helm/www/lambdadelta/basic_2.html

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  10     <title>\lambda\delta home page</title>
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  21     </div>
  22     <div class="head1">cic:/matita/lambdadelta/basic_2/ (core λδ version 2)</div>
  23     <div class="spacer">
  24       <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
  25     </div>
  26     <div xmlns:ld="http://lambdadelta.info/" class="spacer">
  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">107</td>
 148             <td class="snns italic cyan">characters</td>
 149             <td class="snnn right italic cyan">68741</td>
 150             <td class="snns italic cyan">nodes</td>
 151             <td class="ssnn right italic cyan">232342</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">32</td>
 157             <td class="snns italic green">lemmas</td>
 158             <td class="snnn right italic green">272</td>
 159             <td class="snns italic green">total</td>
 160             <td class="ssnn right italic green">304</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">19</td>
 166             <td class="snss italic yellow">defined</td>
 167             <td class="snsn right italic yellow">31</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">2016 March 25.</span>
 186          Relocation with reference transforming maps (rtmap).
 187    </li>
 188     </ul>
 189     <ul xmlns:ld="http://lambdadelta.info/">
 190       <li>
 191         <span class="emph alpha">2015 October 9.</span>
 192          λδ version 2A2 is started.
 193    </li>
 194     </ul>
 195     <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A1">Stage "A1": "Extending the Applicability Condition"</div>
 196     <ul xmlns:ld="http://lambdadelta.info/">
 197       <li>
 198         <span class="emph delta">2015 August 27.</span>
 199          λδ version 2A1 appears too complex and is dismissed.
 200    </li>
 201     </ul>
 202     <ul xmlns:ld="http://lambdadelta.info/">
 203       <li>
 204         <span class="emph gamma">2014 October 28.</span>
 205          λδ version 2A1 is released.
 206    </li>
 207     </ul>
 208     <ul xmlns:ld="http://lambdadelta.info/">
 209       <li>
 210         <span class="emph beta">2014 September 9.</span>
 211          Iterated static type assignment defined (more elegantly)
 212          as a primitive notion.
 213    </li>
 214     </ul>
 215     <ul xmlns:ld="http://lambdadelta.info/">
 216       <li>
 217         <span class="emph beta">2014 June 18.</span>
 218          Preservation of stratified native validity
 219          for context-sensitive computation on terms.
 220    </li>
 221     </ul>
 222     <ul xmlns:ld="http://lambdadelta.info/">
 223       <li>
 224         <span class="emph alpha">2014 June 9.</span>
 225          Strong qrst-normalization
 226          for simply typed terms.
 227    </li>
 228     </ul>
 229     <ul xmlns:ld="http://lambdadelta.info/">
 230       <li>
 231         <span class="emph alpha">2014 April 16.</span>
 232          Lazy equivalence on local environments
 233          added as q-step to rst-computation on closures
 234          (anniversary milestone).
 235    </li>
 236     </ul>
 237     <ul xmlns:ld="http://lambdadelta.info/">
 238       <li>
 239         <span class="emph alpha">2014 January 20.</span>
 240          Parametrized slicing of local environments
 241          comprises both versions of this operation
 242          (one from basic_1, the other used in basic_2 till now).
 243    </li>
 244     </ul>
 245     <ul xmlns:ld="http://lambdadelta.info/">
 246       <li>
 247         <span class="emph alpha">2013 August 7.</span>
 248          Passive support for global environments.
 249    </li>
 250     </ul>
 251     <ul xmlns:ld="http://lambdadelta.info/">
 252       <li>
 253         <span class="emph alpha">2013 July 27.</span>
 254          Reaxiomatized β-reductum as in rt-reduction.
 255    </li>
 256     </ul>
 257     <ul xmlns:ld="http://lambdadelta.info/">
 258       <li>
 259         <span class="emph alpha">2013 July 20.</span>
 260          Context-sensitive strong rt-normalization
 261          for simply typed terms.
 262    </li>
 263     </ul>
 264     <ul xmlns:ld="http://lambdadelta.info/">
 265       <li>
 266         <span class="emph alpha">2013 April 16.</span>
 267          Reaxiomatized substitution and reduction
 268          commute with respect to subclosure
 269          (anniversary milestone).
 270    </li>
 271     </ul>
 272     <ul xmlns:ld="http://lambdadelta.info/">
 273       <li>
 274         <span class="emph alpha">2013 March 16.</span>
 275          Mutual recursive preservation of stratified native validity
 276          for rst-computation on closures.
