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 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; [<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             </para>
 699           </listitem>
 700         </varlistentry>
 701         <varlistentry>
 702           <term>New sequents to prove:</term>
 703           <listitem>
 704             <para>It opens one new sequent for each case. The names of
 705              the new hypotheses are picked by <command>hyps</command>, if
 706              provided. If hyps specifies also a number of hypotheses that
 707              is less than the number of new hypotheses for a new sequent,
 708              then the exceeding hypothesis will be kept as implications in
 709              the conclusion of the sequent.</para>
 710           </listitem>
 711         </varlistentry>
 712       </variablelist>
 713     </para>
 714   </sect1>
 715   <sect1 id="tac_elimType">
 716     <title>elimType</title>
 717     <titleabbrev>elimType</titleabbrev>
 718     <para><userinput>elimType T using th hyps</userinput></para>
 719     <para>
 720       <variablelist>
 721         <varlistentry role="tactic.synopsis">
 722           <term>Synopsis:</term>
 723           <listitem>
 724             <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
 725           </listitem>
 726         </varlistentry>
 727         <varlistentry>
 728           <term>Pre-conditions:</term>
 729           <listitem>
 730             <para><command>T</command> must be an inductive type.</para>
 731           </listitem>
 732         </varlistentry>
 733         <varlistentry>
 734           <term>Action:</term>
 735           <listitem>
 736             <para>TODO (severely bugged now).</para>
 737           </listitem>
 738         </varlistentry>
 739         <varlistentry>
 740           <term>New sequents to prove:</term>
 741           <listitem>
 742             <para>TODO</para>
 743           </listitem>
 744         </varlistentry>
 745       </variablelist>
 746     </para>
 747   </sect1>
 748   <sect1 id="tac_exact">
 749     <title>exact</title>
 750     <titleabbrev>exact</titleabbrev>
 751     <para><userinput>exact p</userinput></para>
 752     <para>
 753       <variablelist>
 754         <varlistentry role="tactic.synopsis">
 755           <term>Synopsis:</term>
 756           <listitem>
 757             <para><emphasis role="bold">exact</emphasis> &sterm;</para>
 758           </listitem>
 759         </varlistentry>
 760         <varlistentry>
 761           <term>Pre-conditions:</term>
 762           <listitem>
 763             <para>The type of <command>p</command> must be convertible
 764              with the conclusion of the current sequent.</para>
 765           </listitem>
 766         </varlistentry>
 767         <varlistentry>
 768           <term>Action:</term>
 769           <listitem>
 770             <para>It closes the current sequent using <command>p</command>.</para>
 771           </listitem>
 772         </varlistentry>
 773         <varlistentry>
 774           <term>New sequents to prove:</term>
 775           <listitem>
 776             <para>None.</para>
 777           </listitem>
 778         </varlistentry>
 779       </variablelist>
 780     </para>
 781   </sect1>
 782   <sect1 id="tac_exists">
 783     <title>exists</title>
 784     <titleabbrev>exists</titleabbrev>
 785     <para><userinput>exists </userinput></para>
 786     <para>
 787       <variablelist>
 788         <varlistentry role="tactic.synopsis">
 789           <term>Synopsis:</term>
 790           <listitem>
 791             <para><emphasis role="bold">exists</emphasis></para>
 792           </listitem>
 793         </varlistentry>
 794         <varlistentry>
 795           <term>Pre-conditions:</term>
 796           <listitem>
 797             <para>The conclusion of the current sequent must be
 798              an inductive type or the application of an inductive type
 799              with at least one constructor.</para>
 800           </listitem>
 801         </varlistentry>
 802         <varlistentry>
 803           <term>Action:</term>
 804           <listitem>
 805             <para>Equivalent to <command>constructor 1</command>.</para>
 806           </listitem>
 807         </varlistentry>
 808         <varlistentry>
 809           <term>New sequents to prove:</term>
 810           <listitem>
 811             <para>It opens a new sequent for each premise of the first
 812              constructor of the inductive type that is the conclusion of the
 813              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 814           </listitem>
 815         </varlistentry>
 816       </variablelist>
 817     </para>
 818   </sect1>
 819   <sect1 id="tac_fail">
 820     <title>fail</title>
 821     <titleabbrev>fail</titleabbrev>
 822     <para><userinput>fail</userinput></para>
 823     <para>
 824       <variablelist>
 825         <varlistentry role="tactic.synopsis">
 826           <term>Synopsis:</term>
 827           <listitem>
