helm/www/lambdadelta/basic_2.html

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  10     <title>\lambda\delta home page</title>
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  22     <div class="head1">cic:/matita/lambdadelta/basic_2/ (core λδ version 2)</div>
  23     <div class="spacer">
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  25     </div>
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  27       <br />
  28     </div>
  29     <div xmlns:ld="http://lambdadelta.info/" class="text">
  30       <table cellpadding="4" cellspacing="0">
  31         <tbody>
  32           <tr>
  33             <td class="snns capitalize italic sky">
  34               <a href="http://lambdadelta.info/index.html">home</a>
  35             </td>
  36             <td class="snns capitalize italic magenta">
  37               <a href="http://lambdadelta.info/news.html">news</a>
  38             </td>
  39             <td class="snns capitalize italic white">
  40               <a href="http://lambdadelta.info/specification.html">specification</a>
  41             </td>
  42             <td class="snnn capitalize italic white">
  43               <br />
  44             </td>
  45             <td class="snnn capitalize italic white">
  46               <br />
  47             </td>
  48             <td class="snns capitalize italic orange">
  49               <a href="http://lambdadelta.info/documentation.html">documentation</a>
  50             </td>
  51             <td class="snns capitalize italic green">
  52               <a href="http://lambdadelta.info/implementation.html">implementation</a>
  53             </td>
  54             <td class="ssnn capitalize italic green">
  55               <br />
  56             </td>
  57           </tr>
  58           <tr>
  59             <td class="snns capitalize sky">
  60               <a href="http://lambdadelta.info/index.html#foreword">foreword</a>
  61             </td>
  62             <td class="snns capitalize magenta">
  63               <a href="http://lambdadelta.info/news.html#milestones">milestones</a>
  64             </td>
  65             <td class="snns capitalize white">
  66               <a href="http://lambdadelta.info/specification.html#v2">version 2</a>
  67             </td>
  68             <td class="snnn capitalize white">(<a href="http://lambdadelta.info/ground_2.html">background</a> - <a href="http://lambdadelta.info/basic_2.html">core</a> - <a href="http://lambdadelta.info/apps_2.html">applications</a>)</td>
  69             <td class="snnn capitalize white">
  70               <br />
  71             </td>
  72             <td class="snns capitalize orange">
  73               <a href="http://lambdadelta.info/documentation.html#v2">version 2</a>
  74             </td>
  75             <td class="snns capitalize green">
  76               <a href="http://lambdadelta.info/implementation.html#helena">helena</a>
  77             </td>
  78             <td class="ssnn capitalize green">
  79               <br />
  80             </td>
  81           </tr>
  82           <tr>
  83             <td class="snss capitalize sky">
  84               <a href="http://lambdadelta.info/index.html#citations">citations</a>
  85             </td>
  86             <td class="snss capitalize magenta">
  87               <a href="http://lambdadelta.info/news.html#visibility">visibility</a>
  88             </td>
  89             <td class="snss capitalize white">
  90               <a href="http://lambdadelta.info/specification.html#v1">version 1</a>
  91             </td>
  92             <td class="snsn capitalize white">(<a href="http://lambdadelta.info/ground_1.html">background</a> - <a href="http://lambdadelta.info/basic_1.html">core</a>)</td>
  93             <td class="snsn capitalize white">(<a href="http://lambdadelta.info/static/matita/lambdadelta/">static HELM directory</a>)</td>
  94             <td class="snss capitalize orange">
  95               <a href="http://lambdadelta.info/documentation.html#v1">version 1</a>
  96             </td>
  97             <td class="snss capitalize green">
  98               <a href="http://lambdadelta.info/implementation.html#lddl">library</a>
  99             </td>
 100             <td class="sssn capitalize green">(<a href="http://lambdadelta.info/static/lddl/">static LDDL directory</a>)</td>
 101           </tr>
 102         </tbody>
 103       </table>
 104     </div>
 105     <!--
 106    <section>System's Syntax and Behavior</section>
 107    <body>This is a summary of the "block structure"
 108          of the System's syntactic items and reductions.
