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<!DOCTYPE html><html><head><meta charset="utf-8"><meta http-equiv="x-ua-compatible" content="ie=edge"><meta property="fb:app_id" content="118554188236439"><meta name="viewport" content="width=device-width, initial-scale=1"><meta name="author" content="Maxim Sokhatsky"><meta name="twitter:site" content="@5HT"><meta name="twitter:creator" content="@5HT"><meta property="og:type" content="website"><meta property="og:image" content="https://avatars.githubusercontent.com/u/17128096?s=400&u=66a63d4cdd9625b2b4b37d724cc00fe6401e5bd8&v=4"><meta name="msapplication-TileColor" content="#ffffff"><meta name="msapplication-TileImage" content="https://anders.groupoid.space/images/ms-icon-144x144.png"><meta name="theme-color" content="#ffffff"><link rel="stylesheet" href="https://anders.groupoid.space/main.css?v=1"><link rel="apple-touch-icon" sizes="57x57" href="https://anders.groupoid.space/images/apple-icon-57x57.png"><link rel="apple-touch-icon" sizes="60x60" href="https://anders.groupoid.space/images/apple-icon-60x60.png"><link rel="apple-touch-icon" sizes="72x72" href="https://anders.groupoid.space/images/apple-icon-72x72.png"><link rel="apple-touch-icon" sizes="76x76" href="https://anders.groupoid.space/images/apple-icon-76x76.png"><link rel="apple-touch-icon" sizes="114x114" href="https://anders.groupoid.space/images/apple-icon-114x114.png"><link rel="apple-touch-icon" sizes="120x120" href="https://anders.groupoid.space/images/apple-icon-120x120.png"><link rel="apple-touch-icon" sizes="144x144" href="https://anders.groupoid.space/images/apple-icon-144x144.png"><link rel="apple-touch-icon" sizes="152x152" href="https://anders.groupoid.space/images/apple-icon-152x152.png"><link rel="apple-touch-icon" sizes="180x180" href="https://anders.groupoid.space/images//apple-icon-180x180.png"><link rel="icon" type="image/png" sizes="192x192" href="https://anders.groupoid.space/images/android-icon-192x192.png"><link rel="icon" type="image/png" sizes="32x32" href="https://anders.groupoid.space/images/favicon-32x32.png"><link rel="icon" type="image/png" sizes="96x96" href="https://anders.groupoid.space/images/favicon-96x96.png"><link rel="icon" type="image/png" sizes="16x16" href="https://anders.groupoid.space/images/favicon-16x16.png"><link rel="manifest" href="https://anders.groupoid.space/images/manifest.json"><style>svg a{fill:blue;stroke:blue}
[data-mml-node="merror"]>g{fill:red;stroke:red}
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</style></head><body class="content"></body></html><html><head><meta property="og:title" content="ANDERS"><meta property="og:description" content="Modal Homotopy Type System"><meta property="og:url" content="https://anders.groupoid.space/"></head></html><title>ANDERS</title><nav><a href='#'>ANDERS</a>
<a href='lib/index.html'>LIB</a>
<a href='spec/index.html'>SPEC</a></nav><article class="main"><div class="exe"><section><h1>MODAL HOMOTOPY TYPE SYSTEM</h1></section><aside><time>Published: 4 JUL 2021</time><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="8.149ex" height="1.593ex" role="img" focusable="false" viewBox="0 -698 3602 704" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2005-TEX-B-1D400" d="M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z"></path><path id="MJX-2005-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2005-TEX-B-1D41D" d="M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z"></path><path id="MJX-2005-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2005-TEX-B-1D42B" d="M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z"></path><path id="MJX-2005-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D400" xlink:href="#MJX-2005-TEX-B-1D400"></use><use data-c="1D427" xlink:href="#MJX-2005-TEX-B-1D427" transform="translate(869,0)"></use><use data-c="1D41D" xlink:href="#MJX-2005-TEX-B-1D41D" transform="translate(1508,0)"></use><use data-c="1D41E" xlink:href="#MJX-2005-TEX-B-1D41E" transform="translate(2147,0)"></use><use data-c="1D42B" xlink:href="#MJX-2005-TEX-B-1D42B" transform="translate(2674,0)"></use><use data-c="1D42C" xlink:href="#MJX-2005-TEX-B-1D42C" transform="translate(3148,0)"></use></g></g></g></g></svg></mjx-container> is a Modal HoTT proof assistant based on: classical <a href="https://raw.githubusercontent.com/michaelt/martin-lof/master/pdfs/Bibliopolis-Book-retypeset-1984.pdf">MLTT-80</a> [14] with 0, 1, 2, W types;
<a href="https://arxiv.org/pdf/1611.02108.pdf">CCHM</a> [4] in CHM flavour as cubical type system with hcomp/transp Kan operations;
HTS strict equality on pretypes; <a href="https://arxiv.org/pdf/1806.05966.pdf">de Rham</a> [19] stack modality primitives.
We tend not to touch general recursive higher inductive schemes yet,
instead we will try to express as much HIT as possible through W, Coequlizer
and HubSpokes Disc in the style of HoTT/Coq homotopy library and Three-HIT theorem.
</p></aside><section><div class="macro"><div class="macro__col"><h3 id="mltt"><b>MLTT</b></h3><ol><li><a href="#univ">UNI</a></li><li><a href="#pi">PI</a></li><li><a href="#sigma">SIGMA</a></li><li><a href="#id">ID</a></li><li><a href="#induction">0,1,2,W</a></li></ol></div><div class="macro__col"><h3 id="homotopy"><b>CCHM</b></h3><ol><li><a href="#univ">UNI</a></li><li><a href="#pi">PI</a></li><li><a href="#sigma">SIGMA</a></li><li><a href="#path">PATH</a></li><li><a href="#glue">GLUE</a></li><li><a href="#induction">HIT</a></li></ol></div><div class="macro__col"><h3 id="hts"><b>HTS</b></h3><ol><li><a href="#univ">UNI</a></li><li><a href="#pi">PI</a></li><li><a href="#sigma">SIGMA</a></li><li><a href="#id">ID</a></li><li><a href="#path">PATH</a></li><li><a href="#glue">GLUE</a></li><li><a href="#induction">0,1,2,W</a></li></ol></div></div><p>The <a href="https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/HTS.pdf">HTS</a> [1]
language proposed by Voevodsky exposes two different presheaf models of type theory:
the inner one is homotopy type system presheaf that models HoTT and the outer one is traditional Martin-Löf
type system presheaf that models set theory with UIP. The motivation behind this doubling is to
have an ability to express semisemplicial types. Theoretical work on merging inner
and outer languages was continued in <a href="https://arxiv.org/pdf/1705.03307.pdf">2LTT</a> [2,3].
</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="12.923ex" height="1.586ex" role="img" focusable="false" viewBox="0 -695 5712 701" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2006-TEX-B-1D408" d="M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z"></path><path id="MJX-2006-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2006-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path><path id="MJX-2006-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2006-TEX-B-1D41A" d="M64 349Q64 399 107 426T255 453Q346 453 402 423T473 341Q478 327 478 310T479 196V77Q493 63 529 62Q549 62 553 57T558 31Q558 9 552 5T514 0H497H481Q375 0 367 56L356 46Q300 -6 210 -6Q130 -6 81 30T32 121Q32 188 111 226T332 272H350V292Q350 313 348 327T337 361T306 391T248 402T194 399H189Q204 376 204 354Q204 327 187 306T134 284Q97 284 81 305T64 349ZM164 121Q164 89 186 67T238 45Q274 45 307 63T346 108L350 117V226H347Q248 218 206 189T164 121Z"></path><path id="MJX-2006-TEX-B-1D425" d="M43 686L134 690Q225 694 226 694H232V62H301V0H292Q274 3 170 3Q67 3 49 0H40V62H109V332Q109 387 109 453T110 534Q110 593 108 605T94 620Q80 624 53 624H40V686H43Z"></path><path id="MJX-2006-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2006-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D408" xlink:href="#MJX-2006-TEX-B-1D408"></use><use data-c="1D427" xlink:href="#MJX-2006-TEX-B-1D427" transform="translate(436,0)"></use><use data-c="1D42C" xlink:href="#MJX-2006-TEX-B-1D42C" transform="translate(1075,0)"></use><use data-c="1D42D" xlink:href="#MJX-2006-TEX-B-1D42D" transform="translate(1529,0)"></use><use data-c="1D41A" xlink:href="#MJX-2006-TEX-B-1D41A" transform="translate(1976,0)"></use><use data-c="1D425" xlink:href="#MJX-2006-TEX-B-1D425" transform="translate(2535,0)"></use><use data-c="1D425" xlink:href="#MJX-2006-TEX-B-1D425" transform="translate(2854,0)"></use><use data-c="1D41A" xlink:href="#MJX-2006-TEX-B-1D41A" transform="translate(3173,0)"></use><use data-c="1D42D" xlink:href="#MJX-2006-TEX-B-1D42D" transform="translate(3732,0)"></use><use data-c="1D422" xlink:href="#MJX-2006-TEX-B-1D422" transform="translate(4179,0)"></use><use data-c="1D428" xlink:href="#MJX-2006-TEX-B-1D428" transform="translate(4498,0)"></use><use data-c="1D427" xlink:href="#MJX-2006-TEX-B-1D427" transform="translate(5073,0)"></use></g></g></g></g></svg></mjx-container>. While we are on our road to Lean-like tactic language, currently
we are at the stage of regular cubical HTS type checker
with CHM-style [5] primitives.
You may try it from Github sources: <a href="https://github.com/groupoid/anders">groupoid/anders</a> or
install through OPAM package manager.
Main commands are check (to check a program) and repl (to enter the proof shell).
</p><code>$ opam install anders</code><p>Anders is fast, idiomatic and educational. We carefully draw the favourite
Lean-compatible syntax to fit 200 LOC in Menhir.
The CHM kernel is 1K LOC. Whole Anders compiles under 2 seconds
and checks all the base library under 1 second [i7-8700].
<b>Anders</b> proof assistant as Homotopy Type System comes
with its own <a href="https://anders.groupoid.space/lib/">Homotopy Library</a>.
</p><code>$ anders help
Anders Proof Assistant version 1.4.0
Copyright © 2021-2022 Groupoid Infinity
https://anders.groupoid.space/lib/
invocation = anders | anders list
list = [] | command list
primitive = zero | one | interval
command = check <filename> | lex <filename>
| parse <filename> | prim primitive <name>
| cubicaltt <filename> | girard
| trace | verbose
| repl | help
</code><br></section><section><h1>SYNTAX</h1><p>The syntax resembles original syntax of the reference CCHM type checker
cubicaltt, is slightly compatible with <b>Lean</b> syntax
and contains the full set of Cubical Agda [10]
primitives (except generic higher inductive schemes).
</p><p>Here is given the mathematical pseudo-code notation
of the language expressions that come immediately after parsing.
