helm/software/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             <para><emphasis role="bold">autobatch</emphasis> &autoparams;</para>
 189           </listitem>
 190         </varlistentry>
 191         <varlistentry>
 192           <term>Pre-conditions:</term>
 193           <listitem>
 194             <para>None, but the tactic may fail finding a proof if every
 195              proof is in the search space that is pruned away. Pruning is
 196              controlled by the optional <command>params</command>.
 197              Moreover, only lemmas whose type signature is a subset of the
 198              signature of the current sequent are considered. The signature of
 199              a sequent is essentially the set of constats appearing in it.
 200            </para>
 201           </listitem>
 202         </varlistentry>
 203         <varlistentry>
 204           <term>Action:</term>
 205           <listitem>
 206             <para>It closes the current sequent by repeated application of
 207              rewriting steps (unless <command>paramodulation</command> is
 208              omitted), hypothesis and lemmas in the library.</para>
 209           </listitem>
 210         </varlistentry>
 211         <varlistentry>
 212           <term>New sequents to prove:</term>
 213           <listitem>
 214             <para>None</para>
 215           </listitem>
 216         </varlistentry>
 217       </variablelist>
 218     </para>
 219   </sect1>
 220   <sect1 id="tac_cases">
 221     <title>cases</title>
 222     <titleabbrev>cases</titleabbrev>
 223     <para><userinput>
 224      cases t hyps
 225     </userinput></para>
 226     <para>
 227       <variablelist>
 228         <varlistentry role="tactic.synopsis">
 229           <term>Synopsis:</term>
 230           <listitem>
 231             <para>
 232              <emphasis role="bold">cases</emphasis>
 233              &term; [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]
 234             </para>
 235           </listitem>
 236         </varlistentry>
 237         <varlistentry>
 238           <term>Pre-conditions:</term>
 239           <listitem>
 240             <para>
 241              <command>t</command> must inhabit an inductive type
 242             </para>
 243           </listitem>
 244         </varlistentry>
 245         <varlistentry>
 246           <term>Action:</term>
 247           <listitem>
 248             <para>
 249              It proceed by cases on <command>t</command>. The new generated
 250              hypothesis in each branch are named according to
 251              <command>hyps</command>.
 252             </para>
 253           </listitem>
 254         </varlistentry>
 255         <varlistentry>
 256           <term>New sequents to prove:</term>
 257           <listitem>
 258             <para>One new sequent for each constructor of the type of
 259              <command>t</command>. Each sequent has a new hypothesis for
 260              each argument of the constructor.</para>
 261           </listitem>
 262         </varlistentry>
 263       </variablelist>
 264     </para>
 265   </sect1>
 266   <sect1 id="tac_clear">
 267     <title>clear</title>
 268     <titleabbrev>clear</titleabbrev>
 269     <para><userinput>
 270      clear H<subscript>1</subscript> ... H<subscript>m</subscript>
 271     </userinput></para>
 272     <para>
 273       <variablelist>
 274         <varlistentry role="tactic.synopsis">
 275           <term>Synopsis:</term>
 276           <listitem>
 277             <para>
 278              <emphasis role="bold">clear</emphasis>
 279              &id; [&id;…]
 280             </para>
 281           </listitem>
 282         </varlistentry>
 283         <varlistentry>
 284           <term>Pre-conditions:</term>
 285           <listitem>
 286             <para>
 287              <command>
 288               H<subscript>1</subscript> ... H<subscript>m</subscript>
 289              </command> must be hypotheses of the
 290              current sequent to prove.
 291             </para>
 292           </listitem>
 293         </varlistentry>
 294         <varlistentry>
 295           <term>Action:</term>
 296           <listitem>
 297             <para>
 298              It hides the hypotheses
 299              <command>
 300               H<subscript>1</subscript> ... H<subscript>m</subscript>
 301              </command> from the current sequent.
 302             </para>
 303           </listitem>
 304         </varlistentry>
 305         <varlistentry>
 306           <term>New sequents to prove:</term>
 307           <listitem>
 308             <para>None</para>
 309           </listitem>
 310         </varlistentry>
 311       </variablelist>
 312     </para>
 313   </sect1>
 314   <sect1 id="tac_clearbody">
 315     <title>clearbody</title>
 316     <titleabbrev>clearbody</titleabbrev>
 317     <para><userinput>clearbody H</userinput></para>
 318     <para>
 319       <variablelist>
 320         <varlistentry role="tactic.synopsis">
 321           <term>Synopsis:</term>
 322           <listitem>
 323             <para><emphasis role="bold">clearbody</emphasis> &id;</para>
 324           </listitem>
 325         </varlistentry>
 326         <varlistentry>
 327           <term>Pre-conditions:</term>
 328           <listitem>
 329             <para><command>H</command> must be an hypothesis of the
 330              current sequent to prove.</para>
 331           </listitem>
 332         </varlistentry>
 333         <varlistentry>
 334           <term>Action:</term>
 335           <listitem>
 336             <para>It hides the definiens of a definition in the current
 337              sequent context. Thus the definition becomes an hypothesis.</para>
 338           </listitem>
 339         </varlistentry>
 340         <varlistentry>
 341           <term>New sequents to prove:</term>
 342           <listitem>
 343             <para>None.</para>
 344           </listitem>
 345         </varlistentry>
 346       </variablelist>
 347     </para>
 348   </sect1>
 349   <sect1 id="tac_compose">
 350     <title>compose</title>
 351     <titleabbrev>compose</titleabbrev>
 352     <para><userinput>compose n t1 with t2 hyps</userinput></para>
 353     <para>
 354       <variablelist>
 355         <varlistentry role="tactic.synopsis">
 356           <term>Synopsis:</term>
 357           <listitem>
 358             <para><emphasis role="bold">compose</emphasis> [&nat;] &sterm; [<emphasis role="bold">with</emphasis> &sterm;] [&intros-spec;]</para>
 359           </listitem>
 360         </varlistentry>
 361         <varlistentry>
 362           <term>Pre-conditions:</term>
 363           <listitem>
 364             <para></para>
 365           </listitem>
 366         </varlistentry>
 367         <varlistentry>
 368           <term>Action:</term>
 369           <listitem>
 370             <para>Composes t1 with t2 in every possible way
 371               <command>n</command> times introducing generated terms
 372               as if <command>intros hyps</command> was issued.</para>
 373             <para>If <command>t1:∀x:A.B[x]</command> and
 374             <command>t2:∀x,y:A.B[x]→B[y]→C[x,y]</command> it generates:
 375              <itemizedlist>
 376                 <listitem>
 377                   <para><command>λx,y:A.t2 x y (t1 x) : ∀x,y:A.B[y]→C[x,y]</command></para>
 378                 </listitem>
 379                 <listitem>
 380                   <para><command>λx,y:A.λH:B[x].t2 x y H (t1 y) : ∀x,y:A.B[x]→C[x,y]
 381                   </command></para>
 382                 </listitem>
 383              </itemizedlist>
 384           </para>
 385           <para>If <command>t2</command> is omitted it composes
 386             <command>t1</command>
 387               with every hypothesis that can be introduced.
