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 (T<subscript>1</subscript> ... T<subscript>n</subscript>)
 500      H as H<subscript>1</subscript> ... H<subscript>m</subscript>
 501     </userinput></para>
 502     <para>
 503       <variablelist>
 504         <varlistentry role="tactic.synopsis">
 505           <term>Synopsis:</term>
 506           <listitem>
 507             <para>
 508              <emphasis role="bold">decompose</emphasis>
 509              [<emphasis role="bold">(</emphasis>
 510              &id;…
 511              <emphasis role="bold">)</emphasis>]
 512              [&id;]
 513              [<emphasis role="bold">as</emphasis> &id;…]
 514             </para>
 515           </listitem>
 516         </varlistentry>
 517         <varlistentry>
 518           <term>Pre-conditions:</term>
 519           <listitem>
 520             <para>
 521              <command>H</command> must inhabit one inductive type among
 522              <command>
 523               T<subscript>1</subscript> ... T<subscript>n</subscript>
 524              </command>
 525              and the types of a predefined list.
 526             </para>
 527           </listitem>
 528         </varlistentry>
 529         <varlistentry>
 530           <term>Action:</term>
 531           <listitem>
 532             <para>
 533              Runs <command>
 534               elim H H<subscript>1</subscript> ... H<subscript>m</subscript>
 535              </command>, clears <command>H</command> and tries to run itself
 536              recursively on each new identifier introduced by
 537              <command>elim</command> in the opened sequents.
 538              If <command>H</command> is not provided tries this operation on
 539              each premise in the current context.
 540             </para>
 541           </listitem>
 542         </varlistentry>
 543         <varlistentry>
 544           <term>New sequents to prove:</term>
 545           <listitem>
 546             <para>
 547              The ones generated by all the <command>elim</command> tactics run.
 548             </para>
 549           </listitem>
 550         </varlistentry>
 551       </variablelist>
 552     </para>
 553   </sect1>
 554   <sect1 id="tac_demodulate">
 555     <title>demodulate</title>
 556     <titleabbrev>demodulate</titleabbrev>
 557     <para><userinput>demodulate</userinput></para>
 558     <para>
 559       <variablelist>
 560         <varlistentry role="tactic.synopsis">
 561           <term>Synopsis:</term>
 562           <listitem>
 563             <para><emphasis role="bold">demodulate</emphasis></para>
 564           </listitem>
 565         </varlistentry>
 566         <varlistentry>
 567           <term>Pre-conditions:</term>
 568           <listitem>
 569             <para>None.</para>
 570           </listitem>
 571         </varlistentry>
 572         <varlistentry>
 573           <term>Action:</term>
 574           <listitem>
 575             <para>&TODO;</para>
 576           </listitem>
 577         </varlistentry>
 578         <varlistentry>
 579           <term>New sequents to prove:</term>
 580           <listitem>
 581             <para>None.</para>
 582           </listitem>
 583         </varlistentry>
 584       </variablelist>
 585     </para>
 586   </sect1>
 587   <sect1 id="tac_destruct">
 588     <title>destruct</title>
 589     <titleabbrev>destruct</titleabbrev>
 590     <para><userinput>destruct p</userinput></para>
 591     <para>
 592       <variablelist>
 593         <varlistentry role="tactic.synopsis">
 594           <term>Synopsis:</term>
 595           <listitem>
 596             <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
 597           </listitem>
 598         </varlistentry>
 599         <varlistentry>
 600           <term>Pre-conditions:</term>
 601           <listitem>
 602             <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>
 603           </listitem>
 604         </varlistentry>
 605         <varlistentry>
 606           <term>Action:</term>
 607           <listitem>
 608             <para>The tactic recursively compare the two sides of the equality
 609              looking for different constructors in corresponding position.
 610              If two of them are found, the tactic closes the current sequent
 611              by proving the absurdity of <command>p</command>. Otherwise
 612              it adds a new hypothesis for each leaf of the formula that
 613              states the equality of the subformulae in the corresponding
 614              positions on the two sides of the equality.
 615             </para>
 616           </listitem>
 617         </varlistentry>
 618         <varlistentry>
 619           <term>New sequents to prove:</term>
 620           <listitem>
 621             <para>None.</para>
 622           </listitem>
 623         </varlistentry>
 624       </variablelist>
 625     </para>
 626   </sect1>
 627   <sect1 id="tac_elim">
 628     <title>elim</title>
 629     <titleabbrev>elim</titleabbrev>
 630     <para><userinput>elim t using th hyps</userinput></para>
 631     <para>
 632       <variablelist>
 633         <varlistentry role="tactic.synopsis">
 634           <term>Synopsis:</term>
 635           <listitem>
 636             <para><emphasis role="bold">elim</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 637           </listitem>
 638         </varlistentry>
 639         <varlistentry>
 640           <term>Pre-conditions:</term>
 641           <listitem>
 642             <para><command>t</command> must inhabit an inductive type and
 643              <command>th</command> must be an elimination principle for that
 644              inductive type. If <command>th</command> is omitted the appropriate
 645              standard elimination principle is chosen.</para>
 646           </listitem>
 647         </varlistentry>
 648         <varlistentry>
 649           <term>Action:</term>
 650           <listitem>
 651             <para>It proceeds by cases on the values of <command>t</command>,
 652              according to the elimination principle <command>th</command>.
