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_cases">
 219     <title>cases</title>
 220     <titleabbrev>cases</titleabbrev>
 221     <para><userinput>
 222      cases t hyps
 223     </userinput></para>
 224     <para>
 225       <variablelist>
 226         <varlistentry role="tactic.synopsis">
 227           <term>Synopsis:</term>
 228           <listitem>
 229             <para>
 230              <emphasis role="bold">cases</emphasis>
 231              &term; [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]
 232             </para>
 233           </listitem>
 234         </varlistentry>
 235         <varlistentry>
 236           <term>Pre-conditions:</term>
 237           <listitem>
 238             <para>
 239              <command>t</command> must inhabit an inductive type
 240             </para>
 241           </listitem>
 242         </varlistentry>
 243         <varlistentry>
 244           <term>Action:</term>
 245           <listitem>
 246             <para>
 247              It proceed by cases on <command>t</command>. The new generated
 248              hypothesis in each branch are named according to
 249              <command>hyps</command>.
 250             </para>
 251           </listitem>
 252         </varlistentry>
 253         <varlistentry>
 254           <term>New sequents to prove:</term>
 255           <listitem>
 256             <para>One new sequent for each constructor of the type of
 257              <command>t</command>. Each sequent has a new hypothesis for
 258              each argument of the constructor.</para>
 259           </listitem>
 260         </varlistentry>
 261       </variablelist>
 262     </para>
 263   </sect1>
 264   <sect1 id="tac_clear">
 265     <title>clear</title>
 266     <titleabbrev>clear</titleabbrev>
 267     <para><userinput>
 268      clear H<subscript>1</subscript> ... H<subscript>m</subscript>
 269     </userinput></para>
 270     <para>
 271       <variablelist>
 272         <varlistentry role="tactic.synopsis">
 273           <term>Synopsis:</term>
 274           <listitem>
 275             <para>
 276              <emphasis role="bold">clear</emphasis>
 277              &id; [&id;…]
 278             </para>
 279           </listitem>
 280         </varlistentry>
 281         <varlistentry>
 282           <term>Pre-conditions:</term>
 283           <listitem>
 284             <para>
 285              <command>
 286               H<subscript>1</subscript> ... H<subscript>m</subscript>
 287              </command> must be hypotheses of the
 288              current sequent to prove.
 289             </para>
 290           </listitem>
 291         </varlistentry>
 292         <varlistentry>
 293           <term>Action:</term>
 294           <listitem>
 295             <para>
 296              It hides the hypotheses
 297              <command>
 298               H<subscript>1</subscript> ... H<subscript>m</subscript>
 299              </command> from the current sequent.
 300             </para>
 301           </listitem>
 302         </varlistentry>
 303         <varlistentry>
 304           <term>New sequents to prove:</term>
 305           <listitem>
 306             <para>None</para>
 307           </listitem>
 308         </varlistentry>
 309       </variablelist>
 310     </para>
 311   </sect1>
 312   <sect1 id="tac_clearbody">
 313     <title>clearbody</title>
 314     <titleabbrev>clearbody</titleabbrev>
 315     <para><userinput>clearbody H</userinput></para>
 316     <para>
 317       <variablelist>
 318         <varlistentry role="tactic.synopsis">
 319           <term>Synopsis:</term>
 320           <listitem>
 321             <para><emphasis role="bold">clearbody</emphasis> &id;</para>
 322           </listitem>
 323         </varlistentry>
 324         <varlistentry>
 325           <term>Pre-conditions:</term>
 326           <listitem>
 327             <para><command>H</command> must be an hypothesis of the
 328              current sequent to prove.</para>
 329           </listitem>
 330         </varlistentry>
 331         <varlistentry>
 332           <term>Action:</term>
 333           <listitem>
 334             <para>It hides the definiens of a definition in the current
 335              sequent context. Thus the definition becomes an hypothesis.</para>
 336           </listitem>
 337         </varlistentry>
 338         <varlistentry>
 339           <term>New sequents to prove:</term>
 340           <listitem>
 341             <para>None.</para>
 342           </listitem>
 343         </varlistentry>
 344       </variablelist>
 345     </para>
 346   </sect1>
 347   <sect1 id="tac_change">
 348     <title>change</title>
 349     <titleabbrev>change</titleabbrev>
 350     <para><userinput>change patt with t</userinput></para>
 351     <para>
 352       <variablelist>
 353         <varlistentry role="tactic.synopsis">
 354           <term>Synopsis:</term>
 355           <listitem>
 356             <para><emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
 357           </listitem>
 358         </varlistentry>
 359         <varlistentry>
 360           <term>Pre-conditions:</term>
 361           <listitem>
 362             <para>Each subterm matched by the pattern must be convertible
 363              with the term <command>t</command> disambiguated in the context
 364              of the matched subterm.</para>
 365           </listitem>
 366         </varlistentry>
 367         <varlistentry>
 368           <term>Action:</term>
 369           <listitem>
 370             <para>It replaces the subterms of the current sequent matched by
 371              <command>patt</command> with the new term <command>t</command>.