 109    </body>
 110    <table name="basic_2_blk"/>
 111    <body>* In terms only.
 112          ** In terms and local environments only.
 113          *** In global environments only.
 114          **** Sort level k in terms only.
 115    </body>
 116 -->
 117     <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="summary">Summary of the Specification <img class="icon37" alt="[spacer]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
 118     </div>
 119     <div xmlns:ld="http://lambdadelta.info/" class="text">Here is a numerical account of the specification's contents
 120          and its timeline.
 121    </div>
 122     <div xmlns:ld="http://lambdadelta.info/" class="text">
 123       <table cellpadding="4" cellspacing="0">
 124         <tbody>
 125           <tr>
 126             <td class="snns capitalize italic gray">category</td>
 127             <td class="snns italic gray">objects</td>
 128             <td class="snnn right italic gray">
 129               <br />
 130             </td>
 131             <td class="snnn italic gray">
 132               <br />
 133             </td>
 134             <td class="snnn right italic gray">
 135               <br />
 136             </td>
 137             <td class="snnn italic gray">
 138               <br />
 139             </td>
 140             <td class="ssnn right italic gray">
 141               <br />
 142             </td>
 143           </tr>
 144           <tr>
 145             <td class="snns capitalize italic cyan">sizes</td>
 146             <td class="snns italic cyan">files</td>
 147             <td class="snnn right italic cyan">102</td>
 148             <td class="snns italic cyan">characters</td>
 149             <td class="snnn right italic cyan">69295</td>
 150             <td class="snns italic cyan">nodes</td>
 151             <td class="ssnn right italic cyan">245853</td>
 152           </tr>
 153           <tr>
 154             <td class="snns capitalize italic green">propositions</td>
 155             <td class="snns italic green">theorems</td>
 156             <td class="snnn right italic green">34</td>
 157             <td class="snns italic green">lemmas</td>
 158             <td class="snnn right italic green">256</td>
 159             <td class="snns italic green">total</td>
 160             <td class="ssnn right italic green">290</td>
 161           </tr>
 162           <tr>
 163             <td class="snss capitalize italic yellow">concepts</td>
 164             <td class="snss italic yellow">declared</td>
 165             <td class="snsn right italic yellow">21</td>
 166             <td class="snss italic yellow">defined</td>
 167             <td class="snsn right italic yellow">29</td>
 168             <td class="snss italic yellow">total</td>
 169             <td class="sssn right italic yellow">50</td>
 170           </tr>
 171         </tbody>
 172       </table>
 173     </div>
 174     <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="B">Stage "B"</div>
 175     <ul xmlns:ld="http://lambdadelta.info/">
 176       <li>
 177         <span class="emph alpha">Ongoing.</span>
 178          Context-sensitive subject equivalence
 179          for native type assignment.
 180    </li>
 181     </ul>
 182     <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A2">Stage "A2": "Extending the Applicability Condition"</div>
 183     <ul xmlns:ld="http://lambdadelta.info/">
 184       <li>
 185         <span class="emph alpha">2015 October 9.</span>
 186          λδ version 2A2 is started.
 187    </li>
 188     </ul>
 189     <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A1">Stage "A1": "Extending the Applicability Condition"</div>
 190     <ul xmlns:ld="http://lambdadelta.info/">
 191       <li>
 192         <span class="emph delta">2015 August 27.</span>
 193          λδ version 2A1 appears too complex and is dismissed.
 194    </li>
 195     </ul>
 196     <ul xmlns:ld="http://lambdadelta.info/">
 197       <li>
 198         <span class="emph gamma">2014 October 28.</span>
 199          λδ version 2A1 is released.
 200    </li>
 201     </ul>
 202     <ul xmlns:ld="http://lambdadelta.info/">
 203       <li>
 204         <span class="emph beta">2014 September 9.</span>
 205          Iterated static type assignment defined (more elegantly)
 206          as a primitive notion.
