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_compose">
 348     <title>compose</title>
 349     <titleabbrev>compose</titleabbrev>
 350     <para><userinput>compose n t1 with t2 hyps</userinput></para>
 351     <para>
 352       <variablelist>
 353         <varlistentry role="tactic.synopsis">
 354           <term>Synopsis:</term>
 355           <listitem>
 356             <para><emphasis role="bold">compose</emphasis> [&nat;] &sterm; [<emphasis role="bold">with</emphasis> &sterm;] [&intros-spec;]</para>
 357           </listitem>
 358         </varlistentry>
 359         <varlistentry>
 360           <term>Pre-conditions:</term>
 361           <listitem>
 362             <para></para>
 363           </listitem>
 364         </varlistentry>
 365         <varlistentry>
 366           <term>Action:</term>
 367           <listitem>
 368             <para>Composes t1 with t2 in every possible way
 369               <command>n</command> times introducing generated terms
 370               as if <command>intros hyps</command> was issued.</para>
 371             <para>If <command>t1:∀x:A.B[x]</command> and
 372             <command>t2:∀x,y:A.B[x]→B[y]→C[x,y]</command> it generates:
 373              <itemizedlist>
 374                 <listitem>
 375                   <para><command>λx,y:A.t2 x y (t1 x) : ∀x,y:A.B[y]→C[x,y]</command></para>
 376                 </listitem>
 377                 <listitem>
 378                   <para><command>λx,y:A.λH:B[x].t2 x y H (t1 y) : ∀x,y:A.B[x]→C[x,y]
 379                   </command></para>
 380                 </listitem>
 381              </itemizedlist>
 382           </para>
 383           <para>If <command>t2</command> is omitted it composes
 384             <command>t1</command>
 385               with every hypothesis that can be introduced.
 386               <command>n</command> iterates the process.</para>
 387           </listitem>
 388         </varlistentry>
 389         <varlistentry>
 390           <term>New sequents to prove:</term>
 391           <listitem>
 392             <para>The same, but with more hypothesis eventually introduced
 393             by the &intros-spec;.</para>
 394           </listitem>
 395         </varlistentry>
 396       </variablelist>
 397     </para>
 398   </sect1>
 399   <sect1 id="tac_change">
 400     <title>change</title>
 401     <titleabbrev>change</titleabbrev>
 402     <para><userinput>change patt with t</userinput></para>
 403     <para>
 404       <variablelist>
 405         <varlistentry role="tactic.synopsis">
 406           <term>Synopsis:</term>
 407           <listitem>
 408             <para><emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
 409           </listitem>
 410         </varlistentry>
 411         <varlistentry>
 412           <term>Pre-conditions:</term>
 413           <listitem>
 414             <para>Each subterm matched by the pattern must be convertible
 415              with the term <command>t</command> disambiguated in the context
 416              of the matched subterm.</para>
 417           </listitem>
 418         </varlistentry>
 419         <varlistentry>
 420           <term>Action:</term>
 421           <listitem>
 422             <para>It replaces the subterms of the current sequent matched by
 423              <command>patt</command> with the new term <command>t</command>.
 424              For each subterm matched by the pattern, <command>t</command> is
 425              disambiguated in the context of the subterm.</para>
 426           </listitem>
 427         </varlistentry>
 428         <varlistentry>
 429           <term>New sequents to prove:</term>
 430           <listitem>
 431             <para>None.</para>
 432           </listitem>
 433         </varlistentry>
 434       </variablelist>
 435     </para>
 436   </sect1>
 437   <sect1 id="tac_constructor">
 438     <title>constructor</title>
 439     <titleabbrev>constructor</titleabbrev>
 440     <para><userinput>constructor n</userinput></para>
 441     <para>
 442       <variablelist>
 443         <varlistentry role="tactic.synopsis">
 444           <term>Synopsis:</term>
 445           <listitem>
 446             <para><emphasis role="bold">constructor</emphasis> &nat;</para>
 447           </listitem>
 448         </varlistentry>
 449         <varlistentry>
 450           <term>Pre-conditions:</term>
 451           <listitem>
 452             <para>The conclusion of the current sequent must be
 453              an inductive type or the application of an inductive type with
 454              at least <command>n</command> constructors.</para>
 455           </listitem>
 456         </varlistentry>
 457         <varlistentry>
 458           <term>Action:</term>
 459           <listitem>
 460             <para>It applies the <command>n</command>-th constructor of the
 461              inductive type of the conclusion of the current sequent.</para>
 462           </listitem>
 463         </varlistentry>
 464         <varlistentry>
 465           <term>New sequents to prove:</term>
 466           <listitem>
 467             <para>It opens a new sequent for each premise of the constructor
 468              that can not be inferred by unification. For more details,
 469              see the <command>apply</command> tactic.</para>
 470           </listitem>
 471         </varlistentry>
 472       </variablelist>
 473     </para>
 474   </sect1>
 475   <sect1 id="tac_contradiction">
 476     <title>contradiction</title>
 477     <titleabbrev>contradiction</titleabbrev>
 478     <para><userinput>contradiction </userinput></para>
 479     <para>
 480       <variablelist>
 481         <varlistentry role="tactic.synopsis">
 482           <term>Synopsis:</term>
 483           <listitem>
 484             <para><emphasis role="bold">contradiction</emphasis></para>
 485           </listitem>
 486         </varlistentry>
 487         <varlistentry>
 488           <term>Pre-conditions:</term>
 489           <listitem>
 490             <para>There must be in the current context an hypothesis of type
 491              <command>False</command>.</para>
 492           </listitem>
 493         </varlistentry>
 494         <varlistentry>
 495           <term>Action:</term>
 496           <listitem>
 497             <para>It closes the current sequent by applying an hypothesis of
 498              type <command>False</command>.</para>
 499           </listitem>
 500         </varlistentry>
 501         <varlistentry>
 502           <term>New sequents to prove:</term>
 503           <listitem>
 504             <para>None</para>
 505           </listitem>
 506         </varlistentry>
 507       </variablelist>
 508     </para>
 509   </sect1>
 510   <sect1 id="tac_cut">
 511     <title>cut</title>
 512     <titleabbrev>cut</titleabbrev>
 513     <para><userinput>cut P as H</userinput></para>
 514     <para>
 515       <variablelist>
 516         <varlistentry role="tactic.synopsis">
 517           <term>Synopsis:</term>
 518           <listitem>
 519             <para><emphasis role="bold">cut</emphasis> &sterm; [<emphasis role="bold">as</emphasis> &id;]</para>
 520           </listitem>
 521         </varlistentry>
 522         <varlistentry>
 523           <term>Pre-conditions:</term>
 524           <listitem>
 525             <para><command>P</command> must have type <command>Prop</command>.</para>
 526           </listitem>
 527         </varlistentry>
 528         <varlistentry>
 529           <term>Action:</term>
 530           <listitem>
 531             <para>It closes the current sequent.</para>
 532           </listitem>
 533         </varlistentry>
 534         <varlistentry>
 535           <term>New sequents to prove:</term>
 536           <listitem>
 537             <para>It opens two new sequents. The first one has an extra
 538              hypothesis <command>H:P</command>. If <command>H</command> is
 539              omitted, the name of the hypothesis is automatically generated.