 277    </li>
 278     </ul>
 279     <ul xmlns:ld="http://lambdadelta.info/">
 280       <li>
 281         <span class="emph alpha">2012 October 16.</span>
 282          Confluence for context-free parallel reduction on closures.
 283    </li>
 284     </ul>
 285     <ul xmlns:ld="http://lambdadelta.info/">
 286       <li>
 287         <span class="emph alpha">2012 July 26.</span>
 288          Term binders polarized to control ζ-reduction (not released).
 289    </li>
 290     </ul>
 291     <ul xmlns:ld="http://lambdadelta.info/">
 292       <li>
 293         <span class="emph alpha">2012 April 16.</span>
 294          Context-sensitive subject equivalence
 295          for atomic arity assignment
 296          (anniversary milestone).
 297    </li>
 298     </ul>
 299     <ul xmlns:ld="http://lambdadelta.info/">
 300       <li>
 301         <span class="emph alpha">2012 March 15.</span>
 302          Context-sensitive strong normalization
 303          for simply typed terms.
 304    </li>
 305     </ul>
 306     <ul xmlns:ld="http://lambdadelta.info/">
 307       <li>
 308         <span class="emph alpha">2012 January 27.</span>
 309          Support for abstract candidates of reducibility.
 310    </li>
 311     </ul>
 312     <ul xmlns:ld="http://lambdadelta.info/">
 313       <li>
 314         <span class="emph alpha">2011 September 21.</span>
 315          Confluence for context-sensitive parallel reduction on terms.
 316    </li>
 317     </ul>
 318     <ul xmlns:ld="http://lambdadelta.info/">
 319       <li>
 320         <span class="emph alpha">2011 September 6.</span>
 321          Confluence for context-free parallel reduction on terms.
 322    </li>
 323     </ul>
 324     <ul xmlns:ld="http://lambdadelta.info/">
 325       <li>
 326         <span class="emph alpha">2011 April 17.</span>
 327          λδ version 2 is started.
 328    </li>
 329     </ul>
 330     <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" />
 331     </div>
 332     <div xmlns:ld="http://lambdadelta.info/" class="text">This table reports the specification's components and their planes.
 333    </div>
 334     <div xmlns:ld="http://lambdadelta.info/" class="text">
 335       <table cellpadding="4" cellspacing="0">
 336         <tbody>
 337           <tr>
 338             <td class="snns top capitalize italic gray">component</td>
 339             <td class="snns top italic gray">plane</td>
 340             <td class="snns top gray">files</td>
 341             <td class="snnn top gray">
 342               <br />
 343             </td>
 344             <td class="snnn top gray">
 345               <br />
 346             </td>
 347             <td class="ssnn top gray">
 348               <br />
 349             </td>
 350           </tr>
 351           <tr>
 352             <td class="snns top capitalize italic green">static typing</td>
 353             <td class="snns top italic green">parameters</td>
 354             <td class="snns top green">sh</td>
 355             <td class="snnn top green">sd</td>
 356             <td class="snnn top green">
 357               <br />
 358             </td>
 359             <td class="ssnn top green">
 360               <br />
 361             </td>
 362           </tr>
 363           <tr>
 364             <td class="nnns top capitalize italic green">
 365               <br />
 366             </td>
 367             <td class="snns top italic green">restricted ref. for local env.</td>
 368             <td class="snns top green">lsubr ( ? ⫃ ? )</td>
 369             <td class="snnn top green">lsubr_length lsubr_drops lsubr_lsubr</td>
 370             <td class="snnn top green">
 371               <br />
 372             </td>
 373             <td class="ssnn top green">
 374               <br />
 375             </td>
 376           </tr>
 377           <tr>
 378             <td class="nnns top capitalize italic green">
 379               <br />
 380             </td>
 381             <td class="snns top italic green">ranged equivalence for closures</td>
 382             <td class="snns top green">freq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )</td>
 383             <td class="snnn top green">freq_freq</td>
 384             <td class="snnn top green">
 385               <br />
 386             </td>
 387             <td class="ssnn top green">
 388               <br />
 389             </td>
 390           </tr>
 391           <tr>
 392             <td class="nnns top capitalize italic green">
 393               <br />
 394             </td>
 395             <td class="snns top italic green">context-sensitive free variables</td>
 396             <td class="snns top green">frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )</td>
 397             <td class="snnn top green">frees_weight frees_lreq frees_frees</td>
 398             <td class="snnn top green">
 399               <br />
 400             </td>
 401             <td class="ssnn top green">
 402               <br />
 403             </td>
 404           </tr>
 405           <tr>
 406             <td class="snns top capitalize italic grass">s-computation</td>
 407             <td class="snns top italic grass">iterated structural successor for closures</td>
 408             <td class="snns top grass">fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )</td>
