matita/help/C/sec_tactics.xml

   1
   2 <!-- ============ Tactics ====================== -->
   3 <chapter id="sec_tactics">
   4   <title>Tactics</title>
   5
   6   <sect1 id="tactics_quickref">
   7     <title>Quick reference card</title>
   8     <para>
   9       &tacticref;
  10     </para>
  11   </sect1>
  12
  13   <sect1 id="tac_absurd">
  14     <title>absurd</title>
  15     <titleabbrev>absurd</titleabbrev>
  16     <para><userinput>absurd P</userinput></para>
  17      <para>
  18       <variablelist>
  19         <varlistentry role="tactic.synopsis">
  20           <term>Synopsis:</term>
  21           <listitem>
  22             <para><emphasis role="bold">absurd</emphasis> &sterm;</para>
  23           </listitem>
  24         </varlistentry>
  25         <varlistentry>
  26           <term>Pre-conditions:</term>
  27           <listitem>
  28             <para><command>P</command> must have type <command>Prop</command>.</para>
  29           </listitem>
  30         </varlistentry>
  31         <varlistentry>
  32           <term>Action:</term>
  33           <listitem>
  34             <para>It closes the current sequent by eliminating an
  35              absurd term.</para>
  36           </listitem>
  37         </varlistentry>
  38         <varlistentry>
  39           <term>New sequents to prove:</term>
  40           <listitem>
  41             <para>It opens two new sequents of conclusion <command>P</command>
  42              and <command>¬P</command>.</para>
  43           </listitem>
  44         </varlistentry>
  45       </variablelist>
  46      </para>
  47   </sect1>
  48   <sect1 id="tac_apply">
  49     <title>apply</title>
  50     <titleabbrev>apply</titleabbrev>
  51     <para><userinput>apply t</userinput></para>
  52     <para>
  53       <variablelist>
  54         <varlistentry role="tactic.synopsis">
  55           <term>Synopsis:</term>
  56           <listitem>
  57             <para><emphasis role="bold">apply</emphasis> &sterm;</para>
  58           </listitem>
  59         </varlistentry>
  60         <varlistentry>
  61           <term>Pre-conditions:</term>
  62           <listitem>
  63             <para><command>t</command> must have type
  64              <command>T<subscript>1</subscript> → ... →
  65               T<subscript>n</subscript> → G</command>
  66              where <command>G</command> can be unified with the conclusion
  67              of the current sequent.</para>
  68           </listitem>
  69         </varlistentry>
  70         <varlistentry>
  71           <term>Action:</term>
  72           <listitem>
  73             <para>It closes the current sequent by applying <command>t</command> to <command>n</command> implicit arguments (that become new sequents).</para>
  74           </listitem>
  75         </varlistentry>
  76         <varlistentry>
  77           <term>New sequents to prove:</term>
  78           <listitem>
  79             <para>It opens a new sequent for each premise
  80              <command>T<subscript>i</subscript></command> that is not
  81              instantiated by unification. <command>T<subscript>i</subscript></command> is
  82              the conclusion of the <command>i</command>-th new sequent to
  83              prove.</para>
  84           </listitem>
  85         </varlistentry>
  86       </variablelist>
  87     </para>
  88   </sect1>
  89   <sect1 id="tac_applyS">
  90     <title>applyS</title>
  91     <titleabbrev>applyS</titleabbrev>
  92     <para><userinput>applyS t auto_params</userinput></para>
  93     <para>
  94       <variablelist>
  95         <varlistentry role="tactic.synopsis">
  96           <term>Synopsis:</term>
  97           <listitem>
  98             <para><emphasis role="bold">applyS</emphasis> &sterm; &autoparams;</para>
  99           </listitem>
 100         </varlistentry>
 101         <varlistentry>
 102           <term>Pre-conditions:</term>
 103           <listitem>
 104             <para><command>t</command> must have type
 105              <command>T<subscript>1</subscript> → ... →
 106               T<subscript>n</subscript> → G</command>.</para>
 107           </listitem>
 108         </varlistentry>
 109         <varlistentry>
 110           <term>Action:</term>
 111           <listitem>
 112             <para><command>applyS</command> is useful when
 113              <command>apply</command> fails because the current goal
 114              and the conclusion of the applied theorems are extensionally
 115              equivalent up to instantiation of metavariables, but cannot
 116              be unified. E.g. the goal is <command>P(n*O+m)</command> and
 117              the theorem to be applied proves <command>∀m.P(m+O)</command>.
 118             </para>
 119             <para>
 120              It tries to automatically rewrite the current goal using
 121              <link linkend="tac_auto">auto paramodulation</link>
 122              to make it unifiable with <command>G</command>.
 123              Then it closes the current sequent by applying
 124              <command>t</command> to <command>n</command>
 125              implicit arguments (that become new sequents).
 126              The <command>auto_params</command> parameters are passed
 127              directly to <command>auto paramodulation</command>.
 128             </para>
 129           </listitem>
 130         </varlistentry>
 131         <varlistentry>
 132           <term>New sequents to prove:</term>
 133           <listitem>
 134             <para>It opens a new sequent for each premise
 135              <command>T<subscript>i</subscript></command> that is not
 136              instantiated by unification. <command>T<subscript>i</subscript></command> is
 137              the conclusion of the <command>i</command>-th new sequent to
 138              prove.</para>
 139           </listitem>
 140         </varlistentry>
 141       </variablelist>
 142     </para>
 143   </sect1>
 144   <sect1 id="tac_assumption">
 145     <title>assumption</title>
 146     <titleabbrev>assumption</titleabbrev>
 147     <para><userinput>assumption </userinput></para>
 148     <para>
 149       <variablelist>
 150         <varlistentry role="tactic.synopsis">
 151           <term>Synopsis:</term>
 152           <listitem>
 153             <para><emphasis role="bold">assumption</emphasis></para>
 154           </listitem>
 155         </varlistentry>
 156         <varlistentry>
 157           <term>Pre-conditions:</term>
 158           <listitem>
 159             <para>There must exist an hypothesis whose type can be unified with
 160              the conclusion of the current sequent.</para>
 161           </listitem>
 162         </varlistentry>
 163         <varlistentry>
 164           <term>Action:</term>
 165           <listitem>
 166             <para>It closes the current sequent exploiting an hypothesis.</para>
 167           </listitem>
 168         </varlistentry>
 169         <varlistentry>
 170           <term>New sequents to prove:</term>
 171           <listitem>
 172             <para>None</para>
 173           </listitem>
 174         </varlistentry>
 175       </variablelist>
 176     </para>
 177   </sect1>
 178   <sect1 id="tac_auto">
 179     <title>auto</title>
 180     <titleabbrev>auto</titleabbrev>
 181     <para><userinput>auto params</userinput></para>
 182     <para>
 183       <variablelist>
 184         <varlistentry role="tactic.synopsis">
 185           <term>Synopsis:</term>
 186           <listitem>
 187             <para><emphasis role="bold">auto</emphasis> &autoparams;</para>
 188           </listitem>
 189         </varlistentry>
 190         <varlistentry>
 191           <term>Pre-conditions:</term>
 192           <listitem>
 193             <para>None, but the tactic may fail finding a proof if every
 194              proof is in the search space that is pruned away. Pruning is
 195              controlled by the optional <command>params</command>.