 828             <para><emphasis role="bold">fail</emphasis></para>
 829           </listitem>
 830         </varlistentry>
 831         <varlistentry>
 832           <term>Pre-conditions:</term>
 833           <listitem>
 834             <para>None.</para>
 835           </listitem>
 836         </varlistentry>
 837         <varlistentry>
 838           <term>Action:</term>
 839           <listitem>
 840             <para>This tactic always fail.</para>
 841           </listitem>
 842         </varlistentry>
 843         <varlistentry>
 844           <term>New sequents to prove:</term>
 845           <listitem>
 846             <para>N.A.</para>
 847           </listitem>
 848         </varlistentry>
 849       </variablelist>
 850     </para>
 851   </sect1>
 852   <sect1 id="tac_fold">
 853     <title>fold</title>
 854     <titleabbrev>fold</titleabbrev>
 855     <para><userinput>fold red t patt</userinput></para>
 856     <para>
 857       <variablelist>
 858         <varlistentry role="tactic.synopsis">
 859           <term>Synopsis:</term>
 860           <listitem>
 861             <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
 862           </listitem>
 863         </varlistentry>
 864         <varlistentry>
 865           <term>Pre-conditions:</term>
 866           <listitem>
 867             <para>The pattern must not specify the wanted term.</para>
 868           </listitem>
 869         </varlistentry>
 870         <varlistentry>
 871           <term>Action:</term>
 872           <listitem>
 873             <para>First of all it locates all the subterms matched by
 874              <command>patt</command>. In the context of each matched subterm
 875              it disambiguates the term <command>t</command> and reduces it
 876              to its <command>red</command> normal form; then it replaces with
 877              <command>t</command> every occurrence of the normal form in the
 878              matched subterm.</para>
 879           </listitem>
 880         </varlistentry>
 881         <varlistentry>
 882           <term>New sequents to prove:</term>
 883           <listitem>
 884             <para>None.</para>
 885           </listitem>
 886         </varlistentry>
 887       </variablelist>
 888     </para>
 889   </sect1>
 890   <sect1 id="tac_fourier">
 891     <title>fourier</title>
 892     <titleabbrev>fourier</titleabbrev>
 893     <para><userinput>fourier </userinput></para>
 894     <para>
 895       <variablelist>
 896         <varlistentry role="tactic.synopsis">
 897           <term>Synopsis:</term>
 898           <listitem>
 899             <para><emphasis role="bold">fourier</emphasis></para>
 900           </listitem>
 901         </varlistentry>
 902         <varlistentry>
 903           <term>Pre-conditions:</term>
 904           <listitem>
 905             <para>The conclusion of the current sequent must be a linear
 906              inequation over real numbers taken from standard library of
 907              Coq. Moreover the inequations in the hypotheses must imply the
 908              inequation in the conclusion of the current sequent.</para>
 909           </listitem>
 910         </varlistentry>
 911         <varlistentry>
 912           <term>Action:</term>
 913           <listitem>
 914             <para>It closes the current sequent by applying the Fourier method.</para>
 915           </listitem>
 916         </varlistentry>
 917         <varlistentry>
 918           <term>New sequents to prove:</term>
 919           <listitem>
 920             <para>None.</para>
 921           </listitem>
 922         </varlistentry>
 923       </variablelist>
 924     </para>
 925   </sect1>
 926   <sect1 id="tac_fwd">
 927     <title>fwd</title>
 928     <titleabbrev>fwd</titleabbrev>
 929     <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
 930     <para>
 931       <variablelist>
 932         <varlistentry role="tactic.synopsis">
 933           <term>Synopsis:</term>
 934           <listitem>
 935             <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
 936           </listitem>
 937         </varlistentry>
 938         <varlistentry>
 939           <term>Pre-conditions:</term>
 940           <listitem>
 941             <para>
 942                The type of <command>H</command> must be the premise of a
 943                forward simplification theorem.
 944             </para>
 945           </listitem>
 946         </varlistentry>
 947         <varlistentry>
 948           <term>Action:</term>
 949           <listitem>
 950             <para>
 951              This tactic is under development.
 952              It simplifies the current context by removing
 953              <command>H</command> using the following methods:
 954              forward application (by <command>lapply</command>) of a suitable
 955              simplification theorem, chosen automatically, of which the type
 956              of <command>H</command> is a premise,
 957              decomposition (by <command>decompose</command>),
 958              rewriting (by <command>rewrite</command>).
