily be expressed as Nix configuration, and because NixOS operates as a function of your configuration, the only way to modify your system would be to modify your config.&lt;/p&gt;
&lt;p&gt;I figured that kind of workflow would be the solution to a problem I&#39;d been having with distro-setup where, over time, my scripts would drift from the state of my system because I&#39;d neglect to update them as I modified things. By default, however, with NixOS, this undesirable state becomes harder to reach, because, again, by default, you must update your configuration if you want to modify your system. Pretty neat, right!?&lt;/p&gt;
&lt;p&gt;Nonetheless, I still kinda held off on doing all this because I &lt;em&gt;really&lt;/em&gt; didn&#39;t want to learn a whole new abstraction and the common sentiment of Nix being a particularly obtuse language with terrible error messages and a steep learning curve only put me off further. But as I continued to consider the potential benefits (lower likelihood of drift and easier reproducibility), I decided to give it a proper shot. I opted to start by learning the meanings of what I think are the most common terms:&lt;/p&gt;
&lt;div data-callout-metadata=&quot;&quot; class=&quot;callout &quot; data-callout=&quot;info&quot;&gt;&lt;div class=&quot;callout-title&quot;&gt;&lt;div class=&quot;callout-title-inner&quot;&gt;Info&lt;/div&gt;&lt;/div&gt;
&lt;div class=&quot;callout-content&quot;&gt;&lt;ul&gt;
&lt;li&gt;The Nix language can be thought of as &lt;a href=&quot;https://nix.dev/tutorials/nix-language.html#:~:text=Note-,If,similar&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;JSON with functions&lt;/a&gt;.&lt;/li&gt;
&lt;li&gt;A &lt;strong&gt;revision&lt;/strong&gt; is a Nix term for a Git commit hash. Most commonly, revisions tend to refer to a specific Git commit hash from the Nixpkgs repo, but it can also refer to a commit hash from any Git repo.&lt;/li&gt;
&lt;li&gt;A &lt;strong&gt;derivation&lt;/strong&gt; is a build recipe/blueprint for a &amp;quot;package&amp;quot;. With a derivation, we describe how to build a &amp;quot;package&amp;quot;: what inputs it needs, what commands to run, and where the output should be placed.&lt;br /&gt;
- I put the term &amp;quot;&lt;strong&gt;package&lt;/strong&gt;&amp;quot; in quotes above because in Nix, the term is used pretty loosely. Usually a package refers to something like a program from a repository or something, but in Nix a package may refer to:&lt;br /&gt;
- a derivation itself (confusing, I know).&lt;br /&gt;
- the result of a derivation.&lt;br /&gt;
- a program or tool (like usual).&lt;br /&gt;
- a library (like those &lt;code&gt;libutils&lt;/code&gt; or whatever packages you&#39;d typically spy in a distribution&#39;s repo).&lt;br /&gt;
- a font, config, man pages, really any software artifact I think.&lt;/li&gt;
&lt;li&gt;The Nix &amp;quot;&lt;strong&gt;store&lt;/strong&gt;&amp;quot;, distinct from &lt;code&gt;/usr/bin&lt;/code&gt;, is located at &lt;code&gt;/nix/store&lt;/code&gt; and it is where everything that Nix builds (&amp;quot;packages&amp;quot;) is stored. Items are stored at a unique path based on the hash of their inputs. This way different versions of an item can exist without collision.&lt;/li&gt;
&lt;li&gt;A &lt;strong&gt;channel&lt;/strong&gt; is a named, stable URL that points to a revision, the latest one among a collection of revisions like &lt;code&gt;nixos-24.11&lt;/code&gt; or &lt;code&gt;nixos-unstable&lt;/code&gt;. It is similar to a Git repo where you can subscribe to receive updates when new commits are pushed through. We point a channel to its latest revision using &lt;code&gt;nix-channel --update&lt;/code&gt; or &lt;code&gt;nixos-rebuild --upgrade&lt;/code&gt;. Channels make it easy to get started with Nix and NixOS.
&lt;ul&gt;
&lt;li&gt;However, it is important to note that this mechanism isn&#39;t reproducible in the same way a &lt;code&gt;package.json&lt;/code&gt; without a &lt;code&gt;package-lock.json&lt;/code&gt; isn&#39;t reproducible. Channels can point to different revisions across machines. In other words, two systems with the same channel-based config could be working with different revisions of the same channel.&lt;/li&gt;
&lt;li&gt;Channels only ever work with the latest revision they are aware. There is no way to rollback or pin to a revision as far as I know&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;An &lt;strong&gt;overlay&lt;/strong&gt; is a mechanism for extending a package set without needing to fork the package set. It allows for the overriding of a package in a package set or adding a new package.&lt;/li&gt;
&lt;li&gt;An &lt;strong&gt;override&lt;/strong&gt; allows for rebuilding a package with different dependencies or options. For example, you want to build a package but do so with a specific flag enabled or swap out an out-of-date dependency with something more recent, etc.&lt;/li&gt;
&lt;li&gt;A &lt;strong&gt;NixOS module&lt;/strong&gt; is a reusable piece of system (NixOS) configuration like in &lt;code&gt;/etc/nixos&lt;/code&gt;, having a &lt;code&gt;configuration.nix&lt;/code&gt; and a &lt;code&gt;hardware-configuration.nix&lt;/code&gt; and importing the latter in the former. Both are modules.&lt;/li&gt;
&lt;li&gt;A &lt;strong&gt;generation&lt;/strong&gt; is a bootable snapshot of your system configuration. Generations are listed for you to select from before booting. Every time you rebuild, NixOS creates a new generation and adds it as a boot entry.&lt;/li&gt;
&lt;li&gt;A &lt;strong&gt;profile&lt;/strong&gt; is a collection of installed packages and configuration for a given user or system.&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;&lt;/div&gt;&lt;p&gt;I also took the time to get at least a partial intuition for what a &lt;strong&gt;flake&lt;/strong&gt; is and how it differs from a &lt;strong&gt;channel&lt;/strong&gt;. The benefits it offers in terms of reproducibility were the main reason I opted to have my config be flake-based as you can see in &lt;a href=&quot;https://github.com/OlaoluwaM/nixos-config&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;the repo&lt;/a&gt;.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;[!aside] Why &amp;quot;reproducibility&amp;quot;&lt;br /&gt;
I value reproducibility because I like being able to get back up and running quickly should anything happen to my device and I need to spin up a new one, or need to do a fresh install. Like with my games, I hate losing my progress :). Even on Fedora, which supports version upgrades, I prefer to do a fresh install as it reduces the chances of things not working right.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;blockquote&gt;
&lt;p&gt;[!aside] What is a &amp;quot;flake&amp;quot;&lt;br /&gt;
A &lt;strong&gt;flake&lt;/strong&gt; is a relatively new approach to defining nix projects. A flake can be thought of as any directory with a &lt;code&gt;flake.nix&lt;/code&gt; and &lt;code&gt;flake.lock&lt;/code&gt; at its root. In a &lt;code&gt;flake.nix&lt;/code&gt; file we define two things: &lt;code&gt;inputs&lt;/code&gt; and &lt;code&gt;outputs&lt;/code&gt;. Like in a function, the &lt;code&gt;inputs&lt;/code&gt; are the dependencies of the flake, the things required to generate the output. The &lt;code&gt;outputs&lt;/code&gt; can be multiple things, a package, a development environment, a library, a NixOS configuration or any other kind of software artifact. Flakes offer us more reproducibility thanks to the &lt;code&gt;flake.lock&lt;/code&gt; file where dependencies, &lt;code&gt;inputs&lt;/code&gt; are explicitly &amp;quot;pinned&amp;quot; similar to what a package lock file does in any other programming language&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;AI has also been a huge boon, particularly as a collaborator. I am playing with all of this on a VM (as anyone should) before I opt to replace Fedora on &lt;a class=&quot;internal-link&quot; target=&quot;&quot; data-note-icon=&quot;&quot; href=&quot;https://thunk.blog/posts/dotfiles-reorg-a-journey/&quot;&gt;Boreas&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;You may also be wondering: &amp;quot;hmm, if you&#39;re using Fedora, why not use one of the atomic spins?&amp;quot; Well, anon, that&#39;s because atomic spins don&#39;t provide the same reproducibility guarantees as NixOS. From what I understand, with atomic spins, we&#39;d all more or less start with the exact same OS base version, but that&#39;s it. Changes are still made imperatively using package managers, and though those changes are underpinned by a reproducible base image, transactional updates (via &lt;code&gt;rpm-ostree&lt;/code&gt;), rollbacks, etc…, the potential for drift between my setup scripts and system state is still high.&lt;/p&gt;
&lt;p&gt;So here we are, NixOS. I&#39;ve been with Fedora since v34, but now it&#39;s time for a change. I&#39;ve got &lt;a href=&quot;https://github.com/OlaoluwaM/nixos-config/pull/1&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;this PR&lt;/a&gt; that migrates my DE/WM-agnostic packages onto Nix. Take a look, and let me know what you think!&lt;/p&gt;
&lt;p&gt;&lt;a class=&quot;internal-link&quot; target=&quot;&quot; data-note-icon=&quot;&quot; href=&quot;https://thunk.blog/posts/nixos-the-journey-continues-part-2/&quot;&gt;Part 2!&lt;/a&gt;&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/nixos-the-start-of-something-new-part-1/"/>
  </entry><entry>
    <title>NixOS, the last milestone (part 3)</title>
    <updated>2026-05-21T11:44:00.000Z</updated>
    <id>https://thunk.blog/posts/nixos-the-last-milestone-part-3/</id>
