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