 196              Moreover, only lemmas whose type signature is a subset of the
 197              signature of the current sequent are considered. The signature of
 198              a sequent is ...&TODO;</para>
 199           </listitem>
 200         </varlistentry>
 201         <varlistentry>
 202           <term>Action:</term>
 203           <listitem>
 204             <para>It closes the current sequent by repeated application of
 205              rewriting steps (unless <command>paramodulation</command> is
 206              omitted), hypothesis and lemmas in the library.</para>
 207           </listitem>
 208         </varlistentry>
 209         <varlistentry>
 210           <term>New sequents to prove:</term>
 211           <listitem>
 212             <para>None</para>
 213           </listitem>
 214         </varlistentry>
 215       </variablelist>
 216     </para>
 217   </sect1>
 218   <sect1 id="tac_clear">
 219     <title>clear</title>
 220     <titleabbrev>clear</titleabbrev>
 221     <para><userinput>
 222      clear H<subscript>1</subscript> ... H<subscript>m</subscript>
 223     </userinput></para>
 224     <para>
 225       <variablelist>
 226         <varlistentry role="tactic.synopsis">
 227           <term>Synopsis:</term>
 228           <listitem>
 229             <para>
 230              <emphasis role="bold">clear</emphasis>
 231              &id; [&id;…]
 232             </para>
 233           </listitem>
 234         </varlistentry>
 235         <varlistentry>
 236           <term>Pre-conditions:</term>
 237           <listitem>
 238             <para>
 239              <command>
 240               H<subscript>1</subscript> ... H<subscript>m</subscript>
 241              </command> must be hypotheses of the
 242              current sequent to prove.
 243             </para>
 244           </listitem>
 245         </varlistentry>
 246         <varlistentry>
 247           <term>Action:</term>
 248           <listitem>
 249             <para>
 250              It hides the hypotheses
 251              <command>
 252               H<subscript>1</subscript> ... H<subscript>m</subscript>
 253              </command> from the current sequent.
 254             </para>
 255           </listitem>
 256         </varlistentry>
 257         <varlistentry>
 258           <term>New sequents to prove:</term>
 259           <listitem>
 260             <para>None</para>
 261           </listitem>
 262         </varlistentry>
 263       </variablelist>
 264     </para>
 265   </sect1>
 266   <sect1 id="tac_clearbody">
 267     <title>clearbody</title>
 268     <titleabbrev>clearbody</titleabbrev>
 269     <para><userinput>clearbody H</userinput></para>
 270     <para>
 271       <variablelist>
 272         <varlistentry role="tactic.synopsis">
 273           <term>Synopsis:</term>
 274           <listitem>
 275             <para><emphasis role="bold">clearbody</emphasis> &id;</para>
 276           </listitem>
 277         </varlistentry>
 278         <varlistentry>
 279           <term>Pre-conditions:</term>
 280           <listitem>
 281             <para><command>H</command> must be an hypothesis of the
 282              current sequent to prove.</para>
 283           </listitem>
 284         </varlistentry>
 285         <varlistentry>
 286           <term>Action:</term>
 287           <listitem>
 288             <para>It hides the definiens of a definition in the current
 289              sequent context. Thus the definition becomes an hypothesis.</para>
 290           </listitem>
 291         </varlistentry>
 292         <varlistentry>
 293           <term>New sequents to prove:</term>
 294           <listitem>
 295             <para>None.</para>
 296           </listitem>
 297         </varlistentry>
 298       </variablelist>
 299     </para>
 300   </sect1>
 301   <sect1 id="tac_change">
 302     <title>change</title>
 303     <titleabbrev>change</titleabbrev>
 304     <para><userinput>change patt with t</userinput></para>
 305     <para>
 306       <variablelist>
 307         <varlistentry role="tactic.synopsis">
 308           <term>Synopsis:</term>
 309           <listitem>
 310             <para><emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
 311           </listitem>
 312         </varlistentry>
 313         <varlistentry>
 314           <term>Pre-conditions:</term>
 315           <listitem>
 316             <para>Each subterm matched by the pattern must be convertible
 317              with the term <command>t</command> disambiguated in the context
 318              of the matched subterm.</para>
 319           </listitem>
 320         </varlistentry>
 321         <varlistentry>
 322           <term>Action:</term>
 323           <listitem>
 324             <para>It replaces the subterms of the current sequent matched by
 325              <command>patt</command> with the new term <command>t</command>.
 326              For each subterm matched by the pattern, <command>t</command> is
 327              disambiguated in the context of the subterm.</para>
 328           </listitem>
 329         </varlistentry>
 330         <varlistentry>
 331           <term>New sequents to prove:</term>
 332           <listitem>
 333             <para>None.</para>
 334           </listitem>
 335         </varlistentry>
 336       </variablelist>
 337     </para>
 338   </sect1>
 339   <sect1 id="tac_constructor">
 340     <title>constructor</title>
 341     <titleabbrev>constructor</titleabbrev>
 342     <para><userinput>constructor n</userinput></para>
 343     <para>
 344       <variablelist>
 345         <varlistentry role="tactic.synopsis">
 346           <term>Synopsis:</term>
 347           <listitem>
 348             <para><emphasis role="bold">constructor</emphasis> &nat;</para>
 349           </listitem>
 350         </varlistentry>
 351         <varlistentry>
 352           <term>Pre-conditions:</term>
 353           <listitem>
 354             <para>The conclusion of the current sequent must be
 355              an inductive type or the application of an inductive type with
 356              at least <command>n</command> constructors.</para>
 357           </listitem>
 358         </varlistentry>
 359         <varlistentry>
 360           <term>Action:</term>
 361           <listitem>
 362             <para>It applies the <command>n</command>-th constructor of the
 363              inductive type of the conclusion of the current sequent.</para>
 364           </listitem>
 365         </varlistentry>
 366         <varlistentry>
 367           <term>New sequents to prove:</term>
 368           <listitem>
 369             <para>It opens a new sequent for each premise of the constructor
 370              that can not be inferred by unification. For more details,
 371              see the <command>apply</command> tactic.</para>
 372           </listitem>
 373         </varlistentry>
 374       </variablelist>
 375     </para>
 376   </sect1>
 377   <sect1 id="tac_contradiction">
 378     <title>contradiction</title>
 379     <titleabbrev>contradiction</titleabbrev>
 380     <para><userinput>contradiction </userinput></para>
 381     <para>
 382       <variablelist>
 383         <varlistentry role="tactic.synopsis">
 384           <term>Synopsis:</term>
 385           <listitem>
 386             <para><emphasis role="bold">contradiction</emphasis></para>
 387           </listitem>
 388         </varlistentry>
 389         <varlistentry>
 390           <term>Pre-conditions:</term>
 391           <listitem>
 392             <para>There must be in the current context an hypothesis of type
 393              <command>False</command>.</para>
 394           </listitem>
 395         </varlistentry>
 396         <varlistentry>
 397           <term>Action:</term>
 398           <listitem>
 399             <para>It closes the current sequent by applying an hypothesis of
 400              type <command>False</command>.</para>
 401           </listitem>
 402         </varlistentry>
 403         <varlistentry>
 404           <term>New sequents to prove:</term>
 405           <listitem>
 406             <para>None</para>
 407           </listitem>
 408         </varlistentry>
 409       </variablelist>
 410     </para>
 411   </sect1>
 412   <sect1 id="tac_cut">
 413     <title>cut</title>
 414     <titleabbrev>cut</titleabbrev>
 415     <para><userinput>cut P as H</userinput></para>
 416     <para>
 417       <variablelist>
 418         <varlistentry role="tactic.synopsis">
 419           <term>Synopsis:</term>
 420           <listitem>
 421             <para><emphasis role="bold">cut</emphasis> &sterm; [<emphasis role="bold">as</emphasis> &id;]</para>
 422           </listitem>
 423         </varlistentry>
 424         <varlistentry>
 425           <term>Pre-conditions:</term>
 426           <listitem>
 427             <para><command>P</command> must have type <command>Prop</command>.</para>
 428           </listitem>
 429         </varlistentry>
 430         <varlistentry>
 431           <term>Action:</term>
 432           <listitem>
 433             <para>It closes the current sequent.</para>
 434           </listitem>
 435         </varlistentry>
 436         <varlistentry>
 437           <term>New sequents to prove:</term>
 438           <listitem>
 439             <para>It opens two new sequents. The first one has an extra
 440              hypothesis <command>H:P</command>. If <command>H</command> is
 441              omitted, the name of the hypothesis is automatically generated.