    <content type="html">&lt;p&gt;Alrighty! Since &lt;a class=&quot;internal-link&quot; target=&quot;&quot; data-note-icon=&quot;&quot; href=&quot;https://thunk.blog/posts/nixos-the-journey-continues-part-2/&quot;&gt;part 2&lt;/a&gt;, I&#39;ve handled the hardware configuration for my NixOS setup. Specifically, the configuration for an Intel CPU and Integrated GPU, and an NVIDIA GPU. My initial plan was to make reusable modules for each component, so I could plug&#39;n&#39;play across hosts. I wanted separate modules for an Intel CPU &amp;amp; GPU, separate modules for an AMD CPU &amp;amp; GPU, and a module for an NVIDIA GPU. However, as it turns out, all that is already covered by the &lt;a href=&quot;https://github.com/NixOS/nixos-hardware&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;nixos-hardware project&lt;/a&gt;. In fact, all I needed to do was find the module corresponding to my host model and just import and hook it up to get all the hardware configuration I was looking for, or at least a baseline to work up from. Pretty neat I say.&lt;/p&gt;
&lt;p&gt;Beyond that I also added configuration for some other miscellaneous items. To check those out, feel free to peruse the &lt;a href=&quot;https://github.com/OlaoluwaM/nixos-config/pull/4&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;PR&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;In part 2, I spoke about how helpful LLMs were in standing all this up. So, I figure it&#39;s only right to now show you, or at least, offer up a snapshot of the kinds of interactions/conversations I have with them within the context of this project.&lt;/p&gt;
&lt;p&gt;On the branch for this work, I started with the following prompt:&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;Alright, the final milestone before I test this in a VM would be setting up hardware integration, things like NVIDIA GPU support, Intel support, AMD (CPU &amp;amp; GPU) support. Boreas has an intel cpu and integrated gpu with an NVIDIA GPU. So it makes sense to directly have options for that in its configurations. However, capturing those options in a module would allow for reusability which I think is much more valuable. I also plan to get a framework in the near/distant future with potentially an AMD CPU and Nvidia GPU, but having separate modules for each makes sense. One for AMD CPU and another for AMD GPU. What do you think? We&#39;re just brainstorming here&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Ignoring the grammar for a bit, I find myself increasingly asking the LLM for its &amp;quot;thoughts&amp;quot; in my prompts. I think the reason I do this is to force brainstorming in some form. To make interactions with it more &amp;quot;interactive&amp;quot;, rather than just telling it what to do and having it &amp;quot;go forth&amp;quot;. I like to think that I am taking advantage of the fact that LLMs have knowledge they&#39;ve gleamed from the huge amount of text and code they are trained on. So it makes sense to want to learn from/query that a bit. LLMs, for all their faults (and the faults of the people behind them), know things. Thus, it only makes sense to want to leverage that knowledge by having them check &lt;em&gt;my&lt;/em&gt; reasoning against their vast repository.&lt;/p&gt;
&lt;p&gt;Nevertheless, that does not mean that their answers should be trusted by default. It does, however, make them useful as a kind of reasoning surface: I can explain what I&#39;m thinking, ask for pushback, and then check the response against real/alternative sources.&lt;/p&gt;
&lt;p&gt;Now, from that initial prompt, the LLM came back with a useful framing:&lt;/p&gt;
&lt;div data-callout-metadata=&quot;&quot; class=&quot;callout &quot; data-callout=&quot;ai-text&quot;&gt;&lt;div class=&quot;callout-title&quot;&gt;&lt;div class=&quot;callout-title-inner&quot;&gt;AI Text&lt;/div&gt;&lt;/div&gt;
&lt;div class=&quot;callout-content&quot;&gt;&lt;p&gt;I think your direction is good: split by hardware capability, not by machine.&lt;/p&gt;
&lt;p&gt;For Boreas, the host config should basically say:&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;nix&quot;&gt;&lt;code class=&quot;language-nix&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;my&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;hardware&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;cpu&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;intel&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;enable&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt; =&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt; true&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt;;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;my&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;hardware&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;gpu&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;intel&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;enable&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt; =&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt; true&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt;;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;my&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;hardware&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;gpu&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;nvidia&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;enable&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt; =&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt; true&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt;;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;my&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;hardware&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;gpu&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;nvidia&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;hybrid&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;enable&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt; =&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt; true&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt;;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Then the reusable modules own the messy details.&lt;/p&gt;
&lt;p&gt;...&lt;/p&gt;
&lt;p&gt;One caveat: this is a good milestone before a VM test for module evaluation, builds, and option shape, but a VM will not meaningfully validate NVIDIA/AMD/Intel hardware behavior. It can tell you “the config composes”; the real machine tells you “the driver stack behaves.” Still worth doing now, just don’t overestimate what the VM proves.&lt;/p&gt;
&lt;/div&gt;&lt;/div&gt;&lt;p&gt;See how it responds like a &amp;quot;coworker&amp;quot; and not a &amp;quot;yes master, of course master&amp;quot; slave. There is also a bit of push back. I then follow up by saying&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;This all tracks with what I was thinking about. And yes, there will absolutely be host specific configuration not suitable for a reusable module, but that&#39;s out of scope for now as those kinds of things tend to pop up as one goes. For the reusable hardware modules, I want to give it a shot myself first, but some inspiration in the form of real-world repo examples would be helpful&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;I like how it feels like a conversation. I&#39;m speaking to what the LLM has told me and suggested. I am looking for code examples I can learn from and indirectly grounding the LLM to actual factual code samples on GitHub in the process.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;[!aside]&lt;br /&gt;
I like getting inspiration from GitHub because a lot of code lives there (it&#39;s current reputation aside). Seems like a great place to learn from others. Pre-AI I&#39;d make good use of the search feature. Now I just describe what I&#39;m looking for and have GitHub Copilot pull up results. It&#39;s a pretty neat practice.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;The LLM then replies with&lt;/p&gt;
&lt;div data-callout-metadata=&quot;&quot; class=&quot;callout &quot; data-callout=&quot;ai-text&quot;&gt;&lt;div class=&quot;callout-title&quot;&gt;&lt;div class=&quot;callout-title-inner&quot;&gt;AI Text&lt;/div&gt;&lt;/div&gt;
&lt;div class=&quot;callout-content&quot;&gt;&lt;p&gt;Yep. Here are the examples I’d study.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Best Reference&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;code&gt;nixos-hardware&lt;/code&gt; is the main one: &lt;a href=&quot;https://github.com/NixOS/nixos-hardware/tree/master/common/cpu&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;common/cpu&lt;/a&gt; and &lt;a href=&quot;https://github.com/NixOS/nixos-hardware/tree/master/common/gpu&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;common/gpu&lt;/a&gt;. What to steal:
&lt;ul&gt;
&lt;li&gt;Intel CPU has both &lt;code&gt;default.nix&lt;/code&gt; and &lt;code&gt;cpu-only.nix&lt;/code&gt;; the default imports CPU + Intel GPU, but the CPU-only escape hatch exists.&lt;/li&gt;
&lt;li&gt;AMD CPU is tiny: it mostly sets &lt;code&gt;hardware.cpu.amd.updateMicrocode&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;AMD GPU uses &lt;code&gt;mkDefault&lt;/code&gt; for &lt;code&gt;modesetting&lt;/code&gt;, &lt;code&gt;hardware.graphics.enable&lt;/code&gt;, &lt;code&gt;enable32Bit&lt;/code&gt;, and &lt;code&gt;hardware.amdgpu.initrd.enable&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;Intel GPU is richer: it exposes options for &lt;code&gt;i915&lt;/code&gt; vs &lt;code&gt;xe&lt;/code&gt;, compute runtime, media runtime, VAAPI driver, and kernel-version assertions.&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;That last point is the big architectural lesson.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Other Useful Patterns&lt;/strong&gt;&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;nix&quot;&gt;&lt;code class=&quot;language-nix&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;nixos&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;system&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;gpu&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;span style=&quot;color:#9CDCFE&quot;&gt;maker&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt; =&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; &quot;amd&quot;&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt; |&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; &quot;intel&quot;&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt; |&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; &quot;nvidia&quot;&lt;/span&gt;&lt;span style=&quot;color:#F44747&quot;&gt;;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;...&lt;/p&gt;