 540              The second sequent has conclusion <command>P</command> and
 541              hypotheses the hypotheses of the current sequent to prove.</para>
 542           </listitem>
 543         </varlistentry>
 544       </variablelist>
 545     </para>
 546   </sect1>
 547   <sect1 id="tac_decompose">
 548     <title>decompose</title>
 549     <titleabbrev>decompose</titleabbrev>
 550     <para><userinput>
 551      decompose as H<subscript>1</subscript> ... H<subscript>m</subscript>
 552     </userinput></para>
 553     <para>
 554       <variablelist>
 555         <varlistentry role="tactic.synopsis">
 556           <term>Synopsis:</term>
 557           <listitem>
 558             <para>
 559              <emphasis role="bold">decompose</emphasis>
 560              [<emphasis role="bold">as</emphasis> &id;…]
 561             </para>
 562           </listitem>
 563         </varlistentry>
 564         <varlistentry>
 565           <term>Pre-conditions:</term>
 566           <listitem>
 567             <para>None.</para>
 568           </listitem>
 569         </varlistentry>
 570         <varlistentry>
 571           <term>Action:</term>
 572           <listitem>
 573             <para>
 574              For each each premise <command>H</command> of type
 575              <command>T</command> in the current context where
 576              <command>T</command> is a non-recursive inductive type without
 577              right parameters and of sort Prop or CProp, the tactic runs
 578              <command>
 579               elim H as H<subscript>1</subscript> ... H<subscript>m</subscript>
 580              </command>, clears <command>H</command>  and runs itself
 581              recursively on each new premise introduced by
 582              <command>elim</command> in the opened sequents.
 583             </para>
 584           </listitem>
 585         </varlistentry>
 586         <varlistentry>
 587           <term>New sequents to prove:</term>
 588           <listitem>
 589             <para>
 590              The ones generated by all the <command>elim</command> tactics run.
 591             </para>
 592           </listitem>
 593         </varlistentry>
 594       </variablelist>
 595     </para>
 596   </sect1>
 597   <sect1 id="tac_demodulate">
 598     <title>demodulate</title>
 599     <titleabbrev>demodulate</titleabbrev>
 600     <para><userinput>demodulate auto_params</userinput></para>
 601     <para>
 602       <variablelist>
 603         <varlistentry role="tactic.synopsis">
 604           <term>Synopsis:</term>
 605           <listitem>
 606             <para><emphasis role="bold">demodulate</emphasis> &autoparams;</para>
 607           </listitem>
 608         </varlistentry>
 609         <varlistentry>
 610           <term>Pre-conditions:</term>
 611           <listitem>
 612             <para>None.</para>
 613           </listitem>
 614         </varlistentry>
 615         <varlistentry>
 616           <term>Action:</term>
 617           <listitem>
 618             <para>&TODO;</para>
 619           </listitem>
 620         </varlistentry>
 621         <varlistentry>
 622           <term>New sequents to prove:</term>
 623           <listitem>
 624             <para>None.</para>
 625           </listitem>
 626         </varlistentry>
 627       </variablelist>
 628     </para>
 629   </sect1>
 630   <sect1 id="tac_destruct">
 631     <title>destruct</title>
 632     <titleabbrev>destruct</titleabbrev>
 633     <para><userinput>destruct p</userinput></para>
 634     <para>
 635       <variablelist>
 636         <varlistentry role="tactic.synopsis">
 637           <term>Synopsis:</term>
 638           <listitem>
 639             <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
 640           </listitem>
 641         </varlistentry>
 642         <varlistentry>
 643           <term>Pre-conditions:</term>
 644           <listitem>
 645             <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>
 646           </listitem>
 647         </varlistentry>
 648         <varlistentry>
 649           <term>Action:</term>
 650           <listitem>
 651             <para>The tactic recursively compare the two sides of the equality
 652              looking for different constructors in corresponding position.
 653              If two of them are found, the tactic closes the current sequent
 654              by proving the absurdity of <command>p</command>. Otherwise
 655              it adds a new hypothesis for each leaf of the formula that
 656              states the equality of the subformulae in the corresponding
 657              positions on the two sides of the equality.
 658             </para>
 659           </listitem>
 660         </varlistentry>
 661         <varlistentry>
 662           <term>New sequents to prove:</term>
 663           <listitem>
 664             <para>None.</para>
 665           </listitem>
 666         </varlistentry>
 667       </variablelist>
 668     </para>
 669   </sect1>
 670   <sect1 id="tac_elim">
 671     <title>elim</title>
 672     <titleabbrev>elim</titleabbrev>
 673     <para><userinput>elim 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">elim</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 inhabit an inductive type and
 686              <command>th</command> must be an elimination principle for that
 687              inductive type. If <command>th</command> is omitted the appropriate
 688              standard elimination principle is chosen.</para>
 689           </listitem>
 690         </varlistentry>
 691         <varlistentry>
 692           <term>Action:</term>
 693           <listitem>
 694             <para>It proceeds by cases on the values of <command>t</command>,
 695              according to the elimination principle <command>th</command>.