 653             </para>
 654           </listitem>
 655         </varlistentry>
 656         <varlistentry>
 657           <term>New sequents to prove:</term>
 658           <listitem>
 659             <para>It opens one new sequent for each case. The names of
 660              the new hypotheses are picked by <command>hyps</command>, if
 661              provided. If hyps specifies also a number of hypotheses that
 662              is less than the number of new hypotheses for a new sequent,
 663              then the exceeding hypothesis will be kept as implications in
 664              the conclusion of the sequent.</para>
 665           </listitem>
 666         </varlistentry>
 667       </variablelist>
 668     </para>
 669   </sect1>
 670   <sect1 id="tac_elimType">
 671     <title>elimType</title>
 672     <titleabbrev>elimType</titleabbrev>
 673     <para><userinput>elimType T using th hyps</userinput></para>
 674     <para>
 675       <variablelist>
 676         <varlistentry role="tactic.synopsis">
 677           <term>Synopsis:</term>
 678           <listitem>
 679             <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 680           </listitem>
 681         </varlistentry>
 682         <varlistentry>
 683           <term>Pre-conditions:</term>
 684           <listitem>
 685             <para><command>T</command> must be an inductive type.</para>
 686           </listitem>
 687         </varlistentry>
 688         <varlistentry>
 689           <term>Action:</term>
 690           <listitem>
 691             <para>TODO (severely bugged now).</para>
 692           </listitem>
 693         </varlistentry>
 694         <varlistentry>
 695           <term>New sequents to prove:</term>
 696           <listitem>
 697             <para>TODO</para>
 698           </listitem>
 699         </varlistentry>
 700       </variablelist>
 701     </para>
 702   </sect1>
 703   <sect1 id="tac_exact">
 704     <title>exact</title>
 705     <titleabbrev>exact</titleabbrev>
 706     <para><userinput>exact p</userinput></para>
 707     <para>
 708       <variablelist>
 709         <varlistentry role="tactic.synopsis">
 710           <term>Synopsis:</term>
 711           <listitem>
 712             <para><emphasis role="bold">exact</emphasis> &sterm;</para>
 713           </listitem>
 714         </varlistentry>
 715         <varlistentry>
 716           <term>Pre-conditions:</term>
 717           <listitem>
 718             <para>The type of <command>p</command> must be convertible
 719              with the conclusion of the current sequent.</para>
 720           </listitem>
 721         </varlistentry>
 722         <varlistentry>
 723           <term>Action:</term>
 724           <listitem>
 725             <para>It closes the current sequent using <command>p</command>.</para>
 726           </listitem>
 727         </varlistentry>
 728         <varlistentry>
 729           <term>New sequents to prove:</term>
 730           <listitem>
 731             <para>None.</para>
 732           </listitem>
 733         </varlistentry>
 734       </variablelist>
 735     </para>
 736   </sect1>
 737   <sect1 id="tac_exists">
 738     <title>exists</title>
 739     <titleabbrev>exists</titleabbrev>
 740     <para><userinput>exists </userinput></para>
 741     <para>
 742       <variablelist>
 743         <varlistentry role="tactic.synopsis">
 744           <term>Synopsis:</term>
 745           <listitem>
 746             <para><emphasis role="bold">exists</emphasis></para>
 747           </listitem>
 748         </varlistentry>
 749         <varlistentry>
 750           <term>Pre-conditions:</term>
 751           <listitem>
 752             <para>The conclusion of the current sequent must be
 753              an inductive type or the application of an inductive type
 754              with at least one constructor.</para>
 755           </listitem>
 756         </varlistentry>
 757         <varlistentry>
 758           <term>Action:</term>
 759           <listitem>
 760             <para>Equivalent to <command>constructor 1</command>.</para>
 761           </listitem>
 762         </varlistentry>
 763         <varlistentry>
 764           <term>New sequents to prove:</term>
 765           <listitem>
 766             <para>It opens a new sequent for each premise of the first
 767              constructor of the inductive type that is the conclusion of the
 768              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 769           </listitem>
 770         </varlistentry>
 771       </variablelist>
 772     </para>
 773   </sect1>
 774   <sect1 id="tac_fail">
 775     <title>fail</title>
 776     <titleabbrev>fail</titleabbrev>
 777     <para><userinput>fail</userinput></para>
 778     <para>
 779       <variablelist>
 780         <varlistentry role="tactic.synopsis">
 781           <term>Synopsis:</term>
 782           <listitem>
 783             <para><emphasis role="bold">fail</emphasis></para>
 784           </listitem>
 785         </varlistentry>
 786         <varlistentry>
 787           <term>Pre-conditions:</term>
 788           <listitem>
 789             <para>None.</para>
 790           </listitem>
 791         </varlistentry>
 792         <varlistentry>
 793           <term>Action:</term>
 794           <listitem>
 795             <para>This tactic always fail.</para>
 796           </listitem>
 797         </varlistentry>
 798         <varlistentry>
 799           <term>New sequents to prove:</term>
 800           <listitem>
 801             <para>N.A.</para>
 802           </listitem>
 803         </varlistentry>
 804       </variablelist>
 805     </para>
 806   </sect1>
 807   <sect1 id="tac_fold">
 808     <title>fold</title>
 809     <titleabbrev>fold</titleabbrev>
 810     <para><userinput>fold red t patt</userinput></para>
 811     <para>
 812       <variablelist>
 813         <varlistentry role="tactic.synopsis">
 814           <term>Synopsis:</term>
 815           <listitem>
 816             <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
 817           </listitem>
 818         </varlistentry>
 819         <varlistentry>
 820           <term>Pre-conditions:</term>
 821           <listitem>
 822             <para>The pattern must not specify the wanted term.</para>
 823           </listitem>
 824         </varlistentry>
 825         <varlistentry>
 826           <term>Action:</term>
 827           <listitem>
 828             <para>First of all it locates all the subterms matched by
 829              <command>patt</command>. In the context of each matched subterm
 830              it disambiguates the term <command>t</command> and reduces it
 831              to its <command>red</command> normal form; then it replaces with