&lt;/div&gt;&lt;/div&gt;&lt;p&gt;Now, this part might sound a little hand-wavy, but the &lt;code&gt;nixos-hardware&lt;/code&gt; it mentions piqued my interest. It sounded like something to take a look into, and so I did. What I found was something that already handles what I was trying to do. Something I could integrate rather than try to replicate, so that&#39;s what I did. I replied by saying&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;Or maybe I should just use nixos-hardware outright. No sense duplicating work. Or have my own integration based on it&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Then from there we got around to integrating &lt;code&gt;nixos-hardware&lt;/code&gt;. Fortunately, there is a module for the 2022 version of my host device that still holds with the 2023 version, so I could use it without any modifications. It would have been cooler/nicer if the LLM straight up told me from the start, &amp;quot;No bro, just use nixos-hardware&amp;quot;. I think if it truly understood what it is I was trying to do, it would have made that suggestion. Then again, perhaps if I had asked it to suggest the best/most conventional way of going about what I wanted to do, it might have made the &amp;quot;nixos-hardware&amp;quot; suggestion more immediately.&lt;/p&gt;
&lt;p&gt;I think this interaction is a good reminder that whatever &amp;quot;understanding&amp;quot; we think these systems have is brittle and prompt-dependent. Codex only had &lt;code&gt;nixos-hardware&lt;/code&gt; as &lt;em&gt;an&lt;/em&gt; answer to my query despite it being &lt;em&gt;the&lt;/em&gt; answer to what I was trying to accomplish. It didn&#39;t and perhaps wouldn&#39;t have made the connection unless I set it up to do so. But that&#39;s fine because I was there to do that. For me, this is yet another strong case for keeping a human in the loop: the model can surface options, but someone still has to recognize which option actually fits the job.&lt;/p&gt;
&lt;p&gt;Nevertheless, from there, we worked to port over things like cron jobs to systemd timers, handle dotfiles symlinking via home-manager, and improve the overall cohesion of the configuration, which just means consolidating and having less duplication.&lt;/p&gt;
&lt;p&gt;Since writing this blog post I have two other branches based off this one for refinements from testing and Hyprland ricing. I plan to make PRs for those and may even write accompanying blog posts if there&#39;s anything to share.&lt;/p&gt;
&lt;p&gt;With that, thanks for stopping by. Till next time!&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/nixos-the-last-milestone-part-3/"/>
  </entry><entry>
    <title>NixOS, the journey continues (part 2)</title>
    <updated>2026-05-21T11:04:00.000Z</updated>
    <id>https://thunk.blog/posts/nixos-the-journey-continues-part-2/</id>
    <content type="html">&lt;p&gt;Since &lt;a class=&quot;internal-link&quot; target=&quot;&quot; data-note-icon=&quot;&quot; href=&quot;https://thunk.blog/posts/nixos-the-start-of-something-new-part-1/&quot;&gt;part 1&lt;/a&gt;, I have done work to configure and make configurable the GUI for my budding NixOS setup. If you&#39;re interested in the code, the PR for it can be found &lt;a href=&quot;https://github.com/OlaoluwaM/nixos-config/pull/3&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;here&lt;/a&gt;. Comments and feedback welcome!&lt;/p&gt;
&lt;p&gt;Now, getting into some detail, AI, specifically Codex, has been really valuable thus far with this project. It&#39;s been a useful collaborator in discovering config options, understanding how they work (standalone or with others), figuring out best practices, and reviewing my work. One particularly interesting application has been using GitHub Copilot to search for real-world examples of derivations that I could draw inspiration from. I mainly did this to help with writing derivations for things not available in Nixpkgs; these can be found in the &lt;a href=&quot;https://github.com/OlaoluwaM/nixos-config/tree/main/pkgs&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;pkgs&lt;/code&gt;&lt;/a&gt; directory.&lt;/p&gt;
&lt;p&gt;It turns out that there are different strategies for writing a derivation depending on the kind of software you&#39;re looking to package. So far, I&#39;ve got one for a binary program, a Go program, and a Python program. None of them look like the others. Codex was useful in ensuring I was on the right track and in offering suggestions.&lt;/p&gt;
&lt;p&gt;Generally, being able to prompt Codex with &amp;quot;could you take a look at this, am I on the right track?&amp;quot;, or &amp;quot;is this the best way to do this?&amp;quot; took a lot of frustration out of things. I think having these kinds of interactions is a great way to learn and get ramped up quickly because you aren&#39;t just letting the AI do all the work; you&#39;re pair programming with it. I, for one, believe this approach has helped me get up to speed at a much faster pace than I would have otherwise. I&#39;m still fuzzy on a couple of things, like &lt;code&gt;inherit&lt;/code&gt; for instance, but I have come away with a better understanding of and greater appreciation for the configurability and level of control Nix offers.&lt;/p&gt;
&lt;p&gt;The work behind the PR also taught me a few things:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Using &lt;code&gt;lib.fakeHash&lt;/code&gt; in place of hashes you want NixOS to correct/fill in for you. This is relevant to my project because I&#39;m working on my setup on a Fedora system and then testing it in a VM.&lt;/li&gt;
&lt;li&gt;Some of the different ways a derivation can be made to package software and the different options available to simplify the process based on the kind of software you want to package. For instance, with a Go program, you may want to use something like &lt;code&gt;buildGoModule&lt;/code&gt;; for Python, you may want to reach for &lt;code&gt;python3Packages.buildPythonApplication&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;Derivations by default attempt to do a &lt;code&gt;make build&lt;/code&gt;, but you can instruct them not to do that by setting the &lt;code&gt;dontBuild&lt;/code&gt; option to &lt;code&gt;true&lt;/code&gt;. This was useful with the &lt;code&gt;rxfetch&lt;/code&gt; derivation since &lt;a href=&quot;https://github.com/mngshm/rxfetch&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;its repo&lt;/a&gt; already contained the binary.&lt;/li&gt;
&lt;li&gt;You can section off configuration into modules and create interfaces to create a more focused public &amp;quot;API&amp;quot; for those configurations. See &lt;a href=&quot;https://github.com/OlaoluwaM/nixos-config/pull/3/changes#diff-feca8763973a8b955b20039acae4cf20ae097b032f86eafaf2ca6d066d538ab8&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;desktop.nix&lt;/code&gt;&lt;/a&gt;.&lt;/li&gt;
&lt;li&gt;Nix supports imperative scripts using &lt;a href=&quot;https://nixos.asia/en/writeShellApplication&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;writeShellApplication&lt;/code&gt;&lt;/a&gt;. Scripts written in this way won&#39;t be run automatically but act more like shell scripts defined in a &lt;code&gt;.zshrc&lt;/code&gt; or &lt;code&gt;.bashrc&lt;/code&gt; file. They will be made available for you to invoke in your shell.&lt;/li&gt;
&lt;li&gt;Not everything can or should be configured with Nix. DE settings, sure, cloning stuff from GitHub? Setting up an SSH connection with a server? Probably not, unless you wish to do so with a Nix script. My intuition for this is still nascent so I don&#39;t really have a concrete heuristic to share.&lt;/li&gt;
&lt;li&gt;Whatever you have configured in &lt;code&gt;dconf&lt;/code&gt; in GNOME can probably be configured in Nix&lt;/li&gt;
&lt;li&gt;It&#39;s just like software engineering. Group and encapsulate related configurations. Expose them using abstract interfaces. The Nix language has support for defining your own options.&lt;/li&gt;
&lt;li&gt;Nix flakes have this requirement where files/modules must be checked into Git before they are evaluated. To circumvent this, you can use the &lt;code&gt;git add -N&lt;/code&gt; command, which signals the intent to check in a file.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;From here, I plan on creating modules for integrating with NVIDIA GPUs, AMD hardware (CPU &amp;amp; GPU), and Intel. I think that will be the last major piece before I start testing this in my VM. Let&#39;s see how it goes. Till next time!&lt;/p&gt;
&lt;p&gt;&lt;a class=&quot;internal-link&quot; target=&quot;&quot; data-note-icon=&quot;&quot; href=&quot;https://thunk.blog/posts/nixos-the-last-milestone-part-3/&quot;&gt;Part 3&lt;/a&gt;&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/nixos-the-journey-continues-part-2/"/>
  </entry><entry>
    <title>Implementing Redis INFO in Haskell</title>
    <updated>2026-06-16T17:38:00.000Z</updated>
    <id>https://thunk.blog/posts/implementing-redis-info-in-haskell/</id>
