matita/help/C/sec_tactics.xml

   1
   2 <!-- ============ Tactics ====================== -->
   3 <chapter id="sec_tactics">
   4  <title>Tactics</title>
   5
   6   <sect1 id="tac_absurd">
   7     <title>absurd &lt;term&gt;</title>
   8     <titleabbrev>absurd</titleabbrev>
   9     <para><userinput>absurd P</userinput></para>
  10      <para>
  11       <variablelist>
  12         <varlistentry>
  13           <term>Pre-conditions:</term>
  14           <listitem>
  15             <para><command>P</command> must have type <command>Prop</command>.</para>
  16           </listitem>
  17         </varlistentry>
  18         <varlistentry>
  19           <term>Action:</term>
  20           <listitem>
  21             <para>it closes the current sequent by eliminating an
  22              absurd term.</para>
  23           </listitem>
  24         </varlistentry>
  25         <varlistentry>
  26           <term>New sequents to prove:</term>
  27           <listitem>
  28             <para>it opens two new sequents of conclusion <command>P</command>
  29              and <command>¬P</command>.</para>
  30           </listitem>
  31         </varlistentry>
  32       </variablelist>
  33      </para>
  34   </sect1>
  35   <sect1 id="tac_apply">
  36     <title>apply &lt;term&gt;</title>
  37     <titleabbrev>apply</titleabbrev>
  38     <para><userinput>apply t</userinput></para>
  39     <para>
  40       <variablelist>
  41         <varlistentry>
  42           <term>Pre-conditions:</term>
  43           <listitem>
  44             <para><command>t</command> must have type
  45              <command>T<subscript>1</subscript> → ... →
  46               T<subscript>n</subscript> → G</command>
  47              where <command>G</command> can be unified with the conclusion
  48              of the current sequent.</para>
  49           </listitem>
  50         </varlistentry>
  51         <varlistentry>
  52           <term>Action:</term>
  53           <listitem>
  54             <para>it closes the current sequent by applying <command>t</command> to <command>n</command> implicit arguments (that become new sequents).</para>
  55           </listitem>
  56         </varlistentry>
  57         <varlistentry>
  58           <term>New sequents to prove:</term>
  59           <listitem>
  60             <para>it opens a new sequent for each premise
  61              <command>T<subscript>i</subscript></command> that is not
  62              instantiated by unification. <command>T<subscript>i</subscript></command> is
  63              the conclusion of the <command>i</command>-th new sequent to
  64              prove.</para>
  65           </listitem>
  66         </varlistentry>
  67       </variablelist>
  68     </para>
  69   </sect1>
  70   <sect1 id="tac_assumption">
  71     <title>assumption</title>
  72     <titleabbrev>assumption</titleabbrev>
  73     <para><userinput>assumption </userinput></para>
  74     <para>
  75       <variablelist>
  76         <varlistentry>
  77           <term>Pre-conditions:</term>
  78           <listitem>
  79             <para>there must exist an hypothesis whose type can be unified with
  80              the conclusion of the current sequent.</para>
  81           </listitem>
  82         </varlistentry>
  83         <varlistentry>
  84           <term>Action:</term>
  85           <listitem>
  86             <para>it closes the current sequent exploiting an hypothesis.</para>
  87           </listitem>
  88         </varlistentry>
  89         <varlistentry>
  90           <term>New sequents to prove:</term>
  91           <listitem>
  92             <para>none</para>
  93           </listitem>
  94         </varlistentry>
  95       </variablelist>
  96     </para>
  97   </sect1>
  98   <sect1 id="tac_auto">
  99     <title>auto [depth=&lt;int&gt;] [width=&lt;int&gt;] [paramodulation] [full]</title>
 100     <titleabbrev>auto</titleabbrev>
 101     <para><userinput>auto depth=d width=w paramodulation full</userinput></para>
 102     <para>
 103       <variablelist>
 104         <varlistentry>
 105           <term>Pre-conditions:</term>
 106           <listitem>
 107             <para>none, but the tactic may fail finding a proof if every
 108              proof is in the search space that is pruned away. Pruning is
 109              controlled by <command>d</command> and <command>w</command>.