 388               <command>n</command> iterates the process.</para>
 389           </listitem>
 390         </varlistentry>
 391         <varlistentry>
 392           <term>New sequents to prove:</term>
 393           <listitem>
 394             <para>The same, but with more hypothesis eventually introduced
 395             by the &intros-spec;.</para>
 396           </listitem>
 397         </varlistentry>
 398       </variablelist>
 399     </para>
 400   </sect1>
 401   <sect1 id="tac_change">
 402     <title>change</title>
 403     <titleabbrev>change</titleabbrev>
 404     <para><userinput>change patt with t</userinput></para>
 405     <para>
 406       <variablelist>
 407         <varlistentry role="tactic.synopsis">
 408           <term>Synopsis:</term>
 409           <listitem>
 410             <para><emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
 411           </listitem>
 412         </varlistentry>
 413         <varlistentry>
 414           <term>Pre-conditions:</term>
 415           <listitem>
 416             <para>Each subterm matched by the pattern must be convertible
 417              with the term <command>t</command> disambiguated in the context
 418              of the matched subterm.</para>
 419           </listitem>
 420         </varlistentry>
 421         <varlistentry>
 422           <term>Action:</term>
 423           <listitem>
 424             <para>It replaces the subterms of the current sequent matched by
 425              <command>patt</command> with the new term <command>t</command>.
 426              For each subterm matched by the pattern, <command>t</command> is
 427              disambiguated in the context of the subterm.</para>
 428           </listitem>
 429         </varlistentry>
 430         <varlistentry>
 431           <term>New sequents to prove:</term>
 432           <listitem>
 433             <para>None.</para>
 434           </listitem>
 435         </varlistentry>
 436       </variablelist>
 437     </para>
 438   </sect1>
 439   <sect1 id="tac_constructor">
 440     <title>constructor</title>
 441     <titleabbrev>constructor</titleabbrev>
 442     <para><userinput>constructor n</userinput></para>
 443     <para>
 444       <variablelist>
 445         <varlistentry role="tactic.synopsis">
 446           <term>Synopsis:</term>
 447           <listitem>
 448             <para><emphasis role="bold">constructor</emphasis> &nat;</para>
 449           </listitem>
 450         </varlistentry>
 451         <varlistentry>
 452           <term>Pre-conditions:</term>
 453           <listitem>
 454             <para>The conclusion of the current sequent must be
 455              an inductive type or the application of an inductive type with
 456              at least <command>n</command> constructors.</para>
 457           </listitem>
 458         </varlistentry>
 459         <varlistentry>
 460           <term>Action:</term>
 461           <listitem>
 462             <para>It applies the <command>n</command>-th constructor of the
 463              inductive type of the conclusion of the current sequent.</para>
 464           </listitem>
 465         </varlistentry>
 466         <varlistentry>
 467           <term>New sequents to prove:</term>
 468           <listitem>
 469             <para>It opens a new sequent for each premise of the constructor
 470              that can not be inferred by unification. For more details,
 471              see the <command>apply</command> tactic.</para>
 472           </listitem>
 473         </varlistentry>
 474       </variablelist>
 475     </para>
 476   </sect1>
 477   <sect1 id="tac_contradiction">
 478     <title>contradiction</title>
 479     <titleabbrev>contradiction</titleabbrev>
 480     <para><userinput>contradiction </userinput></para>
 481     <para>
 482       <variablelist>
 483         <varlistentry role="tactic.synopsis">
 484           <term>Synopsis:</term>
 485           <listitem>
 486             <para><emphasis role="bold">contradiction</emphasis></para>
 487           </listitem>
 488         </varlistentry>
 489         <varlistentry>
 490           <term>Pre-conditions:</term>
 491           <listitem>
 492             <para>There must be in the current context an hypothesis of type
 493              <command>False</command>.</para>
 494           </listitem>
 495         </varlistentry>
 496         <varlistentry>
 497           <term>Action:</term>
 498           <listitem>
 499             <para>It closes the current sequent by applying an hypothesis of
 500              type <command>False</command>.</para>
 501           </listitem>
 502         </varlistentry>
 503         <varlistentry>
 504           <term>New sequents to prove:</term>
 505           <listitem>
 506             <para>None</para>
 507           </listitem>
 508         </varlistentry>
 509       </variablelist>
 510     </para>
 511   </sect1>
 512   <sect1 id="tac_cut">
 513     <title>cut</title>
 514     <titleabbrev>cut</titleabbrev>
 515     <para><userinput>cut P as H</userinput></para>
 516     <para>
 517       <variablelist>
 518         <varlistentry role="tactic.synopsis">
 519           <term>Synopsis:</term>
 520           <listitem>
 521             <para><emphasis role="bold">cut</emphasis> &sterm; [<emphasis role="bold">as</emphasis> &id;]</para>
 522           </listitem>
 523         </varlistentry>
 524         <varlistentry>
 525           <term>Pre-conditions:</term>
 526           <listitem>
 527             <para><command>P</command> must have type <command>Prop</command>.</para>
 528           </listitem>
 529         </varlistentry>
 530         <varlistentry>
 531           <term>Action:</term>
 532           <listitem>
 533             <para>It closes the current sequent.</para>
 534           </listitem>
 535         </varlistentry>
 536         <varlistentry>
 537           <term>New sequents to prove:</term>
 538           <listitem>
 539             <para>It opens two new sequents. The first one has an extra
 540              hypothesis <command>H:P</command>. If <command>H</command> is
 541              omitted, the name of the hypothesis is automatically generated.
 542              The second sequent has conclusion <command>P</command> and
 543              hypotheses the hypotheses of the current sequent to prove.</para>
 544           </listitem>
 545         </varlistentry>
 546       </variablelist>
 547     </para>
 548   </sect1>
 549   <sect1 id="tac_decompose">
 550     <title>decompose</title>
 551     <titleabbrev>decompose</titleabbrev>
 552     <para><userinput>
 553      decompose as H<subscript>1</subscript> ... H<subscript>m</subscript>
 554     </userinput></para>
 555     <para>
 556       <variablelist>
 557         <varlistentry role="tactic.synopsis">
 558           <term>Synopsis:</term>
 559           <listitem>
 560             <para>
 561              <emphasis role="bold">decompose</emphasis>
 562              [<emphasis role="bold">as</emphasis> &id;…]
 563             </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>
 576              For each each premise <command>H</command> of type
 577              <command>T</command> in the current context where
 578              <command>T</command> is a non-recursive inductive type without
 579              right parameters and of sort Prop or CProp, the tactic runs
 580              <command>
 581               elim H as H<subscript>1</subscript> ... H<subscript>m</subscript>
 582              </command>, clears <command>H</command>  and runs itself
 583              recursively on each new premise introduced by
 584              <command>elim</command> in the opened sequents.