 207    </li>
 208     </ul>
 209     <ul xmlns:ld="http://lambdadelta.info/">
 210       <li>
 211         <span class="emph beta">2014 June 18.</span>
 212          Preservation of stratified native validity
 213          for context-sensitive computation on terms.
 214    </li>
 215     </ul>
 216     <ul xmlns:ld="http://lambdadelta.info/">
 217       <li>
 218         <span class="emph alpha">2014 June 9.</span>
 219          Strong qrst-normalization
 220          for simply typed terms.
 221    </li>
 222     </ul>
 223     <ul xmlns:ld="http://lambdadelta.info/">
 224       <li>
 225         <span class="emph alpha">2014 April 16.</span>
 226          Lazy equivalence on local environments
 227          added as q-step to rst-computation on closures
 228          (anniversary milestone).
 229    </li>
 230     </ul>
 231     <ul xmlns:ld="http://lambdadelta.info/">
 232       <li>
 233         <span class="emph alpha">2014 January 20.</span>
 234          Parametrized slicing of local environments
 235          comprises both versions of this operation
 236          (one from basic_1, the other used in basic_2 till now).
 237    </li>
 238     </ul>
 239     <ul xmlns:ld="http://lambdadelta.info/">
 240       <li>
 241         <span class="emph alpha">2013 August 7.</span>
 242          Passive support for global environments.
 243    </li>
 244     </ul>
 245     <ul xmlns:ld="http://lambdadelta.info/">
 246       <li>
 247         <span class="emph alpha">2013 July 27.</span>
 248          Reaxiomatized β-reductum as in rt-reduction.
 249    </li>
 250     </ul>
 251     <ul xmlns:ld="http://lambdadelta.info/">
 252       <li>
 253         <span class="emph alpha">2013 July 20.</span>
 254          Context-sensitive strong rt-normalization
 255          for simply typed terms.
 256    </li>
 257     </ul>
 258     <ul xmlns:ld="http://lambdadelta.info/">
 259       <li>
 260         <span class="emph alpha">2013 April 16.</span>
 261          Reaxiomatized substitution and reduction
 262          commute with respect to subclosure
 263          (anniversary milestone).
 264    </li>
 265     </ul>
 266     <ul xmlns:ld="http://lambdadelta.info/">
 267       <li>
 268         <span class="emph alpha">2013 March 16.</span>
 269          Mutual recursive preservation of stratified native validity
 270          for rst-computation on closures.
 271    </li>
 272     </ul>
 273     <ul xmlns:ld="http://lambdadelta.info/">
 274       <li>
 275         <span class="emph alpha">2012 October 16.</span>
 276          Confluence for context-free parallel reduction on closures.
 277    </li>
 278     </ul>
 279     <ul xmlns:ld="http://lambdadelta.info/">
 280       <li>
 281         <span class="emph alpha">2012 July 26.</span>
 282          Term binders polarized to control ζ-reduction (not released).
 283    </li>
 284     </ul>
 285     <ul xmlns:ld="http://lambdadelta.info/">
 286       <li>
 287         <span class="emph alpha">2012 April 16.</span>
 288          Context-sensitive subject equivalence
 289          for atomic arity assignment
 290          (anniversary milestone).
 291    </li>
 292     </ul>
 293     <ul xmlns:ld="http://lambdadelta.info/">
 294       <li>
 295         <span class="emph alpha">2012 March 15.</span>
 296          Context-sensitive strong normalization
 297          for simply typed terms.
 298    </li>
 299     </ul>
 300     <ul xmlns:ld="http://lambdadelta.info/">
 301       <li>
 302         <span class="emph alpha">2012 January 27.</span>
 303          Support for abstract candidates of reducibility.
 304    </li>
 305     </ul>
 306     <ul xmlns:ld="http://lambdadelta.info/">
 307       <li>
 308         <span class="emph alpha">2011 September 21.</span>
 309          Confluence for context-sensitive parallel reduction on terms.
 310    </li>
 311     </ul>
 312     <ul xmlns:ld="http://lambdadelta.info/">
 313       <li>
 314         <span class="emph alpha">2011 September 6.</span>
 315          Confluence for context-free parallel reduction on terms.
 316    </li>
 317     </ul>
 318     <ul xmlns:ld="http://lambdadelta.info/">
 319       <li>
 320         <span class="emph alpha">2011 April 17.</span>
 321          λδ version 2 is started.