 372              For each subterm matched by the pattern, <command>t</command> is
 373              disambiguated in the context of the subterm.</para>
 374           </listitem>
 375         </varlistentry>
 376         <varlistentry>
 377           <term>New sequents to prove:</term>
 378           <listitem>
 379             <para>None.</para>
 380           </listitem>
 381         </varlistentry>
 382       </variablelist>
 383     </para>
 384   </sect1>
 385   <sect1 id="tac_constructor">
 386     <title>constructor</title>
 387     <titleabbrev>constructor</titleabbrev>
 388     <para><userinput>constructor n</userinput></para>
 389     <para>
 390       <variablelist>
 391         <varlistentry role="tactic.synopsis">
 392           <term>Synopsis:</term>
 393           <listitem>
 394             <para><emphasis role="bold">constructor</emphasis> &nat;</para>
 395           </listitem>
 396         </varlistentry>
 397         <varlistentry>
 398           <term>Pre-conditions:</term>
 399           <listitem>
 400             <para>The conclusion of the current sequent must be
 401              an inductive type or the application of an inductive type with
 402              at least <command>n</command> constructors.</para>
 403           </listitem>
 404         </varlistentry>
 405         <varlistentry>
 406           <term>Action:</term>
 407           <listitem>
 408             <para>It applies the <command>n</command>-th constructor of the
 409              inductive type of the conclusion of the current sequent.</para>
 410           </listitem>
 411         </varlistentry>
 412         <varlistentry>
 413           <term>New sequents to prove:</term>
 414           <listitem>
 415             <para>It opens a new sequent for each premise of the constructor
 416              that can not be inferred by unification. For more details,
 417              see the <command>apply</command> tactic.</para>
 418           </listitem>
 419         </varlistentry>
 420       </variablelist>
 421     </para>
 422   </sect1>
 423   <sect1 id="tac_contradiction">
 424     <title>contradiction</title>
 425     <titleabbrev>contradiction</titleabbrev>
 426     <para><userinput>contradiction </userinput></para>
 427     <para>
 428       <variablelist>
 429         <varlistentry role="tactic.synopsis">
 430           <term>Synopsis:</term>
 431           <listitem>
 432             <para><emphasis role="bold">contradiction</emphasis></para>
 433           </listitem>
 434         </varlistentry>
 435         <varlistentry>
 436           <term>Pre-conditions:</term>
 437           <listitem>
 438             <para>There must be in the current context an hypothesis of type
 439              <command>False</command>.</para>
 440           </listitem>
 441         </varlistentry>
 442         <varlistentry>
 443           <term>Action:</term>
 444           <listitem>
 445             <para>It closes the current sequent by applying an hypothesis of
 446              type <command>False</command>.</para>
 447           </listitem>
 448         </varlistentry>
 449         <varlistentry>
 450           <term>New sequents to prove:</term>
 451           <listitem>
 452             <para>None</para>
 453           </listitem>
 454         </varlistentry>
 455       </variablelist>
 456     </para>
 457   </sect1>
 458   <sect1 id="tac_cut">
 459     <title>cut</title>
 460     <titleabbrev>cut</titleabbrev>
 461     <para><userinput>cut P as H</userinput></para>
 462     <para>
 463       <variablelist>
 464         <varlistentry role="tactic.synopsis">
 465           <term>Synopsis:</term>
 466           <listitem>
 467             <para><emphasis role="bold">cut</emphasis> &sterm; [<emphasis role="bold">as</emphasis> &id;]</para>
 468           </listitem>
 469         </varlistentry>
 470         <varlistentry>
 471           <term>Pre-conditions:</term>
 472           <listitem>
 473             <para><command>P</command> must have type <command>Prop</command>.</para>
 474           </listitem>
 475         </varlistentry>
 476         <varlistentry>
 477           <term>Action:</term>
 478           <listitem>
 479             <para>It closes the current sequent.</para>
 480           </listitem>
 481         </varlistentry>
 482         <varlistentry>
 483           <term>New sequents to prove:</term>
 484           <listitem>
 485             <para>It opens two new sequents. The first one has an extra
 486              hypothesis <command>H:P</command>. If <command>H</command> is
 487              omitted, the name of the hypothesis is automatically generated.
 488              The second sequent has conclusion <command>P</command> and
 489              hypotheses the hypotheses of the current sequent to prove.</para>
 490           </listitem>
 491         </varlistentry>
 492       </variablelist>
 493     </para>
 494   </sect1>
 495   <sect1 id="tac_decompose">
 496     <title>decompose</title>
 497     <titleabbrev>decompose</titleabbrev>
 498     <para><userinput>
 499      decompose as H<subscript>1</subscript> ... H<subscript>m</subscript>
 500     </userinput></para>
 501     <para>
 502       <variablelist>
 503         <varlistentry role="tactic.synopsis">
 504           <term>Synopsis:</term>
 505           <listitem>
 506             <para>
 507              <emphasis role="bold">decompose</emphasis>
 508              [<emphasis role="bold">as</emphasis> &id;…]
 509             </para>
 510           </listitem>
 511         </varlistentry>
 512         <varlistentry>
 513           <term>Pre-conditions:</term>
 514           <listitem>
 515             <para>None.</para>
 516           </listitem>
 517         </varlistentry>
 518         <varlistentry>
 519           <term>Action:</term>
 520           <listitem>
 521             <para>
 522              For each each premise <command>H</command>
 523              of type <command>T</command> in the current context
 524              where <command>T</command> is a non-recursive inductive type
 525              withour right parameters, the tactic runs
 526              <command>
 527               elim H as H<subscript>1</subscript> ... H<subscript>m</subscript>
 528              </command>, clears <command>H</command>  and runs itself
 529              recursively on each new premise introduced by
 530              <command>elim</command> in the opened sequents.
 531             </para>
 532           </listitem>
 533         </varlistentry>
 534         <varlistentry>
 535           <term>New sequents to prove:</term>
 536           <listitem>
 537             <para>
 538              The ones generated by all the <command>elim</command> tactics run.
 539             </para>
 540           </listitem>
 541         </varlistentry>
 542       </variablelist>
 543     </para>
 544   </sect1>
 545   <sect1 id="tac_demodulate">
 546     <title>demodulate</title>
 547     <titleabbrev>demodulate</titleabbrev>
 548     <para><userinput>demodulate</userinput></para>
 549     <para>
 550       <variablelist>
 551         <varlistentry role="tactic.synopsis">
 552           <term>Synopsis:</term>
 553           <listitem>
 554             <para><emphasis role="bold">demodulate</emphasis></para>
 555           </listitem>
 556         </varlistentry>
 557         <varlistentry>
 558           <term>Pre-conditions:</term>
 559           <listitem>
 560             <para>None.</para>
 561           </listitem>
 562         </varlistentry>
 563         <varlistentry>
 564           <term>Action:</term>
 565           <listitem>
 566             <para>&TODO;</para>
 567           </listitem>
 568         </varlistentry>
 569         <varlistentry>
 570           <term>New sequents to prove:</term>
 571           <listitem>
 572             <para>None.</para>
 573           </listitem>
 574         </varlistentry>
 575       </variablelist>
 576     </para>
 577   </sect1>
 578   <sect1 id="tac_destruct">
 579     <title>destruct</title>
 580     <titleabbrev>destruct</titleabbrev>
 581     <para><userinput>destruct p</userinput></para>
 582     <para>
 583       <variablelist>
 584         <varlistentry role="tactic.synopsis">
 585           <term>Synopsis:</term>
 586           <listitem>
 587             <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
 588           </listitem>
 589         </varlistentry>
 590         <varlistentry>
 591           <term>Pre-conditions:</term>
 592           <listitem>
 593             <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>
 594           </listitem>
 595         </varlistentry>
 596         <varlistentry>
 597           <term>Action:</term>
 598           <listitem>
 599             <para>The tactic recursively compare the two sides of the equality
 600              looking for different constructors in corresponding position.
 601              If two of them are found, the tactic closes the current sequent
 602              by proving the absurdity of <command>p</command>. Otherwise
 603              it adds a new hypothesis for each leaf of the formula that
 604              states the equality of the subformulae in the corresponding
 605              positions on the two sides of the equality.