    <content type="html">&lt;p&gt;In a &lt;a class=&quot;internal-link&quot; target=&quot;&quot; data-note-icon=&quot;&quot; href=&quot;https://thunk.blog/posts/yes-cabal-custom-scripts-do-have-uses/&quot;&gt;previous blog post&lt;/a&gt;, I talked about how I used Cabal&#39;s custom setup script to generate a module with build info at compile time. As mentioned in that post, the custom setup script work was but a piece of what went into implementing the &lt;code&gt;INFO&lt;/code&gt; command for my Redis server. Let&#39;s talk about it.&lt;/p&gt;
&lt;p&gt;As always, if you want to peruse the code, feel free to check out the accompanying &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/pull/1&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;PR&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;Ok. Implementing &lt;code&gt;INFO&lt;/code&gt;.&lt;/p&gt;
&lt;p&gt;The Redis &lt;a href=&quot;https://redis.io/docs/latest/commands/info/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;INFO&lt;/code&gt;&lt;/a&gt; command was actually rather straightforward to build out, at least, relative to some of the others like &lt;code&gt;SET&lt;/code&gt; or &lt;code&gt;BGSAVE&lt;/code&gt;. For context, &lt;code&gt;INFO&lt;/code&gt; is a command for retrieving data on the current state of a Redis server. Thus, its implementation can be broken down into roughly two parts:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Finding the information to be displayed&lt;/li&gt;
&lt;li&gt;Displaying said information in the appropriate format&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;First, I will say that the &amp;quot;Finding the information&amp;quot; piece was definitely the more involved of the two. &amp;quot;Displaying the information&amp;quot; was far more mechanical. Once I got the overall structure for the command down – &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/pull/1/changes#diff-210325193069ec43b1d2d9516eded15e9b9bcb4b40e649786f5a439bb96d3e9b&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;Info.hs&lt;/code&gt;&lt;/a&gt; as the central coordinator/dispatcher, with a module per INFO section handling the retrieval and formatting of its own data – things fell into place. For example, for the &lt;code&gt;Server&lt;/code&gt; section, I have a &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/pull/1/changes#diff-d8e0740f46b3dc0de82ea0f8f2eb282ee87bb262a9fe862d532ddef507ea57f1&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;Server.hs&lt;/code&gt;&lt;/a&gt; module specifically for defining, organizing and formatting all the data relevant to that section. Likewise for the &lt;code&gt;Replication&lt;/code&gt; section there is &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/pull/1/changes#diff-1eea0c2579bf0bc14fc191a2d6ac4f6bdd094dcc96d4feb02ee0b848791df29a&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;Replication.hs&lt;/code&gt;&lt;/a&gt; module, and so on.&lt;/p&gt;
&lt;p&gt;&lt;code&gt;Info.hs&lt;/code&gt; is a command module. Command modules are responsible for:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Defining the handler for their associated command request&lt;/li&gt;
&lt;li&gt;Defining the domain types for their associated command request&lt;/li&gt;
&lt;li&gt;Defining how to parse their associated raw RESP request (more specifically, an intermediate representation of the raw request) into their defined domain types&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;I find things easier to manage when related stuff is co-located like this. For &lt;code&gt;INFO&lt;/code&gt;, the Redis docs define its request as optionally containing one or more output section filters. There are also some special section filter keywords like &lt;code&gt;all&lt;/code&gt;, &lt;code&gt;everything&lt;/code&gt;, and &lt;code&gt;default&lt;/code&gt;.&lt;/p&gt;
&lt;p&gt;Per my testing with an official Redis server via Docker, &lt;code&gt;all == everything&lt;/code&gt;. Perhaps there are options that make them behave differently, but I opted not to pry into that right now. If a section filter is omitted or, for some reason, they are all invalid, then &lt;code&gt;default&lt;/code&gt; is assumed.&lt;/p&gt;
&lt;p&gt;In &lt;code&gt;Info.hs&lt;/code&gt;, the &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info.hs#L43&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;mkInfoCmdArg&lt;/code&gt;&lt;/a&gt; function parses an intermediate representation of the raw Redis request (a list of bulkstrings) into the appropriate domain type: &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info.hs#L35&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;InfoCmdArg&lt;/code&gt;&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;Modeling the &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info.hs#L39&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;individual sections as a sum type&lt;/a&gt; was a choice. I &lt;em&gt;could&lt;/em&gt; have gone with strings, but types made all the subsequent pattern matching easier, particularly the dispatching on &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info.hs#L109&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;line 109&lt;/a&gt;. Plus, there are a finite number of section keywords anyways and they probably won&#39;t be changing anytime soon, so it wasn&#39;t a big lift.&lt;/p&gt;
&lt;p&gt;&lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info.hs#L92&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;handleInfo&lt;/code&gt; on line 92&lt;/a&gt; is the, well, handler for this command, synonymous with an API request handler in your favorite backend framework. All command modules have one. In &lt;code&gt;Info.hs&lt;/code&gt; all it does, besides gathering the necessary information, is:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;Process the parsed section filters (the actual parsing happens elsewhere using &lt;code&gt;mkInfoCmdArg&lt;/code&gt;), which boils down to handling the special section filters I mentioned earlier. See &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info.hs#L121&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;processInfoSectionsToShow&lt;/code&gt;&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;For each section to display, generate its section string; a part of the final response. The function that handles this comes from the module for that section (e.g. &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info/Replication.hs#L46&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;genReplicationInfoSectionStr&lt;/code&gt;&lt;/a&gt; and &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info/Server.hs#L192&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;genServerInfoSectionStr&lt;/code&gt;&lt;/a&gt;)&lt;/li&gt;
&lt;li&gt;Put all the section strings together&lt;/li&gt;
&lt;li&gt;Send the response&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;That&#39;s it.&lt;/p&gt;
&lt;p&gt;Now, to pass the challenge step, all I &lt;em&gt;actually&lt;/em&gt; needed was for my server to be able to handle an &lt;code&gt;INFO replication&lt;/code&gt; request and output just the &amp;quot;role&amp;quot; field specifically. But, the go-getter I am, I figured why not take a stab at the &amp;quot;Server&amp;quot; section too. Turns out the &amp;quot;Server&amp;quot; section&#39;s got hands:&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;https://thunk.blog/img/user/Extras/Assets/redis-info-server-got-hands-meme.jpg&quot; alt=&quot;redis-info-server-got-hands-meme.jpg&quot; /&gt;&lt;/p&gt;
&lt;p&gt;Implementing the &amp;quot;Server&amp;quot; section was an ok learning experience, but it took way longer than expected, mostly because I had to understand what its different fields were about and how their values were sourced. This meant a lot of navigating through the Redis codebase and I am not particularly well versed in C. GitHub Copilot was immensely helpful in getting me up to speed.&lt;/p&gt;
&lt;p&gt;Some items were trivial to source once I understood what I was looking at. Also, once I got my Cabal custom script thingy working, &lt;code&gt;redis_git_sha1&lt;/code&gt;, &lt;code&gt;redis_git_dirty&lt;/code&gt;, and &lt;code&gt;redis_build_id&lt;/code&gt; became straightforward as well. On the other hand, there were some items I couldn&#39;t or didn&#39;t want to spend the time trying to implement, fields like &lt;code&gt;lru_clock&lt;/code&gt;, &lt;code&gt;hz&lt;/code&gt;, and &lt;code&gt;configured_hz&lt;/code&gt;. Those I just stubbed. My implementation isn&#39;t 100% faithful to the official anyways. I added comments to all the &lt;code&gt;Server&lt;/code&gt; section fields, so feel free to check those out and perhaps point out any misconceptions I may have had.&lt;/p&gt;
&lt;p&gt;Next up on the list was testing. I won&#39;t go into much detail on how I&#39;ve set up my test harness since that&#39;s out of scope for this blog post. Though, if you&#39;re curious you can check &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/test/Redis/Test.hs&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Test.hs&lt;/a&gt;. I can also write something up about it if there is enough interest. In any case, what I do want to touch on are the tests for the &lt;code&gt;INFO&lt;/code&gt; command specifically.&lt;/p&gt;
&lt;p&gt;To start, my personal philosophy on testing is as follows&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;Write tests that make you confident that your system works as expected. &amp;quot;Works&amp;quot; here doesn&#39;t necessarily mean internals, though there are cases where ensuring the internals could be important. I, however, consider &amp;quot;works&amp;quot; to mean that &amp;quot;from the perspective of the user, the software does what it is supposed to.&amp;quot; As in, given some input or scenario we get the expected outcome. What happens in the middle is rarely of importance. Essentially, &amp;quot;what happens&amp;quot; &amp;gt; &amp;quot;how&amp;quot;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;For the &lt;code&gt;INFO&lt;/code&gt; command, it seemed like snapshot/golden testing would work best to achieve what I described above. The issue, though, was that parts of the &lt;code&gt;INFO&lt;/code&gt; response would change on each request or over time. Fields like &lt;code&gt;redis_git_sha1&lt;/code&gt;, &lt;code&gt;redis_build_id&lt;/code&gt;, &lt;code&gt;uptime&lt;/code&gt;, etc… I like to let my tests run in as close to a production environment as is reasonable. I don&#39;t like to mock, but mocking here to avoid snapshot churn seemed justifiable.&lt;/p&gt;