 442              The second sequent has conclusion <command>P</command> and
 443              hypotheses the hypotheses of the current sequent to prove.</para>
 444           </listitem>
 445         </varlistentry>
 446       </variablelist>
 447     </para>
 448   </sect1>
 449   <sect1 id="tac_decompose">
 450     <title>decompose</title>
 451     <titleabbrev>decompose</titleabbrev>
 452     <para><userinput>
 453      decompose (T<subscript>1</subscript> ... T<subscript>n</subscript>)
 454      H as H<subscript>1</subscript> ... H<subscript>m</subscript>
 455     </userinput></para>
 456     <para>
 457       <variablelist>
 458         <varlistentry role="tactic.synopsis">
 459           <term>Synopsis:</term>
 460           <listitem>
 461             <para>
 462              <emphasis role="bold">decompose</emphasis>
 463              [<emphasis role="bold">(</emphasis>
 464              &id;…
 465              <emphasis role="bold">)</emphasis>]
 466              [&id;]
 467              [<emphasis role="bold">as</emphasis> &id;…]
 468             </para>
 469           </listitem>
 470         </varlistentry>
 471         <varlistentry>
 472           <term>Pre-conditions:</term>
 473           <listitem>
 474             <para>
 475              <command>H</command> must inhabit one inductive type among
 476              <command>
 477               T<subscript>1</subscript> ... T<subscript>n</subscript>
 478              </command>
 479              and the types of a predefined list.
 480             </para>
 481           </listitem>
 482         </varlistentry>
 483         <varlistentry>
 484           <term>Action:</term>
 485           <listitem>
 486             <para>
 487              Runs <command>
 488               elim H H<subscript>1</subscript> ... H<subscript>m</subscript>
 489              </command>, clears <command>H</command> and tries to run itself
 490              recursively on each new identifier introduced by
 491              <command>elim</command> in the opened sequents.
 492              If <command>H</command> is not provided tries this operation on
 493              each premise in the current context.
 494             </para>
 495           </listitem>
 496         </varlistentry>
 497         <varlistentry>
 498           <term>New sequents to prove:</term>
 499           <listitem>
 500             <para>
 501              The ones generated by all the <command>elim</command> tactics run.
 502             </para>
 503           </listitem>
 504         </varlistentry>
 505       </variablelist>
 506     </para>
 507   </sect1>
 508   <sect1 id="tac_demodulate">
 509     <title>demodulate</title>
 510     <titleabbrev>demodulate</titleabbrev>
 511     <para><userinput>demodulate</userinput></para>
 512     <para>
 513       <variablelist>
 514         <varlistentry role="tactic.synopsis">
 515           <term>Synopsis:</term>
 516           <listitem>
 517             <para><emphasis role="bold">demodulate</emphasis></para>
 518           </listitem>
 519         </varlistentry>
 520         <varlistentry>
 521           <term>Pre-conditions:</term>
 522           <listitem>
 523             <para>None.</para>
 524           </listitem>
 525         </varlistentry>
 526         <varlistentry>
 527           <term>Action:</term>
 528           <listitem>
 529             <para>&TODO;</para>
 530           </listitem>
 531         </varlistentry>
 532         <varlistentry>
 533           <term>New sequents to prove:</term>
 534           <listitem>
 535             <para>None.</para>
 536           </listitem>
 537         </varlistentry>
 538       </variablelist>
 539     </para>
 540   </sect1>
 541   <sect1 id="tac_destruct">
 542     <title>destruct</title>
 543     <titleabbrev>destruct</titleabbrev>
 544     <para><userinput>destruct p</userinput></para>
 545     <para>
 546       <variablelist>
 547         <varlistentry role="tactic.synopsis">
 548           <term>Synopsis:</term>
 549           <listitem>
 550             <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
 551           </listitem>
 552         </varlistentry>
 553         <varlistentry>
 554           <term>Pre-conditions:</term>
 555           <listitem>
 556             <para><command>p</command> must have type <command>E<subscript>1</subscript> = E<subscript>2</subscript></command> where the two sides of the equality are possibly applied constructors of an inductive type.</para>
 557           </listitem>
 558         </varlistentry>
 559         <varlistentry>
 560           <term>Action:</term>
 561           <listitem>
 562             <para>The tactic recursively compare the two sides of the equality
 563              looking for different constructors in corresponding position.
 564              If two of them are found, the tactic closes the current sequent
 565              by proving the absurdity of <command>p</command>. Otherwise
 566              it adds a new hypothesis for each leaf of the formula that
 567              states the equality of the subformulae in the corresponding
 568              positions on the two sides of the equality.
 569             </para>
 570           </listitem>
 571         </varlistentry>
 572         <varlistentry>
 573           <term>New sequents to prove:</term>
 574           <listitem>
 575             <para>None.</para>
 576           </listitem>
 577         </varlistentry>
 578       </variablelist>
 579     </para>
 580   </sect1>
 581   <sect1 id="tac_elim">
 582     <title>elim</title>
 583     <titleabbrev>elim</titleabbrev>
 584     <para><userinput>elim t using th hyps</userinput></para>
 585     <para>
 586       <variablelist>
 587         <varlistentry role="tactic.synopsis">
 588           <term>Synopsis:</term>
 589           <listitem>
 590             <para><emphasis role="bold">elim</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 591           </listitem>
 592         </varlistentry>
 593         <varlistentry>
 594           <term>Pre-conditions:</term>
 595           <listitem>
 596             <para><command>t</command> must inhabit an inductive type and
 597              <command>th</command> must be an elimination principle for that
 598              inductive type. If <command>th</command> is omitted the appropriate
 599              standard elimination principle is chosen.</para>
 600           </listitem>
 601         </varlistentry>
 602         <varlistentry>
 603           <term>Action:</term>
 604           <listitem>
 605             <para>It proceeds by cases on the values of <command>t</command>,
 606              according to the elimination principle <command>th</command>.