 606             </para>
 607           </listitem>
 608         </varlistentry>
 609         <varlistentry>
 610           <term>New sequents to prove:</term>
 611           <listitem>
 612             <para>None.</para>
 613           </listitem>
 614         </varlistentry>
 615       </variablelist>
 616     </para>
 617   </sect1>
 618   <sect1 id="tac_elim">
 619     <title>elim</title>
 620     <titleabbrev>elim</titleabbrev>
 621     <para><userinput>elim t using th hyps</userinput></para>
 622     <para>
 623       <variablelist>
 624         <varlistentry role="tactic.synopsis">
 625           <term>Synopsis:</term>
 626           <listitem>
 627             <para><emphasis role="bold">elim</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 628           </listitem>
 629         </varlistentry>
 630         <varlistentry>
 631           <term>Pre-conditions:</term>
 632           <listitem>
 633             <para><command>t</command> must inhabit an inductive type and
 634              <command>th</command> must be an elimination principle for that
 635              inductive type. If <command>th</command> is omitted the appropriate
 636              standard elimination principle is chosen.</para>
 637           </listitem>
 638         </varlistentry>
 639         <varlistentry>
 640           <term>Action:</term>
 641           <listitem>
 642             <para>It proceeds by cases on the values of <command>t</command>,
 643              according to the elimination principle <command>th</command>.
 644             </para>
 645           </listitem>
 646         </varlistentry>
 647         <varlistentry>
 648           <term>New sequents to prove:</term>
 649           <listitem>
 650             <para>It opens one new sequent for each case. The names of
 651              the new hypotheses are picked by <command>hyps</command>, if
 652              provided. If hyps specifies also a number of hypotheses that
 653              is less than the number of new hypotheses for a new sequent,
 654              then the exceeding hypothesis will be kept as implications in
 655              the conclusion of the sequent.</para>
 656           </listitem>
 657         </varlistentry>
 658       </variablelist>
 659     </para>
 660   </sect1>
 661   <sect1 id="tac_elimType">
 662     <title>elimType</title>
 663     <titleabbrev>elimType</titleabbrev>
 664     <para><userinput>elimType T using th hyps</userinput></para>
 665     <para>
 666       <variablelist>
 667         <varlistentry role="tactic.synopsis">
 668           <term>Synopsis:</term>
 669           <listitem>
 670             <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 671           </listitem>
 672         </varlistentry>
 673         <varlistentry>
 674           <term>Pre-conditions:</term>
 675           <listitem>
 676             <para><command>T</command> must be an inductive type.</para>
 677           </listitem>
 678         </varlistentry>
 679         <varlistentry>
 680           <term>Action:</term>
 681           <listitem>
 682             <para>TODO (severely bugged now).</para>
 683           </listitem>
 684         </varlistentry>
 685         <varlistentry>
 686           <term>New sequents to prove:</term>
 687           <listitem>
 688             <para>TODO</para>
 689           </listitem>
 690         </varlistentry>
 691       </variablelist>
 692     </para>
 693   </sect1>
 694   <sect1 id="tac_exact">
 695     <title>exact</title>
 696     <titleabbrev>exact</titleabbrev>
 697     <para><userinput>exact p</userinput></para>
 698     <para>
 699       <variablelist>
 700         <varlistentry role="tactic.synopsis">
 701           <term>Synopsis:</term>
 702           <listitem>
 703             <para><emphasis role="bold">exact</emphasis> &sterm;</para>
 704           </listitem>
 705         </varlistentry>
 706         <varlistentry>
 707           <term>Pre-conditions:</term>
 708           <listitem>
 709             <para>The type of <command>p</command> must be convertible
 710              with the conclusion of the current sequent.</para>
 711           </listitem>
 712         </varlistentry>
 713         <varlistentry>
 714           <term>Action:</term>
 715           <listitem>
 716             <para>It closes the current sequent using <command>p</command>.</para>
 717           </listitem>
 718         </varlistentry>
 719         <varlistentry>
 720           <term>New sequents to prove:</term>
 721           <listitem>
 722             <para>None.</para>
 723           </listitem>
 724         </varlistentry>
 725       </variablelist>
 726     </para>
 727   </sect1>
 728   <sect1 id="tac_exists">
 729     <title>exists</title>
 730     <titleabbrev>exists</titleabbrev>
 731     <para><userinput>exists </userinput></para>
 732     <para>
 733       <variablelist>
 734         <varlistentry role="tactic.synopsis">
 735           <term>Synopsis:</term>
 736           <listitem>
 737             <para><emphasis role="bold">exists</emphasis></para>
 738           </listitem>
 739         </varlistentry>
 740         <varlistentry>
 741           <term>Pre-conditions:</term>
 742           <listitem>
 743             <para>The conclusion of the current sequent must be
 744              an inductive type or the application of an inductive type
 745              with at least one constructor.</para>
 746           </listitem>
 747         </varlistentry>
 748         <varlistentry>
 749           <term>Action:</term>
 750           <listitem>
 751             <para>Equivalent to <command>constructor 1</command>.</para>
 752           </listitem>
 753         </varlistentry>
 754         <varlistentry>
 755           <term>New sequents to prove:</term>
 756           <listitem>
 757             <para>It opens a new sequent for each premise of the first
 758              constructor of the inductive type that is the conclusion of the
 759              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 760           </listitem>
 761         </varlistentry>
 762       </variablelist>
 763     </para>
 764   </sect1>
 765   <sect1 id="tac_fail">
 766     <title>fail</title>
 767     <titleabbrev>fail</titleabbrev>
 768     <para><userinput>fail</userinput></para>
 769     <para>
 770       <variablelist>
 771         <varlistentry role="tactic.synopsis">
 772           <term>Synopsis:</term>
 773           <listitem>
 774             <para><emphasis role="bold">fail</emphasis></para>
 775           </listitem>
 776         </varlistentry>
 777         <varlistentry>
 778           <term>Pre-conditions:</term>
 779           <listitem>
 780             <para>None.</para>
 781           </listitem>
 782         </varlistentry>
 783         <varlistentry>
 784           <term>Action:</term>
 785           <listitem>
 786             <para>This tactic always fail.</para>
 787           </listitem>
 788         </varlistentry>
 789         <varlistentry>
 790           <term>New sequents to prove:</term>
 791           <listitem>
 792             <para>N.A.</para>
 793           </listitem>
 794         </varlistentry>
 795       </variablelist>
 796     </para>
 797   </sect1>
 798   <sect1 id="tac_fold">
 799     <title>fold</title>
 800     <titleabbrev>fold</titleabbrev>
 801     <para><userinput>fold red t patt</userinput></para>
 802     <para>
 803       <variablelist>
 804         <varlistentry role="tactic.synopsis">
 805           <term>Synopsis:</term>
 806           <listitem>
 807             <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
 808           </listitem>
 809         </varlistentry>
 810         <varlistentry>