&lt;p&gt;Well, mocking is a bit of a misnomer, what I decided to do was &lt;em&gt;mask&lt;/em&gt; the volatile values with a string while in a test environment, which I signal using environment variables. You can check out the &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/test/Redis/Commands/Info/golden/default-info.txt&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;snapshots in the PR&lt;/a&gt; to see this in action. The function that handles this is called &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/blob/b1632a3cdf5dee654039bdc1a262072b396b9cf4/codecrafters-redis/src/Redis/Commands/Info/Common.hs#L13&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;maskFieldWhenNecessary&lt;/code&gt;&lt;/a&gt;. Pretty neat!&lt;/p&gt;
&lt;p&gt;I think those are all the interesting bits in this PR. I&#39;ve done some other things like building out &lt;a href=&quot;https://github.com/OlaoluwaM/redis-haskell/tree/0113c8433f3ce9ba1c180ed9c557e2b14b089c8f/codecrafters-redis/src/Redis/RDB&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;RDB&lt;/a&gt; support, but that predated this PR-blogpost style of writing I&#39;ve got going on here. Implementing that was way more involved and far more thought-provoking. If y&#39;all would be interested in a blog post about that, let me know!&lt;/p&gt;
&lt;p&gt;That&#39;s it. Thank you for reading. Till next time!&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/implementing-redis-info-in-haskell/"/>
  </entry><entry>
    <title>IO in Haskell, an epiphany</title>
    <updated>2026-05-21T11:45:00.000Z</updated>
    <id>https://thunk.blog/posts/io-in-haskell-an-epiphany/</id>
    <content type="html">&lt;p&gt;It seems to me like &lt;code&gt;IO&lt;/code&gt; is Haskell&#39;s way of modelling/representing the general concept of a statement (an action/impure operation, an operation with side effects that may or may not return a value). We know why statements are important. They allow our programs do useful things at the cost of non-determinism (changes to the world state tend not be consistent across invocations) and ordering (we tend to require that effects occur in a certain order).&lt;/p&gt;
&lt;p&gt;In mainstream languages, there is often little to no effort to meaningfully distinguish between statements and expressions (at least in my experience), as a result, expressions and statements tend to be interleave-able allowing any sub-program the ability to perform arbitrary side effects. Haskell, on the other hand, seems to have taken a stance against this by making explicit the notion of a statement at the type level through the &lt;code&gt;IO&lt;/code&gt; type (it&#39;s technically a type constructor but for the sake of explanation we&#39;ll consider it a type). This way, the implicit interleaving of statements and expressions becomes non-standard while the explicit demarcation of statements and expressions becomes enforceable by the type system.&lt;/p&gt;
&lt;p&gt;Though, this isn&#39;t to say they can&#39;t be mixed in any way as expressions can still return a type of &lt;code&gt;IO a&lt;/code&gt; while &lt;code&gt;IO&lt;/code&gt; actions can also trigger the evaluation of expressions, but in standard, idiomatic Haskell, the extent of this mixing is very strictly controlled and purposefully limited though escape hatches do exist for those who know what they&#39;re doing.&lt;/p&gt;
&lt;p&gt;With all that said, it might be accurate to consider Haskell&#39;s &lt;code&gt;IO a&lt;/code&gt; not as a value like &lt;code&gt;String&lt;/code&gt;, &lt;code&gt;Either e a&lt;/code&gt;, or &lt;code&gt;Int&lt;/code&gt;, but rather as a representation/description of an action/statement that when executed at runtime (given the current world state) yields a value of type &lt;code&gt;a&lt;/code&gt; potentially performing side effects along the way. This perspective also aligns somewhat with how &lt;code&gt;IO&lt;/code&gt; is defined as a state function (&lt;code&gt;s -&amp;gt; (a, s)&lt;/code&gt;) under the hood&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/io-in-haskell-an-epiphany/"/>
  </entry><entry>
    <title>How to produce multiple executables from a stack project</title>
    <updated>2026-05-21T11:46:00.000Z</updated>
    <id>https://thunk.blog/posts/how-to-produce-multiple-executables-from-a-stack-project/</id>
    <content type="html">&lt;p&gt;Ever wondered how to produce multiple executables from your &lt;code&gt;stack&lt;/code&gt; project? Neither did I until I found myself unable to do so in an experiment I was playing with. I&#39;ve figured it out now though, and for future reference, whenever things of this nature come up you&#39;ll probably want to look at how it&#39;s done in Cabal then extrapolate for Stack making use of the &lt;a href=&quot;https://github.com/sol/hpack&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;hpack repository&lt;/a&gt; for reference, or just use Cabal. I don&#39;t want to yet cause I&#39;m in too deep&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;[!aside] Aside&lt;br /&gt;
It is common for you to hear executables be referred to as &amp;quot;&lt;em&gt;targets&lt;/em&gt;&amp;quot;.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Anyway, with that said, When you want to generate multiple executables from your &lt;code&gt;stack&lt;/code&gt; project, you do so by defining additional entries under the &lt;code&gt;executables&lt;/code&gt; stanza in your &lt;code&gt;package.yaml&lt;/code&gt; file. The snippet below is an example. In it, we&#39;ve defined two executables that our project may produce. You&#39;ll notice that the &lt;code&gt;main&lt;/code&gt; module for each entry is under the same directory.&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;yaml&quot;&gt;&lt;code class=&quot;language-yaml&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;executables&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  initial-exe&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    main&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;Main.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    source-dirs&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;app&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;5&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    ghc-options&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;6&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-threaded&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;7&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-rtsopts&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;8&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-with-rtsopts=-N&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;9&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    dependencies&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;10&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;package-name&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;11&quot;&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;12&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  scenario1&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;13&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    main&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;Scenario1.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;14&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    source-dirs&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;app&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;15&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    other-modules&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: [] &lt;/span&gt;&lt;span style=&quot;color:#6A9955&quot;&gt;# Prevents hpack from auto-discovering `main` modules listed in the other executable entry within the same directory. If we don&#39;t add this, stack will try to compile those modules while compiling the project to create this executable&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;16&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    ghc-options&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;17&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-fprof-auto&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;18&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-threaded&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;19&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-rtsopts&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;20&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-with-rtsopts=-N&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;21&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    dependencies&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;22&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;package-name&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Note that in this case, the module in &lt;code&gt;app/Scenario1.hs&lt;/code&gt; must be named &lt;code&gt;Main&lt;/code&gt;, not &lt;code&gt;Scenario1&lt;/code&gt;. This is because GHC expects the entry point module to be named &lt;code&gt;Main&lt;/code&gt; and export a &lt;code&gt;main&lt;/code&gt; function. You can work around this using the &lt;a href=&quot;https://stackoverflow.com/questions/14238729/producing-multiple-executables-from-single-project&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;-main-is&lt;/code&gt; ghc-option&lt;/a&gt; like so&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;yaml&quot;&gt;&lt;code class=&quot;language-yaml&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;executables&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  initial-exe&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    main&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;Main.