 959              <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
 960              are passed to the tactics <command>fwd</command> invokes, as
 961              names for the premise they introduce.
 962             </para>
 963           </listitem>
 964         </varlistentry>
 965         <varlistentry>
 966           <term>New sequents to prove:</term>
 967           <listitem>
 968             <para>
 969              The ones opened by the tactics <command>fwd</command> invokes.
 970             </para>
 971           </listitem>
 972         </varlistentry>
 973       </variablelist>
 974     </para>
 975   </sect1>
 976   <sect1 id="tac_generalize">
 977     <title>generalize</title>
 978     <titleabbrev>generalize</titleabbrev>
 979     <para><userinput>generalize patt as H</userinput></para>
 980     <para>
 981       <variablelist>
 982         <varlistentry role="tactic.synopsis">
 983           <term>Synopsis:</term>
 984           <listitem>
 985             <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
 986           </listitem>
 987         </varlistentry>
 988         <varlistentry>
 989           <term>Pre-conditions:</term>
 990           <listitem>
 991             <para>All the terms matched by <command>patt</command> must be
 992              convertible and close in the context of the current sequent.</para>
 993           </listitem>
 994         </varlistentry>
 995         <varlistentry>
 996           <term>Action:</term>
 997           <listitem>
 998             <para>It closes the current sequent by applying a stronger
 999              lemma that is proved using the new generated sequent.</para>
1000           </listitem>
1001         </varlistentry>
1002         <varlistentry>
1003           <term>New sequents to prove:</term>
1004           <listitem>
1005             <para>It opens a new sequent where the current sequent conclusion
1006              <command>G</command> is generalized to
1007              <command>∀x.G{x/t}</command> where <command>{x/t}</command>
1008              is a notation for the replacement with <command>x</command> of all
1009              the occurrences of the term <command>t</command> matched by
1010              <command>patt</command>. If <command>patt</command> matches no
1011              subterm then <command>t</command> is defined as the
1012              <command>wanted</command> part of the pattern.</para>
1013           </listitem>
1014         </varlistentry>
1015       </variablelist>
1016     </para>
1017   </sect1>
1018   <sect1 id="tac_id">
1019     <title>id</title>
1020     <titleabbrev>id</titleabbrev>
1021     <para><userinput>id </userinput></para>
1022     <para>
1023       <variablelist>
1024         <varlistentry role="tactic.synopsis">
1025           <term>Synopsis:</term>
1026           <listitem>
1027             <para><emphasis role="bold">id</emphasis></para>
1028           </listitem>
1029         </varlistentry>
1030         <varlistentry>
1031           <term>Pre-conditions:</term>
1032           <listitem>
1033             <para>None.</para>
1034           </listitem>
1035         </varlistentry>
1036         <varlistentry>
1037           <term>Action:</term>
1038           <listitem>
1039             <para>This identity tactic does nothing without failing.</para>
1040           </listitem>
1041         </varlistentry>
1042         <varlistentry>
1043           <term>New sequents to prove:</term>
1044           <listitem>
1045             <para>None.</para>
1046           </listitem>
1047         </varlistentry>
1048       </variablelist>
1049     </para>
1050   </sect1>
1051   <sect1 id="tac_intro">
1052     <title>intro</title>
1053     <titleabbrev>intro</titleabbrev>
1054     <para><userinput>intro H</userinput></para>
1055     <para>
1056       <variablelist>
1057         <varlistentry role="tactic.synopsis">
1058           <term>Synopsis:</term>
1059           <listitem>
1060             <para><emphasis role="bold">intro</emphasis> [&id;]</para>
1061           </listitem>
1062         </varlistentry>
1063         <varlistentry>
1064           <term>Pre-conditions:</term>
1065           <listitem>
1066             <para>The conclusion of the sequent to prove must be an implication
1067              or a universal quantification.</para>
1068           </listitem>
1069         </varlistentry>
1070         <varlistentry>
1071           <term>Action:</term>
1072           <listitem>
1073             <para>It applies the right introduction rule for implication,
1074              closing the current sequent.</para>
1075           </listitem>
1076         </varlistentry>
1077         <varlistentry>
1078           <term>New sequents to prove:</term>
1079           <listitem>
1080             <para>It opens a new sequent to prove adding to the hypothesis