 811           <term>Pre-conditions:</term>
 812           <listitem>
 813             <para>The pattern must not specify the wanted term.</para>
 814           </listitem>
 815         </varlistentry>
 816         <varlistentry>
 817           <term>Action:</term>
 818           <listitem>
 819             <para>First of all it locates all the subterms matched by
 820              <command>patt</command>. In the context of each matched subterm
 821              it disambiguates the term <command>t</command> and reduces it
 822              to its <command>red</command> normal form; then it replaces with
 823              <command>t</command> every occurrence of the normal form in the
 824              matched subterm.</para>
 825           </listitem>
 826         </varlistentry>
 827         <varlistentry>
 828           <term>New sequents to prove:</term>
 829           <listitem>
 830             <para>None.</para>
 831           </listitem>
 832         </varlistentry>
 833       </variablelist>
 834     </para>
 835   </sect1>
 836   <sect1 id="tac_fourier">
 837     <title>fourier</title>
 838     <titleabbrev>fourier</titleabbrev>
 839     <para><userinput>fourier </userinput></para>
 840     <para>
 841       <variablelist>
 842         <varlistentry role="tactic.synopsis">
 843           <term>Synopsis:</term>
 844           <listitem>
 845             <para><emphasis role="bold">fourier</emphasis></para>
 846           </listitem>
 847         </varlistentry>
 848         <varlistentry>
 849           <term>Pre-conditions:</term>
 850           <listitem>
 851             <para>The conclusion of the current sequent must be a linear
 852              inequation over real numbers taken from standard library of
 853              Coq. Moreover the inequations in the hypotheses must imply the
 854              inequation in the conclusion of the current sequent.</para>
 855           </listitem>
 856         </varlistentry>
 857         <varlistentry>
 858           <term>Action:</term>
 859           <listitem>
 860             <para>It closes the current sequent by applying the Fourier method.</para>
 861           </listitem>
 862         </varlistentry>
 863         <varlistentry>
 864           <term>New sequents to prove:</term>
 865           <listitem>
 866             <para>None.</para>
 867           </listitem>
 868         </varlistentry>
 869       </variablelist>
 870     </para>
 871   </sect1>
 872   <sect1 id="tac_fwd">
 873     <title>fwd</title>
 874     <titleabbrev>fwd</titleabbrev>
 875     <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
 876     <para>
 877       <variablelist>
 878         <varlistentry role="tactic.synopsis">
 879           <term>Synopsis:</term>
 880           <listitem>
 881             <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
 882           </listitem>
 883         </varlistentry>
 884         <varlistentry>
 885           <term>Pre-conditions:</term>
 886           <listitem>
 887             <para>
 888                The type of <command>H</command> must be the premise of a
 889                forward simplification theorem.
 890             </para>
 891           </listitem>
 892         </varlistentry>
 893         <varlistentry>
 894           <term>Action:</term>
 895           <listitem>
 896             <para>
 897              This tactic is under development.
 898              It simplifies the current context by removing
 899              <command>H</command> using the following methods:
 900              forward application (by <command>lapply</command>) of a suitable
 901              simplification theorem, chosen automatically, of which the type
 902              of <command>H</command> is a premise,
 903              decomposition (by <command>decompose</command>),
 904              rewriting (by <command>rewrite</command>).
 905              <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
 906              are passed to the tactics <command>fwd</command> invokes, as
 907              names for the premise they introduce.
 908             </para>
 909           </listitem>
 910         </varlistentry>
 911         <varlistentry>
 912           <term>New sequents to prove:</term>
 913           <listitem>
 914             <para>
 915              The ones opened by the tactics <command>fwd</command> invokes.
 916             </para>
 917           </listitem>
 918         </varlistentry>
 919       </variablelist>
 920     </para>
 921   </sect1>
 922   <sect1 id="tac_generalize">
 923     <title>generalize</title>
 924     <titleabbrev>generalize</titleabbrev>
 925     <para><userinput>generalize patt as H</userinput></para>
 926     <para>
 927       <variablelist>
 928         <varlistentry role="tactic.synopsis">
 929           <term>Synopsis:</term>
 930           <listitem>
 931             <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
 932           </listitem>
 933         </varlistentry>
 934         <varlistentry>
 935           <term>Pre-conditions:</term>
 936           <listitem>
 937             <para>All the terms matched by <command>patt</command> must be
 938              convertible and close in the context of the current sequent.</para>
 939           </listitem>
 940         </varlistentry>
 941         <varlistentry>
 942           <term>Action:</term>
 943           <listitem>
 944             <para>It closes the current sequent by applying a stronger
 945              lemma that is proved using the new generated sequent.</para>
 946           </listitem>
 947         </varlistentry>
 948         <varlistentry>
 949           <term>New sequents to prove:</term>
 950           <listitem>
 951             <para>It opens a new sequent where the current sequent conclusion
 952              <command>G</command> is generalized to
 953              <command>∀x.G{x/t}</command> where <command>{x/t}</command>
 954              is a notation for the replacement with <command>x</command> of all
 955              the occurrences of the term <command>t</command> matched by
 956              <command>patt</command>. If <command>patt</command> matches no
 957              subterm then <command>t</command> is defined as the
 958              <command>wanted</command> part of the pattern.</para>
 959           </listitem>
 960         </varlistentry>
 961       </variablelist>
 962     </para>
 963   </sect1>
 964   <sect1 id="tac_id">
 965     <title>id</title>
 966     <titleabbrev>id</titleabbrev>
 967     <para><userinput>id </userinput></para>
 968     <para>
 969       <variablelist>
 970         <varlistentry role="tactic.synopsis">
 971           <term>Synopsis:</term>
 972           <listitem>
 973             <para><emphasis role="bold">id</emphasis></para>
 974           </listitem>
 975         </varlistentry>
 976         <varlistentry>
 977           <term>Pre-conditions:</term>
 978           <listitem>
 979             <para>None.</para>
 980           </listitem>
 981         </varlistentry>
 982         <varlistentry>
 983           <term>Action:</term>
 984           <listitem>
 985             <para>This identity tactic does nothing without failing.</para>
 986           </listitem>
 987         </varlistentry>
 988         <varlistentry>
 989           <term>New sequents to prove:</term>
 990           <listitem>
 991             <para>None.</para>
 992           </listitem>
 993         </varlistentry>
 994       </variablelist>
 995     </para>
 996   </sect1>
 997   <sect1 id="tac_intro">
 998     <title>intro</title>
 999     <titleabbrev>intro</titleabbrev>
1000     <para><userinput>intro H</userinput></para>
1001     <para>
1002       <variablelist>