 832              <command>t</command> every occurrence of the normal form in the
 833              matched subterm.</para>
 834           </listitem>
 835         </varlistentry>
 836         <varlistentry>
 837           <term>New sequents to prove:</term>
 838           <listitem>
 839             <para>None.</para>
 840           </listitem>
 841         </varlistentry>
 842       </variablelist>
 843     </para>
 844   </sect1>
 845   <sect1 id="tac_fourier">
 846     <title>fourier</title>
 847     <titleabbrev>fourier</titleabbrev>
 848     <para><userinput>fourier </userinput></para>
 849     <para>
 850       <variablelist>
 851         <varlistentry role="tactic.synopsis">
 852           <term>Synopsis:</term>
 853           <listitem>
 854             <para><emphasis role="bold">fourier</emphasis></para>
 855           </listitem>
 856         </varlistentry>
 857         <varlistentry>
 858           <term>Pre-conditions:</term>
 859           <listitem>
 860             <para>The conclusion of the current sequent must be a linear
 861              inequation over real numbers taken from standard library of
 862              Coq. Moreover the inequations in the hypotheses must imply the
 863              inequation in the conclusion of the current sequent.</para>
 864           </listitem>
 865         </varlistentry>
 866         <varlistentry>
 867           <term>Action:</term>
 868           <listitem>
 869             <para>It closes the current sequent by applying the Fourier method.</para>
 870           </listitem>
 871         </varlistentry>
 872         <varlistentry>
 873           <term>New sequents to prove:</term>
 874           <listitem>
 875             <para>None.</para>
 876           </listitem>
 877         </varlistentry>
 878       </variablelist>
 879     </para>
 880   </sect1>
 881   <sect1 id="tac_fwd">
 882     <title>fwd</title>
 883     <titleabbrev>fwd</titleabbrev>
 884     <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
 885     <para>
 886       <variablelist>
 887         <varlistentry role="tactic.synopsis">
 888           <term>Synopsis:</term>
 889           <listitem>
 890             <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
 891           </listitem>
 892         </varlistentry>
 893         <varlistentry>
 894           <term>Pre-conditions:</term>
 895           <listitem>
 896             <para>
 897                The type of <command>H</command> must be the premise of a
 898                forward simplification theorem.
 899             </para>
 900           </listitem>
 901         </varlistentry>
 902         <varlistentry>
 903           <term>Action:</term>
 904           <listitem>
 905             <para>
 906              This tactic is under development.
 907              It simplifies the current context by removing
 908              <command>H</command> using the following methods:
 909              forward application (by <command>lapply</command>) of a suitable
 910              simplification theorem, chosen automatically, of which the type
 911              of <command>H</command> is a premise,
 912              decomposition (by <command>decompose</command>),
 913              rewriting (by <command>rewrite</command>).
 914              <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
 915              are passed to the tactics <command>fwd</command> invokes, as
 916              names for the premise they introduce.
 917             </para>
 918           </listitem>
 919         </varlistentry>
 920         <varlistentry>
 921           <term>New sequents to prove:</term>
 922           <listitem>
 923             <para>
 924              The ones opened by the tactics <command>fwd</command> invokes.
 925             </para>
 926           </listitem>
 927         </varlistentry>
 928       </variablelist>
 929     </para>
 930   </sect1>
 931   <sect1 id="tac_generalize">
 932     <title>generalize</title>
 933     <titleabbrev>generalize</titleabbrev>
 934     <para><userinput>generalize patt as H</userinput></para>
 935     <para>
 936       <variablelist>
 937         <varlistentry role="tactic.synopsis">
 938           <term>Synopsis:</term>
 939           <listitem>
 940             <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
 941           </listitem>
 942         </varlistentry>
 943         <varlistentry>
 944           <term>Pre-conditions:</term>
 945           <listitem>
 946             <para>All the terms matched by <command>patt</command> must be
 947              convertible and close in the context of the current sequent.</para>
 948           </listitem>
 949         </varlistentry>
 950         <varlistentry>
 951           <term>Action:</term>
 952           <listitem>
 953             <para>It closes the current sequent by applying a stronger
 954              lemma that is proved using the new generated sequent.</para>
 955           </listitem>
 956         </varlistentry>
 957         <varlistentry>
 958           <term>New sequents to prove:</term>
 959           <listitem>
 960             <para>It opens a new sequent where the current sequent conclusion
 961              <command>G</command> is generalized to
 962              <command>∀x.G{x/t}</command> where <command>{x/t}</command>
 963              is a notation for the replacement with <command>x</command> of all
 964              the occurrences of the term <command>t</command> matched by
 965              <command>patt</command>. If <command>patt</command> matches no
 966              subterm then <command>t</command> is defined as the
 967              <command>wanted</command> part of the pattern.</para>
 968           </listitem>
 969         </varlistentry>
 970       </variablelist>
 971     </para>
 972   </sect1>
 973   <sect1 id="tac_id">
 974     <title>id</title>
 975     <titleabbrev>id</titleabbrev>
 976     <para><userinput>id </userinput></para>
 977     <para>
 978       <variablelist>
 979         <varlistentry role="tactic.synopsis">
 980           <term>Synopsis:</term>
 981           <listitem>
 982             <para><emphasis role="bold">id</emphasis></para>
 983           </listitem>
 984         </varlistentry>
 985         <varlistentry>
 986           <term>Pre-conditions:</term>
 987           <listitem>
 988             <para>None.</para>
 989           </listitem>
 990         </varlistentry>
 991         <varlistentry>
 992           <term>Action:</term>
 993           <listitem>
 994             <para>This identity tactic does nothing without failing.</para>
 995           </listitem>
 996         </varlistentry>
 997         <varlistentry>
 998           <term>New sequents to prove:</term>
 999           <listitem>