 110              Moreover, only lemmas whose type signature is a subset of the
 111              signature of the current sequent are considered. The signature of
 112              a sequent is ...TODO</para>
 113           </listitem>
 114         </varlistentry>
 115         <varlistentry>
 116           <term>Action:</term>
 117           <listitem>
 118             <para>it closes the current sequent by repeated application of
 119              rewriting steps (unless <command>paramodulation</command> is
 120              omitted), hypothesis and lemmas in the library.</para>
 121           </listitem>
 122         </varlistentry>
 123         <varlistentry>
 124           <term>New sequents to prove:</term>
 125           <listitem>
 126             <para>none</para>
 127           </listitem>
 128         </varlistentry>
 129       </variablelist>
 130     </para>
 131   </sect1>
 132   <sect1 id="tac_clear">
 133     <title>clear &lt;id&gt;</title>
 134     <titleabbrev>clear</titleabbrev>
 135     <para><userinput>clear H</userinput></para>
 136     <para>
 137       <variablelist>
 138         <varlistentry>
 139           <term>Pre-conditions:</term>
 140           <listitem>
 141             <para><command>H</command> must be an hypothesis of the
 142              current sequent to prove.</para>
 143           </listitem>
 144         </varlistentry>
 145         <varlistentry>
 146           <term>Action:</term>
 147           <listitem>
 148             <para>it hides the hypothesis <command>H</command> from the
 149              current sequent.</para>
 150           </listitem>
 151         </varlistentry>
 152         <varlistentry>
 153           <term>New sequents to prove:</term>
 154           <listitem>
 155             <para>none</para>
 156           </listitem>
 157         </varlistentry>
 158       </variablelist>
 159     </para>
 160   </sect1>
 161   <sect1 id="tac_clearbody">
 162     <title>clearbody &lt;id&gt;</title>
 163     <titleabbrev>clearbody</titleabbrev>
 164     <para><userinput>clearbody H</userinput></para>
 165     <para>
 166       <variablelist>
 167         <varlistentry>
 168           <term>Pre-conditions:</term>
 169           <listitem>
 170             <para><command>H</command> must be an hypothesis of the
 171              current sequent to prove.</para>
 172           </listitem>
 173         </varlistentry>
 174         <varlistentry>
 175           <term>Action:</term>
 176           <listitem>
 177             <para>it hides the definiens of a definition in the current
 178              sequent context. Thus the definition becomes an hypothesis.</para>
 179           </listitem>
 180         </varlistentry>
 181         <varlistentry>
 182           <term>New sequents to prove:</term>
 183           <listitem>
 184             <para>none.</para>
 185           </listitem>
 186         </varlistentry>
 187       </variablelist>
 188     </para>
 189   </sect1>
 190   <sect1 id="tac_change">
 191     <title>change &lt;pattern&gt; with &lt;term&gt;</title>
 192     <titleabbrev>change</titleabbrev>
 193     <para><userinput>change patt with t</userinput></para>
 194     <para>
 195       <variablelist>
 196         <varlistentry>
 197           <term>Pre-conditions:</term>
 198           <listitem>
 199             <para>each subterm matched by the pattern must be convertible
 200              with the term <command>t</command> disambiguated in the context
 201              of the matched subterm.</para>
 202           </listitem>
 203         </varlistentry>
 204         <varlistentry>
 205           <term>Action:</term>
 206           <listitem>
 207             <para>it replaces the subterms of the current sequent matched by
 208              <command>patt</command> with the new term <command>t</command>.
 209              For each subterm matched by the pattern, <command>t</command> is
 210              disambiguated in the context of the subterm.</para>
 211           </listitem>
 212         </varlistentry>
 213         <varlistentry>
 214           <term>New sequents to prove:</term>
 215           <listitem>
 216             <para>none.</para>
 217           </listitem>
 218         </varlistentry>
 219       </variablelist>
 220     </para>
 221   </sect1>
 222   <sect1 id="tac_constructor">
 223     <title>constructor &lt;int&gt;</title>
 224     <titleabbrev>constructor</titleabbrev>
 225     <para><userinput>constructor n</userinput></para>
 226     <para>
 227       <variablelist>
 228         <varlistentry>
 229           <term>Pre-conditions:</term>
 230           <listitem>
 231             <para>the conclusion of the current sequent must be
 232              an inductive type or the application of an inductive type with
 233              at least <command>n</command> constructors.</para>
 234           </listitem>
 235         </varlistentry>
 236         <varlistentry>
 237           <term>Action:</term>
 238           <listitem>
 239             <para>it applies the <command>n</command>-th constructor of the
 240              inductive type of the conclusion of the current sequent.</para>
 241           </listitem>
 242         </varlistentry>
 243         <varlistentry>
 244           <term>New sequents to prove:</term>
 245           <listitem>
 246             <para>it opens a new sequent for each premise of the constructor
 247              that can not be inferred by unification. For more details,
 248              see the <command>apply</command> tactic.</para>
 249           </listitem>
 250         </varlistentry>
 251       </variablelist>
 252     </para>
 253   </sect1>
 254   <sect1 id="tac_contradiction">
 255     <title>contradiction</title>
 256     <titleabbrev>contradiction</titleabbrev>
 257     <para><userinput>contradiction </userinput></para>
 258     <para>
 259       <variablelist>
 260         <varlistentry>
 261           <term>Pre-conditions:</term>
 262           <listitem>
 263             <para>there must be in the current context an hypothesis of type
 264              <command>False</command>.</para>
 265           </listitem>
 266         </varlistentry>
 267         <varlistentry>
 268           <term>Action:</term>
 269           <listitem>
 270             <para>it closes the current sequent by applying an hypothesis of
 271              type <command>False</command>.</para>
 272           </listitem>
 273         </varlistentry>
 274         <varlistentry>
 275           <term>New sequents to prove:</term>
 276           <listitem>
 277             <para>none</para>
 278           </listitem>
 279         </varlistentry>
 280       </variablelist>
 281     </para>
 282   </sect1>
 283   <sect1 id="tac_cut">
 284     <title>cut &lt;term&gt; [as &lt;id&gt;]</title>
 285     <titleabbrev>cut</titleabbrev>
 286     <para><userinput>cut P as H</userinput></para>
 287     <para>
 288       <variablelist>
 289         <varlistentry>
 290           <term>Pre-conditions:</term>
 291           <listitem>
 292             <para><command>P</command> must have type <command>Prop</command>.</para>
 293           </listitem>
 294         </varlistentry>
 295         <varlistentry>
 296           <term>Action:</term>
 297           <listitem>
 298             <para>it closes the current sequent.</para>
 299           </listitem>
 300         </varlistentry>
 301         <varlistentry>
 302           <term>New sequents to prove:</term>
 303           <listitem>
 304             <para>it opens two new sequents. The first one has an extra
 305              hypothesis <command>H:P</command>. If <command>H</command> is
 306              omitted, the name of the hypothesis is automatically generated.