 696             </para>
 697           </listitem>
 698         </varlistentry>
 699         <varlistentry>
 700           <term>New sequents to prove:</term>
 701           <listitem>
 702             <para>It opens one new sequent for each case. The names of
 703              the new hypotheses are picked by <command>hyps</command>, if
 704              provided. If hyps specifies also a number of hypotheses that
 705              is less than the number of new hypotheses for a new sequent,
 706              then the exceeding hypothesis will be kept as implications in
 707              the conclusion of the sequent.</para>
 708           </listitem>
 709         </varlistentry>
 710       </variablelist>
 711     </para>
 712   </sect1>
 713   <sect1 id="tac_elimType">
 714     <title>elimType</title>
 715     <titleabbrev>elimType</titleabbrev>
 716     <para><userinput>elimType T using th hyps</userinput></para>
 717     <para>
 718       <variablelist>
 719         <varlistentry role="tactic.synopsis">
 720           <term>Synopsis:</term>
 721           <listitem>
 722             <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 723           </listitem>
 724         </varlistentry>
 725         <varlistentry>
 726           <term>Pre-conditions:</term>
 727           <listitem>
 728             <para><command>T</command> must be an inductive type.</para>
 729           </listitem>
 730         </varlistentry>
 731         <varlistentry>
 732           <term>Action:</term>
 733           <listitem>
 734             <para>TODO (severely bugged now).</para>
 735           </listitem>
 736         </varlistentry>
 737         <varlistentry>
 738           <term>New sequents to prove:</term>
 739           <listitem>
 740             <para>TODO</para>
 741           </listitem>
 742         </varlistentry>
 743       </variablelist>
 744     </para>
 745   </sect1>
 746   <sect1 id="tac_exact">
 747     <title>exact</title>
 748     <titleabbrev>exact</titleabbrev>
 749     <para><userinput>exact p</userinput></para>
 750     <para>
 751       <variablelist>
 752         <varlistentry role="tactic.synopsis">
 753           <term>Synopsis:</term>
 754           <listitem>
 755             <para><emphasis role="bold">exact</emphasis> &sterm;</para>
 756           </listitem>
 757         </varlistentry>
 758         <varlistentry>
 759           <term>Pre-conditions:</term>
 760           <listitem>
 761             <para>The type of <command>p</command> must be convertible
 762              with the conclusion of the current sequent.</para>
 763           </listitem>
 764         </varlistentry>
 765         <varlistentry>
 766           <term>Action:</term>
 767           <listitem>
 768             <para>It closes the current sequent using <command>p</command>.</para>
 769           </listitem>
 770         </varlistentry>
 771         <varlistentry>
 772           <term>New sequents to prove:</term>
 773           <listitem>
 774             <para>None.</para>
 775           </listitem>
 776         </varlistentry>
 777       </variablelist>
 778     </para>
 779   </sect1>
 780   <sect1 id="tac_exists">
 781     <title>exists</title>
 782     <titleabbrev>exists</titleabbrev>
 783     <para><userinput>exists </userinput></para>
 784     <para>
 785       <variablelist>
 786         <varlistentry role="tactic.synopsis">
 787           <term>Synopsis:</term>
 788           <listitem>
 789             <para><emphasis role="bold">exists</emphasis></para>
 790           </listitem>
 791         </varlistentry>
 792         <varlistentry>
 793           <term>Pre-conditions:</term>
 794           <listitem>
 795             <para>The conclusion of the current sequent must be
 796              an inductive type or the application of an inductive type
 797              with at least one constructor.</para>
 798           </listitem>
 799         </varlistentry>
 800         <varlistentry>
 801           <term>Action:</term>
 802           <listitem>
 803             <para>Equivalent to <command>constructor 1</command>.</para>
 804           </listitem>
 805         </varlistentry>
 806         <varlistentry>
 807           <term>New sequents to prove:</term>
 808           <listitem>
 809             <para>It opens a new sequent for each premise of the first
 810              constructor of the inductive type that is the conclusion of the
 811              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 812           </listitem>
 813         </varlistentry>
 814       </variablelist>
 815     </para>
 816   </sect1>
 817   <sect1 id="tac_fail">
 818     <title>fail</title>
 819     <titleabbrev>fail</titleabbrev>
 820     <para><userinput>fail</userinput></para>
 821     <para>
 822       <variablelist>
 823         <varlistentry role="tactic.synopsis">
 824           <term>Synopsis:</term>
 825           <listitem>
 826             <para><emphasis role="bold">fail</emphasis></para>
 827           </listitem>
 828         </varlistentry>
 829         <varlistentry>
 830           <term>Pre-conditions:</term>
 831           <listitem>
 832             <para>None.</para>
 833           </listitem>
 834         </varlistentry>
 835         <varlistentry>
 836           <term>Action:</term>
 837           <listitem>
 838             <para>This tactic always fail.</para>
 839           </listitem>
 840         </varlistentry>
 841         <varlistentry>
 842           <term>New sequents to prove:</term>
 843           <listitem>
 844             <para>N.A.</para>
 845           </listitem>
 846         </varlistentry>
 847       </variablelist>
 848     </para>
 849   </sect1>
 850   <sect1 id="tac_fold">
 851     <title>fold</title>
 852     <titleabbrev>fold</titleabbrev>
 853     <para><userinput>fold red t patt</userinput></para>
 854     <para>
 855       <variablelist>
 856         <varlistentry role="tactic.synopsis">
 857           <term>Synopsis:</term>
 858           <listitem>
 859             <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
 860           </listitem>
 861         </varlistentry>
 862         <varlistentry>
 863           <term>Pre-conditions:</term>
 864           <listitem>
 865             <para>The pattern must not specify the wanted term.</para>
 866           </listitem>
 867         </varlistentry>
 868         <varlistentry>
 869           <term>Action:</term>
 870           <listitem>
 871             <para>First of all it locates all the subterms matched by
 872              <command>patt</command>. In the context of each matched subterm
 873              it disambiguates the term <command>t</command> and reduces it
 874              to its <command>red</command> normal form; then it replaces with
 875              <command>t</command> every occurrence of the normal form in the
 876              matched subterm.</para>
 877           </listitem>
 878         </varlistentry>
 879         <varlistentry>
 880           <term>New sequents to prove:</term>
 881           <listitem>
 882             <para>None.</para>
 883           </listitem>
 884         </varlistentry>
 885       </variablelist>
 886     </para>
 887   </sect1>