 322    </li>
 323     </ul>
 324     <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="structure">Logical Structure of the Specification <img class="icon37" alt="[spacer]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
 325     </div>
 326     <div xmlns:ld="http://lambdadelta.info/" class="text">This table reports the specification's components and their planes.
 327    </div>
 328     <div xmlns:ld="http://lambdadelta.info/" class="text">
 329       <table cellpadding="4" cellspacing="0">
 330         <tbody>
 331           <tr>
 332             <td class="snns top capitalize italic gray">component</td>
 333             <td class="snns top italic gray">plane</td>
 334             <td class="snns top gray">files</td>
 335             <td class="snnn top gray">
 336               <br />
 337             </td>
 338             <td class="snnn top gray">
 339               <br />
 340             </td>
 341             <td class="ssnn top gray">
 342               <br />
 343             </td>
 344           </tr>
 345           <tr>
 346             <td class="snns top capitalize italic green">static typing</td>
 347             <td class="snns top italic green">parameters</td>
 348             <td class="snns top green">sh</td>
 349             <td class="snnn top green">sd</td>
 350             <td class="snnn top green">
 351               <br />
 352             </td>
 353             <td class="ssnn top green">
 354               <br />
 355             </td>
 356           </tr>
 357           <tr>
 358             <td class="nnns top capitalize italic green">
 359               <br />
 360             </td>
 361             <td class="snns top italic green">restricted ref. for local env.</td>
 362             <td class="snns top green">lsubr ( ? ⫃ ? )</td>
 363             <td class="snnn top green">lsubr_length lsubr_drops lsubr_lsubr</td>
 364             <td class="snnn top green">
 365               <br />
 366             </td>
 367             <td class="ssnn top green">
 368               <br />
 369             </td>
 370           </tr>
 371           <tr>
 372             <td class="nnns top capitalize italic green">
 373               <br />
 374             </td>
 375             <td class="snns top italic green">ranged equivalence for closures</td>
 376             <td class="snns top green">freq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )</td>
 377             <td class="snnn top green">freq_freq</td>
 378             <td class="snnn top green">
 379               <br />
 380             </td>
 381             <td class="ssnn top green">
 382               <br />
 383             </td>
 384           </tr>
 385           <tr>
 386             <td class="nnns top capitalize italic green">
 387               <br />
 388             </td>
 389             <td class="snns top italic green">context-sensitive free variables</td>
 390             <td class="snns top green">frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )</td>
 391             <td class="snnn top green">frees_weight frees_lreq frees_frees</td>
 392             <td class="snnn top green">
 393               <br />
 394             </td>
 395             <td class="ssnn top green">
 396               <br />
 397             </td>
 398           </tr>
 399           <tr>
 400             <td class="snns top capitalize italic grass">s-computation</td>
 401             <td class="snns top italic grass" />
 402             <td class="snns top grass">
 403               <br />
 404             </td>
 405             <td class="snnn top grass">
 406               <br />
 407             </td>
 408             <td class="snnn top grass">
 409               <br />
 410             </td>
 411             <td class="ssnn top grass">
 412               <br />
 413             </td>
 414           </tr>
 415           <tr>
 416             <td class="snns top capitalize italic yellow">s-transition</td>
 417             <td class="snns top italic yellow">structural successor for closures</td>
 418             <td class="snns top yellow">fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )</td>
 419             <td class="snnn top yellow">fquq_length fquq_weight</td>
 420             <td class="snnn top yellow">
 421               <br />
 422             </td>
 423             <td class="ssnn top yellow">
 424               <br />
 425             </td>
 426           </tr>
 427           <tr>
 428             <td class="nnns top capitalize italic yellow">
 429               <br />