 307              The second sequent has conclusion <command>P</command> and
 308              hypotheses the hypotheses of the current sequent to prove.</para>
 309           </listitem>
 310         </varlistentry>
 311       </variablelist>
 312     </para>
 313   </sect1>
 314   <sect1 id="tac_decompose">
 315     <title>decompose [&lt;ident list&gt;] &lt;ident&gt; [&lt;intros_spec&gt;]</title>
 316     <titleabbrev>decompose</titleabbrev>
 317     <para><userinput>decompose ???</userinput></para>
 318     <para>
 319       <variablelist>
 320         <varlistentry>
 321           <term>Pre-conditions:</term>
 322           <listitem>
 323             <para>TODO.</para>
 324           </listitem>
 325         </varlistentry>
 326         <varlistentry>
 327           <term>Action:</term>
 328           <listitem>
 329             <para>TODO.</para>
 330           </listitem>
 331         </varlistentry>
 332         <varlistentry>
 333           <term>New sequents to prove:</term>
 334           <listitem>
 335             <para>TODO.</para>
 336           </listitem>
 337         </varlistentry>
 338       </variablelist>
 339     </para>
 340   </sect1>
 341   <sect1 id="tac_discriminate">
 342     <title>discriminate &lt;term&gt;</title>
 343     <titleabbrev>discriminate</titleabbrev>
 344     <para><userinput>discriminate p</userinput></para>
 345     <para>
 346       <variablelist>
 347         <varlistentry>
 348           <term>Pre-conditions:</term>
 349           <listitem>
 350             <para><command>p</command> must have type <command>K t<subscript>1</subscript> ... t<subscript>n</subscript> = K' t'<subscript>1</subscript> ... t'<subscript>m</subscript></command> where <command>K</command> and <command>K'</command> must be different constructors of the same inductive type and each argument list can be empty if
 351 its constructor takes no arguments.</para>
 352           </listitem>
 353         </varlistentry>
 354         <varlistentry>
 355           <term>Action:</term>
 356           <listitem>
 357             <para>it closes the current sequent by proving the absurdity of
 358              <command>p</command>.</para>
 359           </listitem>
 360         </varlistentry>
 361         <varlistentry>
 362           <term>New sequents to prove:</term>
 363           <listitem>
 364             <para>none.</para>
 365           </listitem>
 366         </varlistentry>
 367       </variablelist>
 368     </para>
 369   </sect1>
 370   <sect1 id="tac_elim">
 371     <title>elim &lt;term&gt; [using &lt;term&gt;] [&lt;intros_spec&gt;]</title>
 372     <titleabbrev>elim</titleabbrev>
 373     <para><userinput>elim t using th hyps</userinput></para>
 374     <para>
 375       <variablelist>
 376         <varlistentry>
 377           <term>Pre-conditions:</term>
 378           <listitem>
 379             <para><command>t</command> must inhabit an inductive type and
 380              <command>th</command> must be an elimination principle for that
 381              inductive type. If <command>th</command> is omitted the appropriate
 382              standard elimination principle is chosen.</para>
 383           </listitem>
 384         </varlistentry>
 385         <varlistentry>
 386           <term>Action:</term>
 387           <listitem>
 388             <para>it proceeds by cases on the values of <command>t</command>,
 389              according to the elimination principle <command>th</command>.
 390             </para>
 391           </listitem>
 392         </varlistentry>
 393         <varlistentry>
 394           <term>New sequents to prove:</term>
 395           <listitem>
 396             <para>it opens one new sequent for each case. The names of
 397              the new hypotheses are picked by <command>hyps</command>, if
 398              provided.</para>
 399           </listitem>
 400         </varlistentry>
 401       </variablelist>
 402     </para>
 403   </sect1>
 404   <sect1 id="tac_elimType">
 405     <title>elimType &lt;term&gt; [using &lt;term&gt;] [&lt;intros_spec&gt;]</title>
 406     <titleabbrev>elimType</titleabbrev>
 407     <para><userinput>elimType T using th hyps</userinput></para>
 408     <para>
 409       <variablelist>
 410         <varlistentry>
 411           <term>Pre-conditions:</term>
 412           <listitem>
 413             <para><command>T</command> must be an inductive type.</para>
 414           </listitem>
 415         </varlistentry>
 416         <varlistentry>
 417           <term>Action:</term>
 418           <listitem>
 419             <para>TODO (severely bugged now).</para>
 420           </listitem>
 421         </varlistentry>
 422         <varlistentry>
 423           <term>New sequents to prove:</term>
 424           <listitem>
 425             <para>TODO</para>
 426           </listitem>
 427         </varlistentry>
 428       </variablelist>
 429     </para>
 430   </sect1>
 431   <sect1 id="tac_exact">
 432     <title>exact &lt;term&gt;</title>
 433     <titleabbrev>exact</titleabbrev>
 434     <para><userinput>exact p</userinput></para>
 435     <para>
 436       <variablelist>
 437         <varlistentry>
 438           <term>Pre-conditions:</term>
 439           <listitem>
 440             <para>the type of <command>p</command> must be convertible
 441              with the conclusion of the current sequent.</para>
 442           </listitem>
 443         </varlistentry>
 444         <varlistentry>
 445           <term>Action:</term>
 446           <listitem>
 447             <para>it closes the current sequent using <command>p</command>.</para>
 448           </listitem>
 449         </varlistentry>
 450         <varlistentry>
 451           <term>New sequents to prove:</term>
 452           <listitem>
 453             <para>none.</para>
 454           </listitem>
 455         </varlistentry>
 456       </variablelist>
 457     </para>
 458   </sect1>