 888   <sect1 id="tac_fourier">
 889     <title>fourier</title>
 890     <titleabbrev>fourier</titleabbrev>
 891     <para><userinput>fourier </userinput></para>
 892     <para>
 893       <variablelist>
 894         <varlistentry role="tactic.synopsis">
 895           <term>Synopsis:</term>
 896           <listitem>
 897             <para><emphasis role="bold">fourier</emphasis></para>
 898           </listitem>
 899         </varlistentry>
 900         <varlistentry>
 901           <term>Pre-conditions:</term>
 902           <listitem>
 903             <para>The conclusion of the current sequent must be a linear
 904              inequation over real numbers taken from standard library of
 905              Coq. Moreover the inequations in the hypotheses must imply the
 906              inequation in the conclusion of the current sequent.</para>
 907           </listitem>
 908         </varlistentry>
 909         <varlistentry>
 910           <term>Action:</term>
 911           <listitem>
 912             <para>It closes the current sequent by applying the Fourier method.</para>
 913           </listitem>
 914         </varlistentry>
 915         <varlistentry>
 916           <term>New sequents to prove:</term>
 917           <listitem>
 918             <para>None.</para>
 919           </listitem>
 920         </varlistentry>
 921       </variablelist>
 922     </para>
 923   </sect1>
 924   <sect1 id="tac_fwd">
 925     <title>fwd</title>
 926     <titleabbrev>fwd</titleabbrev>
 927     <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
 928     <para>
 929       <variablelist>
 930         <varlistentry role="tactic.synopsis">
 931           <term>Synopsis:</term>
 932           <listitem>
 933             <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
 934           </listitem>
 935         </varlistentry>
 936         <varlistentry>
 937           <term>Pre-conditions:</term>
 938           <listitem>
 939             <para>
 940                The type of <command>H</command> must be the premise of a
 941                forward simplification theorem.
 942             </para>
 943           </listitem>
 944         </varlistentry>
 945         <varlistentry>
 946           <term>Action:</term>
 947           <listitem>
 948             <para>
 949              This tactic is under development.
 950              It simplifies the current context by removing
 951              <command>H</command> using the following methods:
 952              forward application (by <command>lapply</command>) of a suitable
 953              simplification theorem, chosen automatically, of which the type
 954              of <command>H</command> is a premise,
 955              decomposition (by <command>decompose</command>),
 956              rewriting (by <command>rewrite</command>).
 957              <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
 958              are passed to the tactics <command>fwd</command> invokes, as
 959              names for the premise they introduce.
 960             </para>
 961           </listitem>
 962         </varlistentry>
 963         <varlistentry>
 964           <term>New sequents to prove:</term>
 965           <listitem>
 966             <para>
 967              The ones opened by the tactics <command>fwd</command> invokes.
 968             </para>
 969           </listitem>
 970         </varlistentry>
 971       </variablelist>
 972     </para>
 973   </sect1>
 974   <sect1 id="tac_generalize">
 975     <title>generalize</title>
 976     <titleabbrev>generalize</titleabbrev>
 977     <para><userinput>generalize patt as H</userinput></para>
 978     <para>
 979       <variablelist>
 980         <varlistentry role="tactic.synopsis">
 981           <term>Synopsis:</term>
 982           <listitem>
 983             <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
 984           </listitem>
 985         </varlistentry>
 986         <varlistentry>
 987           <term>Pre-conditions:</term>
 988           <listitem>
 989             <para>All the terms matched by <command>patt</command> must be
 990              convertible and close in the context of the current sequent.</para>
 991           </listitem>
 992         </varlistentry>
 993         <varlistentry>
 994           <term>Action:</term>
 995           <listitem>
 996             <para>It closes the current sequent by applying a stronger
 997              lemma that is proved using the new generated sequent.</para>
 998           </listitem>
 999         </varlistentry>
1000         <varlistentry>
1001           <term>New sequents to prove:</term>
1002           <listitem>
1003             <para>It opens a new sequent where the current sequent conclusion
1004              <command>G</command> is generalized to
1005              <command>∀x.G{x/t}</command> where <command>{x/t}</command>
1006              is a notation for the replacement with <command>x</command> of all
1007              the occurrences of the term <command>t</command> matched by
1008              <command>patt</command>. If <command>patt</command> matches no
1009              subterm then <command>t</command> is defined as the
1010              <command>wanted</command> part of the pattern.</para>
1011           </listitem>
1012         </varlistentry>
1013       </variablelist>
1014     </para>
1015   </sect1>
1016   <sect1 id="tac_id">
1017     <title>id</title>
1018     <titleabbrev>id</titleabbrev>
1019     <para><userinput>id </userinput></para>
1020     <para>
1021       <variablelist>
1022         <varlistentry role="tactic.synopsis">
1023           <term>Synopsis:</term>
1024           <listitem>
1025             <para><emphasis role="bold">id</emphasis></para>
1026           </listitem>
1027         </varlistentry>
1028         <varlistentry>
1029           <term>Pre-conditions:</term>
1030           <listitem>
1031             <para>None.</para>
1032           </listitem>
1033         </varlistentry>
1034         <varlistentry>
1035           <term>Action:</term>
1036           <listitem>
1037             <para>This identity tactic does nothing without failing.</para>
1038           </listitem>
1039         </varlistentry>
1040         <varlistentry>
1041           <term>New sequents to prove:</term>
1042           <listitem>
1043             <para>None.</para>
1044           </listitem>
1045         </varlistentry>
1046       </variablelist>
1047     </para>
1048   </sect1>
1049   <sect1 id="tac_intro">
1050     <title>intro</title>
1051     <titleabbrev>intro</titleabbrev>
1052     <para><userinput>intro H</userinput></para>
1053     <para>
1054       <variablelist>
1055         <varlistentry role="tactic.synopsis">
1056           <term>Synopsis:</term>
1057           <listitem>
1058             <para><emphasis role="bold">intro</emphasis> [&id;]</para>
1059           </listitem>
1060         </varlistentry>
1061         <varlistentry>
1062           <term>Pre-conditions:</term>
1063           <listitem>
1064             <para>The conclusion of the sequent to prove must be an implication
1065              or a universal quantification.</para>
1066           </listitem>