 607             </para>
 608           </listitem>
 609         </varlistentry>
 610         <varlistentry>
 611           <term>New sequents to prove:</term>
 612           <listitem>
 613             <para>It opens one new sequent for each case. The names of
 614              the new hypotheses are picked by <command>hyps</command>, if
 615              provided. If hyps specifies also a number of hypotheses that
 616              is less than the number of new hypotheses for a new sequent,
 617              then the exceeding hypothesis will be kept as implications in
 618              the conclusion of the sequent.</para>
 619           </listitem>
 620         </varlistentry>
 621       </variablelist>
 622     </para>
 623   </sect1>
 624   <sect1 id="tac_elimType">
 625     <title>elimType</title>
 626     <titleabbrev>elimType</titleabbrev>
 627     <para><userinput>elimType T using th hyps</userinput></para>
 628     <para>
 629       <variablelist>
 630         <varlistentry role="tactic.synopsis">
 631           <term>Synopsis:</term>
 632           <listitem>
 633             <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 634           </listitem>
 635         </varlistentry>
 636         <varlistentry>
 637           <term>Pre-conditions:</term>
 638           <listitem>
 639             <para><command>T</command> must be an inductive type.</para>
 640           </listitem>
 641         </varlistentry>
 642         <varlistentry>
 643           <term>Action:</term>
 644           <listitem>
 645             <para>TODO (severely bugged now).</para>
 646           </listitem>
 647         </varlistentry>
 648         <varlistentry>
 649           <term>New sequents to prove:</term>
 650           <listitem>
 651             <para>TODO</para>
 652           </listitem>
 653         </varlistentry>
 654       </variablelist>
 655     </para>
 656   </sect1>
 657   <sect1 id="tac_exact">
 658     <title>exact</title>
 659     <titleabbrev>exact</titleabbrev>
 660     <para><userinput>exact p</userinput></para>
 661     <para>
 662       <variablelist>
 663         <varlistentry role="tactic.synopsis">
 664           <term>Synopsis:</term>
 665           <listitem>
 666             <para><emphasis role="bold">exact</emphasis> &sterm;</para>
 667           </listitem>
 668         </varlistentry>
 669         <varlistentry>
 670           <term>Pre-conditions:</term>
 671           <listitem>
 672             <para>The type of <command>p</command> must be convertible
 673              with the conclusion of the current sequent.</para>
 674           </listitem>
 675         </varlistentry>
 676         <varlistentry>
 677           <term>Action:</term>
 678           <listitem>
 679             <para>It closes the current sequent using <command>p</command>.</para>
 680           </listitem>
 681         </varlistentry>
 682         <varlistentry>
 683           <term>New sequents to prove:</term>
 684           <listitem>
 685             <para>None.</para>
 686           </listitem>
 687         </varlistentry>
 688       </variablelist>
 689     </para>
 690   </sect1>
 691   <sect1 id="tac_exists">
 692     <title>exists</title>
 693     <titleabbrev>exists</titleabbrev>
 694     <para><userinput>exists </userinput></para>
 695     <para>
 696       <variablelist>
 697         <varlistentry role="tactic.synopsis">
 698           <term>Synopsis:</term>
 699           <listitem>
 700             <para><emphasis role="bold">exists</emphasis></para>
 701           </listitem>
 702         </varlistentry>
 703         <varlistentry>
 704           <term>Pre-conditions:</term>
 705           <listitem>
 706             <para>The conclusion of the current sequent must be
 707              an inductive type or the application of an inductive type
 708              with at least one constructor.</para>
 709           </listitem>
 710         </varlistentry>
 711         <varlistentry>
 712           <term>Action:</term>
 713           <listitem>
 714             <para>Equivalent to <command>constructor 1</command>.</para>
 715           </listitem>
 716         </varlistentry>
 717         <varlistentry>
 718           <term>New sequents to prove:</term>
 719           <listitem>
 720             <para>It opens a new sequent for each premise of the first
 721              constructor of the inductive type that is the conclusion of the
 722              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 723           </listitem>
 724         </varlistentry>
 725       </variablelist>
 726     </para>
 727   </sect1>
 728   <sect1 id="tac_fail">
 729     <title>fail</title>
 730     <titleabbrev>fail</titleabbrev>
 731     <para><userinput>fail</userinput></para>
 732     <para>
 733       <variablelist>
 734         <varlistentry role="tactic.synopsis">
 735           <term>Synopsis:</term>
 736           <listitem>
 737             <para><emphasis role="bold">fail</emphasis></para>
 738           </listitem>
 739         </varlistentry>
 740         <varlistentry>
 741           <term>Pre-conditions:</term>
 742           <listitem>
 743             <para>None.</para>
 744           </listitem>
 745         </varlistentry>
 746         <varlistentry>
 747           <term>Action:</term>
 748           <listitem>
 749             <para>This tactic always fail.</para>
 750           </listitem>
 751         </varlistentry>
 752         <varlistentry>
 753           <term>New sequents to prove:</term>
 754           <listitem>
 755             <para>N.A.</para>
 756           </listitem>
 757         </varlistentry>
 758       </variablelist>
 759     </para>
 760   </sect1>
 761   <sect1 id="tac_fold">
 762     <title>fold</title>
 763     <titleabbrev>fold</titleabbrev>
 764     <para><userinput>fold red t patt</userinput></para>
 765     <para>
 766       <variablelist>
 767         <varlistentry role="tactic.synopsis">
 768           <term>Synopsis:</term>
 769           <listitem>
 770             <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
 771           </listitem>
 772         </varlistentry>
 773         <varlistentry>
 774           <term>Pre-conditions:</term>
 775           <listitem>
 776             <para>The pattern must not specify the wanted term.</para>
 777           </listitem>
 778         </varlistentry>
 779         <varlistentry>
 780           <term>Action:</term>
 781           <listitem>
 782             <para>First of all it locates all the subterms matched by
 783              <command>patt</command>. In the context of each matched subterm
 784              it disambiguates the term <command>t</command> and reduces it
 785              to its <command>red</command> normal form; then it replaces with
 786              <command>t</command> every occurrence of the normal form in the
 787              matched subterm.</para>
 788           </listitem>
 789         </varlistentry>
 790         <varlistentry>
 791           <term>New sequents to prove:</term>
 792           <listitem>
 793             <para>None.</para>
 794           </listitem>
 795         </varlistentry>
 796       </variablelist>
 797     </para>
 798   </sect1>
 799   <sect1 id="tac_fourier">
 800     <title>fourier</title>
 801     <titleabbrev>fourier</titleabbrev>
 802     <para><userinput>fourier </userinput></para>
 803     <para>
 804       <variablelist>
 805         <varlistentry role="tactic.synopsis">
 806           <term>Synopsis:</term>
 807           <listitem>
 808             <para><emphasis role="bold">fourier</emphasis></para>
 809           </listitem>
 810         </varlistentry>
 811         <varlistentry>
 812           <term>Pre-conditions:</term>
 813           <listitem>
 814             <para>The conclusion of the current sequent must be a linear
 815              inequation over real numbers taken from standard library of
 816              Coq. Moreover the inequations in the hypotheses must imply the
 817              inequation in the conclusion of the current sequent.</para>
 818           </listitem>
 819         </varlistentry>
 820         <varlistentry>
 821           <term>Action:</term>
 822           <listitem>
 823             <para>It closes the current sequent by applying the Fourier method.</para>
 824           </listitem>
 825         </varlistentry>
 826         <varlistentry>
 827           <term>New sequents to prove:</term>
 828           <listitem>
 829             <para>None.</para>
 830           </listitem>
 831         </varlistentry>
 832       </variablelist>
 833     </para>
 834   </sect1>
 835   <sect1 id="tac_fwd">
 836     <title>fwd</title>
 837     <titleabbrev>fwd</titleabbrev>
 838     <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
 839     <para>
 840       <variablelist>
 841         <varlistentry role="tactic.synopsis">
 842           <term>Synopsis:</term>
 843           <listitem>
 844             <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
 845           </listitem>
 846         </varlistentry>
 847         <varlistentry>
 848           <term>Pre-conditions:</term>
 849           <listitem>
 850             <para>
 851                The type of <command>H</command> must be the premise of a
 852                forward simplification theorem.