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    source-dirs&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;app&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;5&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    ghc-options&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;6&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-threaded&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;7&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-rtsopts&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;8&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-with-rtsopts=-N&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;9&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    dependencies&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;10&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;package-name&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;11&quot;&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;12&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  scenario1&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;13&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    main&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;Scenario1.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;14&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    source-dirs&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;app&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;15&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    other-modules&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: [] &lt;/span&gt;&lt;span style=&quot;color:#6A9955&quot;&gt;# Prevents hpack from auto-discovering `main` modules listed in the other executable entry within the same directory. If we don&#39;t add this, stack will try to compile those modules while compiling the project to create this executable&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;16&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    ghc-options&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;17&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-fprof-auto&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;18&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-threaded&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;19&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-rtsopts&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;20&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-main-is Scenario1&lt;/span&gt;&lt;span style=&quot;color:#6A9955&quot;&gt; # This allows us avoid needing to set the module name to `Main` in `Scenario1.hs`&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;21&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-with-rtsopts=-N&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;22&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    dependencies&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;23&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;package-name&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Personally, I prefer to &lt;a href=&quot;https://stackoverflow.com/questions/54150751/how-to-define-multiple-executables-main-modules-with-stack-hpack&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;give each executable its own directory&lt;/a&gt;. This way, there&#39;s no need to manually set the &lt;code&gt;other-modules&lt;/code&gt; field of each executable entry to &lt;code&gt;[]&lt;/code&gt;, plus I think it allows for more modularity/greater separation of concerns; if you&#39;ve got different executables, you probably have a different use case for each. Should you decide to go with this approach, you&#39;d structure your directory like so:&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;markdown&quot;&gt;&lt;code class=&quot;language-markdown&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;.&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;├── app &amp;#x3C;- Holds one executable module&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;│   └── Main.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;├── scenario1 &amp;#x3C;- Holds another executable module&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;5&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;│   └── Main.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;6&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;└── src &amp;#x3C;- project lib&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Then your &lt;code&gt;package.yaml&lt;/code&gt; configuration ought to look something like&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;yaml&quot;&gt;&lt;code class=&quot;language-yaml&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;executables&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  initial-exe&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    main&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;Main.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    source-dirs&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;app&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;5&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    ghc-options&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;6&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-threaded&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;7&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-rtsopts&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;8&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-with-rtsopts=-N&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;9&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    dependencies&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;10&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;package-name&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;11&quot;&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;12&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  scenario1&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;13&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    main&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;Scenario1.hs&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;14&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    source-dirs&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;scenario1&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;15&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    ghc-options&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;16&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-fprof-auto&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;17&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-threaded&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;18&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-rtsopts&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;19&quot;&gt;&lt;span style=&quot;color:#6A9955&quot;&gt;      # - -main-is Scenario1&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;20&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;-with-rtsopts=-N&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;21&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    dependencies&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;22&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;package-name&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;By placing the &lt;code&gt;main&lt;/code&gt; module of each of our executables in a separate directory, we no longer need to add the &lt;code&gt;other-modules: []&lt;/code&gt; field to each entry since &lt;code&gt;cabal&lt;/code&gt; (which &lt;code&gt;stack&lt;/code&gt; uses under the hood) becomes smart enough to know that the &lt;code&gt;Main.hs&lt;/code&gt; module in the &lt;code&gt;app&lt;/code&gt; directory has no relation to the &lt;code&gt;Scenario1.hs&lt;/code&gt; module in the &lt;code&gt;scenario1&lt;/code&gt; directory.&lt;/p&gt;