 459   <sect1 id="tac_exists">
 460     <title>exists</title>
 461     <titleabbrev>exists</titleabbrev>
 462     <para><userinput>exists </userinput></para>
 463     <para>
 464       <variablelist>
 465         <varlistentry>
 466           <term>Pre-conditions:</term>
 467           <listitem>
 468             <para>the conclusion of the current sequent must be
 469              an inductive type or the application of an inductive type
 470              with at least one constructor.</para>
 471           </listitem>
 472         </varlistentry>
 473         <varlistentry>
 474           <term>Action:</term>
 475           <listitem>
 476             <para>equivalent to <command>constructor 1</command>.</para>
 477           </listitem>
 478         </varlistentry>
 479         <varlistentry>
 480           <term>New sequents to prove:</term>
 481           <listitem>
 482             <para>it opens a new sequent for each premise of the first
 483              constructor of the inductive type that is the conclusion of the
 484              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 485           </listitem>
 486         </varlistentry>
 487       </variablelist>
 488     </para>
 489   </sect1>
 490   <sect1 id="tac_fail">
 491     <title>fail </title>
 492     <titleabbrev>failt</titleabbrev>
 493     <para><userinput>fail</userinput></para>
 494     <para>
 495       <variablelist>
 496         <varlistentry>
 497           <term>Pre-conditions:</term>
 498           <listitem>
 499             <para>none.</para>
 500           </listitem>
 501         </varlistentry>
 502         <varlistentry>
 503           <term>Action:</term>
 504           <listitem>
 505             <para>this tactic always fail.</para>
 506           </listitem>
 507         </varlistentry>
 508         <varlistentry>
 509           <term>New sequents to prove:</term>
 510           <listitem>
 511             <para>N.A.</para>
 512           </listitem>
 513         </varlistentry>
 514       </variablelist>
 515     </para>
 516   </sect1>
 517   <sect1 id="tac_fold">
 518     <title>fold &lt;reduction_kind&gt; &lt;term&gt; &lt;pattern&gt;</title>
 519     <titleabbrev>fold</titleabbrev>
 520     <para><userinput>fold red t patt</userinput></para>
 521     <para>
 522       <variablelist>
 523         <varlistentry>
 524           <term>Pre-conditions:</term>
 525           <listitem>
 526             <para>the pattern must not specify the wanted term.</para>
 527           </listitem>
 528         </varlistentry>
 529         <varlistentry>
 530           <term>Action:</term>
 531           <listitem>
 532             <para>first of all it locates all the subterms matched by
 533              <command>patt</command>. In the context of each matched subterm
 534              it disambiguates the term <command>t</command> and reduces it
 535              to its <command>red</command> normal form; then it replaces with
 536              <command>t</command> every occurrence of the normal form in the
 537              matched subterm.</para>
 538           </listitem>
 539         </varlistentry>
 540         <varlistentry>
 541           <term>New sequents to prove:</term>
 542           <listitem>
 543             <para>none.</para>
 544           </listitem>
 545         </varlistentry>
 546       </variablelist>
 547     </para>
 548   </sect1>
 549   <sect1 id="tac_fourier">
 550     <title>fourier</title>
 551     <titleabbrev>fourier</titleabbrev>
 552     <para><userinput>fourier </userinput></para>
 553     <para>
 554       <variablelist>
 555         <varlistentry>
 556           <term>Pre-conditions:</term>
 557           <listitem>
 558             <para>the conclusion of the current sequent must be a linear
 559              inequation over real numbers taken from standard library of
 560              Coq. Moreover the inequations in the hypotheses must imply the
 561              inequation in the conclusion of the current sequent.</para>
 562           </listitem>
 563         </varlistentry>
 564         <varlistentry>
 565           <term>Action:</term>
 566           <listitem>
 567             <para>it closes the current sequent by applying the Fourier method.</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_fwd">
 580     <title>fwd &lt;ident&gt; [&lt;ident list&gt;]</title>
 581     <titleabbrev>fwd</titleabbrev>
 582     <para><userinput>fwd ...TODO</userinput></para>
 583     <para>
 584       <variablelist>
 585         <varlistentry>
 586           <term>Pre-conditions:</term>
 587           <listitem>
 588             <para>TODO.</para>
 589           </listitem>
 590         </varlistentry>
 591         <varlistentry>
 592           <term>Action:</term>
 593           <listitem>
 594             <para>TODO.</para>
 595           </listitem>
 596         </varlistentry>
 597         <varlistentry>
 598           <term>New sequents to prove:</term>
 599           <listitem>
 600             <para>TODO.</para>
 601           </listitem>
 602         </varlistentry>
 603       </variablelist>
 604     </para>
 605   </sect1>
 606   <sect1 id="tac_generalize">
 607     <title>generalize &lt;pattern&gt; [as &lt;id&gt;]</title>
 608     <titleabbrev>generalize</titleabbrev>
 609     <para><userinput>generalize patt as H</userinput></para>
 610     <para>
 611       <variablelist>
 612         <varlistentry>
 613           <term>Pre-conditions:</term>
 614           <listitem>
 615             <para>all the terms matched by <command>patt</command> must be
 616              convertible and close in the context of the current sequent.</para>
 617           </listitem>
 618         </varlistentry>
 619         <varlistentry>
 620           <term>Action:</term>
 621           <listitem>
 622             <para>it closes the current sequent by applying a stronger
 623              lemma that is proved using the new generated sequent.</para>