 430             </td>
 431             <td class="nnns top italic yellow">
 432               <br />
 433             </td>
 434             <td class="snns top yellow">fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )</td>
 435             <td class="snnn top yellow">fqu_length fqu_weight</td>
 436             <td class="snnn top yellow">
 437               <br />
 438             </td>
 439             <td class="ssnn top yellow">
 440               <br />
 441             </td>
 442           </tr>
 443           <tr>
 444             <td class="snns top capitalize italic orange">relocation</td>
 445             <td class="snns top italic orange">generic slicing for local environments</td>
 446             <td class="snns top orange">drops_vector ( ⬇*[?,?] ? ≡ ? )</td>
 447             <td class="snnn top orange">
 448               <br />
 449             </td>
 450             <td class="snnn top orange">
 451               <br />
 452             </td>
 453             <td class="ssnn top orange">
 454               <br />
 455             </td>
 456           </tr>
 457           <tr>
 458             <td class="nnns top capitalize italic orange">
 459               <br />
 460             </td>
 461             <td class="nnns top italic orange">
 462               <br />
 463             </td>
 464             <td class="snns top orange">drops ( ⬇*[?,?] ? ≡ ? )</td>
 465             <td class="snnn top orange">drops_lstar drops_weight drops_length drops_ceq drops_lexs drops_lreq drops_drops</td>
 466             <td class="snnn top orange">
 467               <br />
 468             </td>
 469             <td class="ssnn top orange">
 470               <br />
 471             </td>
 472           </tr>
 473           <tr>
 474             <td class="nnns top capitalize italic orange">
 475               <br />
 476             </td>
 477             <td class="snns top italic orange">generic relocation for terms</td>
 478             <td class="snns top orange">lifts_vector ( ⬆*[?] ? ≡ ? )</td>
 479             <td class="snnn top orange">lifts_lifts_vector</td>
 480             <td class="snnn top orange">
 481               <br />
 482             </td>
 483             <td class="ssnn top orange">
 484               <br />
 485             </td>
 486           </tr>
 487           <tr>
 488             <td class="nnns top capitalize italic orange">
 489               <br />
 490             </td>
 491             <td class="nnns top italic orange">
 492               <br />
 493             </td>
 494             <td class="snns top orange">lifts ( ⬆*[?] ? ≡ ? )</td>
 495             <td class="snnn top orange">lifts_simple lifts_weight lifts_lifts</td>
 496             <td class="snnn top orange">
 497               <br />
 498             </td>
 499             <td class="ssnn top orange">
 500               <br />
 501             </td>
 502           </tr>
 503           <tr>
 504             <td class="nnns top capitalize italic orange">
 505               <br />
 506             </td>
 507             <td class="snns top italic orange">ranged equivalence for local environments</td>
 508             <td class="snns top orange">lreq ( ? ≡[?] ? )</td>
 509             <td class="snnn top orange">lreq_length lreq_lreq</td>
 510             <td class="snnn top orange">
 511               <br />
 512             </td>
 513             <td class="ssnn top orange">
 514               <br />
 515             </td>
 516           </tr>
 517           <tr>
 518             <td class="nnns top capitalize italic orange">
 519               <br />
 520             </td>
 521             <td class="snns top italic orange">generic entrywise extension of context-sensitive relations for terma</td>
 522             <td class="snns top orange">lexs ( ? ⦻*[?,?,?] ? )</td>
 523             <td class="snnn top orange">lexs_length lexs_lexs</td>
 524             <td class="snnn top orange">
 525               <br />
 526             </td>
 527             <td class="ssnn top orange">
 528               <br />
 529             </td>
 530           </tr>
 531           <tr>
 532             <td class="snns top capitalize italic red">grammar</td>
 533             <td class="snns top italic red">append for local environments</td>
 534             <td class="snns top red">append ( ? @@ ? )</td>
 535             <td class="snnn top red">append_length</td>
 536             <td class="snnn top red">
 537               <br />
 538             </td>
 539             <td class="ssnn top red">