 409             <td class="snnn top grass">fqus_weight fqus_drops fqus_fqup fqus_fqus</td>
 410             <td class="snnn top grass">
 411               <br />
 412             </td>
 413             <td class="ssnn top grass">
 414               <br />
 415             </td>
 416           </tr>
 417           <tr>
 418             <td class="nnns top capitalize italic grass">
 419               <br />
 420             </td>
 421             <td class="nnns top italic grass">
 422               <br />
 423             </td>
 424             <td class="snns top grass">fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )</td>
 425             <td class="snnn top grass">fqup_weight fqup_drops fqup_fqup</td>
 426             <td class="snnn top grass">
 427               <br />
 428             </td>
 429             <td class="ssnn top grass">
 430               <br />
 431             </td>
 432           </tr>
 433           <tr>
 434             <td class="snns top capitalize italic yellow">s-transition</td>
 435             <td class="snns top italic yellow">structural successor for closures</td>
 436             <td class="snns top yellow">fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )</td>
 437             <td class="snnn top yellow">fquq_length fquq_weight</td>
 438             <td class="snnn top yellow">
 439               <br />
 440             </td>
 441             <td class="ssnn top yellow">
 442               <br />
 443             </td>
 444           </tr>
 445           <tr>
 446             <td class="nnns top capitalize italic yellow">
 447               <br />
 448             </td>
 449             <td class="nnns top italic yellow">
 450               <br />
 451             </td>
 452             <td class="snns top yellow">fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )</td>
 453             <td class="snnn top yellow">fqu_length fqu_weight</td>
 454             <td class="snnn top yellow">
 455               <br />
 456             </td>
 457             <td class="ssnn top yellow">
 458               <br />
 459             </td>
 460           </tr>
 461           <tr>
 462             <td class="snns top capitalize italic orange">relocation</td>
 463             <td class="snns top italic orange">generic slicing for local environments</td>
 464             <td class="snns top orange">drops_vector ( ⬇*[?,?] ? ≡ ? )</td>
 465             <td class="snnn top orange">
 466               <br />
 467             </td>
 468             <td class="snnn top orange">
 469               <br />
 470             </td>
 471             <td class="ssnn top orange">
 472               <br />
 473             </td>
 474           </tr>
 475           <tr>
 476             <td class="nnns top capitalize italic orange">
 477               <br />
 478             </td>
 479             <td class="nnns top italic orange">
 480               <br />
 481             </td>
 482             <td class="snns top orange">drops ( ⬇*[?,?] ? ≡ ? )</td>
 483             <td class="snnn top orange">drops_lstar drops_weight drops_length drops_ceq drops_lexs drops_lreq drops_drops</td>
 484             <td class="snnn top orange">
 485               <br />
 486             </td>
 487             <td class="ssnn top orange">
 488               <br />
 489             </td>
 490           </tr>
 491           <tr>
 492             <td class="nnns top capitalize italic orange">
 493               <br />
 494             </td>
 495             <td class="snns top italic orange">generic relocation for terms</td>
 496             <td class="snns top orange">lifts_vector ( ⬆*[?] ? ≡ ? )</td>
 497             <td class="snnn top orange">lifts_lifts_vector</td>
 498             <td class="snnn top orange">
 499               <br />
 500             </td>
 501             <td class="ssnn top orange">
 502               <br />
 503             </td>
 504           </tr>
 505           <tr>
 506             <td class="nnns top capitalize italic orange">
 507               <br />
 508             </td>
 509             <td class="nnns top italic orange">
 510               <br />
 511             </td>
 512             <td class="snns top orange">lifts ( ⬆*[?] ? ≡ ? )</td>
 513             <td class="snnn top orange">lifts_simple lifts_weight lifts_lifts</td>
 514             <td class="snnn top orange">
 515               <br />
 516             </td>
 517             <td class="ssnn top orange">
 518               <br />
 519             </td>
 520           </tr>
 521           <tr>
 522             <td class="nnns top capitalize italic orange">
 523               <br />
 524             </td>
 525             <td class="snns top italic orange">ranged equivalence for local environments</td>
 526             <td class="snns top orange">lreq ( ? ≡[?] ? )</td>
 527             <td class="snnn top orange">lreq_length lreq_lreq</td>
 528             <td class="snnn top orange">
 529               <br />
 530             </td>
 531             <td class="ssnn top orange">
 532               <br />
 533             </td>
 534           </tr>
 535           <tr>
 536             <td class="nnns top capitalize italic orange">
 537               <br />
 538             </td>
 539             <td class="snns top italic orange">generic entrywise extension of context-sensitive relations for terma</td>