 624           </listitem>
 625         </varlistentry>
 626         <varlistentry>
 627           <term>New sequents to prove:</term>
 628           <listitem>
 629             <para>it opens a new sequent where the current sequent conclusion
 630              <command>G</command> is generalized to
 631              <command>∀x.G{x/t}</command> where <command>{x/t}</command>
 632              is a notation for the replacement with <command>x</command> of all
 633              the occurrences of the term <command>t</command> matched by
 634              <command>patt</command>. If <command>patt</command> matches no
 635              subterm then <command>t</command> is defined as the
 636              <command>wanted</command> part of the pattern.</para>
 637           </listitem>
 638         </varlistentry>
 639       </variablelist>
 640     </para>
 641   </sect1>
 642   <sect1 id="tac_id">
 643     <title>id</title>
 644     <titleabbrev>id</titleabbrev>
 645     <para><userinput>id </userinput></para>
 646     <para>
 647       <variablelist>
 648         <varlistentry>
 649           <term>Pre-conditions:</term>
 650           <listitem>
 651             <para>none.</para>
 652           </listitem>
 653         </varlistentry>
 654         <varlistentry>
 655           <term>Action:</term>
 656           <listitem>
 657             <para>this identity tactic does nothing without failing.</para>
 658           </listitem>
 659         </varlistentry>
 660         <varlistentry>
 661           <term>New sequents to prove:</term>
 662           <listitem>
 663             <para>none.</para>
 664           </listitem>
 665         </varlistentry>
 666       </variablelist>
 667     </para>
 668   </sect1>
 669   <sect1 id="tac_injection">
 670     <title>injection &lt;term&gt;</title>
 671     <titleabbrev>injection</titleabbrev>
 672     <para><userinput>injection p</userinput></para>
 673     <para>
 674       <variablelist>
 675         <varlistentry>
 676           <term>Pre-conditions:</term>
 677           <listitem>
 678             <para><command>p</command> must have type <command>K t<subscript>1</subscript> ... t<subscript>n</subscript> = K t'<subscript>1</subscript> ... t'<subscript>n</subscript></command> where both argument lists are empty if
 679 <command>K</command> takes no arguments.</para>
 680           </listitem>
 681         </varlistentry>
 682         <varlistentry>
 683           <term>Action:</term>
 684           <listitem>
 685             <para>it derives new hypotheses by injectivity of
 686              <command>K</command>.</para>
 687           </listitem>
 688         </varlistentry>
 689         <varlistentry>
 690           <term>New sequents to prove:</term>
 691           <listitem>
 692             <para>the new sequent to prove is equal to the current sequent
 693              with the additional hypotheses
 694              <command>t<subscript>1</subscript>=t'<subscript>1</subscript></command> ... <command>t<subscript>n</subscript>=t'<subscript>n</subscript></command>.</para>
 695           </listitem>
 696         </varlistentry>
 697       </variablelist>
 698     </para>
 699   </sect1>
 700   <sect1 id="tac_intro">
 701     <title>intro [&lt;ident&gt;]</title>
 702     <titleabbrev>intro</titleabbrev>
 703     <para><userinput>intro H</userinput></para>
 704     <para>
 705       <variablelist>
 706         <varlistentry>
 707           <term>Pre-conditions:</term>
 708           <listitem>
 709             <para>the conclusion of the sequent to prove must be an implication
 710              or a universal quantification.</para>
 711           </listitem>
 712         </varlistentry>
 713         <varlistentry>
 714           <term>Action:</term>
 715           <listitem>
 716             <para>it applies the right introduction rule for implication,
 717              closing the current sequent.</para>
 718           </listitem>
 719         </varlistentry>
 720         <varlistentry>
 721           <term>New sequents to prove:</term>
 722           <listitem>
 723             <para>it opens a new sequent to prove adding to the hypothesis
 724              the antecedent of the implication and setting the conclusion
 725              to the consequent of the implicaiton. The name of the new
 726              hypothesis is <command>H</command> if provided; otherwise it
 727              is automatically generated.</para>
 728           </listitem>
 729         </varlistentry>
 730       </variablelist>
 731     </para>
 732   </sect1>
 733   <sect1 id="tac_intros">
 734     <title>intros &lt;intros_spec&gt;</title>
 735     <titleabbrev>intros</titleabbrev>
 736     <para><userinput>intros hyps</userinput></para>
 737     <para>
 738       <variablelist>
 739         <varlistentry>
 740           <term>Pre-conditions:</term>
 741           <listitem>
 742             <para>If <command>hyps</command> specifies a number of hypotheses
 743              to introduce, then the conclusion of the current sequent must
 744              be formed by at least that number of imbricated implications
 745              or universal quantifications.</para>
 746           </listitem>
 747         </varlistentry>
 748         <varlistentry>
 749           <term>Action:</term>
 750           <listitem>
 751             <para>it applies several times the right introduction rule for
 752              implication, closing the current sequent.</para>
 753           </listitem>
 754         </varlistentry>
 755         <varlistentry>
 756           <term>New sequents to prove:</term>
 757           <listitem>
 758             <para>it opens a new sequent to prove adding a number of new
 759              hypotheses equal to the number of new hypotheses requested.
 760              If the user does not request a precise number of new hypotheses,
 761              it adds as many hypotheses as possible.