1000             <para>None.</para>
1001           </listitem>
1002         </varlistentry>
1003       </variablelist>
1004     </para>
1005   </sect1>
1006   <sect1 id="tac_intro">
1007     <title>intro</title>
1008     <titleabbrev>intro</titleabbrev>
1009     <para><userinput>intro H</userinput></para>
1010     <para>
1011       <variablelist>
1012         <varlistentry role="tactic.synopsis">
1013           <term>Synopsis:</term>
1014           <listitem>
1015             <para><emphasis role="bold">intro</emphasis> [&id;]</para>
1016           </listitem>
1017         </varlistentry>
1018         <varlistentry>
1019           <term>Pre-conditions:</term>
1020           <listitem>
1021             <para>The conclusion of the sequent to prove must be an implication
1022              or a universal quantification.</para>
1023           </listitem>
1024         </varlistentry>
1025         <varlistentry>
1026           <term>Action:</term>
1027           <listitem>
1028             <para>It applies the right introduction rule for implication,
1029              closing the current sequent.</para>
1030           </listitem>
1031         </varlistentry>
1032         <varlistentry>
1033           <term>New sequents to prove:</term>
1034           <listitem>
1035             <para>It opens a new sequent to prove adding to the hypothesis
1036              the antecedent of the implication and setting the conclusion
1037              to the consequent of the implicaiton. The name of the new
1038              hypothesis is <command>H</command> if provided; otherwise it
1039              is automatically generated.</para>
1040           </listitem>
1041         </varlistentry>
1042       </variablelist>
1043     </para>
1044   </sect1>
1045   <sect1 id="tac_intros">
1046     <title>intros</title>
1047     <titleabbrev>intros</titleabbrev>
1048     <para><userinput>intros hyps</userinput></para>
1049     <para>
1050       <variablelist>
1051         <varlistentry role="tactic.synopsis">
1052           <term>Synopsis:</term>
1053           <listitem>
1054             <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
1055           </listitem>
1056         </varlistentry>
1057         <varlistentry>
1058           <term>Pre-conditions:</term>
1059           <listitem>
1060             <para>If <command>hyps</command> specifies a number of hypotheses
1061              to introduce, then the conclusion of the current sequent must
1062              be formed by at least that number of imbricated implications
1063              or universal quantifications.</para>
1064           </listitem>
1065         </varlistentry>
1066         <varlistentry>
1067           <term>Action:</term>
1068           <listitem>
1069             <para>It applies several times the right introduction rule for
1070              implication, closing the current sequent.</para>
1071           </listitem>
1072         </varlistentry>
1073         <varlistentry>
1074           <term>New sequents to prove:</term>
1075           <listitem>
1076             <para>It opens a new sequent to prove adding a number of new
1077              hypotheses equal to the number of new hypotheses requested.
1078              If the user does not request a precise number of new hypotheses,
1079              it adds as many hypotheses as possible.
1080              The name of each new hypothesis is either popped from the
1081              user provided list of names, or it is automatically generated when
1082              the list is (or becomes) empty.</para>
1083           </listitem>
1084         </varlistentry>
1085       </variablelist>
1086     </para>
1087   </sect1>
1088   <sect1 id="tac_inversion">
1089     <title>inversion</title>
1090     <titleabbrev>inversion</titleabbrev>
1091     <para><userinput>inversion t</userinput></para>
1092     <para>
1093       <variablelist>
1094         <varlistentry role="tactic.synopsis">
1095           <term>Synopsis:</term>
1096           <listitem>
1097             <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
1098           </listitem>
1099         </varlistentry>
1100         <varlistentry>
1101           <term>Pre-conditions:</term>
1102           <listitem>
1103             <para>The type of the term <command>t</command> must be an inductive
1104              type or the application of an inductive type.</para>
1105           </listitem>
1106         </varlistentry>
1107         <varlistentry>
1108           <term>Action:</term>
1109           <listitem>
1110             <para>It proceeds by cases on <command>t</command> paying attention
1111              to the constraints imposed by the actual &quot;right arguments&quot;
1112              of the inductive type.</para>
1113           </listitem>
1114         </varlistentry>
1115         <varlistentry>
1116           <term>New sequents to prove:</term>
1117           <listitem>
1118             <para>It opens one new sequent to prove for each case in the
1119              definition of the type of <command>t</command>. With respect to
1120              a simple elimination, each new sequent has additional hypotheses
1121              that states the equalities of the &quot;right parameters&quot;
1122              of the inductive type with terms originally present in the
1123              sequent to prove.</para>
1124           </listitem>
1125         </varlistentry>
1126       </variablelist>
1127     </para>
1128   </sect1>
1129   <sect1 id="tac_lapply">
1130     <title>lapply</title>
1131     <titleabbrev>lapply</titleabbrev>
1132     <para><userinput>
1133      lapply linear depth=d t
1134      to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
1135     </userinput></para>
1136     <para>
1137       <variablelist>
1138         <varlistentry role="tactic.synopsis">
1139           <term>Synopsis:</term>
1140           <listitem>
1141             <para>
1142              <emphasis role="bold">lapply</emphasis>
1143              [<emphasis role="bold">linear</emphasis>]
1144              [<emphasis role="bold">depth=</emphasis>&nat;]
1145              &sterm;
1146              [<emphasis role="bold">to</emphasis>
1147               &sterm;
1148               [<emphasis role="bold">,</emphasis>&sterm;…]
1149              ]
1150              [<emphasis role="bold">as</emphasis> &id;]
1151             </para>
1152           </listitem>
1153         </varlistentry>
1154         <varlistentry>
1155           <term>Pre-conditions:</term>
1156           <listitem>
1157             <para>
1158              <command>t</command> must have at least <command>d</command>
1159              independent premises and <command>n</command> must not be
1160              greater than <command>d</command>.