 585             </para>
 586           </listitem>
 587         </varlistentry>
 588         <varlistentry>
 589           <term>New sequents to prove:</term>
 590           <listitem>
 591             <para>
 592              The ones generated by all the <command>elim</command> tactics run.
 593             </para>
 594           </listitem>
 595         </varlistentry>
 596       </variablelist>
 597     </para>
 598   </sect1>
 599   <sect1 id="tac_demodulate">
 600     <title>demodulate</title>
 601     <titleabbrev>demodulate</titleabbrev>
 602     <para><userinput>demodulate auto_params</userinput></para>
 603     <para>
 604       <variablelist>
 605         <varlistentry role="tactic.synopsis">
 606           <term>Synopsis:</term>
 607           <listitem>
 608             <para><emphasis role="bold">demodulate</emphasis> &autoparams;</para>
 609           </listitem>
 610         </varlistentry>
 611         <varlistentry>
 612           <term>Pre-conditions:</term>
 613           <listitem>
 614             <para>None.</para>
 615           </listitem>
 616         </varlistentry>
 617         <varlistentry>
 618           <term>Action:</term>
 619           <listitem>
 620             <para>&TODO;</para>
 621           </listitem>
 622         </varlistentry>
 623         <varlistentry>
 624           <term>New sequents to prove:</term>
 625           <listitem>
 626             <para>None.</para>
 627           </listitem>
 628         </varlistentry>
 629       </variablelist>
 630     </para>
 631   </sect1>
 632   <sect1 id="tac_destruct">
 633     <title>destruct</title>
 634     <titleabbrev>destruct</titleabbrev>
 635     <para><userinput>destruct p</userinput></para>
 636     <para>
 637       <variablelist>
 638         <varlistentry role="tactic.synopsis">
 639           <term>Synopsis:</term>
 640           <listitem>
 641             <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
 642           </listitem>
 643         </varlistentry>
 644         <varlistentry>
 645           <term>Pre-conditions:</term>
 646           <listitem>
 647             <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>
 648           </listitem>
 649         </varlistentry>
 650         <varlistentry>
 651           <term>Action:</term>
 652           <listitem>
 653             <para>The tactic recursively compare the two sides of the equality
 654              looking for different constructors in corresponding position.
 655              If two of them are found, the tactic closes the current sequent
 656              by proving the absurdity of <command>p</command>. Otherwise
 657              it adds a new hypothesis for each leaf of the formula that
 658              states the equality of the subformulae in the corresponding
 659              positions on the two sides of the equality.
 660             </para>
 661           </listitem>
 662         </varlistentry>
 663         <varlistentry>
 664           <term>New sequents to prove:</term>
 665           <listitem>
 666             <para>None.</para>
 667           </listitem>
 668         </varlistentry>
 669       </variablelist>
 670     </para>
 671   </sect1>
 672   <sect1 id="tac_elim">
 673     <title>elim</title>
 674     <titleabbrev>elim</titleabbrev>
 675     <para><userinput>elim t pattern using th hyps</userinput></para>
 676     <para>
 677       <variablelist>
 678         <varlistentry role="tactic.synopsis">
 679           <term>Synopsis:</term>
 680           <listitem>
 681             <para><emphasis role="bold">elim</emphasis> &sterm; &pattern; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 682           </listitem>
 683         </varlistentry>
 684         <varlistentry>
 685           <term>Pre-conditions:</term>
 686           <listitem>
 687             <para><command>t</command> must inhabit an inductive type and
 688              <command>th</command> must be an elimination principle for that
 689              inductive type. If <command>th</command> is omitted the appropriate
 690              standard elimination principle is chosen.</para>
 691           </listitem>
 692         </varlistentry>
 693         <varlistentry>
 694           <term>Action:</term>
 695           <listitem>
 696             <para>It proceeds by cases on the values of <command>t</command>,
 697              according to the elimination principle <command>th</command>.
 698              The induction predicate is restricted by
 699              <command>pattern</command>. In particular, if some hypothesis
 700              is listed in <command>pattern</command>, the hypothesis is
 701              generalized and cleared before eliminating <command>t</command>
 702             </para>
 703           </listitem>
 704         </varlistentry>
 705         <varlistentry>
 706           <term>New sequents to prove:</term>
 707           <listitem>
 708             <para>It opens one new sequent for each case. The names of
 709              the new hypotheses are picked by <command>hyps</command>, if
 710              provided. If hyps specifies also a number of hypotheses that
 711              is less than the number of new hypotheses for a new sequent,
 712              then the exceeding hypothesis will be kept as implications in
 713              the conclusion of the sequent.</para>
 714           </listitem>
 715         </varlistentry>
 716       </variablelist>
 717     </para>
 718   </sect1>
 719   <sect1 id="tac_elimType">
 720     <title>elimType</title>
 721     <titleabbrev>elimType</titleabbrev>
 722     <para><userinput>elimType T using th hyps</userinput></para>
 723     <para>
 724       <variablelist>
 725         <varlistentry role="tactic.synopsis">
 726           <term>Synopsis:</term>
 727           <listitem>
 728             <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 729           </listitem>
 730         </varlistentry>
 731         <varlistentry>
 732           <term>Pre-conditions:</term>
 733           <listitem>
 734             <para><command>T</command> must be an inductive type.</para>
 735           </listitem>
 736         </varlistentry>
 737         <varlistentry>
 738           <term>Action:</term>
 739           <listitem>
 740             <para>TODO (severely bugged now).</para>
 741           </listitem>
 742         </varlistentry>
 743         <varlistentry>
 744           <term>New sequents to prove:</term>
 745           <listitem>
 746             <para>TODO</para>
 747           </listitem>
 748         </varlistentry>
 749       </variablelist>
 750     </para>
 751   </sect1>
 752   <sect1 id="tac_exact">
 753     <title>exact</title>
 754     <titleabbrev>exact</titleabbrev>
 755     <para><userinput>exact p</userinput></para>
 756     <para>
 757       <variablelist>
 758         <varlistentry role="tactic.synopsis">
 759           <term>Synopsis:</term>
 760           <listitem>
 761             <para><emphasis role="bold">exact</emphasis> &sterm;</para>
 762           </listitem>
 763         </varlistentry>
 764         <varlistentry>
 765           <term>Pre-conditions:</term>
 766           <listitem>
 767             <para>The type of <command>p</command> must be convertible
 768              with the conclusion of the current sequent.</para>
 769           </listitem>
 770         </varlistentry>
 771         <varlistentry>
 772           <term>Action:</term>
 773           <listitem>