The core syntax definition of HTS language <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.541ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 681 680" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2007-TEX-N-45" d="M128 619Q121 626 117 628T101 631T58 634H25V680H597V676Q599 670 611 560T625 444V440H585V444Q584 447 582 465Q578 500 570 526T553 571T528 601T498 619T457 629T411 633T353 634Q266 634 251 633T233 622Q233 622 233 621Q232 619 232 497V376H286Q359 378 377 385Q413 401 416 469Q416 471 416 473V493H456V213H416V233Q415 268 408 288T383 317T349 328T297 330Q290 330 286 330H232V196V114Q232 57 237 52Q243 47 289 47H340H391Q428 47 452 50T505 62T552 92T584 146Q594 172 599 200T607 247T612 270V273H652V270Q651 267 632 137T610 3V0H25V46H58Q100 47 109 49T128 61V619Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="45" xlink:href="#MJX-2007-TEX-N-45"></use></g></g></g></g></svg></mjx-container> corresponds
to the type defined in OCaml module:
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data-mjx-texclass="ORD" transform="translate(16827.3,0)"><g data-mml-node="mi"><use data-c="45" xlink:href="#MJX-2008-TEX-N-45"></use></g></g></g></g><g data-mml-node="mtr" transform="translate(0,-15462.3)"><g data-mml-node="mtd" transform="translate(9130.7,0)"></g></g></g></g></g></svg></mjx-container></p><p>Further Menhir BNF notation will be used to describe the top-level language parser
as type checker is written in OCaml.</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.452ex;" xmlns="http://www.w3.org/2000/svg" width="11.33ex" height="2.023ex" role="img" focusable="false" viewBox="0 -694 5008 894" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2009-TEX-B-1D40A" d="M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V338L472 483L634 624H565V686H576Q597 683 728 683Q814 683 829 686H836V624H730L614 524Q507 432 497 422Q496 422 498 418T514 395T553 342T627 241L759 63L805 62H852V0H842Q830 3 701 3Q550 3 526 0H513V62H549Q584 62 584 63Q583 65 486 196T388 328L304 256V62H412V0H400Z"></path><path id="MJX-2009-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2009-TEX-B-1D432" d="M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z"></path><path id="MJX-2009-TEX-B-1D430" d="M624 444Q636 441 722 441Q797 441 800 444H805V382H741L593 11Q592 10 590 8T586 4T584 2T581 0T579 -2T575 -3T571 -3T567 -4T561 -4T553 -4H542Q525 -4 518 6T490 70Q474 110 463 137L415 257L367 137Q357 111 341 72Q320 17 313 7T289 -4H277Q259 -4 253 -2T238 11L90 382H25V444H32Q47 441 140 441Q243 441 261 444H270V382H222L310 164L382 342L366 382H303V444H310Q322 441 407 441Q508 441 523 444H531V382H506Q481 382 481 380Q482 376 529 259T577 142L674 382H617V444H624Z"></path><path id="MJX-2009-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path id="MJX-2009-TEX-B-1D42B" d="M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z"></path><path id="MJX-2009-TEX-B-1D41D" d="M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z"></path><path id="MJX-2009-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D40A" xlink:href="#MJX-2009-TEX-B-1D40A"></use><use data-c="1D41E" xlink:href="#MJX-2009-TEX-B-1D41E" transform="translate(901,0)"></use><use data-c="1D432" xlink:href="#MJX-2009-TEX-B-1D432" transform="translate(1428,0)"></use><use data-c="1D430" xlink:href="#MJX-2009-TEX-B-1D430" transform="translate(2035,0)"></use><use data-c="1D428" xlink:href="#MJX-2009-TEX-B-1D428" transform="translate(2866,0)"></use><use data-c="1D42B" xlink:href="#MJX-2009-TEX-B-1D42B" transform="translate(3441,0)"></use><use data-c="1D41D" xlink:href="#MJX-2009-TEX-B-1D41D" transform="translate(3915,0)"></use><use data-c="1D42C" xlink:href="#MJX-2009-TEX-B-1D42C" transform="translate(4554,0)"></use></g></g></g></g></svg></mjx-container>.
The words of a top-level language (file or repl) consist of keywords or identifiers.
The keywords and special characters are following:</p><code>(, ), [, ], <, >, /, .1, .2, Π, Σ, λ, ×, →,
:, :=, ↦, U, V, ∧, ∨, -, +, @, module, where, import,
option, def, axiom, inc, postulate, theorem, PathP,
transp, hcomp, zero, one, Partial, ouc, interval, W,
sup, Glue, glue, unglue, ind₀, ind₁, ind₂, indᵂ</code><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="13.057ex" height="1.597ex" role="img" focusable="false" viewBox="0 -700 5771 706" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2010-TEX-B-1D408" d="M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z"></path><path id="MJX-2010-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2010-TEX-B-1D41D" d="M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z"></path><path id="MJX-2010-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2010-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2010-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2010-TEX-B-1D41F" d="M308 0Q290 3 172 3Q58 3 49 0H40V62H109V382H42V444H109V503L110 562L112 572Q127 625 178 658T316 699Q318 699 330 699T348 700Q381 698 404 687T436 658T449 629T452 606Q452 576 432 557T383 537Q355 537 335 555T314 605Q314 635 328 649H325Q311 649 293 644T253 618T227 560Q226 555 226 498V444H340V382H232V62H318V0H308Z"></path><path id="MJX-2010-TEX-B-1D42B" d="M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z"></path><path id="MJX-2010-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D408" xlink:href="#MJX-2010-TEX-B-1D408"></use><use data-c="1D427" xlink:href="#MJX-2010-TEX-B-1D427" transform="translate(436,0)"></use><use data-c="1D41D" xlink:href="#MJX-2010-TEX-B-1D41D" transform="translate(1075,0)"></use><use data-c="1D41E" xlink:href="#MJX-2010-TEX-B-1D41E" transform="translate(1714,0)"></use><use data-c="1D427" xlink:href="#MJX-2010-TEX-B-1D427" transform="translate(2241,0)"></use><use data-c="1D42D" xlink:href="#MJX-2010-TEX-B-1D42D" transform="translate(2880,0)"></use><use data-c="1D422" xlink:href="#MJX-2010-TEX-B-1D422" transform="translate(3327,0)"></use><use data-c="1D41F" xlink:href="#MJX-2010-TEX-B-1D41F" transform="translate(3646,0)"></use><use data-c="1D422" xlink:href="#MJX-2010-TEX-B-1D422" transform="translate(3997,0)"></use><use data-c="1D41E" xlink:href="#MJX-2010-TEX-B-1D41E" transform="translate(4316,0)"></use><use data-c="1D42B" xlink:href="#MJX-2010-TEX-B-1D42B" transform="translate(4843,0)"></use><use data-c="1D42C" xlink:href="#MJX-2010-TEX-B-1D42C" transform="translate(5317,0)"></use></g></g></g></g></svg></mjx-container>. Identifiers support UTF-8. Indentifiers couldn’t start
with :, -, →. Sample identifiers:</p><code>¬-of-∨, 1=1, is-?, =, $^~]!005x, ∞, x→Nat, f'</code><br><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="9.604ex" height="1.584ex" role="img" focusable="false" viewBox="0 -694 4245 700" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2011-TEX-B-1D40C" d="M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H305Q316 679 323 667Q330 653 434 414L546 157L658 414Q766 662 773 674Q778 681 788 686H1052V624H944V62H1052V0H1040Q1016 3 874 3T708 0H696V62H804V341L803 618L786 580Q770 543 735 462T671 315Q540 13 536 9Q528 1 507 1Q485 1 477 9Q472 14 408 162T281 457T217 603Q215 603 215 334V62H323V0H314Z"></path><path id="MJX-2011-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path id="MJX-2011-TEX-B-1D41D" d="M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z"></path><path id="MJX-2011-TEX-B-1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z"></path><path id="MJX-2011-TEX-B-1D425" d="M43 686L134 690Q225 694 226 694H232V62H301V0H292Q274 3 170 3Q67 3 49 0H40V62H109V332Q109 387 109 453T110 534Q110 593 108 605T94 620Q80 624 53 624H40V686H43Z"></path><path id="MJX-2011-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2011-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D40C" xlink:href="#MJX-2011-TEX-B-1D40C"></use><use data-c="1D428" xlink:href="#MJX-2011-TEX-B-1D428" transform="translate(1092,0)"></use><use data-c="1D41D" xlink:href="#MJX-2011-TEX-B-1D41D" transform="translate(1667,0)"></use><use data-c="1D42E" xlink:href="#MJX-2011-TEX-B-1D42E" transform="translate(2306,0)"></use><use data-c="1D425" xlink:href="#MJX-2011-TEX-B-1D425" transform="translate(2945,0)"></use><use data-c="1D41E" xlink:href="#MJX-2011-TEX-B-1D41E" transform="translate(3264,0)"></use><use data-c="1D42C" xlink:href="#MJX-2011-TEX-B-1D42C" transform="translate(3791,0)"></use></g></g></g></g></svg></mjx-container>. Modules represent files with declarations. More accurate,
BNF notation of module consists of imports, options and declarations.</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.439ex;" xmlns="http://www.w3.org/2000/svg" width="9.011ex" height="1.991ex" role="img" focusable="false" viewBox="0 -686 3983 880" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2012-TEX-B-1D408" d="M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z"></path><path id="MJX-2012-TEX-B-1D426" d="M40 442Q217 450 218 450H224V365Q226 367 235 378T254 397T278 416T314 435T362 448Q376 450 400 450H406Q503 450 534 393Q545 376 545 370Q545 368 555 379Q611 450 716 450Q774 450 809 434Q850 414 861 379T873 276V213V198V62H942V0H933Q915 3 809 3Q702 3 684 0H675V62H744V194V275Q744 348 735 373T690 399Q645 399 607 370T557 290Q555 281 554 171V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2012-TEX-B-1D429" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path><path id="MJX-2012-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path id="MJX-2012-TEX-B-1D42B" d="M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z"></path><path id="MJX-2012-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2012-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D408" xlink:href="#MJX-2012-TEX-B-1D408"></use><use data-c="1D426" xlink:href="#MJX-2012-TEX-B-1D426" transform="translate(436,0)"></use><use data-c="1D429" xlink:href="#MJX-2012-TEX-B-1D429" transform="translate(1394,0)"></use><use data-c="1D428" xlink:href="#MJX-2012-TEX-B-1D428" transform="translate(2033,0)"></use><use data-c="1D42B" xlink:href="#MJX-2012-TEX-B-1D42B" transform="translate(2608,0)"></use><use data-c="1D42D" xlink:href="#MJX-2012-TEX-B-1D42D" transform="translate(3082,0)"></use><use data-c="1D42C" xlink:href="#MJX-2012-TEX-B-1D42C" transform="translate(3529,0)"></use></g></g></g></g></svg></mjx-container>. The import construction supports file folder
structure (without file extensions) by using reserved symbol /
for hierarchy walking.</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.439ex;" xmlns="http://www.w3.org/2000/svg" width="8.907ex" height="2.014ex" role="img" focusable="false" viewBox="0 -696 3937 890" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2013-TEX-B-1D40E" d="M64 339Q64 431 96 502T182 614T295 675T420 696Q469 696 481 695Q620 680 709 589T798 339Q798 173 697 82T432 -10Q262 -10 163 85T64 339ZM625 454Q618 502 600 538T562 593T515 624T469 639T431 642Q331 642 276 563Q232 493 232 353Q232 315 234 285T244 216T267 148T308 94T372 56Q405 46 432 46Q517 46 567 106T627 267Q631 299 631 353Q631 418 625 454Z"></path><path id="MJX-2013-TEX-B-1D429" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path><path id="MJX-2013-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2013-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2013-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path id="MJX-2013-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2013-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D40E" xlink:href="#MJX-2013-TEX-B-1D40E"></use><use data-c="1D429" xlink:href="#MJX-2013-TEX-B-1D429" transform="translate(864,0)"></use><use data-c="1D42D" xlink:href="#MJX-2013-TEX-B-1D42D" transform="translate(1503,0)"></use><use data-c="1D422" xlink:href="#MJX-2013-TEX-B-1D422" transform="translate(1950,0)"></use><use data-c="1D428" xlink:href="#MJX-2013-TEX-B-1D428" transform="translate(2269,0)"></use><use data-c="1D427" xlink:href="#MJX-2013-TEX-B-1D427" transform="translate(2844,0)"></use><use data-c="1D42C" xlink:href="#MJX-2013-TEX-B-1D42C" transform="translate(3483,0)"></use></g></g></g></g></svg></mjx-container>. Each option holds bool value. Language supports following options:
1) girard (enables U : U);
2) pre-eval (normalization cache);
3) impredicative (infinite hierarchy with impredicativity rule);</p><code>%start <Module.file> file
%start <Module.command> repl
repl : COLON IDENT exp1 EOF | COLON IDENT EOF | exp0 EOF | EOF
file : MODULE IDENT WHERE line* EOF
path : IDENT
line :
| IMPORT path+
| OPTION IDENT IDENT
| declarations</code><p>In Anders you can enable or disable language core types,
adjust syntaxes or tune inner variables of the type checker.