1067         </varlistentry>
1068         <varlistentry>
1069           <term>Action:</term>
1070           <listitem>
1071             <para>It applies the right introduction rule for implication,
1072              closing the current sequent.</para>
1073           </listitem>
1074         </varlistentry>
1075         <varlistentry>
1076           <term>New sequents to prove:</term>
1077           <listitem>
1078             <para>It opens a new sequent to prove adding to the hypothesis
1079              the antecedent of the implication and setting the conclusion
1080              to the consequent of the implicaiton. The name of the new
1081              hypothesis is <command>H</command> if provided; otherwise it
1082              is automatically generated.</para>
1083           </listitem>
1084         </varlistentry>
1085       </variablelist>
1086     </para>
1087   </sect1>
1088   <sect1 id="tac_intros">
1089     <title>intros</title>
1090     <titleabbrev>intros</titleabbrev>
1091     <para><userinput>intros hyps</userinput></para>
1092     <para>
1093       <variablelist>
1094         <varlistentry role="tactic.synopsis">
1095           <term>Synopsis:</term>
1096           <listitem>
1097             <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
1098           </listitem>
1099         </varlistentry>
1100         <varlistentry>
1101           <term>Pre-conditions:</term>
1102           <listitem>
1103             <para>If <command>hyps</command> specifies a number of hypotheses
1104              to introduce, then the conclusion of the current sequent must
1105              be formed by at least that number of imbricated implications
1106              or universal quantifications.</para>
1107           </listitem>
1108         </varlistentry>
1109         <varlistentry>
1110           <term>Action:</term>
1111           <listitem>
1112             <para>It applies several times the right introduction rule for
1113              implication, closing the current sequent.</para>
1114           </listitem>
1115         </varlistentry>
1116         <varlistentry>
1117           <term>New sequents to prove:</term>
1118           <listitem>
1119             <para>It opens a new sequent to prove adding a number of new
1120              hypotheses equal to the number of new hypotheses requested.
1121              If the user does not request a precise number of new hypotheses,
1122              it adds as many hypotheses as possible.
1123              The name of each new hypothesis is either popped from the
1124              user provided list of names, or it is automatically generated when
1125              the list is (or becomes) empty.</para>
1126           </listitem>
1127         </varlistentry>
1128       </variablelist>
1129     </para>
1130   </sect1>
1131   <sect1 id="tac_inversion">
1132     <title>inversion</title>
1133     <titleabbrev>inversion</titleabbrev>
1134     <para><userinput>inversion t</userinput></para>
1135     <para>
1136       <variablelist>
1137         <varlistentry role="tactic.synopsis">
1138           <term>Synopsis:</term>
1139           <listitem>
1140             <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
1141           </listitem>
1142         </varlistentry>
1143         <varlistentry>
1144           <term>Pre-conditions:</term>
1145           <listitem>
1146             <para>The type of the term <command>t</command> must be an inductive
1147              type or the application of an inductive type.</para>
1148           </listitem>
1149         </varlistentry>
1150         <varlistentry>
1151           <term>Action:</term>
1152           <listitem>
1153             <para>It proceeds by cases on <command>t</command> paying attention
1154              to the constraints imposed by the actual &quot;right arguments&quot;
1155              of the inductive type.</para>
1156           </listitem>
1157         </varlistentry>
1158         <varlistentry>
1159           <term>New sequents to prove:</term>
1160           <listitem>
1161             <para>It opens one new sequent to prove for each case in the
1162              definition of the type of <command>t</command>. With respect to
1163              a simple elimination, each new sequent has additional hypotheses
1164              that states the equalities of the &quot;right parameters&quot;
1165              of the inductive type with terms originally present in the
1166              sequent to prove.</para>
1167           </listitem>
1168         </varlistentry>
1169       </variablelist>
1170     </para>
1171   </sect1>
1172   <sect1 id="tac_lapply">
1173     <title>lapply</title>
1174     <titleabbrev>lapply</titleabbrev>
1175     <para><userinput>
1176      lapply linear depth=d t
1177      to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
1178     </userinput></para>
1179     <para>
1180       <variablelist>
1181         <varlistentry role="tactic.synopsis">
1182           <term>Synopsis:</term>
1183           <listitem>
1184             <para>
1185              <emphasis role="bold">lapply</emphasis>
1186              [<emphasis role="bold">linear</emphasis>]
1187              [<emphasis role="bold">depth=</emphasis>&nat;]
1188              &sterm;
1189              [<emphasis role="bold">to</emphasis>
1190               &sterm;
1191               [<emphasis role="bold">,</emphasis>&sterm;…]
1192              ]
1193              [<emphasis role="bold">as</emphasis> &id;]
1194             </para>
1195           </listitem>
1196         </varlistentry>
1197         <varlistentry>
1198           <term>Pre-conditions:</term>
1199           <listitem>
1200             <para>
1201              <command>t</command> must have at least <command>d</command>
1202              independent premises and <command>n</command> must not be
1203              greater than <command>d</command>.
1204             </para>
1205           </listitem>
1206         </varlistentry>
1207         <varlistentry>
1208           <term>Action:</term>
1209           <listitem>
1210             <para>
1211              Invokes <command>letin H ≝ (t ? ... ?)</command>
1212              with enough <command>?</command>'s to reach the
1213              <command>d</command>-th independent premise of
1214              <command>t</command>
1215              (<command>d</command> is maximum if unspecified).
1216              Then istantiates (by <command>apply</command>) with
1217              t<subscript>1</subscript>, ..., t<subscript>n</subscript>
1218              the <command>?</command>'s corresponding to the first
1219              <command>n</command> independent premises of
1220              <command>t</command>.
1221              Usually the other <command>?</command>'s preceding the
1222              <command>n</command>-th independent premise of
1223              <command>t</command> are istantiated as a consequence.
1224              If the <command>linear</command> flag is specified and if
1225              <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
1226              are (applications of) premises in the current context, they are
1227              <command>clear</command>ed.