 540             <td class="snns top orange">lexs ( ? ⦻*[?,?,?] ? )</td>
 541             <td class="snnn top orange">lexs_length lexs_lexs</td>
 542             <td class="snnn top orange">
 543               <br />
 544             </td>
 545             <td class="ssnn top orange">
 546               <br />
 547             </td>
 548           </tr>
 549           <tr>
 550             <td class="snns top capitalize italic red">grammar</td>
 551             <td class="snns top italic red">append for local environments</td>
 552             <td class="snns top red">append ( ? @@ ? )</td>
 553             <td class="snnn top red">append_length</td>
 554             <td class="snnn top red">
 555               <br />
 556             </td>
 557             <td class="ssnn top red">
 558               <br />
 559             </td>
 560           </tr>
 561           <tr>
 562             <td class="nnns top capitalize italic red">
 563               <br />
 564             </td>
 565             <td class="snns top italic red">context-sensitive equivalences for terms</td>
 566             <td class="snns top red">ceq</td>
 567             <td class="snnn top red">ceq_ceq</td>
 568             <td class="snnn top red">
 569               <br />
 570             </td>
 571             <td class="ssnn top red">
 572               <br />
 573             </td>
 574           </tr>
 575           <tr>
 576             <td class="nnns top capitalize italic red">
 577               <br />
 578             </td>
 579             <td class="snns top italic red">same top term structure</td>
 580             <td class="snns top red">tsts ( ? ≂ ? )</td>
 581             <td class="snnn top red">tsts_tsts tsts_vector</td>
 582             <td class="snnn top red">
 583               <br />
 584             </td>
 585             <td class="ssnn top red">
 586               <br />
 587             </td>
 588           </tr>
 589           <tr>
 590             <td class="nnns top capitalize italic red">
 591               <br />
 592             </td>
 593             <td class="snns top italic red">closures</td>
 594             <td class="snns top red">cl_weight ( ♯{?,?,?} )</td>
 595             <td class="snnn top red">cl_restricted_weight ( ♯{?,?} )</td>
 596             <td class="snnn top red">
 597               <br />
 598             </td>
 599             <td class="ssnn top red">
 600               <br />
 601             </td>
 602           </tr>
 603           <tr>
 604             <td class="nnns top capitalize italic red">
 605               <br />
 606             </td>
 607             <td class="snns top italic red">internal syntax</td>
 608             <td class="snns top red">genv</td>
 609             <td class="snnn top red">
 610               <br />
 611             </td>
 612             <td class="snnn top red">
 613               <br />
 614             </td>
 615             <td class="ssnn top red">
 616               <br />
 617             </td>
 618           </tr>
 619           <tr>
 620             <td class="nnns top capitalize italic red">
 621               <br />
 622             </td>
 623             <td class="nnns top italic red">
 624               <br />
 625             </td>
 626             <td class="snns top red">lenv</td>
 627             <td class="snnn top red">lenv_weight ( ♯{?} )</td>
 628             <td class="snnn top red">lenv_length ( |?| )</td>
 629             <td class="ssnn top red">
 630               <br />
 631             </td>
 632           </tr>
 633           <tr>
 634             <td class="nnns top capitalize italic red">
 635               <br />
 636             </td>
 637             <td class="nnns top italic red">
 638               <br />
 639             </td>
 640             <td class="snns top red">term</td>
 641             <td class="snnn top red">term_weight ( ♯{?} )</td>
 642             <td class="snnn top red">term_simple ( 𝐒⦃?⦄ )</td>
 643             <td class="ssnn top red">term_vector ( Ⓐ?.? )</td>
 644           </tr>
 645           <tr>
 646             <td class="nnns top capitalize italic red">
 647               <br />
 648             </td>
 649             <td class="nnns top italic red">
 650               <br />
 651             </td>
 652             <td class="snns top red">item</td>
 653             <td class="snnn top red">
 654               <br />
 655             </td>
 656             <td class="snnn top red">
 657               <br />
 658             </td>
 659             <td class="ssnn top red">
 660               <br />
 661             </td>
 662           </tr>
 663           <tr>
 664             <td class="nnss top capitalize italic red">
 665               <br />
 666             </td>
 667             <td class="snss top italic red">external syntax</td>
 668             <td class="snss top red">aarity</td>
 669             <td class="snsn top red">
 670               <br />
 671             </td>
 672             <td class="snsn top red">
 673               <br />
 674             </td>
 675             <td class="sssn top red">
 676               <br />
 677             </td>
 678           </tr>
 679         </tbody>
 680       </table>
 681     </div>
 682     <div class="spacer">
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