 853             </para>
 854           </listitem>
 855         </varlistentry>
 856         <varlistentry>
 857           <term>Action:</term>
 858           <listitem>
 859             <para>
 860              This tactic is under development.
 861              It simplifies the current context by removing
 862              <command>H</command> using the following methods:
 863              forward application (by <command>lapply</command>) of a suitable
 864              simplification theorem, chosen automatically, of which the type
 865              of <command>H</command> is a premise,
 866              decomposition (by <command>decompose</command>),
 867              rewriting (by <command>rewrite</command>).
 868              <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
 869              are passed to the tactics <command>fwd</command> invokes, as
 870              names for the premise they introduce.
 871             </para>
 872           </listitem>
 873         </varlistentry>
 874         <varlistentry>
 875           <term>New sequents to prove:</term>
 876           <listitem>
 877             <para>
 878              The ones opened by the tactics <command>fwd</command> invokes.
 879             </para>
 880           </listitem>
 881         </varlistentry>
 882       </variablelist>
 883     </para>
 884   </sect1>
 885   <sect1 id="tac_generalize">
 886     <title>generalize</title>
 887     <titleabbrev>generalize</titleabbrev>
 888     <para><userinput>generalize patt as H</userinput></para>
 889     <para>
 890       <variablelist>
 891         <varlistentry role="tactic.synopsis">
 892           <term>Synopsis:</term>
 893           <listitem>
 894             <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
 895           </listitem>
 896         </varlistentry>
 897         <varlistentry>
 898           <term>Pre-conditions:</term>
 899           <listitem>
 900             <para>All the terms matched by <command>patt</command> must be
 901              convertible and close in the context of the current sequent.</para>
 902           </listitem>
 903         </varlistentry>
 904         <varlistentry>
 905           <term>Action:</term>
 906           <listitem>
 907             <para>It closes the current sequent by applying a stronger
 908              lemma that is proved using the new generated sequent.</para>
 909           </listitem>
 910         </varlistentry>
 911         <varlistentry>
 912           <term>New sequents to prove:</term>
 913           <listitem>
 914             <para>It opens a new sequent where the current sequent conclusion
 915              <command>G</command> is generalized to
 916              <command>∀x.G{x/t}</command> where <command>{x/t}</command>
 917              is a notation for the replacement with <command>x</command> of all
 918              the occurrences of the term <command>t</command> matched by
 919              <command>patt</command>. If <command>patt</command> matches no
 920              subterm then <command>t</command> is defined as the
 921              <command>wanted</command> part of the pattern.</para>
 922           </listitem>
 923         </varlistentry>
 924       </variablelist>
 925     </para>
 926   </sect1>
 927   <sect1 id="tac_id">
 928     <title>id</title>
 929     <titleabbrev>id</titleabbrev>
 930     <para><userinput>id </userinput></para>
 931     <para>
 932       <variablelist>
 933         <varlistentry role="tactic.synopsis">
 934           <term>Synopsis:</term>
 935           <listitem>
 936             <para><emphasis role="bold">id</emphasis></para>
 937           </listitem>
 938         </varlistentry>
 939         <varlistentry>
 940           <term>Pre-conditions:</term>
 941           <listitem>
 942             <para>None.</para>
 943           </listitem>
 944         </varlistentry>
 945         <varlistentry>
 946           <term>Action:</term>
 947           <listitem>
 948             <para>This identity tactic does nothing without failing.</para>
 949           </listitem>
 950         </varlistentry>
 951         <varlistentry>
 952           <term>New sequents to prove:</term>
 953           <listitem>
 954             <para>None.</para>
 955           </listitem>
 956         </varlistentry>
 957       </variablelist>
 958     </para>
 959   </sect1>
 960   <sect1 id="tac_intro">
 961     <title>intro</title>
 962     <titleabbrev>intro</titleabbrev>
 963     <para><userinput>intro H</userinput></para>
 964     <para>
 965       <variablelist>
 966         <varlistentry role="tactic.synopsis">
 967           <term>Synopsis:</term>
 968           <listitem>
 969             <para><emphasis role="bold">intro</emphasis> [&id;]</para>
 970           </listitem>
 971         </varlistentry>
 972         <varlistentry>
 973           <term>Pre-conditions:</term>
 974           <listitem>
 975             <para>The conclusion of the sequent to prove must be an implication
 976              or a universal quantification.</para>
 977           </listitem>
 978         </varlistentry>
 979         <varlistentry>
 980           <term>Action:</term>
 981           <listitem>
 982             <para>It applies the right introduction rule for implication,
 983              closing the current sequent.</para>
 984           </listitem>
 985         </varlistentry>
 986         <varlistentry>
 987           <term>New sequents to prove:</term>
 988           <listitem>
 989             <para>It opens a new sequent to prove adding to the hypothesis
 990              the antecedent of the implication and setting the conclusion
 991              to the consequent of the implicaiton. The name of the new
 992              hypothesis is <command>H</command> if provided; otherwise it
 993              is automatically generated.</para>
 994           </listitem>
 995         </varlistentry>
 996       </variablelist>
 997     </para>
 998   </sect1>
 999   <sect1 id="tac_intros">
1000     <title>intros</title>
1001     <titleabbrev>intros</titleabbrev>
1002     <para><userinput>intros hyps</userinput></para>
1003     <para>
1004       <variablelist>
1005         <varlistentry role="tactic.synopsis">
1006           <term>Synopsis:</term>
1007           <listitem>
1008             <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
1009           </listitem>
1010         </varlistentry>
1011         <varlistentry>
1012           <term>Pre-conditions:</term>
1013           <listitem>
1014             <para>If <command>hyps</command> specifies a number of hypotheses
1015              to introduce, then the conclusion of the current sequent must
1016              be formed by at least that number of imbricated implications
1017              or universal quantifications.</para>
1018           </listitem>
1019         </varlistentry>
1020         <varlistentry>
1021           <term>Action:</term>
1022           <listitem>
1023             <para>It applies several times the right introduction rule for
1024              implication, closing the current sequent.</para>
1025           </listitem>
1026         </varlistentry>
1027         <varlistentry>
1028           <term>New sequents to prove:</term>
1029           <listitem>
1030             <para>It opens a new sequent to prove adding a number of new
1031              hypotheses equal to the number of new hypotheses requested.
1032              If the user does not request a precise number of new hypotheses,
1033              it adds as many hypotheses as possible.