1228             </para>
1229           </listitem>
1230         </varlistentry>
1231         <varlistentry>
1232           <term>New sequents to prove:</term>
1233           <listitem>
1234             <para>
1235              The ones opened by the tactics <command>lapply</command> invokes.
1236             </para>
1237           </listitem>
1238         </varlistentry>
1239       </variablelist>
1240     </para>
1241   </sect1>
1242   <sect1 id="tac_left">
1243     <title>left</title>
1244     <titleabbrev>left</titleabbrev>
1245     <para><userinput>left </userinput></para>
1246     <para>
1247       <variablelist>
1248         <varlistentry role="tactic.synopsis">
1249           <term>Synopsis:</term>
1250           <listitem>
1251             <para><emphasis role="bold">left</emphasis></para>
1252           </listitem>
1253         </varlistentry>
1254         <varlistentry>
1255           <term>Pre-conditions:</term>
1256           <listitem>
1257             <para>The conclusion of the current sequent must be
1258              an inductive type or the application of an inductive type
1259              with at least one constructor.</para>
1260           </listitem>
1261         </varlistentry>
1262         <varlistentry>
1263           <term>Action:</term>
1264           <listitem>
1265             <para>Equivalent to <command>constructor 1</command>.</para>
1266           </listitem>
1267         </varlistentry>
1268         <varlistentry>
1269           <term>New sequents to prove:</term>
1270           <listitem>
1271             <para>It opens a new sequent for each premise of the first
1272              constructor of the inductive type that is the conclusion of the
1273              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1274           </listitem>
1275         </varlistentry>
1276       </variablelist>
1277     </para>
1278   </sect1>
1279   <sect1 id="tac_letin">
1280     <title>letin</title>
1281     <titleabbrev>letin</titleabbrev>
1282     <para><userinput>letin x ≝ t</userinput></para>
1283     <para>
1284       <variablelist>
1285         <varlistentry role="tactic.synopsis">
1286           <term>Synopsis:</term>
1287           <listitem>
1288             <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
1289           </listitem>
1290         </varlistentry>
1291         <varlistentry>
1292           <term>Pre-conditions:</term>
1293           <listitem>
1294             <para>None.</para>
1295           </listitem>
1296         </varlistentry>
1297         <varlistentry>
1298           <term>Action:</term>
1299           <listitem>
1300             <para>It adds to the context of the current sequent to prove a new
1301              definition <command>x ≝ t</command>.</para>
1302           </listitem>
1303         </varlistentry>
1304         <varlistentry>
1305           <term>New sequents to prove:</term>
1306           <listitem>
1307             <para>None.</para>
1308           </listitem>
1309         </varlistentry>
1310       </variablelist>
1311     </para>
1312   </sect1>
1313   <sect1 id="tac_normalize">
1314     <title>normalize</title>
1315     <titleabbrev>normalize</titleabbrev>
1316     <para><userinput>normalize patt</userinput></para>
1317     <para>
1318       <variablelist>
1319         <varlistentry role="tactic.synopsis">
1320           <term>Synopsis:</term>
1321           <listitem>
1322             <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
1323           </listitem>
1324         </varlistentry>
1325         <varlistentry>
1326           <term>Pre-conditions:</term>
1327           <listitem>
1328             <para>None.</para>
1329           </listitem>
1330         </varlistentry>
1331         <varlistentry>
1332           <term>Action:</term>
1333           <listitem>
1334             <para>It replaces all the terms matched by <command>patt</command>
1335              with their βδιζ-normal form.</para>
1336           </listitem>
1337         </varlistentry>
1338         <varlistentry>
1339           <term>New sequents to prove:</term>
1340           <listitem>
1341             <para>None.</para>
1342           </listitem>
1343         </varlistentry>
1344       </variablelist>
1345     </para>
1346   </sect1>
1347   <sect1 id="tac_reflexivity">
1348     <title>reflexivity</title>
1349     <titleabbrev>reflexivity</titleabbrev>
1350     <para><userinput>reflexivity </userinput></para>
1351     <para>
1352       <variablelist>
1353         <varlistentry role="tactic.synopsis">
1354           <term>Synopsis:</term>
1355           <listitem>
1356             <para><emphasis role="bold">reflexivity</emphasis></para>
1357           </listitem>
1358         </varlistentry>
1359         <varlistentry>
1360           <term>Pre-conditions:</term>
1361           <listitem>
1362             <para>The conclusion of the current sequent must be
1363              <command>t=t</command> for some term <command>t</command></para>
1364           </listitem>
1365         </varlistentry>
1366         <varlistentry>
1367           <term>Action:</term>
1368           <listitem>
1369             <para>It closes the current sequent by reflexivity
1370              of equality.</para>
1371           </listitem>
1372         </varlistentry>
1373         <varlistentry>
1374           <term>New sequents to prove:</term>
1375           <listitem>
1376             <para>None.</para>
1377           </listitem>
1378         </varlistentry>
1379       </variablelist>
1380     </para>
1381   </sect1>
1382   <sect1 id="tac_replace">
1383     <title>replace</title>
1384     <titleabbrev>change</titleabbrev>
1385     <para><userinput>change patt with t</userinput></para>
1386     <para>
1387       <variablelist>
1388         <varlistentry role="tactic.synopsis">
1389           <term>Synopsis:</term>
1390           <listitem>
1391             <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
1392           </listitem>
1393         </varlistentry>
1394         <varlistentry>
1395           <term>Pre-conditions:</term>
1396           <listitem>
1397             <para>None.</para>
1398           </listitem>
1399         </varlistentry>
1400         <varlistentry>
1401           <term>Action:</term>
1402           <listitem>
1403             <para>It replaces the subterms of the current sequent matched by
1404              <command>patt</command> with the new term <command>t</command>.