1003         <varlistentry role="tactic.synopsis">
1004           <term>Synopsis:</term>
1005           <listitem>
1006             <para><emphasis role="bold">intro</emphasis> [&id;]</para>
1007           </listitem>
1008         </varlistentry>
1009         <varlistentry>
1010           <term>Pre-conditions:</term>
1011           <listitem>
1012             <para>The conclusion of the sequent to prove must be an implication
1013              or a universal quantification.</para>
1014           </listitem>
1015         </varlistentry>
1016         <varlistentry>
1017           <term>Action:</term>
1018           <listitem>
1019             <para>It applies the right introduction rule for implication,
1020              closing the current sequent.</para>
1021           </listitem>
1022         </varlistentry>
1023         <varlistentry>
1024           <term>New sequents to prove:</term>
1025           <listitem>
1026             <para>It opens a new sequent to prove adding to the hypothesis
1027              the antecedent of the implication and setting the conclusion
1028              to the consequent of the implicaiton. The name of the new
1029              hypothesis is <command>H</command> if provided; otherwise it
1030              is automatically generated.</para>
1031           </listitem>
1032         </varlistentry>
1033       </variablelist>
1034     </para>
1035   </sect1>
1036   <sect1 id="tac_intros">
1037     <title>intros</title>
1038     <titleabbrev>intros</titleabbrev>
1039     <para><userinput>intros hyps</userinput></para>
1040     <para>
1041       <variablelist>
1042         <varlistentry role="tactic.synopsis">
1043           <term>Synopsis:</term>
1044           <listitem>
1045             <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
1046           </listitem>
1047         </varlistentry>
1048         <varlistentry>
1049           <term>Pre-conditions:</term>
1050           <listitem>
1051             <para>If <command>hyps</command> specifies a number of hypotheses
1052              to introduce, then the conclusion of the current sequent must
1053              be formed by at least that number of imbricated implications
1054              or universal quantifications.</para>
1055           </listitem>
1056         </varlistentry>
1057         <varlistentry>
1058           <term>Action:</term>
1059           <listitem>
1060             <para>It applies several times the right introduction rule for
1061              implication, closing the current sequent.</para>
1062           </listitem>
1063         </varlistentry>
1064         <varlistentry>
1065           <term>New sequents to prove:</term>
1066           <listitem>
1067             <para>It opens a new sequent to prove adding a number of new
1068              hypotheses equal to the number of new hypotheses requested.
1069              If the user does not request a precise number of new hypotheses,
1070              it adds as many hypotheses as possible.
1071              The name of each new hypothesis is either popped from the
1072              user provided list of names, or it is automatically generated when
1073              the list is (or becomes) empty.</para>
1074           </listitem>
1075         </varlistentry>
1076       </variablelist>
1077     </para>
1078   </sect1>
1079   <sect1 id="tac_inversion">
1080     <title>inversion</title>
1081     <titleabbrev>inversion</titleabbrev>
1082     <para><userinput>inversion t</userinput></para>
1083     <para>
1084       <variablelist>
1085         <varlistentry role="tactic.synopsis">
1086           <term>Synopsis:</term>
1087           <listitem>
1088             <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
1089           </listitem>
1090         </varlistentry>
1091         <varlistentry>
1092           <term>Pre-conditions:</term>
1093           <listitem>
1094             <para>The type of the term <command>t</command> must be an inductive
1095              type or the application of an inductive type.</para>
1096           </listitem>
1097         </varlistentry>
1098         <varlistentry>
1099           <term>Action:</term>
1100           <listitem>
1101             <para>It proceeds by cases on <command>t</command> paying attention
1102              to the constraints imposed by the actual &quot;right arguments&quot;
1103              of the inductive type.</para>
1104           </listitem>
1105         </varlistentry>
1106         <varlistentry>
1107           <term>New sequents to prove:</term>
1108           <listitem>
1109             <para>It opens one new sequent to prove for each case in the
1110              definition of the type of <command>t</command>. With respect to
1111              a simple elimination, each new sequent has additional hypotheses
1112              that states the equalities of the &quot;right parameters&quot;
1113              of the inductive type with terms originally present in the
1114              sequent to prove.</para>
1115           </listitem>
1116         </varlistentry>
1117       </variablelist>
1118     </para>
1119   </sect1>
1120   <sect1 id="tac_lapply">
1121     <title>lapply</title>
1122     <titleabbrev>lapply</titleabbrev>
1123     <para><userinput>
1124      lapply linear depth=d t
1125      to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
1126     </userinput></para>
1127     <para>
1128       <variablelist>
1129         <varlistentry role="tactic.synopsis">
1130           <term>Synopsis:</term>
1131           <listitem>
1132             <para>
1133              <emphasis role="bold">lapply</emphasis>
1134              [<emphasis role="bold">linear</emphasis>]
1135              [<emphasis role="bold">depth=</emphasis>&nat;]
1136              &sterm;
1137              [<emphasis role="bold">to</emphasis>
1138               &sterm;
1139               [<emphasis role="bold">,</emphasis>&sterm;…]
1140              ]
1141              [<emphasis role="bold">as</emphasis> &id;]
1142             </para>
1143           </listitem>
1144         </varlistentry>
1145         <varlistentry>
1146           <term>Pre-conditions:</term>
1147           <listitem>
1148             <para>
1149              <command>t</command> must have at least <command>d</command>
1150              independent premises and <command>n</command> must not be
1151              greater than <command>d</command>.
1152             </para>
1153           </listitem>
1154         </varlistentry>
1155         <varlistentry>
1156           <term>Action:</term>
1157           <listitem>
1158             <para>
1159              Invokes <command>letin H ≝ (t ? ... ?)</command>
1160              with enough <command>?</command>'s to reach the
1161              <command>d</command>-th independent premise of
1162              <command>t</command>
1163              (<command>d</command> is maximum if unspecified).
1164              Then istantiates (by <command>apply</command>) with
1165              t<subscript>1</subscript>, ..., t<subscript>n</subscript>
1166              the <command>?</command>'s corresponding to the first
1167              <command>n</command> independent premises of
1168              <command>t</command>.
1169              Usually the other <command>?</command>'s preceding the
1170              <command>n</command>-th independent premise of
1171              <command>t</command> are istantiated as a consequence.
1172              If the <command>linear</command> flag is specified and if
1173              <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
1174              are (applications of) premises in the current context, they are
1175              <command>clear</command>ed.