1034              The name of each new hypothesis is either popped from the
1035              user provided list of names, or it is automatically generated when
1036              the list is (or becomes) empty.</para>
1037           </listitem>
1038         </varlistentry>
1039       </variablelist>
1040     </para>
1041   </sect1>
1042   <sect1 id="tac_inversion">
1043     <title>inversion</title>
1044     <titleabbrev>inversion</titleabbrev>
1045     <para><userinput>inversion t</userinput></para>
1046     <para>
1047       <variablelist>
1048         <varlistentry role="tactic.synopsis">
1049           <term>Synopsis:</term>
1050           <listitem>
1051             <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
1052           </listitem>
1053         </varlistentry>
1054         <varlistentry>
1055           <term>Pre-conditions:</term>
1056           <listitem>
1057             <para>The type of the term <command>t</command> must be an inductive
1058              type or the application of an inductive type.</para>
1059           </listitem>
1060         </varlistentry>
1061         <varlistentry>
1062           <term>Action:</term>
1063           <listitem>
1064             <para>It proceeds by cases on <command>t</command> paying attention
1065              to the constraints imposed by the actual &quot;right arguments&quot;
1066              of the inductive type.</para>
1067           </listitem>
1068         </varlistentry>
1069         <varlistentry>
1070           <term>New sequents to prove:</term>
1071           <listitem>
1072             <para>It opens one new sequent to prove for each case in the
1073              definition of the type of <command>t</command>. With respect to
1074              a simple elimination, each new sequent has additional hypotheses
1075              that states the equalities of the &quot;right parameters&quot;
1076              of the inductive type with terms originally present in the
1077              sequent to prove.</para>
1078           </listitem>
1079         </varlistentry>
1080       </variablelist>
1081     </para>
1082   </sect1>
1083   <sect1 id="tac_lapply">
1084     <title>lapply</title>
1085     <titleabbrev>lapply</titleabbrev>
1086     <para><userinput>
1087      lapply linear depth=d t
1088      to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
1089     </userinput></para>
1090     <para>
1091       <variablelist>
1092         <varlistentry role="tactic.synopsis">
1093           <term>Synopsis:</term>
1094           <listitem>
1095             <para>
1096              <emphasis role="bold">lapply</emphasis>
1097              [<emphasis role="bold">linear</emphasis>]
1098              [<emphasis role="bold">depth=</emphasis>&nat;]
1099              &sterm;
1100              [<emphasis role="bold">to</emphasis>
1101               &sterm;
1102               [<emphasis role="bold">,</emphasis>&sterm;…]
1103              ]
1104              [<emphasis role="bold">as</emphasis> &id;]
1105             </para>
1106           </listitem>
1107         </varlistentry>
1108         <varlistentry>
1109           <term>Pre-conditions:</term>
1110           <listitem>
1111             <para>
1112              <command>t</command> must have at least <command>d</command>
1113              independent premises and <command>n</command> must not be
1114              greater than <command>d</command>.
1115             </para>
1116           </listitem>
1117         </varlistentry>
1118         <varlistentry>
1119           <term>Action:</term>
1120           <listitem>
1121             <para>
1122              Invokes <command>letin H ≝ (t ? ... ?)</command>
1123              with enough <command>?</command>'s to reach the
1124              <command>d</command>-th independent premise of
1125              <command>t</command>
1126              (<command>d</command> is maximum if unspecified).
1127              Then istantiates (by <command>apply</command>) with
1128              t<subscript>1</subscript>, ..., t<subscript>n</subscript>
1129              the <command>?</command>'s corresponding to the first
1130              <command>n</command> independent premises of
1131              <command>t</command>.
1132              Usually the other <command>?</command>'s preceding the
1133              <command>n</command>-th independent premise of
1134              <command>t</command> are istantiated as a consequence.
1135              If the <command>linear</command> flag is specified and if
1136              <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
1137              are (applications of) premises in the current context, they are
1138              <command>clear</command>ed.
1139             </para>
1140           </listitem>
1141         </varlistentry>
1142         <varlistentry>
1143           <term>New sequents to prove:</term>
1144           <listitem>
1145             <para>
1146              The ones opened by the tactics <command>lapply</command> invokes.
1147             </para>
1148           </listitem>
1149         </varlistentry>
1150       </variablelist>
1151     </para>
1152   </sect1>
1153   <sect1 id="tac_left">
1154     <title>left</title>
1155     <titleabbrev>left</titleabbrev>
1156     <para><userinput>left </userinput></para>
1157     <para>
1158       <variablelist>
1159         <varlistentry role="tactic.synopsis">
1160           <term>Synopsis:</term>
1161           <listitem>
1162             <para><emphasis role="bold">left</emphasis></para>
1163           </listitem>
1164         </varlistentry>
1165         <varlistentry>
1166           <term>Pre-conditions:</term>
1167           <listitem>
1168             <para>The conclusion of the current sequent must be
1169              an inductive type or the application of an inductive type
1170              with at least one constructor.</para>
1171           </listitem>
1172         </varlistentry>
1173         <varlistentry>
1174           <term>Action:</term>
1175           <listitem>
1176             <para>Equivalent to <command>constructor 1</command>.</para>
1177           </listitem>
1178         </varlistentry>
1179         <varlistentry>
1180           <term>New sequents to prove:</term>
1181           <listitem>
1182             <para>It opens a new sequent for each premise of the first
1183              constructor of the inductive type that is the conclusion of the
1184              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1185           </listitem>
1186         </varlistentry>
1187       </variablelist>
1188     </para>
1189   </sect1>
1190   <sect1 id="tac_letin">
1191     <title>letin</title>
1192     <titleabbrev>letin</titleabbrev>
1193     <para><userinput>letin x ≝ t</userinput></para>
1194     <para>
1195       <variablelist>
1196         <varlistentry role="tactic.synopsis">
1197           <term>Synopsis:</term>
1198           <listitem>
1199             <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
1200           </listitem>
1201         </varlistentry>
1202         <varlistentry>
1203           <term>Pre-conditions:</term>
1204           <listitem>
1205             <para>None.</para>
1206           </listitem>
1207         </varlistentry>
1208         <varlistentry>
1209           <term>Action:</term>
1210           <listitem>
1211             <para>It adds to the context of the current sequent to prove a new
1212              definition <command>x ≝ t</command>.</para>
1213           </listitem>
1214         </varlistentry>
1215         <varlistentry>
1216           <term>New sequents to prove:</term>
1217           <listitem>
1218             <para>None.</para>
1219           </listitem>
1220         </varlistentry>
1221       </variablelist>
1222     </para>
1223   </sect1>
1224   <sect1 id="tac_normalize">
1225     <title>normalize</title>
1226     <titleabbrev>normalize</titleabbrev>
1227     <para><userinput>normalize patt</userinput></para>
1228     <para>
1229       <variablelist>
1230         <varlistentry role="tactic.synopsis">
1231           <term>Synopsis:</term>
1232           <listitem>
1233             <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
1234           </listitem>
1235         </varlistentry>
1236         <varlistentry>
1237           <term>Pre-conditions:</term>
1238           <listitem>
1239             <para>None.</para>
1240           </listitem>
1241         </varlistentry>
1242         <varlistentry>
1243           <term>Action:</term>
1244           <listitem>
1245             <para>It replaces all the terms matched by <command>patt</command>
1246              with their βδιζ-normal form.</para>
1247           </listitem>
1248         </varlistentry>
1249         <varlistentry>
1250           <term>New sequents to prove:</term>
1251           <listitem>
1252             <para>None.</para>
1253           </listitem>
1254         </varlistentry>
1255       </variablelist>
1256     </para>
1257   </sect1>
1258   <sect1 id="tac_reduce">
1259     <title>reduce</title>
1260     <titleabbrev>reduce</titleabbrev>
1261     <para><userinput>reduce patt</userinput></para>
1262     <para>
1263       <variablelist>
1264         <varlistentry role="tactic.synopsis">
1265           <term>Synopsis:</term>
1266           <listitem>
1267             <para><emphasis role="bold">reduce</emphasis> &pattern;</para>
1268           </listitem>
1269         </varlistentry>
1270         <varlistentry>
1271           <term>Pre-conditions:</term>
1272           <listitem>
1273             <para>None.</para>
1274           </listitem>
1275         </varlistentry>
1276         <varlistentry>
1277           <term>Action:</term>
1278           <listitem>
1279             <para>It replaces all the terms matched by <command>patt</command>
1280              with their βδιζ-normal form.</para>
1281           </listitem>
1282         </varlistentry>
1283         <varlistentry>
1284           <term>New sequents to prove:</term>
1285           <listitem>
1286             <para>None.</para>
1287           </listitem>
1288         </varlistentry>
1289       </variablelist>
1290     </para>
1291   </sect1>
1292   <sect1 id="tac_reflexivity">
1293     <title>reflexivity</title>
1294     <titleabbrev>reflexivity</titleabbrev>
1295     <para><userinput>reflexivity </userinput></para>
1296     <para>
1297       <variablelist>
1298         <varlistentry role="tactic.synopsis">
1299           <term>Synopsis:</term>
1300           <listitem>
1301             <para><emphasis role="bold">reflexivity</emphasis></para>
1302           </listitem>
1303         </varlistentry>
1304         <varlistentry>
1305           <term>Pre-conditions:</term>
1306           <listitem>
1307             <para>The conclusion of the current sequent must be
1308              <command>t=t</command> for some term <command>t</command></para>
1309           </listitem>
1310         </varlistentry>
1311         <varlistentry>
1312           <term>Action:</term>
1313           <listitem>
1314             <para>It closes the current sequent by reflexivity
1315              of equality.</para>
1316           </listitem>
1317         </varlistentry>
1318         <varlistentry>
1319           <term>New sequents to prove:</term>
1320           <listitem>
1321             <para>None.</para>
1322           </listitem>
1323         </varlistentry>
1324       </variablelist>
1325     </para>
1326   </sect1>
1327   <sect1 id="tac_replace">
1328     <title>replace</title>
1329     <titleabbrev>change</titleabbrev>
1330     <para><userinput>change patt with t</userinput></para>
1331     <para>
1332       <variablelist>
1333         <varlistentry role="tactic.synopsis">
1334           <term>Synopsis:</term>
1335           <listitem>
1336             <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
1337           </listitem>
1338         </varlistentry>
1339         <varlistentry>
1340           <term>Pre-conditions:</term>
1341           <listitem>
1342             <para>None.</para>
1343           </listitem>
1344         </varlistentry>
1345         <varlistentry>
1346           <term>Action:</term>
1347           <listitem>
1348             <para>It replaces the subterms of the current sequent matched by
1349              <command>patt</command> with the new term <command>t</command>.