1405              For each subterm matched by the pattern, <command>t</command> is
1406              disambiguated in the context of the subterm.</para>
1407           </listitem>
1408         </varlistentry>
1409         <varlistentry>
1410           <term>New sequents to prove:</term>
1411           <listitem>
1412             <para>For each matched term <command>t'</command> it opens
1413              a new sequent to prove whose conclusion is
1414              <command>t'=t</command>.</para>
1415           </listitem>
1416         </varlistentry>
1417       </variablelist>
1418     </para>
1419   </sect1>
1420   <sect1 id="tac_rewrite">
1421     <title>rewrite</title>
1422     <titleabbrev>rewrite</titleabbrev>
1423     <para><userinput>rewrite dir p patt</userinput></para>
1424     <para>
1425       <variablelist>
1426         <varlistentry role="tactic.synopsis">
1427           <term>Synopsis:</term>
1428           <listitem>
1429             <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold">&lt;</emphasis>|<emphasis role="bold">&gt;</emphasis>] &sterm; &pattern;</para>
1430           </listitem>
1431         </varlistentry>
1432         <varlistentry>
1433           <term>Pre-conditions:</term>
1434           <listitem>
1435             <para><command>p</command> must be the proof of an equality,
1436              possibly under some hypotheses.</para>
1437           </listitem>
1438         </varlistentry>
1439         <varlistentry>
1440           <term>Action:</term>
1441           <listitem>
1442             <para>It looks in every term matched by <command>patt</command>
1443              for all the occurrences of the
1444              left hand side of the equality that <command>p</command> proves
1445              (resp. the right hand side if <command>dir</command> is
1446              <command>&lt;</command>). Every occurence found is replaced with
1447              the opposite side of the equality.</para>
1448           </listitem>
1449         </varlistentry>
1450         <varlistentry>
1451           <term>New sequents to prove:</term>
1452           <listitem>
1453             <para>It opens one new sequent for each hypothesis of the
1454              equality proved by <command>p</command> that is not closed
1455              by unification.</para>
1456           </listitem>
1457         </varlistentry>
1458       </variablelist>
1459     </para>
1460   </sect1>
1461   <sect1 id="tac_right">
1462     <title>right</title>
1463     <titleabbrev>right</titleabbrev>
1464     <para><userinput>right </userinput></para>
1465     <para>
1466       <variablelist>
1467         <varlistentry role="tactic.synopsis">
1468           <term>Synopsis:</term>
1469           <listitem>
1470             <para><emphasis role="bold">right</emphasis></para>
1471           </listitem>
1472         </varlistentry>
1473         <varlistentry>
1474           <term>Pre-conditions:</term>
1475           <listitem>
1476             <para>The conclusion of the current sequent must be
1477              an inductive type or the application of an inductive type with
1478              at least two constructors.</para>
1479           </listitem>
1480         </varlistentry>
1481         <varlistentry>
1482           <term>Action:</term>
1483           <listitem>
1484             <para>Equivalent to <command>constructor 2</command>.</para>
1485           </listitem>
1486         </varlistentry>
1487         <varlistentry>
1488           <term>New sequents to prove:</term>
1489           <listitem>
1490             <para>It opens a new sequent for each premise of the second
1491              constructor of the inductive type that is the conclusion of the
1492              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1493           </listitem>
1494         </varlistentry>
1495       </variablelist>
1496     </para>
1497   </sect1>
1498   <sect1 id="tac_ring">
1499     <title>ring</title>
1500     <titleabbrev>ring</titleabbrev>
1501     <para><userinput>ring </userinput></para>
1502     <para>
1503       <variablelist>
1504         <varlistentry role="tactic.synopsis">
1505           <term>Synopsis:</term>
1506           <listitem>
1507             <para><emphasis role="bold">ring</emphasis></para>
1508           </listitem>
1509         </varlistentry>
1510         <varlistentry>
1511           <term>Pre-conditions:</term>
1512           <listitem>
1513             <para>The conclusion of the current sequent must be an
1514              equality over Coq's real numbers that can be proved using
1515              the ring properties of the real numbers only.</para>
1516           </listitem>
1517         </varlistentry>
1518         <varlistentry>
1519           <term>Action:</term>
1520           <listitem>
1521             <para>It closes the current sequent veryfying the equality by
1522              means of computation (i.e. this is a reflexive tactic, implemented
1523              exploiting the &quot;two level reasoning&quot; technique).</para>
1524           </listitem>
1525         </varlistentry>
1526         <varlistentry>
1527           <term>New sequents to prove:</term>
1528           <listitem>
1529             <para>None.</para>
1530           </listitem>
1531         </varlistentry>
1532       </variablelist>
1533     </para>
1534   </sect1>
1535   <sect1 id="tac_simplify">
1536     <title>simplify</title>
1537     <titleabbrev>simplify</titleabbrev>
1538     <para><userinput>simplify patt</userinput></para>
1539     <para>
1540       <variablelist>
1541         <varlistentry role="tactic.synopsis">
1542           <term>Synopsis:</term>
1543           <listitem>
1544             <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
1545           </listitem>
1546         </varlistentry>
1547         <varlistentry>
1548           <term>Pre-conditions:</term>
1549           <listitem>
1550             <para>None.</para>
1551           </listitem>
1552         </varlistentry>
1553         <varlistentry>
1554           <term>Action:</term>
1555           <listitem>
1556             <para>It replaces all the terms matched by <command>patt</command>
1557              with other convertible terms that are supposed to be simpler.</para>
1558           </listitem>
1559         </varlistentry>
1560         <varlistentry>
1561           <term>New sequents to prove:</term>
1562           <listitem>
1563             <para>None.</para>
1564           </listitem>
1565         </varlistentry>
1566       </variablelist>
1567     </para>
1568   </sect1>
1569   <sect1 id="tac_split">
1570     <title>split</title>
1571     <titleabbrev>split</titleabbrev>
1572     <para><userinput>split </userinput></para>
1573     <para>
1574       <variablelist>
1575         <varlistentry role="tactic.synopsis">
1576           <term>Synopsis:</term>
1577           <listitem>
1578             <para><emphasis role="bold">split</emphasis></para>
1579           </listitem>
1580         </varlistentry>
1581         <varlistentry>
1582           <term>Pre-conditions:</term>
1583           <listitem>
1584             <para>The conclusion of the current sequent must be
1585              an inductive type or the application of an inductive type with
1586              at least one constructor.</para>
1587           </listitem>
1588         </varlistentry>
1589         <varlistentry>
1590           <term>Action:</term>
1591           <listitem>
1592             <para>Equivalent to <command>constructor 1</command>.</para>
1593           </listitem>
1594         </varlistentry>
1595         <varlistentry>
1596           <term>New sequents to prove:</term>
1597           <listitem>
1598             <para>It opens a new sequent for each premise of the first
1599              constructor of the inductive type that is the conclusion of the
1600              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1601           </listitem>