&lt;p&gt;Now, if we wanted to execute one of these &amp;quot;targets&amp;quot;, we could do&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;bash&quot;&gt;&lt;code class=&quot;language-bash&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#DCDCAA&quot;&gt;stack&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; run&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; scenario1&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt; --&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt; &amp;#x3C;&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;options-to-pass-to-executabl&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;e&gt;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;&lt;a href=&quot;https://docs.haskellstack.org/en/stable/commands/run_command/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;stack run&lt;/code&gt;&lt;/a&gt; will first compile, then run the resulting executable. If you don&#39;t pass an executable name, &lt;code&gt;stack run&lt;/code&gt; will compile and run the first executable listed under your project&#39;s &lt;code&gt;executables&lt;/code&gt; stanza.&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;bash&quot;&gt;&lt;code class=&quot;language-bash&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#DCDCAA&quot;&gt;stack&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; run&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  --&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt; &amp;#x3C;&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;options-to-pass-to-executabl&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;e&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#6A9955&quot;&gt;# Synonymous with&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;5&quot;&gt;&lt;span style=&quot;color:#DCDCAA&quot;&gt;stack&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; run&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; initial-exe&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt; --&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt; &amp;#x3C;&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;options-to-pass-to-executabl&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;e&gt;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;There is also &lt;a href=&quot;https://docs.haskellstack.org/en/stable/commands/exec_command/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;stack exec&lt;/code&gt;&lt;/a&gt;, but it will only run the executable. It does not re-compile it.&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;bash&quot;&gt;&lt;code class=&quot;language-bash&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#DCDCAA&quot;&gt;stack&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; exec&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; scenario1&lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt; --&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt; &amp;#x3C;&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;options-to-pass-to-executabl&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;e&gt;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;If you wanted to just compile/build a specific target without running it, you&#39;d need to use the &lt;a href=&quot;https://docs.haskellstack.org/en/stable/commands/build_command/#target-project-package-component&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;&lt;code&gt;stack build&lt;/code&gt;&lt;/a&gt;&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;bash&quot;&gt;&lt;code class=&quot;language-bash&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#DCDCAA&quot;&gt;stack&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; build&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; my-package:exe:my-executable&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span style=&quot;color:#DCDCAA&quot;&gt;stack&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; build&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; my-package:exe:my-executable-2&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#DCDCAA&quot;&gt;stack&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; build&lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt; other-package-in-proj:exe:exe3&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;And there you have it, now go do stuff&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/how-to-produce-multiple-executables-from-a-stack-project/"/>
  </entry><entry>
    <title>Endianness, WOOT!?</title>
    <updated>2026-05-21T11:46:00.000Z</updated>
    <id>https://thunk.blog/posts/endianness-woot/</id>
    <content type="html">&lt;p&gt;So there I was programming some binary serialization and deserialization logic to add persistence to my Redis server. I was in the zone then the docs I was referencing began to mention this &amp;quot;endian&amp;quot; thing, what the heck does that mean? I mean, I know it has something to do with how bytes are parsed but the rest of that thought was a little fuzzy ngl. It is at this point, dear reader, that I wish to inform you that I did it, for you, I fell right into the rabbit hole so you don&#39;t have to. Here&#39;s what I learned:&lt;/p&gt;
&lt;div data-callout-metadata=&quot;&quot; class=&quot;callout &quot; data-callout=&quot;tldr&quot;&gt;&lt;div class=&quot;callout-title&quot;&gt;&lt;div class=&quot;callout-title-inner&quot;&gt; TLDR&lt;/div&gt;&lt;/div&gt;
&lt;div class=&quot;callout-content&quot;&gt;&lt;p&gt;Assuming the following sequence of bytes:&lt;br /&gt;&lt;code&gt;[0x01, 0x02, 0x03, 0x04]&lt;/code&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Big-endian&lt;/strong&gt; (most significant byte first):&lt;br /&gt;&lt;code&gt;0x01020304 = 16,909,060&lt;/code&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Little-endian&lt;/strong&gt; (least significant byte first):&lt;br /&gt;&lt;code&gt;0x04030201 = 67,305,985&lt;/code&gt;&lt;/p&gt;
&lt;p&gt;Again,&lt;/p&gt;
&lt;p&gt;Reading a 16-bit value from a file:&lt;br /&gt;&lt;code&gt;bytes = [0x12, 0x34]&lt;/code&gt;&lt;/p&gt;
&lt;p&gt;Big-endian system interprets this as:&lt;br /&gt;&lt;code&gt;value = 0x1234 // 4660&lt;/code&gt;&lt;/p&gt;
&lt;p&gt;Little-endian system interprets this as:&lt;br /&gt;&lt;code&gt;value = 0x3412 // 13330&lt;/code&gt;&lt;/p&gt;
&lt;p&gt;If that doesn&#39;t suffice as a complete and holistic explanation, read on&lt;/p&gt;
&lt;/div&gt;&lt;/div&gt;&lt;p&gt;Endianness refers to the direction/order in which computers parse/store sequences of bytes, as the order influences what ends up being interpreted from said bytes.&lt;/p&gt;
&lt;p&gt;Note that this isn&#39;t the same thing as the bit order. AFAIU, endianness does not affect the order in which individual bits are stored and parsed. Bit order is somewhat consistent across systems but there &lt;em&gt;are&lt;/em&gt; instances where bit order &lt;em&gt;does&lt;/em&gt; matter namely in contexts where one finds themselves splitting bytes or doing some bit manipulation on not-so-neat chunks, which tends to be the case when working close to physical mediums/hardware. Ethernet is a good example of this. Nevertheless, with bit order it&#39;s typically alright to assume the largest binary place value as being at the leftmost position like in denary (base 10)&lt;/p&gt;
&lt;p&gt;To better grasp the concept of endianness, it might be best to first discuss the concept of most and least significant byte. You see, one might consider that numerals (&lt;a href=&quot;https://ola.bearblog.dev/numbers-numerals-oh-boy/&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;why am I calling them numerals they&#39;re numbers!?&lt;/a&gt;) are often made up of sub numerals, for example, in base 10 the numeral 912 is made up of the sub numerals 900, 10 and 2. Now which of this sub numerals would you say contributes the most to the whole value of 912, the 900 right? This then would be the most significant decimal numeral while the 2 would be the least significant. See where I&#39;m going with this?&lt;/p&gt;
&lt;p&gt;When dealing with a byte sequence, it&#39;s similar. Within said sequence there exist a byte that contributes most to the overall value represented by said sequence and there exists a byte that contributes the least. The positions of these bytes is determined by the endianness with which we parse such a sequence. If we were to use big endian, then we assume that the most significant byte is the leftmost byte similar to our decimal example. This kind of endianness feels more natural in that it allows us to parse byte sequences from left to right as we would decimal numbers. This kind of endianness is most common in network communications specifically network protocols.&lt;/p&gt;
&lt;p&gt;On the other hand, there is little endian which is the opposite of big endian. With little endian the leftmost byte in a sequence of bytes is considered the &lt;em&gt;least&lt;/em&gt; significant byte while the rightmost byte is considered the most significant. It is like reading decimal numbers from right to left. It&#39;s not as intuitive as big endian, but is commonly used within processors like the ubiquitous &lt;code&gt;x86_64&lt;/code&gt; Intel processor and most modern ARM implementations (though ARM processors can actually operate in either mode).&lt;/p&gt;
&lt;p&gt;The notion of endianness is important because it influences how data, in this case byte sequences, are interpreted. Like our decimal example 912 is representative of a completely different quantity than 219. The former is what we get with big endian whilst the latter is what we interpret using little endian.&lt;/p&gt;
&lt;p&gt;Now that&#39;s all well and good but why should you care about this? Frankly, I don&#39;t know but I reckon this information will in real handy when you find yourself dealing with a web protocol that communicates using a binary format or if you happen to be into systems programming in which case you probably already know all this stuff why are you here?&lt;/p&gt;
&lt;p&gt;Nevertheless, how&#39;s that for a productive Saturday! That wasn&#39;t so bad, right? (I need to get a life…)&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/endianness-woot/"/>
  </entry><entry>
    <title>Dotfiles Reorg</title>
    <updated>2026-05-21T11:46:00.000Z</updated>
    <id>https://thunk.blog/posts/dotfiles-reorg-a-journey/</id>
    <content type="html">&lt;p&gt;&lt;a href=&quot;https://github.com/OlaoluwaM/dotfiles/pull/3&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;I did some spring-cleaning on my dotfiles&lt;/a&gt;. Since I am planning on moving to NixOS, I figured now was as good a time as any to renovate. I&#39;ve had my current setup for a long time. It has served me well, but nothing lasts forever, I suppose.&lt;/p&gt;
&lt;p&gt;The layout up to this point was basically an answer to the question: &amp;quot;what&#39;s the simplest thing I could do?&amp;quot;.&lt;/p&gt;
&lt;p&gt;I first opted to have all my dotfiles grouped by function and listed out at the top-level. Following that, I adopted having a &lt;code&gt;.config&lt;/code&gt; parent directory.&lt;/p&gt;