 762              The name of each new hypothesis is either popped from the
 763              user provided list of names, or it is automatically generated when
 764              the list is (or becomes) empty.</para>
 765           </listitem>
 766         </varlistentry>
 767       </variablelist>
 768     </para>
 769   </sect1>
 770   <sect1 id="tac_inversion">
 771     <title>inversion &lt;term&gt;</title>
 772     <titleabbrev>inversion</titleabbrev>
 773     <para><userinput>inversion t</userinput></para>
 774     <para>
 775       <variablelist>
 776         <varlistentry>
 777           <term>Pre-conditions:</term>
 778           <listitem>
 779             <para>the type of the term <command>t</command> must be an inductive
 780              type or the application of an inductive type.</para>
 781           </listitem>
 782         </varlistentry>
 783         <varlistentry>
 784           <term>Action:</term>
 785           <listitem>
 786             <para>it proceeds by cases on <command>t</command> paying attention
 787              to the constraints imposed by the actual &quot;right arguments&quot;
 788              of the inductive type.</para>
 789           </listitem>
 790         </varlistentry>
 791         <varlistentry>
 792           <term>New sequents to prove:</term>
 793           <listitem>
 794             <para>it opens one new sequent to prove for each case in the
 795              definition of the type of <command>t</command>. With respect to
 796              a simple elimination, each new sequent has additional hypotheses
 797              that states the equalities of the &quot;right parameters&quot;
 798              of the inductive type with terms originally present in the
 799              sequent to prove.</para>
 800           </listitem>
 801         </varlistentry>
 802       </variablelist>
 803     </para>
 804   </sect1>
 805   <sect1 id="tac_lapply">
 806     <title>lapply [depth=&lt;int&gt;] &lt;term&gt; [to &lt;term list] [using &lt;ident&gt;]</title>
 807     <titleabbrev>lapply</titleabbrev>
 808     <para><userinput>lapply ???</userinput></para>
 809     <para>
 810       <variablelist>
 811         <varlistentry>
 812           <term>Pre-conditions:</term>
 813           <listitem>
 814             <para>TODO.</para>
 815           </listitem>
 816         </varlistentry>
 817         <varlistentry>
 818           <term>Action:</term>
 819           <listitem>
 820             <para>TODO.</para>
 821           </listitem>
 822         </varlistentry>
 823         <varlistentry>
 824           <term>New sequents to prove:</term>
 825           <listitem>
 826             <para>TODO.</para>
 827           </listitem>
 828         </varlistentry>
 829       </variablelist>
 830     </para>
 831   </sect1>
 832   <sect1 id="tac_left">
 833     <title>left</title>
 834     <titleabbrev>left</titleabbrev>
 835     <para><userinput>left </userinput></para>
 836     <para>
 837       <variablelist>
 838         <varlistentry>
 839           <term>Pre-conditions:</term>
 840           <listitem>
 841             <para>the conclusion of the current sequent must be
 842              an inductive type or the application of an inductive type
 843              with at least one constructor.</para>
 844           </listitem>
 845         </varlistentry>
 846         <varlistentry>
 847           <term>Action:</term>
 848           <listitem>
 849             <para>equivalent to <command>constructor 1</command>.</para>
 850           </listitem>
 851         </varlistentry>
 852         <varlistentry>
 853           <term>New sequents to prove:</term>
 854           <listitem>
 855             <para>it opens a new sequent for each premise of the first
 856              constructor of the inductive type that is the conclusion of the
 857              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 858           </listitem>
 859         </varlistentry>
 860       </variablelist>
 861     </para>
 862   </sect1>
 863   <sect1 id="tac_letin">
 864     <title>letin &lt;ident&gt; ≝ &lt;term&gt;</title>
 865     <titleabbrev>letin</titleabbrev>
 866     <para><userinput>letin x ≝ t</userinput></para>
 867     <para>
 868       <variablelist>
 869         <varlistentry>
 870           <term>Pre-conditions:</term>
 871           <listitem>
 872             <para>none.</para>
 873           </listitem>
 874         </varlistentry>
 875         <varlistentry>
 876           <term>Action:</term>
 877           <listitem>
 878             <para>it adds to the context of the current sequent to prove a new
 879              definition <command>x ≝ t</command>.</para>
 880           </listitem>
 881         </varlistentry>
 882         <varlistentry>
 883           <term>New sequents to prove:</term>
 884           <listitem>
 885             <para>none.</para>
 886           </listitem>
 887         </varlistentry>
 888       </variablelist>
 889     </para>
 890   </sect1>
 891   <sect1 id="tac_normalize">
 892     <title>normalize &lt;pattern&gt;</title>
 893     <titleabbrev>normalize</titleabbrev>
 894     <para><userinput>normalize patt</userinput></para>
 895     <para>
 896       <variablelist>
 897         <varlistentry>
 898           <term>Pre-conditions:</term>
 899           <listitem>
 900             <para>none.</para>
 901           </listitem>
 902         </varlistentry>
 903         <varlistentry>
 904           <term>Action:</term>
 905           <listitem>
 906             <para>it replaces all the terms matched by <command>patt</command>
 907              with their βδιζ-normal form.</para>
 908           </listitem>
 909         </varlistentry>
 910         <varlistentry>
 911           <term>New sequents to prove:</term>
 912           <listitem>
 913             <para>none.</para>
 914           </listitem>
 915         </varlistentry>
 916       </variablelist>
 917     </para>
 918   </sect1>
 919   <sect1 id="tac_paramodulation">
 920     <title>paramodulation &lt;pattern&gt;</title>
 921     <titleabbrev>paramodulation</titleabbrev>
 922     <para><userinput>paramodulation patt</userinput></para>
 923     <para>
 924       <variablelist>
 925         <varlistentry>
 926           <term>Pre-conditions:</term>
 927           <listitem>
 928             <para>TODO.</para>
 929           </listitem>
 930         </varlistentry>
 931         <varlistentry>
 932           <term>Action:</term>
 933           <listitem>
 934             <para>TODO.</para>
 935           </listitem>
 936         </varlistentry>
 937         <varlistentry>
 938           <term>New sequents to prove:</term>
 939           <listitem>
 940             <para>TODO.</para>
 941           </listitem>
 942         </varlistentry>
 943       </variablelist>
 944     </para>
 945   </sect1>
 946   <sect1 id="tac_reduce">
 947     <title>reduce &lt;pattern&gt;</title>
 948     <titleabbrev>reduce</titleabbrev>
 949     <para><userinput>reduce patt</userinput></para>
 950     <para>
 951       <variablelist>
 952         <varlistentry>
 953           <term>Pre-conditions:</term>
 954           <listitem>
 955             <para>none.</para>
 956           </listitem>
 957         </varlistentry>
 958         <varlistentry>
 959           <term>Action:</term>
 960           <listitem>
 961             <para>it replaces all the terms matched by <command>patt</command>
 962              with their βδιζ-normal form.</para>
 963           </listitem>
 964         </varlistentry>
 965         <varlistentry>
 966           <term>New sequents to prove:</term>
 967           <listitem>
 968             <para>none.</para>
 969           </listitem>
 970         </varlistentry>
 971       </variablelist>
 972     </para>
 973   </sect1>
 974   <sect1 id="tac_reflexivity">
 975     <title>reflexivity</title>
 976     <titleabbrev>reflexivity</titleabbrev>
 977     <para><userinput>reflexivity </userinput></para>
 978     <para>
 979       <variablelist>
 980         <varlistentry>
 981           <term>Pre-conditions:</term>
 982           <listitem>
 983             <para>the conclusion of the current sequent must be
 984              <command>t=t</command> for some term <command>t</command></para>
 985           </listitem>
 986         </varlistentry>
 987         <varlistentry>
 988           <term>Action:</term>
 989           <listitem>
 990             <para>it closes the current sequent by reflexivity
 991              of equality.</para>
 992           </listitem>
 993         </varlistentry>
 994         <varlistentry>
 995           <term>New sequents to prove:</term>
 996           <listitem>
 997             <para>none.</para>
 998           </listitem>
 999         </varlistentry>
1000       </variablelist>
1001     </para>
1002   </sect1>
1003   <sect1 id="tac_replace">
1004     <title>replace &lt;pattern&gt; with &lt;term&gt;</title>
1005     <titleabbrev>change</titleabbrev>
1006     <para><userinput>change patt with t</userinput></para>
1007     <para>
1008       <variablelist>
1009         <varlistentry>
1010           <term>Pre-conditions:</term>
1011           <listitem>
1012             <para>none.</para>
1013           </listitem>
1014         </varlistentry>
1015         <varlistentry>
1016           <term>Action:</term>
1017           <listitem>
1018             <para>it replaces the subterms of the current sequent matched by
1019              <command>patt</command> with the new term <command>t</command>.