1176             </para>
1177           </listitem>
1178         </varlistentry>
1179         <varlistentry>
1180           <term>New sequents to prove:</term>
1181           <listitem>
1182             <para>
1183              The ones opened by the tactics <command>lapply</command> invokes.
1184             </para>
1185           </listitem>
1186         </varlistentry>
1187       </variablelist>
1188     </para>
1189   </sect1>
1190   <sect1 id="tac_left">
1191     <title>left</title>
1192     <titleabbrev>left</titleabbrev>
1193     <para><userinput>left </userinput></para>
1194     <para>
1195       <variablelist>
1196         <varlistentry role="tactic.synopsis">
1197           <term>Synopsis:</term>
1198           <listitem>
1199             <para><emphasis role="bold">left</emphasis></para>
1200           </listitem>
1201         </varlistentry>
1202         <varlistentry>
1203           <term>Pre-conditions:</term>
1204           <listitem>
1205             <para>The conclusion of the current sequent must be
1206              an inductive type or the application of an inductive type
1207              with at least one constructor.</para>
1208           </listitem>
1209         </varlistentry>
1210         <varlistentry>
1211           <term>Action:</term>
1212           <listitem>
1213             <para>Equivalent to <command>constructor 1</command>.</para>
1214           </listitem>
1215         </varlistentry>
1216         <varlistentry>
1217           <term>New sequents to prove:</term>
1218           <listitem>
1219             <para>It opens a new sequent for each premise of the first
1220              constructor of the inductive type that is the conclusion of the
1221              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1222           </listitem>
1223         </varlistentry>
1224       </variablelist>
1225     </para>
1226   </sect1>
1227   <sect1 id="tac_letin">
1228     <title>letin</title>
1229     <titleabbrev>letin</titleabbrev>
1230     <para><userinput>letin x ≝ t</userinput></para>
1231     <para>
1232       <variablelist>
1233         <varlistentry role="tactic.synopsis">
1234           <term>Synopsis:</term>
1235           <listitem>
1236             <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
1237           </listitem>
1238         </varlistentry>
1239         <varlistentry>
1240           <term>Pre-conditions:</term>
1241           <listitem>
1242             <para>None.</para>
1243           </listitem>
1244         </varlistentry>
1245         <varlistentry>
1246           <term>Action:</term>
1247           <listitem>
1248             <para>It adds to the context of the current sequent to prove a new
1249              definition <command>x ≝ t</command>.</para>
1250           </listitem>
1251         </varlistentry>
1252         <varlistentry>
1253           <term>New sequents to prove:</term>
1254           <listitem>
1255             <para>None.</para>
1256           </listitem>
1257         </varlistentry>
1258       </variablelist>
1259     </para>
1260   </sect1>
1261   <sect1 id="tac_normalize">
1262     <title>normalize</title>
1263     <titleabbrev>normalize</titleabbrev>
1264     <para><userinput>normalize patt</userinput></para>
1265     <para>
1266       <variablelist>
1267         <varlistentry role="tactic.synopsis">
1268           <term>Synopsis:</term>
1269           <listitem>
1270             <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
1271           </listitem>
1272         </varlistentry>
1273         <varlistentry>
1274           <term>Pre-conditions:</term>
1275           <listitem>
1276             <para>None.</para>
1277           </listitem>
1278         </varlistentry>
1279         <varlistentry>
1280           <term>Action:</term>
1281           <listitem>
1282             <para>It replaces all the terms matched by <command>patt</command>
1283              with their βδιζ-normal form.</para>
1284           </listitem>
1285         </varlistentry>
1286         <varlistentry>
1287           <term>New sequents to prove:</term>
1288           <listitem>
1289             <para>None.</para>
1290           </listitem>
1291         </varlistentry>
1292       </variablelist>
1293     </para>
1294   </sect1>
1295   <sect1 id="tac_reduce">
1296     <title>reduce</title>
1297     <titleabbrev>reduce</titleabbrev>
1298     <para><userinput>reduce patt</userinput></para>
1299     <para>
1300       <variablelist>
1301         <varlistentry role="tactic.synopsis">
1302           <term>Synopsis:</term>
1303           <listitem>
1304             <para><emphasis role="bold">reduce</emphasis> &pattern;</para>
1305           </listitem>
1306         </varlistentry>
1307         <varlistentry>
1308           <term>Pre-conditions:</term>
1309           <listitem>
1310             <para>None.</para>
1311           </listitem>
1312         </varlistentry>
1313         <varlistentry>
1314           <term>Action:</term>
1315           <listitem>
1316             <para>It replaces all the terms matched by <command>patt</command>
1317              with their βδιζ-normal form.</para>
1318           </listitem>
1319         </varlistentry>
1320         <varlistentry>
1321           <term>New sequents to prove:</term>
1322           <listitem>
1323             <para>None.</para>
1324           </listitem>
1325         </varlistentry>
1326       </variablelist>
1327     </para>
1328   </sect1>
1329   <sect1 id="tac_reflexivity">
1330     <title>reflexivity</title>
1331     <titleabbrev>reflexivity</titleabbrev>
1332     <para><userinput>reflexivity </userinput></para>
1333     <para>
1334       <variablelist>
1335         <varlistentry role="tactic.synopsis">
1336           <term>Synopsis:</term>
1337           <listitem>
1338             <para><emphasis role="bold">reflexivity</emphasis></para>
1339           </listitem>
1340         </varlistentry>
1341         <varlistentry>
1342           <term>Pre-conditions:</term>
1343           <listitem>
1344             <para>The conclusion of the current sequent must be
1345              <command>t=t</command> for some term <command>t</command></para>
1346           </listitem>
1347         </varlistentry>
1348         <varlistentry>
1349           <term>Action:</term>
1350           <listitem>
1351             <para>It closes the current sequent by reflexivity
1352              of equality.</para>
1353           </listitem>
1354         </varlistentry>
1355         <varlistentry>
1356           <term>New sequents to prove:</term>
1357           <listitem>
1358             <para>None.</para>
1359           </listitem>
1360         </varlistentry>
1361       </variablelist>
1362     </para>
1363   </sect1>
1364   <sect1 id="tac_replace">
1365     <title>replace</title>
1366     <titleabbrev>change</titleabbrev>
1367     <para><userinput>change patt with t</userinput></para>
1368     <para>
1369       <variablelist>
1370         <varlistentry role="tactic.synopsis">
1371           <term>Synopsis:</term>
1372           <listitem>
1373             <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
1374           </listitem>
1375         </varlistentry>
1376         <varlistentry>
1377           <term>Pre-conditions:</term>
1378           <listitem>
1379             <para>None.</para>
1380           </listitem>
1381         </varlistentry>
1382         <varlistentry>
1383           <term>Action:</term>
1384           <listitem>
1385             <para>It replaces the subterms of the current sequent matched by
1386              <command>patt</command> with the new term <command>t</command>.