1161             </para>
1162           </listitem>
1163         </varlistentry>
1164         <varlistentry>
1165           <term>Action:</term>
1166           <listitem>
1167             <para>
1168              Invokes <command>letin H ≝ (t ? ... ?)</command>
1169              with enough <command>?</command>'s to reach the
1170              <command>d</command>-th independent premise of
1171              <command>t</command>
1172              (<command>d</command> is maximum if unspecified).
1173              Then istantiates (by <command>apply</command>) with
1174              t<subscript>1</subscript>, ..., t<subscript>n</subscript>
1175              the <command>?</command>'s corresponding to the first
1176              <command>n</command> independent premises of
1177              <command>t</command>.
1178              Usually the other <command>?</command>'s preceding the
1179              <command>n</command>-th independent premise of
1180              <command>t</command> are istantiated as a consequence.
1181              If the <command>linear</command> flag is specified and if
1182              <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
1183              are (applications of) premises in the current context, they are
1184              <command>clear</command>ed.
1185             </para>
1186           </listitem>
1187         </varlistentry>
1188         <varlistentry>
1189           <term>New sequents to prove:</term>
1190           <listitem>
1191             <para>
1192              The ones opened by the tactics <command>lapply</command> invokes.
1193             </para>
1194           </listitem>
1195         </varlistentry>
1196       </variablelist>
1197     </para>
1198   </sect1>
1199   <sect1 id="tac_left">
1200     <title>left</title>
1201     <titleabbrev>left</titleabbrev>
1202     <para><userinput>left </userinput></para>
1203     <para>
1204       <variablelist>
1205         <varlistentry role="tactic.synopsis">
1206           <term>Synopsis:</term>
1207           <listitem>
1208             <para><emphasis role="bold">left</emphasis></para>
1209           </listitem>
1210         </varlistentry>
1211         <varlistentry>
1212           <term>Pre-conditions:</term>
1213           <listitem>
1214             <para>The conclusion of the current sequent must be
1215              an inductive type or the application of an inductive type
1216              with at least one constructor.</para>
1217           </listitem>
1218         </varlistentry>
1219         <varlistentry>
1220           <term>Action:</term>
1221           <listitem>
1222             <para>Equivalent to <command>constructor 1</command>.</para>
1223           </listitem>
1224         </varlistentry>
1225         <varlistentry>
1226           <term>New sequents to prove:</term>
1227           <listitem>
1228             <para>It opens a new sequent for each premise of the first
1229              constructor of the inductive type that is the conclusion of the
1230              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1231           </listitem>
1232         </varlistentry>
1233       </variablelist>
1234     </para>
1235   </sect1>
1236   <sect1 id="tac_letin">
1237     <title>letin</title>
1238     <titleabbrev>letin</titleabbrev>
1239     <para><userinput>letin x ≝ t</userinput></para>
1240     <para>
1241       <variablelist>
1242         <varlistentry role="tactic.synopsis">
1243           <term>Synopsis:</term>
1244           <listitem>
1245             <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
1246           </listitem>
1247         </varlistentry>
1248         <varlistentry>
1249           <term>Pre-conditions:</term>
1250           <listitem>
1251             <para>None.</para>
1252           </listitem>
1253         </varlistentry>
1254         <varlistentry>
1255           <term>Action:</term>
1256           <listitem>
1257             <para>It adds to the context of the current sequent to prove a new
1258              definition <command>x ≝ t</command>.</para>
1259           </listitem>
1260         </varlistentry>
1261         <varlistentry>
1262           <term>New sequents to prove:</term>
1263           <listitem>
1264             <para>None.</para>
1265           </listitem>
1266         </varlistentry>
1267       </variablelist>
1268     </para>
1269   </sect1>
1270   <sect1 id="tac_normalize">
1271     <title>normalize</title>
1272     <titleabbrev>normalize</titleabbrev>
1273     <para><userinput>normalize patt</userinput></para>
1274     <para>
1275       <variablelist>
1276         <varlistentry role="tactic.synopsis">
1277           <term>Synopsis:</term>
1278           <listitem>
1279             <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
1280           </listitem>
1281         </varlistentry>
1282         <varlistentry>
1283           <term>Pre-conditions:</term>
1284           <listitem>
1285             <para>None.</para>
1286           </listitem>
1287         </varlistentry>
1288         <varlistentry>
1289           <term>Action:</term>
1290           <listitem>
1291             <para>It replaces all the terms matched by <command>patt</command>
1292              with their βδιζ-normal form.</para>
1293           </listitem>
1294         </varlistentry>
1295         <varlistentry>
1296           <term>New sequents to prove:</term>
1297           <listitem>
1298             <para>None.</para>
1299           </listitem>
1300         </varlistentry>
1301       </variablelist>
1302     </para>
1303   </sect1>
1304   <sect1 id="tac_reduce">
1305     <title>reduce</title>
1306     <titleabbrev>reduce</titleabbrev>
1307     <para><userinput>reduce patt</userinput></para>
1308     <para>
1309       <variablelist>
1310         <varlistentry role="tactic.synopsis">
1311           <term>Synopsis:</term>
1312           <listitem>
1313             <para><emphasis role="bold">reduce</emphasis> &pattern;</para>
1314           </listitem>
1315         </varlistentry>
1316         <varlistentry>