Here is the example how to setup minimal core able to prove
internalization of MLTT-73 variation (Path
instead of Id and no inductive types, see base library):</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="14.174ex" height="1.586ex" role="img" focusable="false" viewBox="0 -695 6265 701" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2014-TEX-B-1D403" d="M39 624V686H270H310H408Q500 686 545 680T638 649Q768 584 805 438Q817 388 817 338Q817 171 702 75Q628 17 515 2Q504 1 270 0H39V62H147V624H39ZM655 337Q655 370 655 390T650 442T639 494T616 540T580 580T526 607T451 623Q443 624 368 624H298V62H377H387H407Q445 62 472 65T540 83T606 129Q629 156 640 195T653 262T655 337Z"></path><path id="MJX-2014-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2014-TEX-B-1D41C" d="M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z"></path><path id="MJX-2014-TEX-B-1D425" d="M43 686L134 690Q225 694 226 694H232V62H301V0H292Q274 3 170 3Q67 3 49 0H40V62H109V332Q109 387 109 453T110 534Q110 593 108 605T94 620Q80 624 53 624H40V686H43Z"></path><path id="MJX-2014-TEX-B-1D41A" d="M64 349Q64 399 107 426T255 453Q346 453 402 423T473 341Q478 327 478 310T479 196V77Q493 63 529 62Q549 62 553 57T558 31Q558 9 552 5T514 0H497H481Q375 0 367 56L356 46Q300 -6 210 -6Q130 -6 81 30T32 121Q32 188 111 226T332 272H350V292Q350 313 348 327T337 361T306 391T248 402T194 399H189Q204 376 204 354Q204 327 187 306T134 284Q97 284 81 305T64 349ZM164 121Q164 89 186 67T238 45Q274 45 307 63T346 108L350 117V226H347Q248 218 206 189T164 121Z"></path><path id="MJX-2014-TEX-B-1D42B" d="M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z"></path><path id="MJX-2014-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2014-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2014-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path id="MJX-2014-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2014-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D403" xlink:href="#MJX-2014-TEX-B-1D403"></use><use data-c="1D41E" xlink:href="#MJX-2014-TEX-B-1D41E" transform="translate(882,0)"></use><use data-c="1D41C" xlink:href="#MJX-2014-TEX-B-1D41C" transform="translate(1409,0)"></use><use data-c="1D425" xlink:href="#MJX-2014-TEX-B-1D425" transform="translate(1920,0)"></use><use data-c="1D41A" xlink:href="#MJX-2014-TEX-B-1D41A" transform="translate(2239,0)"></use><use data-c="1D42B" xlink:href="#MJX-2014-TEX-B-1D42B" transform="translate(2798,0)"></use><use data-c="1D41A" xlink:href="#MJX-2014-TEX-B-1D41A" transform="translate(3272,0)"></use><use data-c="1D42D" xlink:href="#MJX-2014-TEX-B-1D42D" transform="translate(3831,0)"></use><use data-c="1D422" xlink:href="#MJX-2014-TEX-B-1D422" transform="translate(4278,0)"></use><use data-c="1D428" xlink:href="#MJX-2014-TEX-B-1D428" transform="translate(4597,0)"></use><use data-c="1D427" xlink:href="#MJX-2014-TEX-B-1D427" transform="translate(5172,0)"></use><use data-c="1D42C" xlink:href="#MJX-2014-TEX-B-1D42C" transform="translate(5811,0)"></use></g></g></g></g></svg></mjx-container>. Language supports following top level declarations:
1) <b>axiom</b> (non-computable declaration that breakes normalization);
2) <b>postulate</b> (alternative or inverted axiom that can preserve consistency);
3) <b>definition</b> (almost any explicit term or type in type theory);
5) <b>lemma</b> (helper in big game).
4) <b>theorem</b> (something valuable or complex enough).</p><code>ident : IRREF | IDENT
vars : ident+
lense : LPARENS vars COLON exp1 RPARENS
telescope : lense telescope
params : telescope | []
declarations:
| DEF IDENT params DEFEQ exp1
| DEF IDENT params COLON exp1 DEFEQ exp1
| AXIOM IDENT params COLON exp1</code><br><p>Sample declarations. For example, signature <b>isProp (A : U)</b> of
type <b>U</b> could be defined as normalization-blocking axiom
without proof-term or by providing proof-term as definition.</p><code><span class="h__keyword">axiom</span> isProp <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">:</span> <span class="h__keyword">U</span>
<span class="h__keyword">def</span> isSet <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">:</span> <span class="h__keyword">U</span> <span class="h__symbol">:</span><span class="h__symbol">=</span> Π <span class="h__symbol">(</span>a b <span class="h__symbol">:</span> A<span class="h__symbol">)</span>
<span class="h__symbol">(</span>x y <span class="h__symbol">:</span> <span class="h__keyword">Path</span> A a b<span class="h__symbol">)</span>, <span class="h__keyword">Path</span> <span class="h__symbol">(</span><span class="h__keyword">Path</span> A a b<span class="h__symbol">)</span> x y</code><p>In this example <b>(A : U)</b>, <b>(a b : A)</b> and <b>(x y : Path A a b)</b> are
called telescopes. Each telescope consists of a series of lenses or empty.
Each lense provides a set of variables of the same type. Telescope
defines parameters of a declaration. Types in a telescope,
type of a declaration and a proof-terms are a language expressions <b>exp1</b>.</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.439ex;" xmlns="http://www.w3.org/2000/svg" width="13.344ex" height="2.011ex" role="img" focusable="false" viewBox="0 -695 5898 889" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2015-TEX-B-1D404" d="M723 286Q721 284 700 145T677 3V0H39V62H147V618H39V680H660V676Q662 670 675 552T691 428V424H629V428Q629 429 627 448T618 494T601 541Q574 593 527 605T382 618H374H304V384H336Q338 384 347 384T361 384T376 386T392 390T407 397T421 407T432 423Q442 444 443 482V501H505V205H443V224Q442 258 435 278T411 307T380 318T336 322H304V62H375H394Q429 62 449 62T497 66T541 76T577 95T609 126T632 170T651 232Q661 287 661 289H723V286Z"></path><path id="MJX-2015-TEX-B-1D431" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path><path id="MJX-2015-TEX-B-1D429" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path><path id="MJX-2015-TEX-B-1D42B" d="M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z"></path><path id="MJX-2015-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2015-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path><path id="MJX-2015-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2015-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path id="MJX-2015-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D404" xlink:href="#MJX-2015-TEX-B-1D404"></use><use data-c="1D431" xlink:href="#MJX-2015-TEX-B-1D431" transform="translate(756,0)"></use><use data-c="1D429" xlink:href="#MJX-2015-TEX-B-1D429" transform="translate(1363,0)"></use><use data-c="1D42B" xlink:href="#MJX-2015-TEX-B-1D42B" transform="translate(2002,0)"></use><use data-c="1D41E" xlink:href="#MJX-2015-TEX-B-1D41E" transform="translate(2476,0)"></use><use data-c="1D42C" xlink:href="#MJX-2015-TEX-B-1D42C" transform="translate(3003,0)"></use><use data-c="1D42C" xlink:href="#MJX-2015-TEX-B-1D42C" transform="translate(3457,0)"></use><use data-c="1D422" xlink:href="#MJX-2015-TEX-B-1D422" transform="translate(3911,0)"></use><use data-c="1D428" xlink:href="#MJX-2015-TEX-B-1D428" transform="translate(4230,0)"></use><use data-c="1D427" xlink:href="#MJX-2015-TEX-B-1D427" transform="translate(4805,0)"></use><use data-c="1D42C" xlink:href="#MJX-2015-TEX-B-1D42C" transform="translate(5444,0)"></use></g></g></g></g></svg></mjx-container>. All atomic language expressions are grouped
by four categories:
<b>exp0</b> (pair constructions),
<b>exp1</b> (non neutral constructions),
<b>exp2</b> (path and pi applcations),
<b>exp3</b> (neutral constructions).</p><code>face : LPARENS IDENT IDENT IDENT RPARENS
partial : face+ ARROW exp1</code><br><code>exp0 :
| exp1 COMMA exp0
| exp1</code><br><code>exp1:
| LAM telescope COMMA exp1
| PI telescope COMMA exp1
| SIGMA telescope COMMA exp1
| LSQ IRREF ARROW exp1 RSQ
| LSQ separated_list(COMMA, partial) RSQ
| LT vars GT exp1
| exp2 ARROW exp1
| exp2 PROD exp1
| exp2</code><br><p>The LR parsers demand to define <b>exp1</b> as expressions that
cannot be used (without a parens enclosure) as a right part of
left-associative application for both Path and Pi lambdas.</p><code>exp2 :
| exp2 exp3
| exp2 APPFORMULA exp3
| exp3</code><br><p>Universe indices <b>U<sub>j</sub></b> (inner fibrant) and <b>V<sub>k</sub></b> (outer pretypes)
are using unicode subscript letters that are already processed in lexer.</p><code>exp3:
| HOLE | PRE
| KAN | exp3 FST
| exp3 SND | NEGATE exp3
| exp3 AND exp3 | exp3 OR exp3
| ID exp3 | REF exp3
| IDJ exp3 | INC exp3
| OUC exp3 | PATHP exp3
| TRANSP exp3 exp3 | HCOMP exp3
| PARTIAL exp3 | IDENT LSQ exp0 MAP exp0 RSQ
| LPARENS exp0 RPARENS | IDENT
| LPARENS exp0 RPARENS LSQ exp0 MAP exp0 RSQ</code><br><h1>SEMANTICS</h1><p>The idea is to have a unified layered type checker, so you can disbale/enable
any <b>MLTT</b>-style inference, assign types to universes and enable/disable hierachies.
This will be done by providing linking API for pluggable presheaf modules.
We selected 5 levels of type checker awareness from universes and pure type systems
up to synthetic language of homotopy type theory. Each layer corresponds to its
presheaves with separate configuration for universe hierarchies.</p><code>inductive lang : U
| UNI: cosmos → lang
| PI: pure lang → lang
| SIGMA: total lang → lang
| ID: uip lang → lang
| PATH: homotopy lang → lang
| GLUE: gluening lang → lang
| HIT: hit lang → lang</code><br><p>We want to mention here with homage to its authors all categorical
models of dependent type theory: Comprehension Categories (Grothendieck, Jacobs),
LCCC (Seely), D-Categories and CwA (Cartmell), CwF (Dybjer), C-Systems (Voevodsky),
Natural Models (Awodey). While we can build some transports between them,
we leave this excercise for our <a href="https://groupoid.space">The Cubical Base Library</a> library.</p><p>We will use here the Coquand’s notation
for <a href="https://anders.groupoid.space/mathematics/topoi/presheaf/">Presheaf Type Theories</a> in
terms of restriction maps.</p><br><h2>Universe Hierarchies</h2><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="11.054ex" height="1.597ex" role="img" focusable="false" viewBox="0 -695 4886 706" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2016-TEX-B-1D414" d="M570 686Q588 683 703 683T836 686H845V624H737V420Q737 390 737 345T738 284Q738 205 729 164T689 83Q614 -11 465 -11Q321 -11 240 51T148 207Q147 214 147 421V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V405V370V268Q304 181 311 146T346 87Q387 52 466 52Q642 52 667 195Q668 204 669 415V624H561V686H570Z"></path><path id="MJX-2016-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2016-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2016-TEX-B-1D42F" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path id="MJX-2016-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2016-TEX-B-1D42B" d="M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z"></path><path id="MJX-2016-TEX-B-1D42C" d="M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D414" xlink:href="#MJX-2016-TEX-B-1D414"></use><use data-c="1D427" xlink:href="#MJX-2016-TEX-B-1D427" transform="translate(885,0)"></use><use data-c="1D422" xlink:href="#MJX-2016-TEX-B-1D422" transform="translate(1524,0)"></use><use data-c="1D42F" xlink:href="#MJX-2016-TEX-B-1D42F" transform="translate(1843,0)"></use><use data-c="1D41E" xlink:href="#MJX-2016-TEX-B-1D41E" transform="translate(2450,0)"></use><use data-c="1D42B" xlink:href="#MJX-2016-TEX-B-1D42B" transform="translate(2977,0)"></use><use data-c="1D42C" xlink:href="#MJX-2016-TEX-B-1D42C" transform="translate(3451,0)"></use><use data-c="1D41E" xlink:href="#MJX-2016-TEX-B-1D41E" transform="translate(3905,0)"></use><use data-c="1D42C" xlink:href="#MJX-2016-TEX-B-1D42C" transform="translate(4432,0)"></use></g></g></g></g></svg></mjx-container>. Anders supports hierarchy with two universes:
fibrant (U) and pretypes (V). All universes are bounded with preorder:</p><p style="text-align:center;"><mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.666ex;" xmlns="http://www.w3.org/2000/svg" width="22.022ex" height="2.236ex" role="img" focusable="false" viewBox="0 -694 9733.8 988.2" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2017-TEX-I-1D439" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 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transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D439" xlink:href="#MJX-2017-TEX-I-1D439"></use></g><g data-mml-node="mi" transform="translate(749,0)"><use data-c="1D456" xlink:href="#MJX-2017-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(1094,0)"><use data-c="1D44F" xlink:href="#MJX-2017-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(1523,0)"><use data-c="1D45F" xlink:href="#MJX-2017-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(1974,0)"><use data-c="1D44E" xlink:href="#MJX-2017-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(2503,0)"><use data-c="1D45B" xlink:href="#MJX-2017-TEX-I-1D45B"></use></g><g data-mml-node="msub" transform="translate(3103,0)"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-2017-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(394,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-2017-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(4068.7,0)"><use data-c="227A" xlink:href="#MJX-2017-TEX-N-227A"></use></g><g data-mml-node="mi" transform="translate(5124.5,0)"><use data-c="1D443" xlink:href="#MJX-2017-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(5875.5,0)"><use data-c="1D45F" xlink:href="#MJX-2017-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(6326.5,0)"><use data-c="1D452" xlink:href="#MJX-2017-TEX-I-1D452"></use></g><g data-mml-node="mi" transform="translate(6792.5,0)"><use data-c="1D461" xlink:href="#MJX-2017-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(7153.5,0)"><use data-c="1D466" xlink:href="#MJX-2017-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(7643.5,0)"><use data-c="1D45D" xlink:href="#MJX-2017-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(8146.5,0)"><use data-c="1D452" xlink:href="#MJX-2017-TEX-I-1D452"></use></g><g data-mml-node="msub" transform="translate(8612.5,0)"><g data-mml-node="mi"><use data-c="1D460" xlink:href="#MJX-2017-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-2017-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(9455.8,0)"><use data-c="2C" xlink:href="#MJX-2017-TEX-N-2C"></use></g></g></g></svg></mjx-container></p><p>in which <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2018-TEX-I-1D456" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-2018-TEX-I-1D456"></use></g></g></g></svg></mjx-container> and <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.462ex;" xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2019-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-2019-TEX-I-1D457"></use></g></g></g></svg></mjx-container> are bounded with equation: <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.462ex;" xmlns="http://www.w3.org/2000/svg" width="4.73ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 2090.6 865" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2020-TEX-I-1D456" d="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"></path><path id="MJX-2020-TEX-N-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path id="MJX-2020-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-2020-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(622.8,0)"><use data-c="3C" xlink:href="#MJX-2020-TEX-N-3C"></use></g><g data-mml-node="mi" transform="translate(1678.6,0)"><use data-c="1D457" xlink:href="#MJX-2020-TEX-I-1D457"></use></g></g></g></svg></mjx-container>.