1350              For each subterm matched by the pattern, <command>t</command> is
1351              disambiguated in the context of the subterm.</para>
1352           </listitem>
1353         </varlistentry>
1354         <varlistentry>
1355           <term>New sequents to prove:</term>
1356           <listitem>
1357             <para>For each matched term <command>t'</command> it opens
1358              a new sequent to prove whose conclusion is
1359              <command>t'=t</command>.</para>
1360           </listitem>
1361         </varlistentry>
1362       </variablelist>
1363     </para>
1364   </sect1>
1365   <sect1 id="tac_rewrite">
1366     <title>rewrite</title>
1367     <titleabbrev>rewrite</titleabbrev>
1368     <para><userinput>rewrite dir p patt</userinput></para>
1369     <para>
1370       <variablelist>
1371         <varlistentry role="tactic.synopsis">
1372           <term>Synopsis:</term>
1373           <listitem>
1374             <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold">&lt;</emphasis>|<emphasis role="bold">&gt;</emphasis>] &sterm; &pattern;</para>
1375           </listitem>
1376         </varlistentry>
1377         <varlistentry>
1378           <term>Pre-conditions:</term>
1379           <listitem>
1380             <para><command>p</command> must be the proof of an equality,
1381              possibly under some hypotheses.</para>
1382           </listitem>
1383         </varlistentry>
1384         <varlistentry>
1385           <term>Action:</term>
1386           <listitem>
1387             <para>It looks in every term matched by <command>patt</command>
1388              for all the occurrences of the
1389              left hand side of the equality that <command>p</command> proves
1390              (resp. the right hand side if <command>dir</command> is
1391              <command>&lt;</command>). Every occurence found is replaced with
1392              the opposite side of the equality.</para>
1393           </listitem>
1394         </varlistentry>
1395         <varlistentry>
1396           <term>New sequents to prove:</term>
1397           <listitem>
1398             <para>It opens one new sequent for each hypothesis of the
1399              equality proved by <command>p</command> that is not closed
1400              by unification.</para>
1401           </listitem>
1402         </varlistentry>
1403       </variablelist>
1404     </para>
1405   </sect1>
1406   <sect1 id="tac_right">
1407     <title>right</title>
1408     <titleabbrev>right</titleabbrev>
1409     <para><userinput>right </userinput></para>
1410     <para>
1411       <variablelist>
1412         <varlistentry role="tactic.synopsis">
1413           <term>Synopsis:</term>
1414           <listitem>
1415             <para><emphasis role="bold">right</emphasis></para>
1416           </listitem>
1417         </varlistentry>
1418         <varlistentry>
1419           <term>Pre-conditions:</term>
1420           <listitem>
1421             <para>The conclusion of the current sequent must be
1422              an inductive type or the application of an inductive type with
1423              at least two constructors.</para>
1424           </listitem>
1425         </varlistentry>
1426         <varlistentry>
1427           <term>Action:</term>
1428           <listitem>
1429             <para>Equivalent to <command>constructor 2</command>.</para>
1430           </listitem>
1431         </varlistentry>
1432         <varlistentry>
1433           <term>New sequents to prove:</term>
1434           <listitem>
1435             <para>It opens a new sequent for each premise of the second
1436              constructor of the inductive type that is the conclusion of the
1437              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1438           </listitem>
1439         </varlistentry>
1440       </variablelist>
1441     </para>
1442   </sect1>
1443   <sect1 id="tac_ring">
1444     <title>ring</title>
1445     <titleabbrev>ring</titleabbrev>
1446     <para><userinput>ring </userinput></para>
1447     <para>
1448       <variablelist>
1449         <varlistentry role="tactic.synopsis">
1450           <term>Synopsis:</term>
1451           <listitem>
1452             <para><emphasis role="bold">ring</emphasis></para>
1453           </listitem>
1454         </varlistentry>
1455         <varlistentry>
1456           <term>Pre-conditions:</term>
1457           <listitem>
1458             <para>The conclusion of the current sequent must be an
1459              equality over Coq's real numbers that can be proved using
1460              the ring properties of the real numbers only.</para>
1461           </listitem>
1462         </varlistentry>
1463         <varlistentry>
1464           <term>Action:</term>
1465           <listitem>
1466             <para>It closes the current sequent veryfying the equality by
1467              means of computation (i.e. this is a reflexive tactic, implemented
1468              exploiting the &quot;two level reasoning&quot; technique).</para>
1469           </listitem>
1470         </varlistentry>
1471         <varlistentry>
1472           <term>New sequents to prove:</term>
1473           <listitem>
1474             <para>None.</para>
1475           </listitem>
1476         </varlistentry>
1477       </variablelist>
1478     </para>
1479   </sect1>
1480   <sect1 id="tac_simplify">
1481     <title>simplify</title>
1482     <titleabbrev>simplify</titleabbrev>
1483     <para><userinput>simplify patt</userinput></para>
1484     <para>
1485       <variablelist>
1486         <varlistentry role="tactic.synopsis">
1487           <term>Synopsis:</term>
1488           <listitem>
1489             <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
1490           </listitem>
1491         </varlistentry>
1492         <varlistentry>
1493           <term>Pre-conditions:</term>
1494           <listitem>
1495             <para>None.</para>
1496           </listitem>
1497         </varlistentry>
1498         <varlistentry>
1499           <term>Action:</term>
1500           <listitem>
1501             <para>It replaces all the terms matched by <command>patt</command>
1502              with other convertible terms that are supposed to be simpler.</para>
1503           </listitem>
1504         </varlistentry>
1505         <varlistentry>
1506           <term>New sequents to prove:</term>
1507           <listitem>
1508             <para>None.</para>
1509           </listitem>
1510         </varlistentry>
1511       </variablelist>
1512     </para>
1513   </sect1>
1514   <sect1 id="tac_split">
1515     <title>split</title>
1516     <titleabbrev>split</titleabbrev>
1517     <para><userinput>split </userinput></para>
1518     <para>
1519       <variablelist>
1520         <varlistentry role="tactic.synopsis">
1521           <term>Synopsis:</term>
1522           <listitem>
1523             <para><emphasis role="bold">split</emphasis></para>
1524           </listitem>
1525         </varlistentry>
1526         <varlistentry>
1527           <term>Pre-conditions:</term>
1528           <listitem>
1529             <para>The conclusion of the current sequent must be
1530              an inductive type or the application of an inductive type with
1531              at least one constructor.</para>
1532           </listitem>
1533         </varlistentry>
1534         <varlistentry>
1535           <term>Action:</term>
1536           <listitem>
1537             <para>Equivalent to <command>constructor 1</command>.</para>
1538           </listitem>
1539         </varlistentry>
1540         <varlistentry>
1541           <term>New sequents to prove:</term>
1542           <listitem>
1543             <para>It opens a new sequent for each premise of the first
1544              constructor of the inductive type that is the conclusion of the
1545              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1546           </listitem>