1317           <term>Pre-conditions:</term>
1318           <listitem>
1319             <para>None.</para>
1320           </listitem>
1321         </varlistentry>
1322         <varlistentry>
1323           <term>Action:</term>
1324           <listitem>
1325             <para>It replaces all the terms matched by <command>patt</command>
1326              with their βδιζ-normal form.</para>
1327           </listitem>
1328         </varlistentry>
1329         <varlistentry>
1330           <term>New sequents to prove:</term>
1331           <listitem>
1332             <para>None.</para>
1333           </listitem>
1334         </varlistentry>
1335       </variablelist>
1336     </para>
1337   </sect1>
1338   <sect1 id="tac_reflexivity">
1339     <title>reflexivity</title>
1340     <titleabbrev>reflexivity</titleabbrev>
1341     <para><userinput>reflexivity </userinput></para>
1342     <para>
1343       <variablelist>
1344         <varlistentry role="tactic.synopsis">
1345           <term>Synopsis:</term>
1346           <listitem>
1347             <para><emphasis role="bold">reflexivity</emphasis></para>
1348           </listitem>
1349         </varlistentry>
1350         <varlistentry>
1351           <term>Pre-conditions:</term>
1352           <listitem>
1353             <para>The conclusion of the current sequent must be
1354              <command>t=t</command> for some term <command>t</command></para>
1355           </listitem>
1356         </varlistentry>
1357         <varlistentry>
1358           <term>Action:</term>
1359           <listitem>
1360             <para>It closes the current sequent by reflexivity
1361              of equality.</para>
1362           </listitem>
1363         </varlistentry>
1364         <varlistentry>
1365           <term>New sequents to prove:</term>
1366           <listitem>
1367             <para>None.</para>
1368           </listitem>
1369         </varlistentry>
1370       </variablelist>
1371     </para>
1372   </sect1>
1373   <sect1 id="tac_replace">
1374     <title>replace</title>
1375     <titleabbrev>change</titleabbrev>
1376     <para><userinput>change patt with t</userinput></para>
1377     <para>
1378       <variablelist>
1379         <varlistentry role="tactic.synopsis">
1380           <term>Synopsis:</term>
1381           <listitem>
1382             <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
1383           </listitem>
1384         </varlistentry>
1385         <varlistentry>
1386           <term>Pre-conditions:</term>
1387           <listitem>
1388             <para>None.</para>
1389           </listitem>
1390         </varlistentry>
1391         <varlistentry>
1392           <term>Action:</term>
1393           <listitem>
1394             <para>It replaces the subterms of the current sequent matched by
1395              <command>patt</command> with the new term <command>t</command>.
1396              For each subterm matched by the pattern, <command>t</command> is
1397              disambiguated in the context of the subterm.</para>
1398           </listitem>
1399         </varlistentry>
1400         <varlistentry>
1401           <term>New sequents to prove:</term>
1402           <listitem>
1403             <para>For each matched term <command>t'</command> it opens
1404              a new sequent to prove whose conclusion is
1405              <command>t'=t</command>.</para>
1406           </listitem>
1407         </varlistentry>
1408       </variablelist>
1409     </para>
1410   </sect1>
1411   <sect1 id="tac_rewrite">
1412     <title>rewrite</title>
1413     <titleabbrev>rewrite</titleabbrev>
1414     <para><userinput>rewrite dir p patt</userinput></para>
1415     <para>
1416       <variablelist>
1417         <varlistentry role="tactic.synopsis">
1418           <term>Synopsis:</term>
1419           <listitem>
1420             <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold">&lt;</emphasis>|<emphasis role="bold">&gt;</emphasis>] &sterm; &pattern;</para>
1421           </listitem>
1422         </varlistentry>
1423         <varlistentry>
1424           <term>Pre-conditions:</term>
1425           <listitem>
1426             <para><command>p</command> must be the proof of an equality,
1427              possibly under some hypotheses.</para>
1428           </listitem>
1429         </varlistentry>
1430         <varlistentry>
1431           <term>Action:</term>
1432           <listitem>
1433             <para>It looks in every term matched by <command>patt</command>
1434              for all the occurrences of the
1435              left hand side of the equality that <command>p</command> proves
1436              (resp. the right hand side if <command>dir</command> is
1437              <command>&lt;</command>). Every occurence found is replaced with
1438              the opposite side of the equality.</para>
1439           </listitem>
1440         </varlistentry>
1441         <varlistentry>
1442           <term>New sequents to prove:</term>
1443           <listitem>
1444             <para>It opens one new sequent for each hypothesis of the
1445              equality proved by <command>p</command> that is not closed
1446              by unification.</para>
1447           </listitem>
1448         </varlistentry>
1449       </variablelist>
1450     </para>
1451   </sect1>
1452   <sect1 id="tac_right">
1453     <title>right</title>
1454     <titleabbrev>right</titleabbrev>
1455     <para><userinput>right </userinput></para>
1456     <para>
1457       <variablelist>
1458         <varlistentry role="tactic.synopsis">
1459           <term>Synopsis:</term>
1460           <listitem>
1461             <para><emphasis role="bold">right</emphasis></para>
1462           </listitem>
1463         </varlistentry>
1464         <varlistentry>
1465           <term>Pre-conditions:</term>
1466           <listitem>
1467             <para>The conclusion of the current sequent must be
1468              an inductive type or the application of an inductive type with
1469              at least two constructors.</para>