Large elimination to upper universes is prohibited. This is extendable to Agda model:</p><code>inductive cosmos : U
| prop: nat → cosmos
| fibrant: nat → cosmos
| pretypes: nat → cosmos
| strict: nat → cosmos
| omega: cosmos
| lock: cosmos</code><br><h2>Dependent Types</h2><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="11.351ex" height="1.597ex" role="img" focusable="false" viewBox="0 -700 5017 706" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2021-TEX-B-1D403" d="M39 624V686H270H310H408Q500 686 545 680T638 649Q768 584 805 438Q817 388 817 338Q817 171 702 75Q628 17 515 2Q504 1 270 0H39V62H147V624H39ZM655 337Q655 370 655 390T650 442T639 494T616 540T580 580T526 607T451 623Q443 624 368 624H298V62H377H387H407Q445 62 472 65T540 83T606 129Q629 156 640 195T653 262T655 337Z"></path><path id="MJX-2021-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2021-TEX-B-1D41F" d="M308 0Q290 3 172 3Q58 3 49 0H40V62H109V382H42V444H109V503L110 562L112 572Q127 625 178 658T316 699Q318 699 330 699T348 700Q381 698 404 687T436 658T449 629T452 606Q452 576 432 557T383 537Q355 537 335 555T314 605Q314 635 328 649H325Q311 649 293 644T253 618T227 560Q226 555 226 498V444H340V382H232V62H318V0H308Z"></path><path id="MJX-2021-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2021-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2021-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2021-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D403" xlink:href="#MJX-2021-TEX-B-1D403"></use><use data-c="1D41E" xlink:href="#MJX-2021-TEX-B-1D41E" transform="translate(882,0)"></use><use data-c="1D41F" xlink:href="#MJX-2021-TEX-B-1D41F" transform="translate(1409,0)"></use><use data-c="1D422" xlink:href="#MJX-2021-TEX-B-1D422" transform="translate(1760,0)"></use><use data-c="1D427" xlink:href="#MJX-2021-TEX-B-1D427" transform="translate(2079,0)"></use><use data-c="1D422" xlink:href="#MJX-2021-TEX-B-1D422" transform="translate(2718,0)"></use><use data-c="1D42D" xlink:href="#MJX-2021-TEX-B-1D42D" transform="translate(3037,0)"></use><use data-c="1D422" xlink:href="#MJX-2021-TEX-B-1D422" transform="translate(3484,0)"></use><use data-c="1D428" xlink:href="#MJX-2021-TEX-B-1D428" transform="translate(3803,0)"></use><use data-c="1D427" xlink:href="#MJX-2021-TEX-B-1D427" transform="translate(4378,0)"></use></g></g></g></g></svg></mjx-container> (Type).
A type is interpreted as a presheaf <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2022-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-2022-TEX-I-1D434"></use></g></g></g></svg></mjx-container>, a family of sets <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.691ex" height="1.959ex" role="img" focusable="false" viewBox="0 -716 1189.4 866" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2023-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-2023-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-2023-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(783,-150) scale(0.707)"><use data-c="1D43C" xlink:href="#MJX-2023-TEX-I-1D43C"></use></g></g></g></g></svg></mjx-container> with restriction maps
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A dependent type
B on A is interpreted by a presheaf on category of elements of <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2026-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-2026-TEX-I-1D434"></use></g></g></g></svg></mjx-container>: the objects
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241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path id="MJX-2029-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path><path id="MJX-2029-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-2029-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(827.8,0)"><use data-c="3A" xlink:href="#MJX-2029-TEX-N-3A"></use></g><g data-mml-node="mo" transform="translate(1383.6,0)"><use data-c="28" xlink:href="#MJX-2029-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1772.6,0)"><use data-c="1D43D" xlink:href="#MJX-2029-TEX-I-1D43D"></use></g><g data-mml-node="mo" transform="translate(2405.6,0)"><use data-c="2C" xlink:href="#MJX-2029-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2850.2,0)"><use data-c="1D463" xlink:href="#MJX-2029-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(3335.2,0)"><use data-c="29" xlink:href="#MJX-2029-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4002,0)"><use data-c="2192" xlink:href="#MJX-2029-TEX-N-2192"></use></g><g data-mml-node="mo" transform="translate(5279.8,0)"><use data-c="28" xlink:href="#MJX-2029-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5668.8,0)"><use data-c="1D43C" xlink:href="#MJX-2029-TEX-I-1D43C"></use></g><g data-mml-node="mo" transform="translate(6172.8,0)"><use data-c="2C" xlink:href="#MJX-2029-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(6617.4,0)"><use data-c="1D462" xlink:href="#MJX-2029-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(7189.4,0)"><use data-c="29" xlink:href="#MJX-2029-TEX-N-29"></use></g></g></g></svg></mjx-container> are
maps <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="7.453ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 3294.3 910" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2030-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-2030-TEX-N-3A" d="M78 370Q78 394 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-2030-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(827.8,0)"><use data-c="3A" xlink:href="#MJX-2030-TEX-N-3A"></use></g><g data-mml-node="mi" transform="translate(1383.6,0)"><use data-c="1D43D" xlink:href="#MJX-2030-TEX-I-1D43D"></use></g><g data-mml-node="mo" transform="translate(2294.3,0)"><use data-c="2192" xlink:href="#MJX-2030-TEX-N-2192"></use></g></g></g></svg></mjx-container> such that <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="7.218ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 3190.6 910" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2031-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-2031-TEX-N-3D" d="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"></path><path id="MJX-2031-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-2031-TEX-N-A0" d=""></path><path id="MJX-2031-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-2031-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(762.8,0)"><use data-c="3D" xlink:href="#MJX-2031-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1818.6,0)"><use data-c="1D462" xlink:href="#MJX-2031-TEX-I-1D462"></use></g><g data-mml-node="mtext" transform="translate(2390.6,0)"><use data-c="A0" xlink:href="#MJX-2031-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(2640.6,0)"><use data-c="1D453" xlink:href="#MJX-2031-TEX-I-1D453"></use></g></g></g></svg></mjx-container>. A dependent type B is thus given
by a family of sets <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="6.918ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3057.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2032-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-2032-TEX-N-28" d="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"></path><path id="MJX-2032-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path><path id="MJX-2032-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-2032-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-2032-TEX-N-29" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-2032-TEX-I-1D435"></use></g><g data-mml-node="mo" transform="translate(759,0)"><use data-c="28" xlink:href="#MJX-2032-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1148,0)"><use data-c="1D43C" xlink:href="#MJX-2032-TEX-I-1D43C"></use></g><g data-mml-node="mo" transform="translate(1652,0)"><use data-c="2C" xlink:href="#MJX-2032-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2096.7,0)"><use data-c="1D462" xlink:href="#MJX-2032-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(2668.7,0)"><use data-c="29" xlink:href="#MJX-2032-TEX-N-29"></use></g></g></g></svg></mjx-container> and restriction maps <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="19.457ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8599.9 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2033-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-2033-TEX-N-28" d="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"></path><path id="MJX-2033-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path><path id="MJX-2033-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-2033-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-2033-TEX-N-29" d="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"></path><path id="MJX-2033-TEX-N-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 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transform="translate(2668.7,0)"><use data-c="29" xlink:href="#MJX-2033-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3335.4,0)"><use data-c="2192" xlink:href="#MJX-2033-TEX-N-2192"></use></g><g data-mml-node="mi" transform="translate(4613.2,0)"><use data-c="1D435" xlink:href="#MJX-2033-TEX-I-1D435"></use></g><g data-mml-node="mo" transform="translate(5372.2,0)"><use data-c="28" xlink:href="#MJX-2033-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5761.2,0)"><use data-c="1D43D" xlink:href="#MJX-2033-TEX-I-1D43D"></use></g><g data-mml-node="mo" transform="translate(6394.2,0)"><use data-c="2C" xlink:href="#MJX-2033-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(6838.9,0)"><use data-c="1D462" xlink:href="#MJX-2033-TEX-I-1D462"></use></g><g data-mml-node="mtext" transform="translate(7410.9,0)"><use data-c="A0" xlink:href="#MJX-2033-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(7660.9,0)"><use data-c="1D453" xlink:href="#MJX-2033-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(8210.9,0)"><use data-c="29" xlink:href="#MJX-2033-TEX-N-29"></use></g></g></g></svg></mjx-container>.</p><p>We think of <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2034-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 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46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-2036-TEX-N-28" d="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"></path><path id="MJX-2036-TEX-I-1D465" d="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 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xlink:href="#MJX-2036-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1148,0)"><use data-c="1D465" xlink:href="#MJX-2036-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1720,0)"><use data-c="29" xlink:href="#MJX-2036-TEX-N-29"></use></g></g></g></svg></mjx-container> varying <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="4.877ex" height="1.645ex" role="img" focusable="false" viewBox="0 -716 2155.6 727" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2037-TEX-I-1D465" d="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 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The operation <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="13.105ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5792.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2038-TEX-N-3A0" d="M128 619Q121 626 117 628T101 631T58 634H25V680H724V634H691Q651 633 640 631T622 619V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V634H232V348L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V619Z"></path><path id="MJX-2038-TEX-N-28" d="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"></path><path id="MJX-2038-TEX-I-1D465" d="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"></path><path id="MJX-2038-TEX-N-3A" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-2038-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-2038-TEX-N-29" d="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"></path><path id="MJX-2038-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="3A0" xlink:href="#MJX-2038-TEX-N-3A0"></use></g><g data-mml-node="mo" transform="translate(750,0)"><use data-c="28" xlink:href="#MJX-2038-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1139,0)"><use data-c="1D465" xlink:href="#MJX-2038-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1988.8,0)"><use data-c="3A" xlink:href="#MJX-2038-TEX-N-3A"></use></g><g data-mml-node="mi" transform="translate(2544.6,0)"><use data-c="1D434" xlink:href="#MJX-2038-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(3294.6,0)"><use data-c="29" xlink:href="#MJX-2038-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(3683.6,0)"><use data-c="1D435" xlink:href="#MJX-2038-TEX-I-1D435"></use></g><g data-mml-node="mo" transform="translate(4442.6,0)"><use data-c="28" xlink:href="#MJX-2038-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(4831.6,0)"><use data-c="1D465" xlink:href="#MJX-2038-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(5403.6,0)"><use data-c="29" xlink:href="#MJX-2038-TEX-N-29"></use></g></g></g></svg></mjx-container> generalizes the semantics of