1387              For each subterm matched by the pattern, <command>t</command> is
1388              disambiguated in the context of the subterm.</para>
1389           </listitem>
1390         </varlistentry>
1391         <varlistentry>
1392           <term>New sequents to prove:</term>
1393           <listitem>
1394             <para>For each matched term <command>t'</command> it opens
1395              a new sequent to prove whose conclusion is
1396              <command>t'=t</command>.</para>
1397           </listitem>
1398         </varlistentry>
1399       </variablelist>
1400     </para>
1401   </sect1>
1402   <sect1 id="tac_rewrite">
1403     <title>rewrite</title>
1404     <titleabbrev>rewrite</titleabbrev>
1405     <para><userinput>rewrite dir p patt</userinput></para>
1406     <para>
1407       <variablelist>
1408         <varlistentry role="tactic.synopsis">
1409           <term>Synopsis:</term>
1410           <listitem>
1411             <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold">&lt;</emphasis>|<emphasis role="bold">&gt;</emphasis>] &sterm; &pattern;</para>
1412           </listitem>
1413         </varlistentry>
1414         <varlistentry>
1415           <term>Pre-conditions:</term>
1416           <listitem>
1417             <para><command>p</command> must be the proof of an equality,
1418              possibly under some hypotheses.</para>
1419           </listitem>
1420         </varlistentry>
1421         <varlistentry>
1422           <term>Action:</term>
1423           <listitem>
1424             <para>It looks in every term matched by <command>patt</command>
1425              for all the occurrences of the
1426              left hand side of the equality that <command>p</command> proves
1427              (resp. the right hand side if <command>dir</command> is
1428              <command>&lt;</command>). Every occurence found is replaced with
1429              the opposite side of the equality.</para>
1430           </listitem>
1431         </varlistentry>
1432         <varlistentry>
1433           <term>New sequents to prove:</term>
1434           <listitem>
1435             <para>It opens one new sequent for each hypothesis of the
1436              equality proved by <command>p</command> that is not closed
1437              by unification.</para>
1438           </listitem>
1439         </varlistentry>
1440       </variablelist>
1441     </para>
1442   </sect1>
1443   <sect1 id="tac_right">
1444     <title>right</title>
1445     <titleabbrev>right</titleabbrev>
1446     <para><userinput>right </userinput></para>
1447     <para>
1448       <variablelist>
1449         <varlistentry role="tactic.synopsis">
1450           <term>Synopsis:</term>
1451           <listitem>
1452             <para><emphasis role="bold">right</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
1459              an inductive type or the application of an inductive type with
1460              at least two constructors.</para>
1461           </listitem>
1462         </varlistentry>
1463         <varlistentry>
1464           <term>Action:</term>
1465           <listitem>
1466             <para>Equivalent to <command>constructor 2</command>.</para>
1467           </listitem>
1468         </varlistentry>
1469         <varlistentry>
1470           <term>New sequents to prove:</term>
1471           <listitem>
1472             <para>It opens a new sequent for each premise of the second
1473              constructor of the inductive type that is the conclusion of the
1474              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1475           </listitem>
1476         </varlistentry>
1477       </variablelist>
1478     </para>
1479   </sect1>
1480   <sect1 id="tac_ring">
1481     <title>ring</title>
1482     <titleabbrev>ring</titleabbrev>
1483     <para><userinput>ring </userinput></para>
1484     <para>
1485       <variablelist>
1486         <varlistentry role="tactic.synopsis">
1487           <term>Synopsis:</term>
1488           <listitem>
1489             <para><emphasis role="bold">ring</emphasis></para>
1490           </listitem>
1491         </varlistentry>
1492         <varlistentry>
1493           <term>Pre-conditions:</term>
1494           <listitem>
1495             <para>The conclusion of the current sequent must be an
1496              equality over Coq's real numbers that can be proved using
1497              the ring properties of the real numbers only.</para>
1498           </listitem>
1499         </varlistentry>
1500         <varlistentry>
1501           <term>Action:</term>
1502           <listitem>
1503             <para>It closes the current sequent veryfying the equality by
1504              means of computation (i.e. this is a reflexive tactic, implemented
1505              exploiting the &quot;two level reasoning&quot; technique).</para>
1506           </listitem>
1507         </varlistentry>
1508         <varlistentry>
1509           <term>New sequents to prove:</term>
1510           <listitem>
1511             <para>None.</para>
1512           </listitem>
1513         </varlistentry>
1514       </variablelist>
1515     </para>
1516   </sect1>
1517   <sect1 id="tac_simplify">
1518     <title>simplify</title>
1519     <titleabbrev>simplify</titleabbrev>
1520     <para><userinput>simplify patt</userinput></para>
1521     <para>
1522       <variablelist>
1523         <varlistentry role="tactic.synopsis">
1524           <term>Synopsis:</term>
1525           <listitem>
1526             <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
1527           </listitem>
1528         </varlistentry>
1529         <varlistentry>
1530           <term>Pre-conditions:</term>
1531           <listitem>
1532             <para>None.</para>
1533           </listitem>
1534         </varlistentry>
1535         <varlistentry>
1536           <term>Action:</term>
1537           <listitem>
1538             <para>It replaces all the terms matched by <command>patt</command>
1539              with other convertible terms that are supposed to be simpler.</para>
1540           </listitem>
1541         </varlistentry>
1542         <varlistentry>
1543           <term>New sequents to prove:</term>
1544           <listitem>
1545             <para>None.</para>
1546           </listitem>
1547         </varlistentry>
1548       </variablelist>
1549     </para>
1550   </sect1>
1551   <sect1 id="tac_split">
1552     <title>split</title>
1553     <titleabbrev>split</titleabbrev>
1554     <para><userinput>split </userinput></para>
1555     <para>
1556       <variablelist>
1557         <varlistentry role="tactic.synopsis">
1558           <term>Synopsis:</term>
1559           <listitem>
1560             <para><emphasis role="bold">split</emphasis></para>
1561           </listitem>
1562         </varlistentry>
1563         <varlistentry>
1564           <term>Pre-conditions:</term>
1565           <listitem>
1566             <para>The conclusion of the current sequent must be
1567              an inductive type or the application of an inductive type with
1568              at least one constructor.</para>
1569           </listitem>
1570         </varlistentry>
1571         <varlistentry>
1572           <term>Action:</term>
1573           <listitem>
1574             <para>Equivalent to <command>constructor 1</command>.</para>
1575           </listitem>
1576         </varlistentry>
1577         <varlistentry>
1578           <term>New sequents to prove:</term>
1579           <listitem>
1580             <para>It opens a new sequent for each premise of the first
1581              constructor of the inductive type that is the conclusion of the
1582              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1583           </listitem>