1020              For each subterm matched by the pattern, <command>t</command> is
1021              disambiguated in the context of the subterm.</para>
1022           </listitem>
1023         </varlistentry>
1024         <varlistentry>
1025           <term>New sequents to prove:</term>
1026           <listitem>
1027             <para>for each matched term <command>t'</command> it opens
1028              a new sequent to prove whose conclusion is
1029              <command>t'=t</command>.</para>
1030           </listitem>
1031         </varlistentry>
1032       </variablelist>
1033     </para>
1034   </sect1>
1035   <sect1 id="tac_rewrite">
1036     <title>rewrite {&lt;|&gt;} &lt;term&gt; &lt;pattern&gt;</title>
1037     <titleabbrev>rewrite</titleabbrev>
1038     <para><userinput>rewrite dir p patt</userinput></para>
1039     <para>
1040       <variablelist>
1041         <varlistentry>
1042           <term>Pre-conditions:</term>
1043           <listitem>
1044             <para><command>p</command> must be the proof of an equality,
1045              possibly under some hypotheses.</para>
1046           </listitem>
1047         </varlistentry>
1048         <varlistentry>
1049           <term>Action:</term>
1050           <listitem>
1051             <para>it looks in every term matched by <command>patt</command>
1052              for all the occurrences of the
1053              left hand side of the equality that <command>p</command> proves
1054              (resp. the right hand side if <command>dir</command> is
1055              <command>&lt;</command>). Every occurence found is replaced with
1056              the opposite side of the equality.</para>
1057           </listitem>
1058         </varlistentry>
1059         <varlistentry>
1060           <term>New sequents to prove:</term>
1061           <listitem>
1062             <para>it opens one new sequent for each hypothesis of the
1063              equality proved by <command>p</command> that is not closed
1064              by unification.</para>
1065           </listitem>
1066         </varlistentry>
1067       </variablelist>
1068     </para>
1069   </sect1>
1070   <sect1 id="tac_right">
1071     <title>right</title>
1072     <titleabbrev>right</titleabbrev>
1073     <para><userinput>right </userinput></para>
1074     <para>
1075       <variablelist>
1076         <varlistentry>
1077           <term>Pre-conditions:</term>
1078           <listitem>
1079             <para>the conclusion of the current sequent must be
1080              an inductive type or the application of an inductive type with
1081              at least two constructors.</para>
1082           </listitem>
1083         </varlistentry>
1084         <varlistentry>
1085           <term>Action:</term>
1086           <listitem>
1087             <para>equivalent to <command>constructor 2</command>.</para>
1088           </listitem>
1089         </varlistentry>
1090         <varlistentry>
1091           <term>New sequents to prove:</term>
1092           <listitem>
1093             <para>it opens a new sequent for each premise of the second
1094              constructor of the inductive type that is the conclusion of the
1095              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1096           </listitem>
1097         </varlistentry>
1098       </variablelist>
1099     </para>
1100   </sect1>
1101   <sect1 id="tac_ring">
1102     <title>ring</title>
1103     <titleabbrev>ring</titleabbrev>
1104     <para><userinput>ring </userinput></para>
1105     <para>
1106       <variablelist>
1107         <varlistentry>
1108           <term>Pre-conditions:</term>
1109           <listitem>
1110             <para>the conclusion of the current sequent must be an
1111              equality over Coq's real numbers that can be proved using
1112              the ring properties of the real numbers only.</para>
1113           </listitem>
1114         </varlistentry>
1115         <varlistentry>
1116           <term>Action:</term>
1117           <listitem>
1118             <para>it closes the current sequent veryfying the equality by
1119              means of computation (i.e. this is a reflexive tactic, implemented
1120              exploiting the &quot;two level reasoning&quot; technique).</para>
1121           </listitem>
1122         </varlistentry>
1123         <varlistentry>
1124           <term>New sequents to prove:</term>
1125           <listitem>
1126             <para>none.</para>
1127           </listitem>
1128         </varlistentry>
1129       </variablelist>
1130     </para>
1131   </sect1>
1132   <sect1 id="tac_simplify">
1133     <title>simplify &lt;pattern&gt;</title>
1134     <titleabbrev>simplify</titleabbrev>
1135     <para><userinput>simplify patt</userinput></para>
1136     <para>
1137       <variablelist>
1138         <varlistentry>
1139           <term>Pre-conditions:</term>
1140           <listitem>
1141             <para>none.</para>
1142           </listitem>
1143         </varlistentry>
1144         <varlistentry>
1145           <term>Action:</term>
1146           <listitem>
1147             <para>it replaces all the terms matched by <command>patt</command>
1148              with other convertible terms that are supposed to be simpler.</para>
1149           </listitem>
1150         </varlistentry>
1151         <varlistentry>
1152           <term>New sequents to prove:</term>
1153           <listitem>
1154             <para>none.</para>
1155           </listitem>
1156         </varlistentry>
1157       </variablelist>
1158     </para>
1159   </sect1>
1160   <sect1 id="tac_split">