 540               <br />
 541             </td>
 542           </tr>
 543           <tr>
 544             <td class="nnns top capitalize italic red">
 545               <br />
 546             </td>
 547             <td class="snns top italic red">context-sensitive equivalences for terms</td>
 548             <td class="snns top red">ceq</td>
 549             <td class="snnn top red">ceq_ceq</td>
 550             <td class="snnn top red">
 551               <br />
 552             </td>
 553             <td class="ssnn top red">
 554               <br />
 555             </td>
 556           </tr>
 557           <tr>
 558             <td class="nnns top capitalize italic red">
 559               <br />
 560             </td>
 561             <td class="snns top italic red">same top term structure</td>
 562             <td class="snns top red">tsts ( ? ≂ ? )</td>
 563             <td class="snnn top red">tsts_tsts tsts_vector</td>
 564             <td class="snnn top red">
 565               <br />
 566             </td>
 567             <td class="ssnn top red">
 568               <br />
 569             </td>
 570           </tr>
 571           <tr>
 572             <td class="nnns top capitalize italic red">
 573               <br />
 574             </td>
 575             <td class="snns top italic red">closures</td>
 576             <td class="snns top red">cl_weight ( ♯{?,?,?} )</td>
 577             <td class="snnn top red">cl_restricted_weight ( ♯{?,?} )</td>
 578             <td class="snnn top red">
 579               <br />
 580             </td>
 581             <td class="ssnn top red">
 582               <br />
 583             </td>
 584           </tr>
 585           <tr>
 586             <td class="nnns top capitalize italic red">
 587               <br />
 588             </td>
 589             <td class="snns top italic red">internal syntax</td>
 590             <td class="snns top red">genv</td>
 591             <td class="snnn top red">
 592               <br />
 593             </td>
 594             <td class="snnn top red">
 595               <br />
 596             </td>
 597             <td class="ssnn top red">
 598               <br />
 599             </td>
 600           </tr>
 601           <tr>
 602             <td class="nnns top capitalize italic red">
 603               <br />
 604             </td>
 605             <td class="nnns top italic red">
 606               <br />
 607             </td>
 608             <td class="snns top red">lenv</td>
 609             <td class="snnn top red">lenv_weight ( ♯{?} )</td>
 610             <td class="snnn top red">lenv_length ( |?| )</td>
 611             <td class="ssnn top red">
 612               <br />
 613             </td>
 614           </tr>
 615           <tr>
 616             <td class="nnns top capitalize italic red">
 617               <br />
 618             </td>
 619             <td class="nnns top italic red">
 620               <br />
 621             </td>
 622             <td class="snns top red">term</td>
 623             <td class="snnn top red">term_weight ( ♯{?} )</td>
 624             <td class="snnn top red">term_simple ( 𝐒⦃?⦄ )</td>
 625             <td class="ssnn top red">term_vector ( Ⓐ?.? )</td>
 626           </tr>
 627           <tr>
 628             <td class="nnns top capitalize italic red">
 629               <br />
 630             </td>
 631             <td class="nnns top italic red">
 632               <br />
 633             </td>
 634             <td class="snns top red">item</td>
 635             <td class="snnn top red">
 636               <br />
 637             </td>
 638             <td class="snnn top red">
 639               <br />
 640             </td>
 641             <td class="ssnn top red">
 642               <br />
 643             </td>
 644           </tr>
 645           <tr>
 646             <td class="nnss top capitalize italic red">
 647               <br />
 648             </td>
 649             <td class="snss top italic red">external syntax</td>
 650             <td class="snss top red">aarity</td>
 651             <td class="snsn top red">
 652               <br />
 653             </td>
 654             <td class="snsn top red">
 655               <br />
 656             </td>
 657             <td class="sssn top red">
 658               <br />
 659             </td>
 660           </tr>
 661         </tbody>
 662       </table>
 663     </div>
 664     <div class="spacer">
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