1547         </varlistentry>
1548       </variablelist>
1549     </para>
1550   </sect1>
1551
1552   <sect1 id="tac_subst">
1553     <title>subst</title>
1554     <titleabbrev>subst</titleabbrev>
1555     <para><userinput>subst</userinput></para>
1556     <para>
1557       <variablelist>
1558         <varlistentry role="tactic.synopsis">
1559           <term>Synopsis:</term>
1560           <listitem>
1561             <para><emphasis role="bold">subst</emphasis></para>
1562           </listitem>
1563         </varlistentry>
1564         <varlistentry>
1565           <term>Pre-conditions:</term>
1566           <listitem><para>
1567             None.
1568           </para></listitem>
1569         </varlistentry>
1570         <varlistentry>
1571           <term>Action:</term>
1572           <listitem><para>
1573             For each premise of the form
1574             <command>H: x = t</command> or <command>H: t = x</command>
1575             where <command>x</command> is a local variable and
1576             <command>t</command> does not depend on <command>x</command>,
1577             the tactic rewrites <command>H</command> wherever
1578             <command>x</command> appears clearing <command>H</command> and
1579             <command>x</command> afterwards.
1580           </para></listitem>
1581         </varlistentry>
1582         <varlistentry>
1583           <term>New sequents to prove:</term>
1584           <listitem><para>
1585             The one opened by the applied tactics.
1586           </para></listitem>
1587         </varlistentry>
1588       </variablelist>
1589     </para>
1590   </sect1>
1591   <sect1 id="tac_symmetry">
1592     <title>symmetry</title>
1593     <titleabbrev>symmetry</titleabbrev>
1594     <para>The tactic <command>symmetry</command> </para>
1595     <para><userinput>symmetry </userinput></para>
1596     <para>
1597       <variablelist>
1598         <varlistentry role="tactic.synopsis">
1599           <term>Synopsis:</term>
1600           <listitem>
1601             <para><emphasis role="bold">symmetry</emphasis></para>
1602           </listitem>
1603         </varlistentry>
1604         <varlistentry>
1605           <term>Pre-conditions:</term>
1606           <listitem>
1607             <para>The conclusion of the current proof must be an equality.</para>
1608           </listitem>
1609         </varlistentry>
1610         <varlistentry>
1611           <term>Action:</term>
1612           <listitem>
1613             <para>It swaps the two sides of the equalityusing the symmetric
1614              property.</para>
1615           </listitem>
1616         </varlistentry>
1617         <varlistentry>
1618           <term>New sequents to prove:</term>
1619           <listitem>
1620             <para>None.</para>
1621           </listitem>
1622         </varlistentry>
1623       </variablelist>
1624     </para>
1625   </sect1>
1626   <sect1 id="tac_transitivity">
1627     <title>transitivity</title>
1628     <titleabbrev>transitivity</titleabbrev>
1629     <para><userinput>transitivity t</userinput></para>
1630     <para>
1631       <variablelist>
1632         <varlistentry role="tactic.synopsis">
1633           <term>Synopsis:</term>
1634           <listitem>
1635             <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
1636           </listitem>
1637         </varlistentry>
1638         <varlistentry>
1639           <term>Pre-conditions:</term>
1640           <listitem>
1641             <para>The conclusion of the current proof must be an equality.</para>
1642           </listitem>
1643         </varlistentry>
1644         <varlistentry>
1645           <term>Action:</term>
1646           <listitem>
1647             <para>It closes the current sequent by transitivity of the equality.</para>
1648           </listitem>
1649         </varlistentry>
1650         <varlistentry>
1651           <term>New sequents to prove:</term>
1652           <listitem>
1653             <para>It opens two new sequents <command>l=t</command> and
1654              <command>t=r</command> where <command>l</command> and <command>r</command> are the left and right hand side of the equality in the conclusion of
1655 the current sequent to prove.</para>
1656           </listitem>
1657         </varlistentry>
1658       </variablelist>
1659     </para>
1660   </sect1>
1661   <sect1 id="tac_unfold">
1662     <title>unfold</title>
1663     <titleabbrev>unfold</titleabbrev>
1664     <para><userinput>unfold t patt</userinput></para>
1665     <para>
1666       <variablelist>
1667         <varlistentry role="tactic.synopsis">
1668           <term>Synopsis:</term>
1669           <listitem>
1670             <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
1671           </listitem>
1672         </varlistentry>
1673         <varlistentry>
1674           <term>Pre-conditions:</term>
1675           <listitem>
1676             <para>None.</para>
1677           </listitem>
1678         </varlistentry>
1679         <varlistentry>
1680           <term>Action:</term>
1681           <listitem>
1682             <para>It finds all the occurrences of <command>t</command>
1683              (possibly applied to arguments) in the subterms matched by
1684              <command>patt</command>. Then it δ-expands each occurrence,
1685              also performing β-reduction of the obtained term. If
1686              <command>t</command> is omitted it defaults to each
1687              subterm matched by <command>patt</command>.</para>
1688           </listitem>
1689         </varlistentry>
1690         <varlistentry>
1691           <term>New sequents to prove:</term>
1692           <listitem>
1693             <para>None.</para>
1694           </listitem>
1695         </varlistentry>
1696       </variablelist>
1697     </para>
1698   </sect1>
1699   <sect1 id="tac_whd">
1700     <title>whd</title>
1701     <titleabbrev>whd</titleabbrev>
1702     <para><userinput>whd patt</userinput></para>
1703     <para>
1704       <variablelist>
1705         <varlistentry role="tactic.synopsis">
1706           <term>Synopsis:</term>
1707           <listitem>
1708             <para><emphasis role="bold">whd</emphasis> &pattern;</para>
1709           </listitem>
1710         </varlistentry>
1711         <varlistentry>
1712           <term>Pre-conditions:</term>
1713           <listitem>
1714             <para>None.</para>
1715           </listitem>
1716         </varlistentry>
1717         <varlistentry>
1718           <term>Action:</term>
1719           <listitem>
1720             <para>It replaces all the terms matched by <command>patt</command>
1721              with their βδιζ-weak-head normal form.</para>
1722           </listitem>
1723         </varlistentry>
1724         <varlistentry>
1725           <term>New sequents to prove:</term>
1726           <listitem>
1727             <para>None.</para>
1728           </listitem>
1729         </varlistentry>
1730       </variablelist>
1731     </para>
1732   </sect1>
1733
1734 </chapter>
1735