implication in a Kripke model.</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="11.351ex" height="1.597ex" role="img" focusable="false" viewBox="0 -700 5017 706" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2039-TEX-B-1D403" d="M39 624V686H270H310H408Q500 686 545 680T638 649Q768 584 805 438Q817 388 817 338Q817 171 702 75Q628 17 515 2Q504 1 270 0H39V62H147V624H39ZM655 337Q655 370 655 390T650 442T639 494T616 540T580 580T526 607T451 623Q443 624 368 624H298V62H377H387H407Q445 62 472 65T540 83T606 129Q629 156 640 195T653 262T655 337Z"></path><path id="MJX-2039-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2039-TEX-B-1D41F" d="M308 0Q290 3 172 3Q58 3 49 0H40V62H109V382H42V444H109V503L110 562L112 572Q127 625 178 658T316 699Q318 699 330 699T348 700Q381 698 404 687T436 658T449 629T452 606Q452 576 432 557T383 537Q355 537 335 555T314 605Q314 635 328 649H325Q311 649 293 644T253 618T227 560Q226 555 226 498V444H340V382H232V62H318V0H308Z"></path><path id="MJX-2039-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2039-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2039-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2039-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D403" xlink:href="#MJX-2039-TEX-B-1D403"></use><use data-c="1D41E" xlink:href="#MJX-2039-TEX-B-1D41E" transform="translate(882,0)"></use><use data-c="1D41F" xlink:href="#MJX-2039-TEX-B-1D41F" transform="translate(1409,0)"></use><use data-c="1D422" xlink:href="#MJX-2039-TEX-B-1D422" transform="translate(1760,0)"></use><use data-c="1D427" xlink:href="#MJX-2039-TEX-B-1D427" transform="translate(2079,0)"></use><use data-c="1D422" xlink:href="#MJX-2039-TEX-B-1D422" transform="translate(2718,0)"></use><use data-c="1D42D" xlink:href="#MJX-2039-TEX-B-1D42D" transform="translate(3037,0)"></use><use data-c="1D422" xlink:href="#MJX-2039-TEX-B-1D422" transform="translate(3484,0)"></use><use data-c="1D428" xlink:href="#MJX-2039-TEX-B-1D428" transform="translate(3803,0)"></use><use data-c="1D427" xlink:href="#MJX-2039-TEX-B-1D427" transform="translate(4378,0)"></use></g></g></g></g></svg></mjx-container> (Pi). An element <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="20.769ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 9180.1 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2040-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 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id="MJX-2040-TEX-I-1D465" d="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"></path><path id="MJX-2040-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-2040-TEX-N-29" d="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"></path><path id="MJX-2040-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-2040-TEX-N-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path id="MJX-2040-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D464" xlink:href="#MJX-2040-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(993.8,0)"><use data-c="3A" xlink:href="#MJX-2040-TEX-N-3A"></use></g><g data-mml-node="mo" transform="translate(1549.6,0)"><use data-c="5B" xlink:href="#MJX-2040-TEX-N-5B"></use></g><g data-mml-node="mi" transform="translate(1827.6,0)"><use data-c="3A0" xlink:href="#MJX-2040-TEX-N-3A0"></use></g><g data-mml-node="mo" transform="translate(2577.6,0)"><use data-c="28" xlink:href="#MJX-2040-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2966.6,0)"><use data-c="1D465" xlink:href="#MJX-2040-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(3816.3,0)"><use data-c="3A" xlink:href="#MJX-2040-TEX-N-3A"></use></g><g data-mml-node="mi" transform="translate(4372.1,0)"><use data-c="1D434" xlink:href="#MJX-2040-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(5122.1,0)"><use data-c="29" xlink:href="#MJX-2040-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(5511.1,0)"><use data-c="1D435" xlink:href="#MJX-2040-TEX-I-1D435"></use></g><g data-mml-node="mo" transform="translate(6270.1,0)"><use data-c="28" xlink:href="#MJX-2040-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(6659.1,0)"><use data-c="1D465" xlink:href="#MJX-2040-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(7231.1,0)"><use data-c="29" xlink:href="#MJX-2040-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(7620.1,0)"><use data-c="5D" xlink:href="#MJX-2040-TEX-N-5D"></use></g><g data-mml-node="mo" transform="translate(7898.1,0)"><use data-c="28" xlink:href="#MJX-2040-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(8287.1,0)"><use data-c="1D43C" xlink:href="#MJX-2040-TEX-I-1D43C"></use></g><g data-mml-node="mo" transform="translate(8791.1,0)"><use data-c="29" xlink:href="#MJX-2040-TEX-N-29"></use></g></g></g></svg></mjx-container> is a family
of functions <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.667ex;" xmlns="http://www.w3.org/2000/svg" width="23.309ex" height="2.364ex" role="img" focusable="false" viewBox="0 -750 10302.7 1045" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2041-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 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667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path id="MJX-2045-TEX-N-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 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data-c="2192" xlink:href="#MJX-2045-TEX-N-2192"></use></g><g data-mml-node="mi" transform="translate(3755.1,0)"><use data-c="1D43D" xlink:href="#MJX-2045-TEX-I-1D43D"></use></g></g></g></svg></mjx-container>.</p><code>inductive pure (lang: U) : U
| var: name → nat → pure lang
| pi: name → nat → lang → lang → pure lang
| lambda: name → nat → lang → lang → pure lang
| app: lang → lang → pure lang</code><br><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="11.351ex" height="1.597ex" role="img" focusable="false" viewBox="0 -700 5017 706" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2046-TEX-B-1D403" d="M39 624V686H270H310H408Q500 686 545 680T638 649Q768 584 805 438Q817 388 817 338Q817 171 702 75Q628 17 515 2Q504 1 270 0H39V62H147V624H39ZM655 337Q655 370 655 390T650 442T639 494T616 540T580 580T526 607T451 623Q443 624 368 624H298V62H377H387H407Q445 62 472 65T540 83T606 129Q629 156 640 195T653 262T655 337Z"></path><path id="MJX-2046-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2046-TEX-B-1D41F" d="M308 0Q290 3 172 3Q58 3 49 0H40V62H109V382H42V444H109V503L110 562L112 572Q127 625 178 658T316 699Q318 699 330 699T348 700Q381 698 404 687T436 658T449 629T452 606Q452 576 432 557T383 537Q355 537 335 555T314 605Q314 635 328 649H325Q311 649 293 644T253 618T227 560Q226 555 226 498V444H340V382H232V62H318V0H308Z"></path><path id="MJX-2046-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2046-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2046-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2046-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D403" xlink:href="#MJX-2046-TEX-B-1D403"></use><use data-c="1D41E" xlink:href="#MJX-2046-TEX-B-1D41E" transform="translate(882,0)"></use><use data-c="1D41F" xlink:href="#MJX-2046-TEX-B-1D41F" transform="translate(1409,0)"></use><use data-c="1D422" xlink:href="#MJX-2046-TEX-B-1D422" transform="translate(1760,0)"></use><use data-c="1D427" xlink:href="#MJX-2046-TEX-B-1D427" transform="translate(2079,0)"></use><use data-c="1D422" xlink:href="#MJX-2046-TEX-B-1D422" transform="translate(2718,0)"></use><use data-c="1D42D" xlink:href="#MJX-2046-TEX-B-1D42D" transform="translate(3037,0)"></use><use data-c="1D422" xlink:href="#MJX-2046-TEX-B-1D422" transform="translate(3484,0)"></use><use data-c="1D428" xlink:href="#MJX-2046-TEX-B-1D428" transform="translate(3803,0)"></use><use data-c="1D427" xlink:href="#MJX-2046-TEX-B-1D427" transform="translate(4378,0)"></use></g></g></g></g></svg></mjx-container> (Sigma). The set <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="13.042ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5764.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2047-TEX-N-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path><path id="MJX-2047-TEX-N-28" d="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"></path><path id="MJX-2047-TEX-I-1D465" d="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"></path><path id="MJX-2047-TEX-N-3A" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-2047-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-2047-TEX-N-29" d="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"></path><path id="MJX-2047-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="3A3" xlink:href="#MJX-2047-TEX-N-3A3"></use></g><g data-mml-node="mo" transform="translate(722,0)"><use data-c="28" xlink:href="#MJX-2047-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1111,0)"><use data-c="1D465" xlink:href="#MJX-2047-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1960.8,0)"><use data-c="3A" xlink:href="#MJX-2047-TEX-N-3A"></use></g><g data-mml-node="mi" transform="translate(2516.6,0)"><use data-c="1D434" xlink:href="#MJX-2047-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(3266.6,0)"><use data-c="29" xlink:href="#MJX-2047-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(3655.6,0)"><use data-c="1D435" xlink:href="#MJX-2047-TEX-I-1D435"></use></g><g data-mml-node="mo" transform="translate(4414.6,0)"><use data-c="28" xlink:href="#MJX-2047-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(4803.6,0)"><use data-c="1D465" xlink:href="#MJX-2047-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(5375.6,0)"><use data-c="29" xlink:href="#MJX-2047-TEX-N-29"></use></g></g></g></svg></mjx-container> is the set
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transform="translate(7869.4,0)"><use data-c="29" xlink:href="#MJX-2049-TEX-N-29"></use></g></g></g></svg></mjx-container> and restriction map <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="18.762ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8292.9 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2050-TEX-N-28" d="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"></path><path id="MJX-2050-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 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xlink:href="#MJX-2050-TEX-I-1D463"></use></g><g data-mml-node="mtext" transform="translate(7103.9,0)"><use data-c="A0" xlink:href="#MJX-2050-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(7353.9,0)"><use data-c="1D453" xlink:href="#MJX-2050-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(7903.9,0)"><use data-c="29" xlink:href="#MJX-2050-TEX-N-29"></use></g></g></g></svg></mjx-container>.</p><code>inductive total (lang: U) : U
| sigma: name → lang → lang → total lang
| pair: lang → lang → total lang
| fst: lang → total lang
| snd: lang → total lang</code><br><p>The preseaf configuration with only Pi and Sigma is called <b>MLTT-72</b>.
</p><h2 id="path">Path Equality</h2><p>The fundamental development of equality inside <b>MLTT</b>
provers led us to the notion of ∞-groupoid as spaces.