1602         </varlistentry>
1603       </variablelist>
1604     </para>
1605   </sect1>
1606
1607   <sect1 id="tac_subst">
1608     <title>subst</title>
1609     <titleabbrev>subst</titleabbrev>
1610     <para><userinput>subst</userinput></para>
1611     <para>
1612       <variablelist>
1613         <varlistentry role="tactic.synopsis">
1614           <term>Synopsis:</term>
1615           <listitem>
1616             <para><emphasis role="bold">subst</emphasis></para>
1617           </listitem>
1618         </varlistentry>
1619         <varlistentry>
1620           <term>Pre-conditions:</term>
1621           <listitem><para>
1622             None.
1623           </para></listitem>
1624         </varlistentry>
1625         <varlistentry>
1626           <term>Action:</term>
1627           <listitem><para>
1628             For each premise of the form
1629             <command>H: x = t</command> or <command>H: t = x</command>
1630             where <command>x</command> is a local variable and
1631             <command>t</command> does not depend on <command>x</command>,
1632             the tactic rewrites <command>H</command> wherever
1633             <command>x</command> appears clearing <command>H</command> and
1634             <command>x</command> afterwards.
1635           </para></listitem>
1636         </varlistentry>
1637         <varlistentry>
1638           <term>New sequents to prove:</term>
1639           <listitem><para>
1640             The one opened by the applied tactics.
1641           </para></listitem>
1642         </varlistentry>
1643       </variablelist>
1644     </para>
1645   </sect1>
1646   <sect1 id="tac_symmetry">
1647     <title>symmetry</title>
1648     <titleabbrev>symmetry</titleabbrev>
1649     <para>The tactic <command>symmetry</command> </para>
1650     <para><userinput>symmetry </userinput></para>
1651     <para>
1652       <variablelist>
1653         <varlistentry role="tactic.synopsis">
1654           <term>Synopsis:</term>
1655           <listitem>
1656             <para><emphasis role="bold">symmetry</emphasis></para>
1657           </listitem>
1658         </varlistentry>
1659         <varlistentry>
1660           <term>Pre-conditions:</term>
1661           <listitem>
1662             <para>The conclusion of the current proof must be an equality.</para>
1663           </listitem>
1664         </varlistentry>
1665         <varlistentry>
1666           <term>Action:</term>
1667           <listitem>
1668             <para>It swaps the two sides of the equalityusing the symmetric
1669              property.</para>
1670           </listitem>
1671         </varlistentry>
1672         <varlistentry>
1673           <term>New sequents to prove:</term>
1674           <listitem>
1675             <para>None.</para>
1676           </listitem>
1677         </varlistentry>
1678       </variablelist>
1679     </para>
1680   </sect1>
1681   <sect1 id="tac_transitivity">
1682     <title>transitivity</title>
1683     <titleabbrev>transitivity</titleabbrev>
1684     <para><userinput>transitivity t</userinput></para>
1685     <para>
1686       <variablelist>
1687         <varlistentry role="tactic.synopsis">
1688           <term>Synopsis:</term>
1689           <listitem>
1690             <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
1691           </listitem>
1692         </varlistentry>
1693         <varlistentry>
1694           <term>Pre-conditions:</term>
1695           <listitem>
1696             <para>The conclusion of the current proof must be an equality.</para>
1697           </listitem>
1698         </varlistentry>
1699         <varlistentry>
1700           <term>Action:</term>
1701           <listitem>
1702             <para>It closes the current sequent by transitivity of the equality.</para>
1703           </listitem>
1704         </varlistentry>
1705         <varlistentry>
1706           <term>New sequents to prove:</term>
1707           <listitem>
1708             <para>It opens two new sequents <command>l=t</command> and
1709              <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
1710 the current sequent to prove.</para>
1711           </listitem>
1712         </varlistentry>
1713       </variablelist>
1714     </para>
1715   </sect1>
1716   <sect1 id="tac_unfold">
1717     <title>unfold</title>
1718     <titleabbrev>unfold</titleabbrev>
1719     <para><userinput>unfold t patt</userinput></para>
1720     <para>
1721       <variablelist>
1722         <varlistentry role="tactic.synopsis">
1723           <term>Synopsis:</term>
1724           <listitem>
1725             <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
1726           </listitem>
1727         </varlistentry>
1728         <varlistentry>
1729           <term>Pre-conditions:</term>
1730           <listitem>
1731             <para>None.</para>
1732           </listitem>
1733         </varlistentry>
1734         <varlistentry>
1735           <term>Action:</term>
1736           <listitem>
1737             <para>It finds all the occurrences of <command>t</command>
1738              (possibly applied to arguments) in the subterms matched by
1739              <command>patt</command>. Then it δ-expands each occurrence,
1740              also performing β-reduction of the obtained term. If
1741              <command>t</command> is omitted it defaults to each
1742              subterm matched by <command>patt</command>.</para>
1743           </listitem>
1744         </varlistentry>
1745         <varlistentry>
1746           <term>New sequents to prove:</term>
1747           <listitem>
1748             <para>None.</para>
1749           </listitem>
1750         </varlistentry>
1751       </variablelist>
1752     </para>
1753   </sect1>
1754   <sect1 id="tac_whd">
1755     <title>whd</title>
1756     <titleabbrev>whd</titleabbrev>
1757     <para><userinput>whd patt</userinput></para>
1758     <para>
1759       <variablelist>
1760         <varlistentry role="tactic.synopsis">
1761           <term>Synopsis:</term>
1762           <listitem>
1763             <para><emphasis role="bold">whd</emphasis> &pattern;</para>
1764           </listitem>
1765         </varlistentry>
1766         <varlistentry>
1767           <term>Pre-conditions:</term>
1768           <listitem>
1769             <para>None.</para>
1770           </listitem>
1771         </varlistentry>
1772         <varlistentry>
1773           <term>Action:</term>
1774           <listitem>
1775             <para>It replaces all the terms matched by <command>patt</command>
1776              with their βδιζ-weak-head normal form.</para>
1777           </listitem>
1778         </varlistentry>
1779         <varlistentry>
1780           <term>New sequents to prove:</term>
1781           <listitem>
1782             <para>None.</para>
1783           </listitem>
1784         </varlistentry>
1785       </variablelist>
1786     </para>
1787   </sect1>
1788
1789 </chapter>
1790