&lt;p&gt;It was all well and good until I got a Mac for work. I then had to scope my dotfiles by OS: one for my Mac and one for my Fedora system. I deleted the &lt;code&gt;mac&lt;/code&gt; directory in this PR since I don&#39;t have a Mac anymore (should&#39;ve done that sooner). Now I just have the remaining out-of-place top-level &lt;code&gt;nixos&lt;/code&gt; directory that I use on my work Framework. It shouldn&#39;t be too much effort to move it over, though I&#39;d need to give it a name.&lt;/p&gt;
&lt;p&gt;In any case, if you took a look at the PR, you&#39;ll notice that &lt;code&gt;boreas&lt;/code&gt; is now a top-level directory with a &lt;code&gt;fedora&lt;/code&gt; and &lt;code&gt;nixos&lt;/code&gt; sub-directory. This is me embracing the Nix convention of organizing configuration per-host.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;[!aside]&lt;br /&gt;
&lt;a href=&quot;https://en.wikipedia.org/wiki/Boreas&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Boreas&lt;/a&gt; is the name I have bestowed upon my current device, my 2023 ASUS ROG Zephyrus M16 (the product line has, unfortunately, been abandoned by ASUS AFAICT).&lt;/p&gt;
&lt;p&gt;I will exclusively be adopting names from ancient Greek mythos for my devices.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Moving forward, each host directory will have sub-directories for each OS used or planned for it. This lends itself to dual booting too. In addition to those, there is the option for a sub-directory called &lt;code&gt;common&lt;/code&gt; where stuff that&#39;s, well, common among at least two OS sub-directories is kept and symlinks are created between it and the relevant OS sub-directories. This way, I no longer need to copy-paste to ensure parity across OSes. I can just edit the files in &lt;code&gt;common&lt;/code&gt; and the changes will propagate.&lt;/p&gt;
&lt;p&gt;There&#39;s also some added cognitive benefit to having hosts as the top-level entity and OS directories as sub-directories: co-location. All the dotfiles relevant to a host are grouped together.&lt;/p&gt;
&lt;p&gt;This co-location is what allows for the &lt;code&gt;common&lt;/code&gt; sub-directory I just described. Doing it the other way, with the OS as the top-level entity and the hosts nested within just seemed suboptimal to me. We&#39;d end up with configurations for a given host scattered across different directories. Nope.&lt;/p&gt;
&lt;p&gt;One interesting thing I did with this renovation was make sure to use &amp;quot;relative&amp;quot; symlinks. That way, the symlinks are never tied to any one host filesystem. It makes them portable and independent of the structure of the filesystem outside the repository.&lt;/p&gt;
&lt;p&gt;All in all, I&#39;m happy with how it&#39;s all turned out and hope to get myself over to NixOS soon. Lots of things to learn. Let&#39;s see how it goes. Oh and feel free to add a comment to the PR if you&#39;ve got anything to say!&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/dotfiles-reorg-a-journey/"/>
  </entry><entry>
    <title>Docker Compose variable interpolation...weirdness</title>
    <updated>2026-05-21T11:46:00.000Z</updated>
    <id>https://thunk.blog/posts/docker-compose-variable-interpolation-weirdness/</id>
    <content type="html">&lt;p&gt;Maybe not weirdness but I ran into something rather strange today. It looks like variables used in interpolation don&#39;t get pulled from the env files defined under the &lt;code&gt;env_file&lt;/code&gt; option for a service. Essentially&lt;/p&gt;
&lt;pre class=&quot;shiki dark-plus&quot; style=&quot;background-color:#1E1E1E;color:#D4D4D4&quot; tabindex=&quot;0&quot; data-language=&quot;yaml&quot;&gt;&lt;code class=&quot;language-yaml&quot;&gt;&lt;span class=&quot;line&quot; data-line=&quot;1&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;  db&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;2&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    image&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;postgres:16&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;3&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    env_file&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;4&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;path&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;./backend/.env.something&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;5&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;        required&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;true&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;6&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;path&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;./backend/.env.dev&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;7&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;        required&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;: &lt;/span&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;false&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;8&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    environment&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;9&quot;&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;      POSTGRES_DB=${POSTGRES_DB}&lt;/span&gt;&lt;span style=&quot;color:#6A9955&quot;&gt; # The variable for this will always be pulled from the shell env and .env specifically rather than from the env files defined as part of the service&#39;s `env_file` option&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;10&quot;&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;      POSTGRES_USER=${POSTGRES_USER}&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;11&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    ports&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;12&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;&quot;5432:5432&quot;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;13&quot;&gt;&lt;span style=&quot;color:#569CD6&quot;&gt;    volumes&lt;/span&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;:&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;line&quot; data-line=&quot;14&quot;&gt;&lt;span style=&quot;color:#D4D4D4&quot;&gt;      - &lt;/span&gt;&lt;span style=&quot;color:#CE9178&quot;&gt;postgres_data:/var/lib/postgresql/data&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;As you can probably tell from the example, this means the variables used in interpolations won&#39;t be sourced from the service&#39;s associated &lt;code&gt;env_file&lt;/code&gt; env file(s), but rather will always be sourced from the shell and &lt;em&gt;the&lt;/em&gt; &lt;code&gt;.env&lt;/code&gt; file. So you can&#39;t mix &lt;code&gt;env_file&lt;/code&gt; and &lt;code&gt;environment&lt;/code&gt; with interpolation in the same service definition and expect things to work properly.&lt;/p&gt;
&lt;p&gt;Our options then seem to be:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Have separate &lt;code&gt;.env&lt;/code&gt; files per service with no interpolation, so you&#39;ll have to copy stuff (like port numbers) and remember to update across files&lt;/li&gt;
&lt;li&gt;Put everything in a single &lt;code&gt;.env&lt;/code&gt; file, but prefix accordingly to avoid name-collision and allow for interpolation&lt;/li&gt;
&lt;/ul&gt;
&lt;blockquote&gt;
&lt;p&gt;[!aside] Aside&lt;br /&gt;
Name collisions can happen when different services have the same env var requirements, like a service for a dev db and test db. Both will accept the usual &lt;a href=&quot;https://github.com/docker-library/docs/blob/master/postgres/README.md#environment-variables&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;Postgres env variables&lt;/a&gt;. So if you&#39;re using a single &lt;code&gt;.env&lt;/code&gt; file, you&#39;ll need prefixes to differentiate and avoid collisions&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;A third option might be to put the variables used in interpolations in &lt;em&gt;the&lt;/em&gt; &lt;code&gt;.env&lt;/code&gt; file and everything else goes into the per service &lt;code&gt;env_file(s)&lt;/code&gt;. You&#39;ll still need to watch out for name collisions and prefix accordingly&lt;/p&gt;
&lt;p&gt;What do y&#39;all think? Is there a better option? Since this blog of mine doesn&#39;t support comments, &lt;a href=&quot;https://bsky.app/profile/olamusta.bsky.social&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;@ me on BluSky&lt;/a&gt; with your commentary instead. Link to this post too, so I know what you&#39;re talking about, heh&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/docker-compose-variable-interpolation-weirdness/"/>
  </entry><entry>
    <title>Be deliberate</title>
    <updated>2026-05-21T11:46:00.000Z</updated>
    <id>https://thunk.blog/posts/be-deliberate/</id>
    <content type="html">&lt;p&gt;Reading through the backlog of James Clear&#39;s 3-2-1 newsletter and came across &lt;a href=&quot;https://jamesclear.com/3-2-1/april-6-2023&quot; target=&quot;_blank&quot; class=&quot;external-link&quot;&gt;this&lt;/a&gt;:&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;[!quote] Quote&lt;br /&gt;
&amp;quot;Create your life rather than live it.&amp;quot;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;To me this reads simply as &amp;quot;be deliberate.&amp;quot; Don&#39;t just let yourself drown in mundane routine. Let each act be, as much as possible, a step towards the life you want for yourself&lt;/p&gt;
&lt;p&gt;This can be a difficult thing to do but as is the case with most things, starting one step at a time, little by little, can be immensely helpful. Another way to interpret this quote might be to ask yourself &amp;quot;what actions can I consistently take to create the life I want for myself&amp;quot;.&lt;/p&gt;
&lt;p&gt;Part of this is also introspection to, that is, being able to sit down with yourself and probe for what you really want out of life.&lt;/p&gt;
</content>
    <link href="https://thunk.blog/posts/be-deliberate/"/>
  </entry></feed>