1470           </listitem>
1471         </varlistentry>
1472         <varlistentry>
1473           <term>Action:</term>
1474           <listitem>
1475             <para>Equivalent to <command>constructor 2</command>.</para>
1476           </listitem>
1477         </varlistentry>
1478         <varlistentry>
1479           <term>New sequents to prove:</term>
1480           <listitem>
1481             <para>It opens a new sequent for each premise of the second
1482              constructor of the inductive type that is the conclusion of the
1483              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1484           </listitem>
1485         </varlistentry>
1486       </variablelist>
1487     </para>
1488   </sect1>
1489   <sect1 id="tac_ring">
1490     <title>ring</title>
1491     <titleabbrev>ring</titleabbrev>
1492     <para><userinput>ring </userinput></para>
1493     <para>
1494       <variablelist>
1495         <varlistentry role="tactic.synopsis">
1496           <term>Synopsis:</term>
1497           <listitem>
1498             <para><emphasis role="bold">ring</emphasis></para>
1499           </listitem>
1500         </varlistentry>
1501         <varlistentry>
1502           <term>Pre-conditions:</term>
1503           <listitem>
1504             <para>The conclusion of the current sequent must be an
1505              equality over Coq's real numbers that can be proved using
1506              the ring properties of the real numbers only.</para>
1507           </listitem>
1508         </varlistentry>
1509         <varlistentry>
1510           <term>Action:</term>
1511           <listitem>
1512             <para>It closes the current sequent veryfying the equality by
1513              means of computation (i.e. this is a reflexive tactic, implemented
1514              exploiting the &quot;two level reasoning&quot; technique).</para>
1515           </listitem>
1516         </varlistentry>
1517         <varlistentry>
1518           <term>New sequents to prove:</term>
1519           <listitem>
1520             <para>None.</para>
1521           </listitem>
1522         </varlistentry>
1523       </variablelist>
1524     </para>
1525   </sect1>
1526   <sect1 id="tac_simplify">
1527     <title>simplify</title>
1528     <titleabbrev>simplify</titleabbrev>
1529     <para><userinput>simplify patt</userinput></para>
1530     <para>
1531       <variablelist>
1532         <varlistentry role="tactic.synopsis">
1533           <term>Synopsis:</term>
1534           <listitem>
1535             <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
1536           </listitem>
1537         </varlistentry>
1538         <varlistentry>
1539           <term>Pre-conditions:</term>
1540           <listitem>
1541             <para>None.</para>
1542           </listitem>
1543         </varlistentry>
1544         <varlistentry>
1545           <term>Action:</term>
1546           <listitem>
1547             <para>It replaces all the terms matched by <command>patt</command>
1548              with other convertible terms that are supposed to be simpler.</para>
1549           </listitem>
1550         </varlistentry>
1551         <varlistentry>
1552           <term>New sequents to prove:</term>
1553           <listitem>
1554             <para>None.</para>
1555           </listitem>
1556         </varlistentry>
1557       </variablelist>
1558     </para>
1559   </sect1>
1560   <sect1 id="tac_split">
1561     <title>split</title>
1562     <titleabbrev>split</titleabbrev>
1563     <para><userinput>split </userinput></para>
1564     <para>
1565       <variablelist>
1566         <varlistentry role="tactic.synopsis">
1567           <term>Synopsis:</term>
1568           <listitem>
1569             <para><emphasis role="bold">split</emphasis></para>
1570           </listitem>
1571         </varlistentry>
1572         <varlistentry>
1573           <term>Pre-conditions:</term>
1574           <listitem>
1575             <para>The conclusion of the current sequent must be
1576              an inductive type or the application of an inductive type with
1577              at least one constructor.</para>
1578           </listitem>
1579         </varlistentry>
1580         <varlistentry>
1581           <term>Action:</term>
1582           <listitem>
1583             <para>Equivalent to <command>constructor 1</command>.</para>
1584           </listitem>
1585         </varlistentry>
1586         <varlistentry>
1587           <term>New sequents to prove:</term>
1588           <listitem>
1589             <para>It opens a new sequent for each premise of the first
1590              constructor of the inductive type that is the conclusion of the
1591              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1592           </listitem>
1593         </varlistentry>
1594       </variablelist>
1595     </para>
1596   </sect1>
1597
1598   <sect1 id="tac_subst">
1599     <title>subst</title>
1600     <titleabbrev>subst</titleabbrev>
1601     <para><userinput>subst</userinput></para>
1602     <para>
1603       <variablelist>
1604         <varlistentry role="tactic.synopsis">
1605           <term>Synopsis:</term>
1606           <listitem>
1607             <para><emphasis role="bold">subst</emphasis></para>
1608           </listitem>
1609         </varlistentry>
1610         <varlistentry>
1611           <term>Pre-conditions:</term>
1612           <listitem><para>
1613             None.
1614           </para></listitem>
1615         </varlistentry>
1616         <varlistentry>
1617           <term>Action:</term>
1618           <listitem><para>
1619             For each premise of the form
1620             <command>H: x = t</command> or <command>H: t = x</command>
1621             where <command>x</command> is a local variable and
1622             <command>t</command> does not depend on <command>x</command>,
1623             the tactic rewrites <command>H</command> wherever
1624             <command>x</command> appears clearing <command>H</command> and
1625             <command>x</command> afterwards.