 774             <para>It closes the current sequent using <command>p</command>.</para>
 775           </listitem>
 776         </varlistentry>
 777         <varlistentry>
 778           <term>New sequents to prove:</term>
 779           <listitem>
 780             <para>None.</para>
 781           </listitem>
 782         </varlistentry>
 783       </variablelist>
 784     </para>
 785   </sect1>
 786   <sect1 id="tac_exists">
 787     <title>exists</title>
 788     <titleabbrev>exists</titleabbrev>
 789     <para><userinput>exists </userinput></para>
 790     <para>
 791       <variablelist>
 792         <varlistentry role="tactic.synopsis">
 793           <term>Synopsis:</term>
 794           <listitem>
 795             <para><emphasis role="bold">exists</emphasis></para>
 796           </listitem>
 797         </varlistentry>
 798         <varlistentry>
 799           <term>Pre-conditions:</term>
 800           <listitem>
 801             <para>The conclusion of the current sequent must be
 802              an inductive type or the application of an inductive type
 803              with at least one constructor.</para>
 804           </listitem>
 805         </varlistentry>
 806         <varlistentry>
 807           <term>Action:</term>
 808           <listitem>
 809             <para>Equivalent to <command>constructor 1</command>.</para>
 810           </listitem>
 811         </varlistentry>
 812         <varlistentry>
 813           <term>New sequents to prove:</term>
 814           <listitem>
 815             <para>It opens a new sequent for each premise of the first
 816              constructor of the inductive type that is the conclusion of the
 817              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 818           </listitem>
 819         </varlistentry>
 820       </variablelist>
 821     </para>
 822   </sect1>
 823   <sect1 id="tac_fail">
 824     <title>fail</title>
 825     <titleabbrev>fail</titleabbrev>
 826     <para><userinput>fail</userinput></para>
 827     <para>
 828       <variablelist>
 829         <varlistentry role="tactic.synopsis">
 830           <term>Synopsis:</term>
 831           <listitem>
 832             <para><emphasis role="bold">fail</emphasis></para>
 833           </listitem>
 834         </varlistentry>
 835         <varlistentry>
 836           <term>Pre-conditions:</term>
 837           <listitem>
 838             <para>None.</para>
 839           </listitem>
 840         </varlistentry>
 841         <varlistentry>
 842           <term>Action:</term>
 843           <listitem>
 844             <para>This tactic always fail.</para>
 845           </listitem>
 846         </varlistentry>
 847         <varlistentry>
 848           <term>New sequents to prove:</term>
 849           <listitem>
 850             <para>N.A.</para>
 851           </listitem>
 852         </varlistentry>
 853       </variablelist>
 854     </para>
 855   </sect1>
 856   <sect1 id="tac_fold">
 857     <title>fold</title>
 858     <titleabbrev>fold</titleabbrev>
 859     <para><userinput>fold red t patt</userinput></para>
 860     <para>
 861       <variablelist>
 862         <varlistentry role="tactic.synopsis">
 863           <term>Synopsis:</term>
 864           <listitem>
 865             <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
 866           </listitem>
 867         </varlistentry>
 868         <varlistentry>
 869           <term>Pre-conditions:</term>
 870           <listitem>
 871             <para>The pattern must not specify the wanted term.</para>
 872           </listitem>
 873         </varlistentry>
 874         <varlistentry>
 875           <term>Action:</term>
 876           <listitem>
 877             <para>First of all it locates all the subterms matched by
 878              <command>patt</command>. In the context of each matched subterm
 879              it disambiguates the term <command>t</command> and reduces it
 880              to its <command>red</command> normal form; then it replaces with
 881              <command>t</command> every occurrence of the normal form in the
 882              matched subterm.</para>
 883           </listitem>
 884         </varlistentry>
 885         <varlistentry>
 886           <term>New sequents to prove:</term>
 887           <listitem>
 888             <para>None.</para>
 889           </listitem>
 890         </varlistentry>
 891       </variablelist>
 892     </para>
 893   </sect1>
 894   <sect1 id="tac_fourier">
 895     <title>fourier</title>
 896     <titleabbrev>fourier</titleabbrev>
 897     <para><userinput>fourier </userinput></para>
 898     <para>
 899       <variablelist>
 900         <varlistentry role="tactic.synopsis">
 901           <term>Synopsis:</term>
 902           <listitem>
 903             <para><emphasis role="bold">fourier</emphasis></para>
 904           </listitem>
 905         </varlistentry>
 906         <varlistentry>
 907           <term>Pre-conditions:</term>
 908           <listitem>
 909             <para>The conclusion of the current sequent must be a linear
 910              inequation over real numbers taken from standard library of
 911              Coq. Moreover the inequations in the hypotheses must imply the
 912              inequation in the conclusion of the current sequent.</para>
 913           </listitem>
 914         </varlistentry>
 915         <varlistentry>
 916           <term>Action:</term>
 917           <listitem>
 918             <para>It closes the current sequent by applying the Fourier method.</para>
 919           </listitem>
 920         </varlistentry>
 921         <varlistentry>
 922           <term>New sequents to prove:</term>
 923           <listitem>
 924             <para>None.</para>
 925           </listitem>
 926         </varlistentry>
 927       </variablelist>
 928     </para>
 929   </sect1>
 930   <sect1 id="tac_fwd">
 931     <title>fwd</title>
 932     <titleabbrev>fwd</titleabbrev>
 933     <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
 934     <para>
 935       <variablelist>
 936         <varlistentry role="tactic.synopsis">
 937           <term>Synopsis:</term>
 938           <listitem>
 939             <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
 940           </listitem>
 941         </varlistentry>
 942         <varlistentry>
 943           <term>Pre-conditions:</term>
 944           <listitem>
 945             <para>
 946                The type of <command>H</command> must be the premise of a
 947                forward simplification theorem.
 948             </para>
 949           </listitem>
 950         </varlistentry>
 951         <varlistentry>
 952           <term>Action:</term>
 953           <listitem>
 954             <para>
 955              This tactic is under development.
 956              It simplifies the current context by removing
 957              <command>H</command> using the following methods:
 958              forward application (by <command>lapply</command>) of a suitable
 959              simplification theorem, chosen automatically, of which the type
 960              of <command>H</command> is a premise,
 961              decomposition (by <command>decompose</command>),
 962              rewriting (by <command>rewrite</command>).
 963              <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
 964              are passed to the tactics <command>fwd</command> invokes, as
 965              names for the premise they introduce.