In this way Path identity type appeared in the core
of type checker along with De Morgan algebra on
built-in interval type.</p><code>inductive homotopy (lang: U) : U
| PathP: lang → lang → lang → homotopy lang
| plam: name → lang → lang → homotopy lang
| papp: lang → lang → homotopy lang
| I: homotopy lang
| zero: homotopy lang
| one: homotopy lang
| meet: lang → lang → homotopy lang
| join: lang → lang → homotopy lang
| neg: lang → homotopy lang
| system: lang → homotopy lang
| Partial: lang → homotopy lang
| transp: lang → lang → homotopy lang
| hcomp: lang → homotopy lang
| Sub: lang → homotopy lang
| inc: lang → homotopy lang
| ouc: lang → homotopy lang</code><br><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="11.351ex" height="1.597ex" role="img" focusable="false" viewBox="0 -700 5017 706" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2051-TEX-B-1D403" d="M39 624V686H270H310H408Q500 686 545 680T638 649Q768 584 805 438Q817 388 817 338Q817 171 702 75Q628 17 515 2Q504 1 270 0H39V62H147V624H39ZM655 337Q655 370 655 390T650 442T639 494T616 540T580 580T526 607T451 623Q443 624 368 624H298V62H377H387H407Q445 62 472 65T540 83T606 129Q629 156 640 195T653 262T655 337Z"></path><path id="MJX-2051-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z"></path><path id="MJX-2051-TEX-B-1D41F" d="M308 0Q290 3 172 3Q58 3 49 0H40V62H109V382H42V444H109V503L110 562L112 572Q127 625 178 658T316 699Q318 699 330 699T348 700Q381 698 404 687T436 658T449 629T452 606Q452 576 432 557T383 537Q355 537 335 555T314 605Q314 635 328 649H325Q311 649 293 644T253 618T227 560Q226 555 226 498V444H340V382H232V62H318V0H308Z"></path><path id="MJX-2051-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2051-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2051-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z"></path><path id="MJX-2051-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D403" xlink:href="#MJX-2051-TEX-B-1D403"></use><use data-c="1D41E" xlink:href="#MJX-2051-TEX-B-1D41E" transform="translate(882,0)"></use><use data-c="1D41F" xlink:href="#MJX-2051-TEX-B-1D41F" transform="translate(1409,0)"></use><use data-c="1D422" xlink:href="#MJX-2051-TEX-B-1D422" transform="translate(1760,0)"></use><use data-c="1D427" xlink:href="#MJX-2051-TEX-B-1D427" transform="translate(2079,0)"></use><use data-c="1D422" xlink:href="#MJX-2051-TEX-B-1D422" transform="translate(2718,0)"></use><use data-c="1D42D" xlink:href="#MJX-2051-TEX-B-1D42D" transform="translate(3037,0)"></use><use data-c="1D422" xlink:href="#MJX-2051-TEX-B-1D422" transform="translate(3484,0)"></use><use data-c="1D428" xlink:href="#MJX-2051-TEX-B-1D428" transform="translate(3803,0)"></use><use data-c="1D427" xlink:href="#MJX-2051-TEX-B-1D427" transform="translate(4378,0)"></use></g></g></g></g></svg></mjx-container> (Cubical Presheaf <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="0.88ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 389 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2052-TEX-D-1D540" d="M20 666Q20 676 31 683H358Q369 676 369 666Q369 648 331 648Q288 645 282 632Q278 626 278 341Q278 57 282 50Q286 42 295 40T331 35Q369 35 369 16Q369 6 358 -1H31Q20 4 20 16Q20 35 58 35Q84 37 93 39T107 50Q113 60 113 341Q113 623 107 632Q101 645 58 648Q20 648 20 666ZM249 35Q246 40 246 41T244 54T242 83T242 139V341Q242 632 244 639L249 648H140Q146 634 147 596T149 341Q149 124 148 86T140 35H249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D540" xlink:href="#MJX-2052-TEX-D-1D540"></use></g></g></g></g></svg></mjx-container>).
The identity types modeled with another presheaf, the presheaf on Lawvere
category of distributive lattices (theory of De Morgan algebras) denoted
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transform="translate(882,0)"></use><use data-c="1D41F" xlink:href="#MJX-2055-TEX-B-1D41F" transform="translate(1409,0)"></use><use data-c="1D422" xlink:href="#MJX-2055-TEX-B-1D422" transform="translate(1760,0)"></use><use data-c="1D427" xlink:href="#MJX-2055-TEX-B-1D427" transform="translate(2079,0)"></use><use data-c="1D422" xlink:href="#MJX-2055-TEX-B-1D422" transform="translate(2718,0)"></use><use data-c="1D42D" xlink:href="#MJX-2055-TEX-B-1D42D" transform="translate(3037,0)"></use><use data-c="1D422" xlink:href="#MJX-2055-TEX-B-1D422" transform="translate(3484,0)"></use><use data-c="1D428" xlink:href="#MJX-2055-TEX-B-1D428" transform="translate(3803,0)"></use><use data-c="1D427" xlink:href="#MJX-2055-TEX-B-1D427" transform="translate(4378,0)"></use></g></g></g></g></svg></mjx-container> Properties of <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="0.88ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 389 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2056-TEX-D-1D540" d="M20 666Q20 676 31 683H358Q369 676 369 666Q369 648 331 648Q288 645 282 632Q278 626 278 341Q278 57 282 50Q286 42 295 40T331 35Q369 35 369 16Q369 6 358 -1H31Q20 4 20 16Q20 35 58 35Q84 37 93 39T107 50Q113 60 113 341Q113 623 107 632Q101 645 58 648Q20 648 20 666ZM249 35Q246 40 246 41T244 54T242 83T242 139V341Q242 632 244 639L249 648H140Q146 634 147 596T149 341Q149 124 148 86T140 35H249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D540" xlink:href="#MJX-2056-TEX-D-1D540"></use></g></g></g></g></svg></mjx-container>. The presheaf <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="0.88ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 389 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2057-TEX-D-1D540" d="M20 666Q20 676 31 683H358Q369 676 369 666Q369 648 331 648Q288 645 282 632Q278 626 278 341Q278 57 282 50Q286 42 295 40T331 35Q369 35 369 16Q369 6 358 -1H31Q20 4 20 16Q20 35 58 35Q84 37 93 39T107 50Q113 60 113 341Q113 623 107 632Q101 645 58 648Q20 648 20 666ZM249 35Q246 40 246 41T244 54T242 83T242 139V341Q242 632 244 639L249 648H140Q146 634 147 596T149 341Q149 124 148 86T140 35H249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D540" xlink:href="#MJX-2057-TEX-D-1D540"></use></g></g></g></g></svg></mjx-container>:
i) has to distinct global elements <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2058-TEX-N-30" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-2058-TEX-N-30"></use></g></g></g></svg></mjx-container> and <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2059-TEX-N-31" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-2059-TEX-N-31"></use></g></g></g></svg></mjx-container> (B<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="0.988ex" height="1.065ex" role="img" focusable="false" viewBox="0 -320.9 436.6 470.9" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2060-TEX-N-31" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"></g><g data-mml-node="mn" transform="translate(33,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-2060-TEX-N-31"></use></g></g></g></g></svg></mjx-container>);
ii) <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="0.88ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 389 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2061-TEX-D-1D540" d="M20 666Q20 676 31 683H358Q369 676 369 666Q369 648 331 648Q288 645 282 632Q278 626 278 341Q278 57 282 50Q286 42 295 40T331 35Q369 35 369 16Q369 6 358 -1H31Q20 4 20 16Q20 35 58 35Q84 37 93 39T107 50Q113 60 113 341Q113 623 107 632Q101 645 58 648Q20 648 20 666ZM249 35Q246 40 246 41T244 54T242 83T242 139V341Q242 632 244 639L249 648H140Q146 634 147 596T149 341Q149 124 148 86T140 35H249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D540" xlink:href="#MJX-2061-TEX-D-1D540"></use></g></g></g></g></svg></mjx-container>(I) has a decidable equality for each <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.14ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 504 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2062-TEX-I-1D43C" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43C" xlink:href="#MJX-2062-TEX-I-1D43C"></use></g></g></g></svg></mjx-container> (B<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="0.988ex" height="1.065ex" role="img" focusable="false" viewBox="0 -320.9 436.6 470.9" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2063-TEX-N-32" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"></g><g data-mml-node="mn" transform="translate(33,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-2063-TEX-N-32"></use></g></g></g></g></svg></mjx-container>);
iii) <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="0.88ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 389 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2064-TEX-D-1D540" d="M20 666Q20 676 31 683H358Q369 676 369 666Q369 648 331 648Q288 645 282 632Q278 626 278 341Q278 57 282 50Q286 42 295 40T331 35Q369 35 369 16Q369 6 358 -1H31Q20 4 20 16Q20 35 58 35Q84 37 93 39T107 50Q113 60 113 341Q113 623 107 632Q101 645 58 648Q20 648 20 666ZM249 35Q246 40 246 41T244 54T242 83T242 139V341Q242 632 244 639L249 648H140Q146 634 147 596T149 341Q149 124 148 86T140 35H249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D540" xlink:href="#MJX-2064-TEX-D-1D540"></use></g></g></g></g></svg></mjx-container> is tiny so the path functor <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="8.3ex" height="1.939ex" role="img" focusable="false" viewBox="0 -846 3668.8 857" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2065-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-2065-TEX-N-21A6" d="M95 155V109Q95 83 92 73T75 63Q61 63 58 74T54 130Q54 140 54 180T55 250Q55 421 57 425Q61 437 75 437Q88 437 91 428T95 393V345V270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H95V155Z"></path><path id="MJX-2065-TEX-D-1D540" d="M20 666Q20 676 31 683H358Q369 676 369 666Q369 648 331 648Q288 645 282 632Q278 626 278 341Q278 57 282 50Q286 42 295 40T331 35Q369 35 369 16Q369 6 358 -1H31Q20 4 20 16Q20 35 58 35Q84 37 93 39T107 50Q113 60 113 341Q113 623 107 632Q101 645 58 648Q20 648 20 666ZM249 35Q246 40 246 41T244 54T242 83T242 139V341Q242 632 244 639L249 648H140Q146 634 147 596T149 341Q149 124 148 86T140 35H249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-2065-TEX-I-1D44B"></use></g><g data-mml-node="mo" transform="translate(1129.8,0)"><use data-c="21A6" xlink:href="#MJX-2065-TEX-N-21A6"></use></g><g data-mml-node="msup" transform="translate(2407.6,0)"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-2065-TEX-I-1D44B"></use></g><g data-mml-node="TeXAtom" transform="translate(936.2,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D540" xlink:href="#MJX-2065-TEX-D-1D540"></use></g></g></g></g></g></svg></mjx-container> has right adjoint (B<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="0.988ex" height="1.099ex" role="img" focusable="false" viewBox="0 -320.2 436.6 485.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2066-TEX-N-33" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"></g><g data-mml-node="mn" transform="translate(33,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-2066-TEX-N-33"></use></g></g></g></g></svg></mjx-container>).;
iv) <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="0.88ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 389 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2067-TEX-D-1D540" d="M20 666Q20 676 31 683H358Q369 676 369 666Q369 648 331 648Q288 645 282 632Q278 626 278 341Q278 57 282 50Q286 42 295 40T331 35Q369 35 369 16Q369 6 358 -1H31Q20 4 20 16Q20 35 58 35Q84 37 93 39T107 50Q113 60 113 341Q113 623 107 632Q101 645 58 648Q20 648 20 666ZM249 35Q246 40 246 41T244 54T242 83T242 139V341Q242 632 244 639L249 648H140Q146 634 147 596T149 341Q149 124 148 86T140 35H249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D540" xlink:href="#MJX-2067-TEX-D-1D540"></use></g></g></g></g></svg></mjx-container> has meet and join (connections).</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.452ex;" xmlns="http://www.w3.org/2000/svg" width="19.726ex" height="2.023ex" role="img" focusable="false" viewBox="0 -694 8719 894" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2068-TEX-B-1D408" d="M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z"></path><path id="MJX-2068-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z"></path><path id="MJX-2068-TEX-B-1D42D" d="M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 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data-mml-node="mi"><use data-c="1D408" xlink:href="#MJX-2068-TEX-B-1D408"></use><use data-c="1D427" xlink:href="#MJX-2068-TEX-B-1D427" transform="translate(436,0)"></use><use data-c="1D42D" xlink:href="#MJX-2068-TEX-B-1D42D" transform="translate(1075,0)"></use><use data-c="1D41E" xlink:href="#MJX-2068-TEX-B-1D41E" transform="translate(1522,0)"></use><use data-c="1D42B" xlink:href="#MJX-2068-TEX-B-1D42B" transform="translate(2049,0)"></use><use data-c="1D42F" xlink:href="#MJX-2068-TEX-B-1D42F" transform="translate(2523,0)"></use><use data-c="1D41A" xlink:href="#MJX-2068-TEX-B-1D41A" transform="translate(3130,0)"></use><use data-c="1D425" xlink:href="#MJX-2068-TEX-B-1D425" transform="translate(3689,0)"></use></g><g data-mml-node="mtext" transform="translate(4008,0)"><use data-c="A0" xlink:href="#MJX-2068-TEX-B-A0"></use></g><g data-mml-node="mi" transform="translate(4258,0)"><use data-c="1D40F" xlink:href="#MJX-2068-TEX-B-1D40F"></use><use data-c="1D42B" xlink:href="#MJX-2068-TEX-B-1D42B" transform="translate(786,0)"></use><use data-c="1D41E" xlink:href="#MJX-2068-TEX-B-1D41E" transform="translate(1260,0)"></use><use data-c="1D42D" xlink:href="#MJX-2068-TEX-B-1D42D" transform="translate(1787,0)"></use><use data-c="1D432" xlink:href="#MJX-2068-TEX-B-1D432" transform="translate(2234,0)"></use><use data-c="1D429" xlink:href="#MJX-2068-TEX-B-1D429" transform="translate(2841,0)"></use><use data-c="1D41E" xlink:href="#MJX-2068-TEX-B-1D41E" transform="translate(3480,0)"></use><use data-c="1D42C" xlink:href="#MJX-2068-TEX-B-1D42C" transform="translate(4007,0)"></use></g></g></g></g></svg></mjx-container>. While having pretypes universe <b>V</b> with interval and