1161     <title>split</title>
1162     <titleabbrev>split</titleabbrev>
1163     <para><userinput>split </userinput></para>
1164     <para>
1165       <variablelist>
1166         <varlistentry>
1167           <term>Pre-conditions:</term>
1168           <listitem>
1169             <para>the conclusion of the current sequent must be
1170              an inductive type or the application of an inductive type with
1171              at least one constructor.</para>
1172           </listitem>
1173         </varlistentry>
1174         <varlistentry>
1175           <term>Action:</term>
1176           <listitem>
1177             <para>equivalent to <command>constructor 1</command>.</para>
1178           </listitem>
1179         </varlistentry>
1180         <varlistentry>
1181           <term>New sequents to prove:</term>
1182           <listitem>
1183             <para>it opens a new sequent for each premise of the first
1184              constructor of the inductive type that is the conclusion of the
1185              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1186           </listitem>
1187         </varlistentry>
1188       </variablelist>
1189     </para>
1190   </sect1>
1191   <sect1 id="tac_symmetry">
1192     <title>symmetry</title>
1193     <titleabbrev>symmetry</titleabbrev>
1194     <para>The tactic <command>symmetry</command> </para>
1195     <para><userinput>symmetry </userinput></para>
1196     <para>
1197       <variablelist>
1198         <varlistentry>
1199           <term>Pre-conditions:</term>
1200           <listitem>
1201             <para>the conclusion of the current proof must be an equality.</para>
1202           </listitem>
1203         </varlistentry>
1204         <varlistentry>
1205           <term>Action:</term>
1206           <listitem>
1207             <para>it swaps the two sides of the equalityusing the symmetric
1208              property.</para>
1209           </listitem>
1210         </varlistentry>
1211         <varlistentry>
1212           <term>New sequents to prove:</term>
1213           <listitem>
1214             <para>none.</para>
1215           </listitem>
1216         </varlistentry>
1217       </variablelist>
1218     </para>
1219   </sect1>
1220   <sect1 id="tac_transitivity">
1221     <title>transitivity &lt;term&gt;</title>
1222     <titleabbrev>transitivity</titleabbrev>
1223     <para><userinput>transitivity t</userinput></para>
1224     <para>
1225       <variablelist>
1226         <varlistentry>
1227           <term>Pre-conditions:</term>
1228           <listitem>
1229             <para>the conclusion of the current proof must be an equality.</para>
1230           </listitem>
1231         </varlistentry>
1232         <varlistentry>
1233           <term>Action:</term>
1234           <listitem>
1235             <para>it closes the current sequent by transitivity of the equality.</para>
1236           </listitem>
1237         </varlistentry>
1238         <varlistentry>
1239           <term>New sequents to prove:</term>
1240           <listitem>
1241             <para>it opens two new sequents <command>l=t</command> and
1242              <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
1243 the current sequent to prove.</para>
1244           </listitem>
1245         </varlistentry>
1246       </variablelist>
1247     </para>
1248   </sect1>
1249   <sect1 id="tac_unfold">
1250     <title>unfold [&lt;term&gt;] &lt;pattern&gt;</title>
1251     <titleabbrev>unfold</titleabbrev>
1252     <para><userinput>unfold t patt</userinput></para>
1253     <para>
1254       <variablelist>
1255         <varlistentry>
1256           <term>Pre-conditions:</term>
1257           <listitem>
1258             <para>none.</para>
1259           </listitem>
1260         </varlistentry>
1261         <varlistentry>
1262           <term>Action:</term>
1263           <listitem>
1264             <para>it finds all the occurrences of <command>t</command>
1265              (possibly applied to arguments) in the subterms matched by
1266              <command>patt</command>. Then it δ-expands each occurrence,
1267              also performing β-reduction of the obtained term. If
1268              <command>t</command> is omitted it defaults to each
1269              subterm matched by <command>patt</command>.</para>
1270           </listitem>
1271         </varlistentry>
1272         <varlistentry>
1273           <term>New sequents to prove:</term>
1274           <listitem>
1275             <para>none.</para>
1276           </listitem>
1277         </varlistentry>
1278       </variablelist>
1279     </para>
1280   </sect1>
1281   <sect1 id="tac_whd">
1282     <title>whd &lt;pattern&gt;</title>
1283     <titleabbrev>whd</titleabbrev>
1284     <para><userinput>whd patt</userinput></para>
1285     <para>
1286       <variablelist>
1287         <varlistentry>
1288           <term>Pre-conditions:</term>
1289           <listitem>
1290             <para>none.</para>
1291           </listitem>
1292         </varlistentry>
1293         <varlistentry>
1294           <term>Action:</term>
1295           <listitem>
1296             <para>it replaces all the terms matched by <command>patt</command>
1297              with their βδιζ-weak-head normal form.</para>
1298           </listitem>
1299         </varlistentry>
1300         <varlistentry>
1301           <term>New sequents to prove:</term>
1302           <listitem>
1303             <para>none.</para>
1304           </listitem>
1305         </varlistentry>
1306       </variablelist>
1307     </para>
1308   </sect1>
1309
1310 </chapter>
1311