1626           </para></listitem>
1627         </varlistentry>
1628         <varlistentry>
1629           <term>New sequents to prove:</term>
1630           <listitem><para>
1631             The one opened by the applied tactics.
1632           </para></listitem>
1633         </varlistentry>
1634       </variablelist>
1635     </para>
1636   </sect1>
1637   <sect1 id="tac_symmetry">
1638     <title>symmetry</title>
1639     <titleabbrev>symmetry</titleabbrev>
1640     <para>The tactic <command>symmetry</command> </para>
1641     <para><userinput>symmetry </userinput></para>
1642     <para>
1643       <variablelist>
1644         <varlistentry role="tactic.synopsis">
1645           <term>Synopsis:</term>
1646           <listitem>
1647             <para><emphasis role="bold">symmetry</emphasis></para>
1648           </listitem>
1649         </varlistentry>
1650         <varlistentry>
1651           <term>Pre-conditions:</term>
1652           <listitem>
1653             <para>The conclusion of the current proof must be an equality.</para>
1654           </listitem>
1655         </varlistentry>
1656         <varlistentry>
1657           <term>Action:</term>
1658           <listitem>
1659             <para>It swaps the two sides of the equalityusing the symmetric
1660              property.</para>
1661           </listitem>
1662         </varlistentry>
1663         <varlistentry>
1664           <term>New sequents to prove:</term>
1665           <listitem>
1666             <para>None.</para>
1667           </listitem>
1668         </varlistentry>
1669       </variablelist>
1670     </para>
1671   </sect1>
1672   <sect1 id="tac_transitivity">
1673     <title>transitivity</title>
1674     <titleabbrev>transitivity</titleabbrev>
1675     <para><userinput>transitivity t</userinput></para>
1676     <para>
1677       <variablelist>
1678         <varlistentry role="tactic.synopsis">
1679           <term>Synopsis:</term>
1680           <listitem>
1681             <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
1682           </listitem>
1683         </varlistentry>
1684         <varlistentry>
1685           <term>Pre-conditions:</term>
1686           <listitem>
1687             <para>The conclusion of the current proof must be an equality.</para>
1688           </listitem>
1689         </varlistentry>
1690         <varlistentry>
1691           <term>Action:</term>
1692           <listitem>
1693             <para>It closes the current sequent by transitivity of the equality.</para>
1694           </listitem>
1695         </varlistentry>
1696         <varlistentry>
1697           <term>New sequents to prove:</term>
1698           <listitem>
1699             <para>It opens two new sequents <command>l=t</command> and
1700              <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
1701 the current sequent to prove.</para>
1702           </listitem>
1703         </varlistentry>
1704       </variablelist>
1705     </para>
1706   </sect1>
1707   <sect1 id="tac_unfold">
1708     <title>unfold</title>
1709     <titleabbrev>unfold</titleabbrev>
1710     <para><userinput>unfold t patt</userinput></para>
1711     <para>
1712       <variablelist>
1713         <varlistentry role="tactic.synopsis">
1714           <term>Synopsis:</term>
1715           <listitem>
1716             <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
1717           </listitem>
1718         </varlistentry>
1719         <varlistentry>
1720           <term>Pre-conditions:</term>
1721           <listitem>
1722             <para>None.</para>
1723           </listitem>
1724         </varlistentry>
1725         <varlistentry>
1726           <term>Action:</term>
1727           <listitem>
1728             <para>It finds all the occurrences of <command>t</command>
1729              (possibly applied to arguments) in the subterms matched by
1730              <command>patt</command>. Then it δ-expands each occurrence,
1731              also performing β-reduction of the obtained term. If
1732              <command>t</command> is omitted it defaults to each
1733              subterm matched by <command>patt</command>.</para>
1734           </listitem>
1735         </varlistentry>
1736         <varlistentry>
1737           <term>New sequents to prove:</term>
1738           <listitem>
1739             <para>None.</para>
1740           </listitem>
1741         </varlistentry>
1742       </variablelist>
1743     </para>
1744   </sect1>
1745   <sect1 id="tac_whd">
1746     <title>whd</title>
1747     <titleabbrev>whd</titleabbrev>
1748     <para><userinput>whd patt</userinput></para>
1749     <para>
1750       <variablelist>
1751         <varlistentry role="tactic.synopsis">
1752           <term>Synopsis:</term>
1753           <listitem>
1754             <para><emphasis role="bold">whd</emphasis> &pattern;</para>
1755           </listitem>
1756         </varlistentry>
1757         <varlistentry>
1758           <term>Pre-conditions:</term>
1759           <listitem>
1760             <para>None.</para>
1761           </listitem>
1762         </varlistentry>
1763         <varlistentry>
1764           <term>Action:</term>
1765           <listitem>
1766             <para>It replaces all the terms matched by <command>patt</command>
1767              with their βδιζ-weak-head normal form.</para>
1768           </listitem>
1769         </varlistentry>
1770         <varlistentry>
1771           <term>New sequents to prove:</term>
1772           <listitem>
1773             <para>None.</para>
1774           </listitem>
1775         </varlistentry>
1776       </variablelist>
1777     </para>
1778   </sect1>
1779
1780 </chapter>
1781