associated De Morgan algebra <b>(<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.509ex" height="1.403ex" role="img" focusable="false" viewBox="0 -598 667 620" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2069-TEX-N-2227" d="M318 591Q325 598 333 598Q344 598 348 591Q349 590 414 445T545 151T611 -4Q609 -22 591 -22Q588 -22 586 -21T581 -20T577 -17T575 -13T572 -9T570 -4L333 528L96 -4Q87 -20 80 -21Q78 -22 75 -22Q57 -22 55 -4Q55 2 120 150T251 444T318 591Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="OPEN"><g data-mml-node="mo"><use data-c="2227" xlink:href="#MJX-2069-TEX-N-2227"></use></g></g></g></g></svg></mjx-container>, <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.509ex" height="1.403ex" role="img" focusable="false" viewBox="0 -598 667 620" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2070-TEX-N-2228" d="M55 580Q56 587 61 592T75 598Q86 598 96 580L333 48L570 580Q579 596 586 597Q588 598 591 598Q609 598 611 580Q611 574 546 426T415 132T348 -15Q343 -22 333 -22T318 -15Q317 -14 252 131T121 425T55 580Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="OPEN"><g data-mml-node="mo"><use data-c="2228" xlink:href="#MJX-2070-TEX-N-2228"></use></g></g></g></g></svg></mjx-container>, <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="1.76ex" height="1.505ex" role="img" focusable="false" viewBox="0 -583 778 665" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2071-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-2071-TEX-N-2212"></use></g></g></g></svg></mjx-container>, <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2072-TEX-N-30" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-2072-TEX-N-30"></use></g></g></g></svg></mjx-container>, <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2073-TEX-N-31" d="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"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-2073-TEX-N-31"></use></g></g></g></svg></mjx-container>, <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 361 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2074-TEX-N-49" d="M328 0Q307 3 180 3T32 0H21V46H43Q92 46 106 49T126 60Q128 63 128 342Q128 620 126 623Q122 628 118 630T96 635T43 637H21V683H32Q53 680 180 680T328 683H339V637H317Q268 637 254 634T234 623Q232 620 232 342Q232 63 234 60Q238 55 242 53T264 48T317 46H339V0H328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="49" xlink:href="#MJX-2074-TEX-N-49"></use></g></g></g></g></svg></mjx-container>)</b> is enough to
perform DNF normalization and proving some basic statements about path, including:
contractability of singletons,
homotopy transport, congruence, functional extensionality; it is not enough for
proving β rule for Path type or path composition.</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.439ex;" xmlns="http://www.w3.org/2000/svg" width="25.466ex" height="2.016ex" role="img" focusable="false" viewBox="0 -697 11256 891" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2075-TEX-B-1D406" d="M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z"></path><path id="MJX-2075-TEX-B-1D41E" d="M32 225Q32 332 102 392T272 452H283Q382 452 436 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Generalized transport <b>transp</b> adresses
first problem of deriving the computational β rule for Path types:</p><code><span class="h__keyword">theorem</span> <span class="h__keyword">Path</span>-β <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">(</span>a <span class="h__symbol">:</span> A<span class="h__symbol">)</span> <span class="h__symbol">(</span>C <span class="h__symbol">:</span> D A<span class="h__symbol">)</span> <span class="h__symbol">(</span>d<span class="h__symbol">:</span> C a a <span class="h__symbol">(</span><span class="h__keyword">refl</span> A a<span class="h__symbol">))</span>
<span class="h__symbol">:</span> Equ <span class="h__symbol">(</span>C a a <span class="h__symbol">(</span><span class="h__keyword">refl</span> A a<span class="h__symbol">))</span> d <span class="h__symbol">(</span>J A a C d a <span class="h__symbol">(</span><span class="h__keyword">refl</span> A a<span class="h__symbol">))</span>
<span class="h__symbol">:</span><span class="h__symbol">=</span> λ <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span>, λ <span class="h__symbol">(</span>a <span class="h__symbol">:</span> A<span class="h__symbol">)</span>, λ <span class="h__symbol">(</span>C <span class="h__symbol">:</span> Π <span class="h__symbol">(</span>x <span class="h__symbol">:</span> A<span class="h__symbol">)</span>, Π <span class="h__symbol">(</span>y <span class="h__symbol">:</span> A<span class="h__symbol">)</span>, <span class="h__keyword">PathP</span> <span class="h__symbol">(</span><_> A<span class="h__symbol">)</span> x y <span class="h__symbol">→</span> <span class="h__keyword">U</span><span class="h__symbol">)</span>,
λ <span class="h__symbol">(</span>d <span class="h__symbol">:</span> C a a <span class="h__symbol">(</span><_> a<span class="h__symbol">))</span>, <j> <span class="h__keyword">transp</span> <span class="h__symbol">(</span><_> C a a <span class="h__symbol">(</span><_> a<span class="h__symbol">))</span> -j d
</code><p>Transport is defined on fibrant types (only) and type checker should cover all the cases.
Note that <b>transpⁱ (Pathʲ A v w) φ u₀</b> case is relying on <b>comp</b> operation
which depends on <b>hcomp</b> primitive.
</p><code>transpⁱ N φ u₀ = u₀
transpⁱ U φ A = A
transpⁱ (Π (x : A), B) φ u₀ v = transpⁱ B(x/w) φ (u₀ w(i/0))
transpⁱ (Σ (x : A), B) φ u₀ = (transpⁱ A φ (u₀.1),transpⁱ B(x/v) φ(u₀.2))
transpⁱ (Pathʲ A v w) φ u₀ = 〈j〉compⁱ A [φ ↦ u₀ j, (j=0) ↦ v, (j=1) ↦ w] (u₀ j)
transpⁱ (Glue [φ ↦ (T,w)] A) ψ u₀ = glue [φ(i/1) ↦ t′₁] a′₁ : B(i/1)</code><br><code>w = transpFill⁻ⁱ A φ v, v : A(i/1)
v = transpFillⁱ A φ u₀.1
u : A(j/0), v : A(j/1)
transp⁻ⁱ A φ u = (transpⁱ A(i/1−i) φ u)(i/1−i) : A(i/0)
transpFillⁱ A φ u₀ = transpʲ A(i/i∧j) (φ∨(i=0)) u₀ : A</code><br><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="18.869ex" height="1.586ex" role="img" focusable="false" viewBox="0 -695 8340 701" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2076-TEX-B-1D40F" d="M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H253Q435 686 470 685T536 678Q585 668 621 648T675 605T705 557T718 514T721 483T718 451T704 409T673 362T616 322T530 293Q500 288 399 287H304V62H412V0H400ZM553 475Q553 554 537 582T459 622Q451 623 373 624H298V343H372Q457 344 480 350Q527 362 540 390T553 475Z"></path><path id="MJX-2076-TEX-B-1D41A" d="M64 349Q64 399 107 426T255 453Q346 453 402 423T473 341Q478 327 478 310T479 196V77Q493 63 529 62Q549 62 553 57T558 31Q558 9 552 5T514 0H497H481Q375 0 367 56L356 46Q300 -6 210 -6Q130 -6 81 30T32 121Q32 188 111 226T332 272H350V292Q350 313 348 327T337 361T306 391T248 402T194 399H189Q204 376 204 354Q204 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In order to explicitly define <b>hcomp</b>
we need to specify n-cubes where some faces are missing. Partial
primitives <b>isOne</b>, <b>1=1</b> and <b>UIP</b> on pretypes are
derivable in Anders due to landing strict equality <b>Id</b> in <b>V</b> universe.
The idea is that <b>(Partial A r)</b> is the type of cubes in <b>A</b>
that are only defined when <b>IsOne r</b> holds. <b>(Partial A r)</b>
is a special version of the function space <b>IsOne r → A</b>
with a more extensional equality: two of its elements are considered
judgmentally equal if they represent the same subcube of <b>A</b>.
They are equal whenever they reduce to equal terms for all the
possible assignment of variables that make <b>r</b> equal to <b>1</b>.</p><code><span class="h__keyword">def</span> Partial′ <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">(</span>i <span class="h__symbol">:</span> I<span class="h__symbol">)</span> <span class="h__symbol">:</span><span class="h__symbol">=</span> Partial A i
<span class="h__keyword">def</span> isOne <span class="h__symbol">:</span> I -> <span class="h__keyword">V</span> <span class="h__symbol">:</span><span class="h__symbol">=</span> <span class="h__keyword">Id</span> I<span class="h__keyword"> 1</span>
<span class="h__keyword">def</span><span class="h__keyword"> 1</span><span class="h__symbol">=</span<span class="h__keyword">>1</span> <span class="h__symbol">:</span> isOne<span class="h__keyword"> 1</span> <span class="h__symbol">:</span><span class="h__symbol">=</span> <span class="h__keyword">ref</span><span class="h__keyword"> 1</span>
<span class="h__keyword">def</span> UIP <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">V</span><span class="h__symbol">)</span> <span class="h__symbol">(</span>a b <span class="h__symbol">:</span> A<span class="h__symbol">)</span> <span class="h__symbol">(</span>p q <span class="h__symbol">:</span> <span class="h__keyword">Id</span> A a b<span class="h__symbol">)</span> <span class="h__symbol">:</span> <span class="h__keyword">Id</span> <span class="h__symbol">(</span><span class="h__keyword">Id</span> A a b<span class="h__symbol">)</span> p q <span class="h__symbol">:</span><span class="h__symbol">=</span> <span class="h__keyword">ref</span> p</code><br><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.452ex;" xmlns="http://www.w3.org/2000/svg" width="19.588ex" height="2.029ex" role="img" focusable="false" viewBox="0 -697 8658 897" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2077-TEX-B-1D402" 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For <b>(A : U) (i : I) (Partial A i)</b>
we can define subtype <b>A [ i ↦ u ]</b>. A term of this type is a term of
type A that is definitionally equal to u when <b>(IsOne i)</b> is satisfied.</p><code><span class="h__keyword">def</span> sub′ <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">(</span>i <span class="h__symbol">:</span> I<span class="h__symbol">)</span> <span class="h__symbol">(</span>u <span class="h__symbol">:</span> Partial A i<span class="h__symbol">)</span> <span class="h__symbol">:</span> <span class="h__keyword">V</span> <span class="h__symbol">:</span><span class="h__symbol">=</span> A [i ↦ u ]
<span class="h__keyword">def</span> <span class="h__keyword">inc</span>′ <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">(</span>i <span class="h__symbol">:</span> I<span class="h__symbol">)</span> <span class="h__symbol">(</span>a <span class="h__symbol">:</span> A<span class="h__symbol">)</span> <span class="h__symbol">:</span> A [i ↦ [<span class="h__symbol">(</span>i <span class="h__symbol">=</span> 1<span class="h__symbol">)</span> <span class="h__symbol">→</span> a]] <span class="h__symbol">:</span><span class="h__symbol">=</span> <span class="h__keyword">inc</span> A i a
<span class="h__keyword">def</span> <span class="h__keyword">ouc</span>′ <span class="h__symbol">(</span>A <span class="h__symbol">:</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">(</span>i <span class="h__symbol">:</span> I<span class="h__symbol">)</span> <span class="h__symbol">(</span>u <span class="h__symbol">:</span> Partial A i<span class="h__symbol">)</span> <span class="h__symbol">(</span>a <span class="h__symbol">:</span> A [i ↦ u]<span class="h__symbol">)</span> <span class="h__symbol">:</span> A <span class="h__symbol">:</span><span class="h__symbol">=</span> <span class="h__keyword">ouc</span> a</code><br><p>We have forth and back fusion rules <b>ouc (inc v) = v</b>
and <b>inc (outc v) = v</b>. Moreover, <b>ouc v</b> will reduce to u 1=1 when i=1.</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.455ex;" xmlns="http://www.w3.org/2000/svg" width="30.6ex" height="2.032ex" role="img" focusable="false" viewBox="0 -697 13525 898" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2078-TEX-B-1D407" d="M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z"></path><path id="MJX-2078-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 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Homogeneous composition <b>hcomp</b> is the answer to second problem:
with <b>hcomp</b> and <b>transp</b> one can express path composition,
groupoid, category of groupoids (groupoid interpretation and internalization in type theory).