 966             </para>
 967           </listitem>
 968         </varlistentry>
 969         <varlistentry>
 970           <term>New sequents to prove:</term>
 971           <listitem>
 972             <para>
 973              The ones opened by the tactics <command>fwd</command> invokes.
 974             </para>
 975           </listitem>
 976         </varlistentry>
 977       </variablelist>
 978     </para>
 979   </sect1>
 980   <sect1 id="tac_generalize">
 981     <title>generalize</title>
 982     <titleabbrev>generalize</titleabbrev>
 983     <para><userinput>generalize patt as H</userinput></para>
 984     <para>
 985       <variablelist>
 986         <varlistentry role="tactic.synopsis">
 987           <term>Synopsis:</term>
 988           <listitem>
 989             <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
 990           </listitem>
 991         </varlistentry>
 992         <varlistentry>
 993           <term>Pre-conditions:</term>
 994           <listitem>
 995             <para>All the terms matched by <command>patt</command> must be
 996              convertible and close in the context of the current sequent.</para>
 997           </listitem>
 998         </varlistentry>
 999         <varlistentry>
1000           <term>Action:</term>
1001           <listitem>
1002             <para>It closes the current sequent by applying a stronger
1003              lemma that is proved using the new generated sequent.</para>
1004           </listitem>
1005         </varlistentry>
1006         <varlistentry>
1007           <term>New sequents to prove:</term>
1008           <listitem>
1009             <para>It opens a new sequent where the current sequent conclusion
1010              <command>G</command> is generalized to
1011              <command>∀x.G{x/t}</command> where <command>{x/t}</command>
1012              is a notation for the replacement with <command>x</command> of all
1013              the occurrences of the term <command>t</command> matched by
1014              <command>patt</command>. If <command>patt</command> matches no
1015              subterm then <command>t</command> is defined as the
1016              <command>wanted</command> part of the pattern.</para>
1017           </listitem>
1018         </varlistentry>
1019       </variablelist>
1020     </para>
1021   </sect1>
1022   <sect1 id="tac_id">
1023     <title>id</title>
1024     <titleabbrev>id</titleabbrev>
1025     <para><userinput>id </userinput></para>
1026     <para>
1027       <variablelist>
1028         <varlistentry role="tactic.synopsis">
1029           <term>Synopsis:</term>
1030           <listitem>
1031             <para><emphasis role="bold">id</emphasis></para>
1032           </listitem>
1033         </varlistentry>
1034         <varlistentry>
1035           <term>Pre-conditions:</term>
1036           <listitem>
1037             <para>None.</para>
1038           </listitem>
1039         </varlistentry>
1040         <varlistentry>
1041           <term>Action:</term>
1042           <listitem>
1043             <para>This identity tactic does nothing without failing.</para>
1044           </listitem>
1045         </varlistentry>
1046         <varlistentry>
1047           <term>New sequents to prove:</term>
1048           <listitem>
1049             <para>None.</para>
1050           </listitem>
1051         </varlistentry>
1052       </variablelist>
1053     </para>
1054   </sect1>
1055   <sect1 id="tac_intro">
1056     <title>intro</title>
1057     <titleabbrev>intro</titleabbrev>
1058     <para><userinput>intro H</userinput></para>
1059     <para>
1060       <variablelist>
1061         <varlistentry role="tactic.synopsis">
1062           <term>Synopsis:</term>
1063           <listitem>
1064             <para><emphasis role="bold">intro</emphasis> [&id;]</para>
1065           </listitem>
1066         </varlistentry>
1067         <varlistentry>
1068           <term>Pre-conditions:</term>
1069           <listitem>
1070             <para>The conclusion of the sequent to prove must be an implication
1071              or a universal quantification.</para>
1072           </listitem>
1073         </varlistentry>
1074         <varlistentry>
1075           <term>Action:</term>
1076           <listitem>
1077             <para>It applies the right introduction rule for implication,
1078              closing the current sequent.</para>
1079           </listitem>
1080         </varlistentry>
1081         <varlistentry>
1082           <term>New sequents to prove:</term>
1083           <listitem>
1084             <para>It opens a new sequent to prove adding to the hypothesis
1085              the antecedent of the implication and setting the conclusion
1086              to the consequent of the implicaiton. The name of the new
1087              hypothesis is <command>H</command> if provided; otherwise it
1088              is automatically generated.</para>
1089           </listitem>
1090         </varlistentry>
1091       </variablelist>
1092     </para>
1093   </sect1>
1094   <sect1 id="tac_intros">
1095     <title>intros</title>
1096     <titleabbrev>intros</titleabbrev>
1097     <para><userinput>intros hyps</userinput></para>
1098     <para>
1099       <variablelist>
1100         <varlistentry role="tactic.synopsis">
1101           <term>Synopsis:</term>
1102           <listitem>
1103             <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
1104           </listitem>
1105         </varlistentry>
1106         <varlistentry>
1107           <term>Pre-conditions:</term>
1108           <listitem>
1109             <para>If <command>hyps</command> specifies a number of hypotheses
1110              to introduce, then the conclusion of the current sequent must
1111              be formed by at least that number of imbricated implications
1112              or universal quantifications.</para>
1113           </listitem>
1114         </varlistentry>
1115         <varlistentry>
1116           <term>Action:</term>
1117           <listitem>
1118             <para>It applies several times the right introduction rule for
1119              implication, closing the current sequent.</para>
1120           </listitem>
1121         </varlistentry>
1122         <varlistentry>
1123           <term>New sequents to prove:</term>
1124           <listitem>
1125             <para>It opens a new sequent to prove adding a number of new
1126              hypotheses equal to the number of new hypotheses requested.
1127              If the user does not request a precise number of new hypotheses,
1128              it adds as many hypotheses as possible.
1129              The name of each new hypothesis is either popped from the
1130              user provided list of names, or it is automatically generated when
1131              the list is (or becomes) empty.</para>
1132           </listitem>
1133         </varlistentry>
1134       </variablelist>
1135     </para>
1136   </sect1>
1137   <sect1 id="tac_inversion">
1138     <title>inversion</title>
1139     <titleabbrev>inversion</titleabbrev>
1140     <para><userinput>inversion t</userinput></para>
1141     <para>
1142       <variablelist>
1143         <varlistentry role="tactic.synopsis">
1144           <term>Synopsis:</term>
1145           <listitem>
1146             <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
1147           </listitem>
1148         </varlistentry>
1149         <varlistentry>
1150           <term>Pre-conditions:</term>
1151           <listitem>
1152             <para>The type of the term <command>t</command> must be an inductive
1153              type or the application of an inductive type.</para>
1154           </listitem>
1155         </varlistentry>
1156         <varlistentry>
1157           <term>Action:</term>
1158           <listitem>
1159             <para>It proceeds by cases on <command>t</command> paying attention
1160              to the constraints imposed by the actual &quot;right arguments&quot;
1161              of the inductive type.</para>
1162           </listitem>
1163         </varlistentry>
1164         <varlistentry>
1165           <term>New sequents to prove:</term>
1166           <listitem>
1167             <para>It opens one new sequent to prove for each case in the
1168              definition of the type of <command>t</command>. With respect to
1169              a simple elimination, each new sequent has additional hypotheses
1170              that states the equalities of the &quot;right parameters&quot;
1171              of the inductive type with terms originally present in the
1172              sequent to prove.</para>
1173           </listitem>
1174         </varlistentry>
1175       </variablelist>
1176     </para>
1177   </sect1>
1178   <sect1 id="tac_lapply">
1179     <title>lapply</title>
1180     <titleabbrev>lapply</titleabbrev>
1181     <para><userinput>
1182      lapply linear depth=d t
1183      to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
1184     </userinput></para>
1185     <para>
1186       <variablelist>
1187         <varlistentry role="tactic.synopsis">
1188           <term>Synopsis:</term>
1189           <listitem>
1190             <para>
1191              <emphasis role="bold">lapply</emphasis>
1192              [<emphasis role="bold">linear</emphasis>]
1193              [<emphasis role="bold">depth=</emphasis>&nat;]
1194              &sterm;
1195              [<emphasis role="bold">to</emphasis>
1196               &sterm;
1197               [<emphasis role="bold">,</emphasis>&sterm;…]
1198              ]
1199              [<emphasis role="bold">as</emphasis> &id;]
1200             </para>
1201           </listitem>
1202         </varlistentry>
1203         <varlistentry>
1204           <term>Pre-conditions:</term>
1205           <listitem>
1206             <para>
1207              <command>t</command> must have at least <command>d</command>
1208              independent premises and <command>n</command> must not be
1209              greater than <command>d</command>.