1584         </varlistentry>
1585       </variablelist>
1586     </para>
1587   </sect1>
1588
1589   <sect1 id="tac_subst">
1590     <title>subst</title>
1591     <titleabbrev>subst</titleabbrev>
1592     <para><userinput>subst</userinput></para>
1593     <para>
1594       <variablelist>
1595         <varlistentry role="tactic.synopsis">
1596           <term>Synopsis:</term>
1597           <listitem>
1598             <para><emphasis role="bold">subst</emphasis></para>
1599           </listitem>
1600         </varlistentry>
1601         <varlistentry>
1602           <term>Pre-conditions:</term>
1603           <listitem><para>
1604             None.
1605           </para></listitem>
1606         </varlistentry>
1607         <varlistentry>
1608           <term>Action:</term>
1609           <listitem><para>
1610             For each premise of the form
1611             <command>H: x = t</command> or <command>H: t = x</command>
1612             where <command>x</command> is a local variable and
1613             <command>t</command> does not depend on <command>x</command>,
1614             the tactic rewrites <command>H</command> wherever
1615             <command>x</command> appears clearing <command>H</command> and
1616             <command>x</command> afterwards.
1617           </para></listitem>
1618         </varlistentry>
1619         <varlistentry>
1620           <term>New sequents to prove:</term>
1621           <listitem><para>
1622             The one opened by the applied tactics.
1623           </para></listitem>
1624         </varlistentry>
1625       </variablelist>
1626     </para>
1627   </sect1>
1628   <sect1 id="tac_symmetry">
1629     <title>symmetry</title>
1630     <titleabbrev>symmetry</titleabbrev>
1631     <para>The tactic <command>symmetry</command> </para>
1632     <para><userinput>symmetry </userinput></para>
1633     <para>
1634       <variablelist>
1635         <varlistentry role="tactic.synopsis">
1636           <term>Synopsis:</term>
1637           <listitem>
1638             <para><emphasis role="bold">symmetry</emphasis></para>
1639           </listitem>
1640         </varlistentry>
1641         <varlistentry>
1642           <term>Pre-conditions:</term>
1643           <listitem>
1644             <para>The conclusion of the current proof must be an equality.</para>
1645           </listitem>
1646         </varlistentry>
1647         <varlistentry>
1648           <term>Action:</term>
1649           <listitem>
1650             <para>It swaps the two sides of the equalityusing the symmetric
1651              property.</para>
1652           </listitem>
1653         </varlistentry>
1654         <varlistentry>
1655           <term>New sequents to prove:</term>
1656           <listitem>
1657             <para>None.</para>
1658           </listitem>
1659         </varlistentry>
1660       </variablelist>
1661     </para>
1662   </sect1>
1663   <sect1 id="tac_transitivity">
1664     <title>transitivity</title>
1665     <titleabbrev>transitivity</titleabbrev>
1666     <para><userinput>transitivity t</userinput></para>
1667     <para>
1668       <variablelist>
1669         <varlistentry role="tactic.synopsis">
1670           <term>Synopsis:</term>
1671           <listitem>
1672             <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
1673           </listitem>
1674         </varlistentry>
1675         <varlistentry>
1676           <term>Pre-conditions:</term>
1677           <listitem>
1678             <para>The conclusion of the current proof must be an equality.</para>
1679           </listitem>
1680         </varlistentry>
1681         <varlistentry>
1682           <term>Action:</term>
1683           <listitem>
1684             <para>It closes the current sequent by transitivity of the equality.</para>
1685           </listitem>
1686         </varlistentry>
1687         <varlistentry>
1688           <term>New sequents to prove:</term>
1689           <listitem>
1690             <para>It opens two new sequents <command>l=t</command> and
1691              <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
1692 the current sequent to prove.</para>
1693           </listitem>
1694         </varlistentry>
1695       </variablelist>
1696     </para>
1697   </sect1>
1698   <sect1 id="tac_unfold">
1699     <title>unfold</title>
1700     <titleabbrev>unfold</titleabbrev>
1701     <para><userinput>unfold t patt</userinput></para>
1702     <para>
1703       <variablelist>
1704         <varlistentry role="tactic.synopsis">
1705           <term>Synopsis:</term>
1706           <listitem>
1707             <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
1708           </listitem>
1709         </varlistentry>
1710         <varlistentry>
1711           <term>Pre-conditions:</term>
1712           <listitem>
1713             <para>None.</para>
1714           </listitem>
1715         </varlistentry>
1716         <varlistentry>
1717           <term>Action:</term>
1718           <listitem>
1719             <para>It finds all the occurrences of <command>t</command>
1720              (possibly applied to arguments) in the subterms matched by
1721              <command>patt</command>. Then it δ-expands each occurrence,
1722              also performing β-reduction of the obtained term. If
1723              <command>t</command> is omitted it defaults to each
1724              subterm matched by <command>patt</command>.</para>
1725           </listitem>
1726         </varlistentry>
1727         <varlistentry>
1728           <term>New sequents to prove:</term>
1729           <listitem>
1730             <para>None.</para>
1731           </listitem>
1732         </varlistentry>
1733       </variablelist>
1734     </para>
1735   </sect1>
1736   <sect1 id="tac_whd">
1737     <title>whd</title>
1738     <titleabbrev>whd</titleabbrev>
1739     <para><userinput>whd patt</userinput></para>
1740     <para>
1741       <variablelist>
1742         <varlistentry role="tactic.synopsis">
1743           <term>Synopsis:</term>
1744           <listitem>
1745             <para><emphasis role="bold">whd</emphasis> &pattern;</para>
1746           </listitem>
1747         </varlistentry>
1748         <varlistentry>
1749           <term>Pre-conditions:</term>
1750           <listitem>
1751             <para>None.</para>
1752           </listitem>
1753         </varlistentry>
1754         <varlistentry>
1755           <term>Action:</term>
1756           <listitem>
1757             <para>It replaces all the terms matched by <command>patt</command>
1758              with their βδιζ-weak-head normal form.</para>
1759           </listitem>
1760         </varlistentry>
1761         <varlistentry>
1762           <term>New sequents to prove:</term>
1763           <listitem>
1764             <para>None.</para>
1765           </listitem>
1766         </varlistentry>
1767       </variablelist>
1768     </para>
1769   </sect1>
1770
1771 </chapter>
1772