One of the main roles of homogeneous composition is to be a carrier in [higher]
inductive type constructors for calculating of homotopy colimits and
direct encoding of CW-complexes (or internal models as gluening of pushout cells).</p><code><span class="h__keyword">def</span> comp <span class="h__symbol">(</span>A <span class="h__symbol">:</span> I <span class="h__symbol">→</span> <span class="h__keyword">U</span><span class="h__symbol">)</span> <span class="h__symbol">(</span>r <span class="h__symbol">:</span> I<span class="h__symbol">)</span> <span class="h__symbol">(</span>u <span class="h__symbol">:</span> Π <span class="h__symbol">(</span>i <span class="h__symbol">:</span> I<span class="h__symbol">)</span>, Partial <span class="h__symbol">(</span>A i<span class="h__symbol">)</span> r<span class="h__symbol">)</span> <span class="h__symbol">(</span>u₀ <span class="h__symbol">:</span> <span class="h__symbol">(</span>A 0<span class="h__symbol">)</span>[r ↦ u 0]<span class="h__symbol">)</span> <span class="h__symbol">:</span> A<span class="h__keyword"> 1</span>
<span class="h__symbol">:</span><span class="h__symbol">=</span> <span class="h__keyword">hcomp</span> <span class="h__symbol">(</span>A<span class="h__keyword"> 1</span><span class="h__symbol">)</span> r <span class="h__symbol">(</span>λ <span class="h__symbol">(</span>i <span class="h__symbol">:</span> I<span class="h__symbol">)</span>, [<span class="h__symbol">(</span>r <span class="h__symbol">=</span> 1<span class="h__symbol">)</span> <span class="h__symbol">→</span> <span class="h__keyword">transp</span> <span class="h__symbol">(</span><j> A <span class="h__symbol">(</span>i ∨ j<span class="h__symbol">))</span> i <span class="h__symbol">(</span>u i<span class="h__keyword"> 1</span><span class="h__symbol">=</span<span class="h__keyword">>1</span><span class="h__symbol">)</span>]<span class="h__symbol">)</span>
<span class="h__symbol">(</span><span class="h__keyword">transp</span> <span class="h__symbol">(</span><i> A i<span class="h__symbol">)</span> 0 <span class="h__symbol">(</span><span class="h__keyword">ouc</span> u₀<span class="h__symbol">))</span>
</code><br><p>The type checker equations for <b>hcomp</b> primitive are following:</p><code>hcompⁱ N [φ ↦ 0] 0 = 0
hcompⁱ N [φ ↦ S u] (S u₀) = S (hcompⁱ N [φ ↦ u] u₀)
hcompⁱ U [φ ↦ E] A = Glue [φ ↦ (E(i/1), equivⁱ E(i/1−i))] A
hcompⁱ (Π (x : A), B) [φ ↦ u] u₀ v = hcompⁱ B(x/v) [φ ↦ u v] (u₀ v)
hcompⁱ (Σ (x : A), B) [φ ↦ u] u₀ = (v(i/1), compⁱ B(x/v) [φ ↦ u.2] u₀.2)
hcompⁱ (Pathʲ A v w) [φ ↦ u] u₀ = 〈j〉 hcompⁱ A [ φ ↦ u j, (j = 0) ↦ v, (j = 1) ↦ w ] (u₀ j)
hcompⁱ (Glue [φ ↦ (T,w)] A) [ψ ↦ u] u₀
= glue [φ ↦ t₁] a₁
= glue [φ ↦ u(i/1)] (unglue u(i/1))
= u(i/1) : Glue [φ ↦ (T,w)] A</code><br><code>hfillⁱ A [φ ↦ u] u₀ = hcompʲ A [φ ↦ u(i/i∧j), (i=0) ↦ u₀] u₀ : A
v = hfillⁱ A [φ ↦ u.1] u₀.1
t₁ = u(i/1) : T
a₁ = unglue u(i/1) : A
glue [φ ↦ t₁] a1 = t₁ : T</code><br><h2 id="id">Strict Equality</h2><p>To avoid conflicts with path equalities which live in fibrant universes
strict equalities live in pretypes universes.</p><code>inductive strict (lang: U) : U
| Id: name → lang → total lang
| ref: lang → lang → total lang
| idJ: lang → lang → lang → total lang</code><br><p>We use strict equality in <b>Anders</b> for pretypes and partial elements which live in <b>V</b>.
The presheaf configuration with Pi, Sigma and Id is called <b>MLTT-73</b>.
The presheaf configuration with Pi, Sigma, Id and Path is called <b>HTS</b>.
</p><h2 id="glue">Glue Types</h2><p>The main purpose of Glue types is to construct a cube where some faces
have been replaced by equivalent types. This is analogous to how hcomp
lets us replace some faces of a cube by composing it with other cubes, but for
Glue types you can compose with equivalences instead of paths.
This implies the univalence principle and it is what lets us
transport along paths built out of equivalences.
</p><code>inductive gluening (lang: U) : U
| Glue: lang → lang → lang → gluening lang
| glue: lang → lang → gluening lang
| unglue: lang → lang → gluening lang</code><br><h2 id="induction">Generic Higher Inductive Schemes</h2><p>The further development of induction inside <b>MLTT</b> provers led
to the theory of polynomial functors and well-founded trees,
known in programming languages as inductive types.</p><p>Inductive types could be encoded in PTS/Cedile using non-recursive
representation of Bohm-Berarducci schemes or with categorical impredicative
encoding by Steve Awodey.</p><p>Anders currently don’t support Lean-compatible generic inductive schemes
definition but in case it will the following AST occurs:
</p><code>inductive tele (A: U) : U | emp: tele A | tel: name → A → tele A → tele A
inductive branch (A: U) : U | br: name → branch A | args: list name → A → branch A
inductive label (A: U) : U | lab: name → label A | t: tele A → label A
inductive hit (lang: U)
| form: name → tele lang → list (label lang) → hit lang
| ctor: name → list lang → hit lang
| htor: name → list lang → list lang → lang → hit lang
| case: name → lang → list (branch lang) → hit lang
</code><p>So instead of generic inductive schemes Anders supports well-founded
trees (W-types). Basic data types like List, Nat, Fin, Vec are
implemented as W-types in base library. As for higher inductive types
Anders has Three-HIT basis (Coequalizer, HubSpoke and Colimit) to
express other HIT.</p><p>The non-well-founded trees (M-types) or infinite coinductive trees
are useful for modeling infinite processes and are part
of Milner’s Pi-calculus. Coinductive streams could be
found in many <b>MLTT</b> base libraries.
</p><div style="text-align:center;padding-top:8px;"><img src="https://anders.groupoid.space/images/pdf.jpg" width=35>
<a href="https://groupoid.github.io/anders/doc/anders.pdf">ANDERS.PDF</a>
</div></section><section><h1>Bibliography</h1><br></section><section><h2>HTS</h2><p>1) <a href="https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/HTS.pdf">A simple type system with two identity types</a> [Voevodsky].
2) <a href="https://arxiv.org/pdf/1705.03307.pdf">Two-level type theory and applications</a> [Annenkov, Capriotti, Kraus, Sattler].
3) <a href="https://types21.liacs.nl/download/syntax-for-two-level-type-theory/">Syntax for two-level type theory</a> [Bonacina, Ahrens].
</p><h2>CCHM</h2><p>4) <a href="https://arxiv.org/pdf/1611.02108.pdf">Cubical Type Theory: a constructive interpretation of the univalence axiom</a> [Cohen, Coquand, Huber, Mörtberg];
5) <a href="https://arxiv.org/pdf/1802.01170.pdf">On Higher Inductive Types in Cubical Type Theory</a> [Coquand, Huber, Mörtberg];
6) <a href="https://arxiv.org/pdf/1607.04156.pdf">Canonicity for Cubical Type Theory</a> [Huber];
7) <a href="https://arxiv.org/pdf/1902.06572.pdf">Canonicity and homotopy canonicity for cubical type theory</a> [Coquand, Huber, Sattler];
8) <a href="https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf">Cubical Synthetic Homotopy Theory</a> [Mörtberg, Pujet];
9) <a href="https://staff.math.su.se/anders.mortberg/papers/unifying.pdf">Unifying Cubical Models of Univalent Type Theory</a> [Cavallo, Mörtberg, Swan];
10) <a href="https://staff.math.su.se/anders.mortberg/papers/cubicalagda.pdf">Cubical Agda: A Dependently Typed PL with Univalence and HITs</a> [Vezzosi, Mörtberg, Abel];
11) <a href="https://simhu.github.io/misc/hcomp.pdf">A Cubical Type Theory for Higher Inductive Types</a> [Huber];
12) <a href="https://drops.dagstuhl.de/opus/volltexte/2019/10532/pdf/LIPIcs-FSCD-2019-25.pdf">Gluing for type theory</a> [Kaposi, Huber, Sattler].
13) <a href="https://www.cambridge.org/core/services/aop-cambridge-core/content/view/ECB3FE6B4A0B19AED2D3A2D785C38AF9/S0960129521000311a.pdf/cubical-methods-in-homotopy-type-theory-and-univalent-foundations.pdf">Cubical Methods in HoTT/UF</a> [Mörtberg].
</p><h2>MLTT</h2><p>14) <a href="https://raw.githubusercontent.com/michaelt/martin-lof/master/pdfs/Bibliopolis-Book-retypeset-1984.pdf">Intuitionistic Type Theory</a> [Martin-Löf];
15) <a href="http://archive-pml.github.io/martin-lof/pdfs/An-Intuitionistic-Theory-of-Types-Predicative-Part-1975.pdf">An intuitionistic theory of types: predicative part.</a> [Martin-Löf];
16) <a href="http://www.ens-lyon.fr/denif/data/martin_lof_prog/1990/contenu/book.pdf">Programming in Martin-Löf’s Type Theory</a> [Nordström, Petersson, Smith];
17) <a href="http://www.cse.chalmers.se/~bengt/papers/GKminiTT.pdf">A simple type-theoretic language: Mini-TT</a> [Coquand, Kinoshita, Nordström, Takeyama];
</p><h2>Modal HoTT</h2><p>18) <a href="https://arxiv.org/pdf/1310.7930v1.pdf">Differential cohomology in a cohesive ∞-topos</a> [Schreiber].
19) <a href="https://arxiv.org/pdf/1806.05966.pdf">Cartan Geometry in Modal Homotopy Type Theory</a> [Cherubini].
20) <a href="https://hott-uf.github.io/2017/abstracts/cohesivett.pdf">Differential Cohesive Type Theory</a> [Gross, Licata, New, Paykin, Riley, Shulman, Cherubini].
21) <a href="https://arxiv.org/abs/1509.07584">Brouwer’s fixed-point theorem in real-cohesive homotopy type theory</a> [Shulman].
</p><center><br>🧊 <br><br><br>
</center><p>This work was sponsored by <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.452ex;" xmlns="http://www.w3.org/2000/svg" width="9.844ex" height="2.025ex" role="img" focusable="false" viewBox="0 -695 4351 895" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2079-TEX-B-1D40C" d="M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H305Q316 679 323 667Q330 653 434 414L546 157L658 414Q766 662 773 674Q778 681 788 686H1052V624H944V62H1052V0H1040Q1016 3 874 3T708 0H696V62H804V341L803 618L786 580Q770 543 735 462T671 315Q540 13 536 9Q528 1 507 1Q485 1 477 9Q472 14 408 162T281 457T217 603Q215 603 215 334V62H323V0H314Z"></path><path id="MJX-2079-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-2079-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 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