1210             </para>
1211           </listitem>
1212         </varlistentry>
1213         <varlistentry>
1214           <term>Action:</term>
1215           <listitem>
1216             <para>
1217              Invokes <command>letin H ≝ (t ? ... ?)</command>
1218              with enough <command>?</command>'s to reach the
1219              <command>d</command>-th independent premise of
1220              <command>t</command>
1221              (<command>d</command> is maximum if unspecified).
1222              Then istantiates (by <command>apply</command>) with
1223              t<subscript>1</subscript>, ..., t<subscript>n</subscript>
1224              the <command>?</command>'s corresponding to the first
1225              <command>n</command> independent premises of
1226              <command>t</command>.
1227              Usually the other <command>?</command>'s preceding the
1228              <command>n</command>-th independent premise of
1229              <command>t</command> are istantiated as a consequence.
1230              If the <command>linear</command> flag is specified and if
1231              <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
1232              are (applications of) premises in the current context, they are
1233              <command>clear</command>ed.
1234             </para>
1235           </listitem>
1236         </varlistentry>
1237         <varlistentry>
1238           <term>New sequents to prove:</term>
1239           <listitem>
1240             <para>
1241              The ones opened by the tactics <command>lapply</command> invokes.
1242             </para>
1243           </listitem>
1244         </varlistentry>
1245       </variablelist>
1246     </para>
1247   </sect1>
1248   <sect1 id="tac_left">
1249     <title>left</title>
1250     <titleabbrev>left</titleabbrev>
1251     <para><userinput>left </userinput></para>
1252     <para>
1253       <variablelist>
1254         <varlistentry role="tactic.synopsis">
1255           <term>Synopsis:</term>
1256           <listitem>
1257             <para><emphasis role="bold">left</emphasis></para>
1258           </listitem>
1259         </varlistentry>
1260         <varlistentry>
1261           <term>Pre-conditions:</term>
1262           <listitem>
1263             <para>The conclusion of the current sequent must be
1264              an inductive type or the application of an inductive type
1265              with at least one constructor.</para>
1266           </listitem>
1267         </varlistentry>
1268         <varlistentry>
1269           <term>Action:</term>
1270           <listitem>
1271             <para>Equivalent to <command>constructor 1</command>.</para>
1272           </listitem>
1273         </varlistentry>
1274         <varlistentry>
1275           <term>New sequents to prove:</term>
1276           <listitem>
1277             <para>It opens a new sequent for each premise of the first
1278              constructor of the inductive type that is the conclusion of the
1279              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1280           </listitem>
1281         </varlistentry>
1282       </variablelist>
1283     </para>
1284   </sect1>
1285   <sect1 id="tac_letin">
1286     <title>letin</title>
1287     <titleabbrev>letin</titleabbrev>
1288     <para><userinput>letin x ≝ t</userinput></para>
1289     <para>
1290       <variablelist>
1291         <varlistentry role="tactic.synopsis">
1292           <term>Synopsis:</term>
1293           <listitem>
1294             <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
1295           </listitem>
1296         </varlistentry>
1297         <varlistentry>
1298           <term>Pre-conditions:</term>
1299           <listitem>
1300             <para>None.</para>
1301           </listitem>
1302         </varlistentry>
1303         <varlistentry>
1304           <term>Action:</term>
1305           <listitem>
1306             <para>It adds to the context of the current sequent to prove a new
1307              definition <command>x ≝ t</command>.</para>
1308           </listitem>
1309         </varlistentry>
1310         <varlistentry>
1311           <term>New sequents to prove:</term>
1312           <listitem>
1313             <para>None.</para>
1314           </listitem>
1315         </varlistentry>
1316       </variablelist>
1317     </para>
1318   </sect1>
1319   <sect1 id="tac_normalize">
1320     <title>normalize</title>
1321     <titleabbrev>normalize</titleabbrev>
1322     <para><userinput>normalize patt</userinput></para>
1323     <para>
1324       <variablelist>
1325         <varlistentry role="tactic.synopsis">
1326           <term>Synopsis:</term>
1327           <listitem>
1328             <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
1329           </listitem>
1330         </varlistentry>
1331         <varlistentry>
1332           <term>Pre-conditions:</term>
1333           <listitem>
1334             <para>None.</para>
1335           </listitem>
1336         </varlistentry>
1337         <varlistentry>
1338           <term>Action:</term>
1339           <listitem>
1340             <para>It replaces all the terms matched by <command>patt</command>
1341              with their βδιζ-normal form.</para>
1342           </listitem>
1343         </varlistentry>
1344         <varlistentry>
1345           <term>New sequents to prove:</term>
1346           <listitem>
1347             <para>None.</para>
1348           </listitem>
1349         </varlistentry>
1350       </variablelist>
1351     </para>
1352   </sect1>
1353   <sect1 id="tac_reflexivity">
1354     <title>reflexivity</title>
1355     <titleabbrev>reflexivity</titleabbrev>
1356     <para><userinput>reflexivity </userinput></para>
1357     <para>
1358       <variablelist>
1359         <varlistentry role="tactic.synopsis">
1360           <term>Synopsis:</term>
1361           <listitem>
1362             <para><emphasis role="bold">reflexivity</emphasis></para>
1363           </listitem>
1364         </varlistentry>
1365         <varlistentry>
1366           <term>Pre-conditions:</term>
1367           <listitem>
1368             <para>The conclusion of the current sequent must be
1369              <command>t=t</command> for some term <command>t</command></para>
1370           </listitem>
1371         </varlistentry>
1372         <varlistentry>
1373           <term>Action:</term>
1374           <listitem>
1375             <para>It closes the current sequent by reflexivity
1376              of equality.</para>
1377           </listitem>
1378         </varlistentry>
1379         <varlistentry>
1380           <term>New sequents to prove:</term>
1381           <listitem>
1382             <para>None.</para>
1383           </listitem>
1384         </varlistentry>
1385       </variablelist>
1386     </para>
1387   </sect1>
1388   <sect1 id="tac_replace">
1389     <title>replace</title>
1390     <titleabbrev>change</titleabbrev>
1391     <para><userinput>change patt with t</userinput></para>
1392     <para>
1393       <variablelist>
1394         <varlistentry role="tactic.synopsis">
1395           <term>Synopsis:</term>
1396           <listitem>
1397             <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
1398           </listitem>
1399         </varlistentry>
1400         <varlistentry>
1401           <term>Pre-conditions:</term>
1402           <listitem>
1403             <para>None.</para>
1404           </listitem>
1405         </varlistentry>
1406         <varlistentry>
1407           <term>Action:</term>
1408           <listitem>
1409             <para>It replaces the subterms of the current sequent matched by
1410              <command>patt</command> with the new term <command>t</command>.
1411              For each subterm matched by the pattern, <command>t</command> is
1412              disambiguated in the context of the subterm.</para>
1413           </listitem>
1414         </varlistentry>
1415         <varlistentry>
1416           <term>New sequents to prove:</term>
1417           <listitem>
1418             <para>For each matched term <command>t'</command> it opens
1419              a new sequent to prove whose conclusion is
1420              <command>t'=t</command>.</para>
1421           </listitem>
1422         </varlistentry>
1423       </variablelist>
1424     </para>
1425   </sect1>
1426   <sect1 id="tac_rewrite">
1427     <title>rewrite</title>
1428     <titleabbrev>rewrite</titleabbrev>
1429     <para><userinput>rewrite dir p patt</userinput></para>
1430     <para>
1431       <variablelist>
1432         <varlistentry role="tactic.synopsis">
1433           <term>Synopsis:</term>
1434           <listitem>
1435             <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold">&lt;</emphasis>|<emphasis role="bold">&gt;</emphasis>] &sterm; &pattern;</para>
1436           </listitem>
1437         </varlistentry>
1438         <varlistentry>
1439           <term>Pre-conditions:</term>
1440           <listitem>
1441             <para><command>p</command> must be the proof of an equality,
1442              possibly under some hypotheses.</para>
1443           </listitem>
1444         </varlistentry>
1445         <varlistentry>
1446           <term>Action:</term>
1447           <listitem>
1448             <para>It looks in every term matched by <command>patt</command>
1449              for all the occurrences of the
1450              left hand side of the equality that <command>p</command> proves
1451              (resp. the right hand side if <command>dir</command> is
1452              <command>&lt;</command>). Every occurence found is replaced with
1453              the opposite side of the equality.</para>
1454           </listitem>
1455         </varlistentry>
1456         <varlistentry>
1457           <term>New sequents to prove:</term>
1458           <listitem>
1459             <para>It opens one new sequent for each hypothesis of the
1460              equality proved by <command>p</command> that is not closed
1461              by unification.</para>
1462           </listitem>
1463         </varlistentry>
1464       </variablelist>
1465     </para>
1466   </sect1>
1467   <sect1 id="tac_right">
1468     <title>right</title>
1469     <titleabbrev>right</titleabbrev>
1470     <para><userinput>right </userinput></para>
1471     <para>
1472       <variablelist>
1473         <varlistentry role="tactic.synopsis">
1474           <term>Synopsis:</term>
1475           <listitem>
1476             <para><emphasis role="bold">right</emphasis></para>
1477           </listitem>
1478         </varlistentry>
1479         <varlistentry>
1480           <term>Pre-conditions:</term>
1481           <listitem>
1482             <para>The conclusion of the current sequent must be
1483              an inductive type or the application of an inductive type with
1484              at least two constructors.</para>
1485           </listitem>
1486         </varlistentry>
1487         <varlistentry>
1488           <term>Action:</term>
1489           <listitem>
1490             <para>Equivalent to <command>constructor 2</command>.</para>
1491           </listitem>
1492         </varlistentry>
1493         <varlistentry>
1494           <term>New sequents to prove:</term>
1495           <listitem>
1496             <para>It opens a new sequent for each premise of the second
1497              constructor of the inductive type that is the conclusion of the
1498              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1499           </listitem>
1500         </varlistentry>
1501       </variablelist>
1502     </para>
1503   </sect1>
1504   <sect1 id="tac_ring">
1505     <title>ring</title>
1506     <titleabbrev>ring</titleabbrev>
1507     <para><userinput>ring </userinput></para>
1508     <para>
1509       <variablelist>
1510         <varlistentry role="tactic.synopsis">
1511           <term>Synopsis:</term>
1512           <listitem>
1513             <para><emphasis role="bold">ring</emphasis></para>
1514           </listitem>
1515         </varlistentry>
1516         <varlistentry>
1517           <term>Pre-conditions:</term>
1518           <listitem>
1519             <para>The conclusion of the current sequent must be an
1520              equality over Coq's real numbers that can be proved using
1521              the ring properties of the real numbers only.</para>
1522           </listitem>
1523         </varlistentry>
1524         <varlistentry>
1525           <term>Action:</term>
1526           <listitem>
1527             <para>It closes the current sequent veryfying the equality by
1528              means of computation (i.e. this is a reflexive tactic, implemented
1529              exploiting the &quot;two level reasoning&quot; technique).</para>
1530           </listitem>
1531         </varlistentry>
1532         <varlistentry>
1533           <term>New sequents to prove:</term>
1534           <listitem>
1535             <para>None.</para>
1536           </listitem>
1537         </varlistentry>
1538       </variablelist>
1539     </para>
1540   </sect1>
1541   <sect1 id="tac_simplify">
1542     <title>simplify</title>
1543     <titleabbrev>simplify</titleabbrev>
1544     <para><userinput>simplify patt</userinput></para>
1545     <para>
1546       <variablelist>
1547         <varlistentry role="tactic.synopsis">
1548           <term>Synopsis:</term>
1549           <listitem>
1550             <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
1551           </listitem>
1552         </varlistentry>
1553         <varlistentry>
1554           <term>Pre-conditions:</term>
1555           <listitem>
1556             <para>None.</para>
1557           </listitem>
1558         </varlistentry>
1559         <varlistentry>
1560           <term>Action:</term>
1561           <listitem>
1562             <para>It replaces all the terms matched by <command>patt</command>
1563              with other convertible terms that are supposed to be simpler.</para>
1564           </listitem>
1565         </varlistentry>
1566         <varlistentry>
1567           <term>New sequents to prove:</term>
1568           <listitem>
1569             <para>None.</para>
1570           </listitem>
1571         </varlistentry>
1572       </variablelist>
1573     </para>
1574   </sect1>
1575   <sect1 id="tac_split">
1576     <title>split</title>
1577     <titleabbrev>split</titleabbrev>
1578     <para><userinput>split </userinput></para>
1579     <para>
1580       <variablelist>
1581         <varlistentry role="tactic.synopsis">
1582           <term>Synopsis:</term>
1583           <listitem>
1584             <para><emphasis role="bold">split</emphasis></para>
1585           </listitem>
1586         </varlistentry>
1587         <varlistentry>
1588           <term>Pre-conditions:</term>
1589           <listitem>
1590             <para>The conclusion of the current sequent must be
1591              an inductive type or the application of an inductive type with
1592              at least one constructor.</para>
1593           </listitem>
1594         </varlistentry>
1595         <varlistentry>
1596           <term>Action:</term>
1597           <listitem>
1598             <para>Equivalent to <command>constructor 1</command>.</para>
1599           </listitem>
1600         </varlistentry>
1601         <varlistentry>
1602           <term>New sequents to prove:</term>
1603           <listitem>
1604             <para>It opens a new sequent for each premise of the first
1605              constructor of the inductive type that is the conclusion of the
1606              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1607           </listitem>
1608         </varlistentry>
1609       </variablelist>
1610     </para>
1611   </sect1>
1612
1613   <sect1 id="tac_subst">
1614     <title>subst</title>
1615     <titleabbrev>subst</titleabbrev>
1616     <para><userinput>subst</userinput></para>
1617     <para>
1618       <variablelist>
1619         <varlistentry role="tactic.synopsis">
1620           <term>Synopsis:</term>
1621           <listitem>
1622             <para><emphasis role="bold">subst</emphasis></para>
1623           </listitem>
1624         </varlistentry>
1625         <varlistentry>
1626           <term>Pre-conditions:</term>
1627           <listitem><para>
1628             None.
1629           </para></listitem>
1630         </varlistentry>
1631         <varlistentry>
1632           <term>Action:</term>
1633           <listitem><para>
1634             For each premise of the form
1635             <command>H: x = t</command> or <command>H: t = x</command>
1636             where <command>x</command> is a local variable and
1637             <command>t</command> does not depend on <command>x</command>,
1638             the tactic rewrites <command>H</command> wherever
1639             <command>x</command> appears clearing <command>H</command> and
1640             <command>x</command> afterwards.
1641           </para></listitem>
1642         </varlistentry>
1643         <varlistentry>
1644           <term>New sequents to prove:</term>
1645           <listitem><para>
1646             The one opened by the applied tactics.
1647           </para></listitem>
1648         </varlistentry>
1649       </variablelist>
1650     </para>
1651   </sect1>
1652   <sect1 id="tac_symmetry">
1653     <title>symmetry</title>
1654     <titleabbrev>symmetry</titleabbrev>
1655     <para>The tactic <command>symmetry</command> </para>
1656     <para><userinput>symmetry </userinput></para>
1657     <para>
1658       <variablelist>
1659         <varlistentry role="tactic.synopsis">
1660           <term>Synopsis:</term>
1661           <listitem>
1662             <para><emphasis role="bold">symmetry</emphasis></para>
1663           </listitem>
1664         </varlistentry>
1665         <varlistentry>
1666           <term>Pre-conditions:</term>
1667           <listitem>
1668             <para>The conclusion of the current proof must be an equality.</para>
1669           </listitem>
1670         </varlistentry>
1671         <varlistentry>
1672           <term>Action:</term>
1673           <listitem>
1674             <para>It swaps the two sides of the equalityusing the symmetric
1675              property.</para>
1676           </listitem>
1677         </varlistentry>
1678         <varlistentry>
1679           <term>New sequents to prove:</term>
1680           <listitem>
1681             <para>None.</para>
1682           </listitem>
1683         </varlistentry>
1684       </variablelist>
1685     </para>
1686   </sect1>
1687   <sect1 id="tac_transitivity">
1688     <title>transitivity</title>
1689     <titleabbrev>transitivity</titleabbrev>
1690     <para><userinput>transitivity t</userinput></para>
1691     <para>
1692       <variablelist>
1693         <varlistentry role="tactic.synopsis">
1694           <term>Synopsis:</term>
1695           <listitem>
1696             <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
1697           </listitem>
1698         </varlistentry>
1699         <varlistentry>
1700           <term>Pre-conditions:</term>
1701           <listitem>
1702             <para>The conclusion of the current proof must be an equality.</para>
1703           </listitem>
1704         </varlistentry>
1705         <varlistentry>
1706           <term>Action:</term>
1707           <listitem>
1708             <para>It closes the current sequent by transitivity of the equality.</para>
1709           </listitem>
1710         </varlistentry>
1711         <varlistentry>
1712           <term>New sequents to prove:</term>
1713           <listitem>
1714             <para>It opens two new sequents <command>l=t</command> and
1715              <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
1716 the current sequent to prove.</para>
1717           </listitem>
1718         </varlistentry>
1719       </variablelist>
1720     </para>
1721   </sect1>
1722   <sect1 id="tac_unfold">
1723     <title>unfold</title>
1724     <titleabbrev>unfold</titleabbrev>
1725     <para><userinput>unfold t patt</userinput></para>
1726     <para>
1727       <variablelist>
1728         <varlistentry role="tactic.synopsis">
1729           <term>Synopsis:</term>
1730           <listitem>
1731             <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
1732           </listitem>
1733         </varlistentry>
1734         <varlistentry>
1735           <term>Pre-conditions:</term>
1736           <listitem>
1737             <para>None.</para>
1738           </listitem>
1739         </varlistentry>
1740         <varlistentry>
1741           <term>Action:</term>
1742           <listitem>
1743             <para>It finds all the occurrences of <command>t</command>
1744              (possibly applied to arguments) in the subterms matched by
1745              <command>patt</command>. Then it δ-expands each occurrence,
1746              also performing β-reduction of the obtained term. If
1747              <command>t</command> is omitted it defaults to each
1748              subterm matched by <command>patt</command>.</para>
1749           </listitem>
1750         </varlistentry>
1751         <varlistentry>
1752           <term>New sequents to prove:</term>
1753           <listitem>
1754             <para>None.</para>
1755           </listitem>
1756         </varlistentry>
1757       </variablelist>
1758     </para>
1759   </sect1>
1760   <sect1 id="tac_whd">
1761     <title>whd</title>
1762     <titleabbrev>whd</titleabbrev>
1763     <para><userinput>whd patt</userinput></para>
1764     <para>
1765       <variablelist>
1766         <varlistentry role="tactic.synopsis">
1767           <term>Synopsis:</term>
1768           <listitem>
1769             <para><emphasis role="bold">whd</emphasis> &pattern;</para>
1770           </listitem>
1771         </varlistentry>
1772         <varlistentry>
1773           <term>Pre-conditions:</term>
1774           <listitem>
1775             <para>None.</para>
1776           </listitem>
1777         </varlistentry>
1778         <varlistentry>
1779           <term>Action:</term>
1780           <listitem>
1781             <para>It replaces all the terms matched by <command>patt</command>
1782              with their βδιζ-weak-head normal form.</para>
1783           </listitem>
1784         </varlistentry>
1785         <varlistentry>
1786           <term>New sequents to prove:</term>
1787           <listitem>
1788             <para>None.</para>
1789           </listitem>
1790         </varlistentry>
1791       </variablelist>
1792     </para>
1793   </sect1>
1794
1795 </chapter>
1796