1081              the antecedent of the implication and setting the conclusion
1082              to the consequent of the implicaiton. The name of the new
1083              hypothesis is <command>H</command> if provided; otherwise it
1084              is automatically generated.</para>
1085           </listitem>
1086         </varlistentry>
1087       </variablelist>
1088     </para>
1089   </sect1>
1090   <sect1 id="tac_intros">
1091     <title>intros</title>
1092     <titleabbrev>intros</titleabbrev>
1093     <para><userinput>intros hyps</userinput></para>
1094     <para>
1095       <variablelist>
1096         <varlistentry role="tactic.synopsis">
1097           <term>Synopsis:</term>
1098           <listitem>
1099             <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
1100           </listitem>
1101         </varlistentry>
1102         <varlistentry>
1103           <term>Pre-conditions:</term>
1104           <listitem>
1105             <para>If <command>hyps</command> specifies a number of hypotheses
1106              to introduce, then the conclusion of the current sequent must
1107              be formed by at least that number of imbricated implications
1108              or universal quantifications.</para>
1109           </listitem>
1110         </varlistentry>
1111         <varlistentry>
1112           <term>Action:</term>
1113           <listitem>
1114             <para>It applies several times the right introduction rule for
1115              implication, closing the current sequent.</para>
1116           </listitem>
1117         </varlistentry>
1118         <varlistentry>
1119           <term>New sequents to prove:</term>
1120           <listitem>
1121             <para>It opens a new sequent to prove adding a number of new
1122              hypotheses equal to the number of new hypotheses requested.
1123              If the user does not request a precise number of new hypotheses,
1124              it adds as many hypotheses as possible.
1125              The name of each new hypothesis is either popped from the
1126              user provided list of names, or it is automatically generated when
1127              the list is (or becomes) empty.</para>
1128           </listitem>
1129         </varlistentry>
1130       </variablelist>
1131     </para>
1132   </sect1>
1133   <sect1 id="tac_inversion">
1134     <title>inversion</title>
1135     <titleabbrev>inversion</titleabbrev>
1136     <para><userinput>inversion t</userinput></para>
1137     <para>
1138       <variablelist>
1139         <varlistentry role="tactic.synopsis">
1140           <term>Synopsis:</term>
1141           <listitem>
1142             <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
1143           </listitem>
1144         </varlistentry>
1145         <varlistentry>
1146           <term>Pre-conditions:</term>
1147           <listitem>
1148             <para>The type of the term <command>t</command> must be an inductive
1149              type or the application of an inductive type.</para>
1150           </listitem>
1151         </varlistentry>
1152         <varlistentry>
1153           <term>Action:</term>
1154           <listitem>
1155             <para>It proceeds by cases on <command>t</command> paying attention
1156              to the constraints imposed by the actual &quot;right arguments&quot;
1157              of the inductive type.</para>
1158           </listitem>
1159         </varlistentry>
1160         <varlistentry>
1161           <term>New sequents to prove:</term>
1162           <listitem>
1163             <para>It opens one new sequent to prove for each case in the
1164              definition of the type of <command>t</command>. With respect to
1165              a simple elimination, each new sequent has additional hypotheses
1166              that states the equalities of the &quot;right parameters&quot;
1167              of the inductive type with terms originally present in the
1168              sequent to prove.</para>
1169           </listitem>
1170         </varlistentry>
1171       </variablelist>
1172     </para>
1173   </sect1>
1174   <sect1 id="tac_lapply">
1175     <title>lapply</title>
1176     <titleabbrev>lapply</titleabbrev>
1177     <para><userinput>
1178      lapply linear depth=d t
1179      to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
1180     </userinput></para>
1181     <para>
1182       <variablelist>
1183         <varlistentry role="tactic.synopsis">
1184           <term>Synopsis:</term>
1185           <listitem>
1186             <para>
1187              <emphasis role="bold">lapply</emphasis>
1188              [<emphasis role="bold">linear</emphasis>]
1189              [<emphasis role="bold">depth=</emphasis>&nat;]
1190              &sterm;
1191              [<emphasis role="bold">to</emphasis>
1192               &sterm;
1193               [<emphasis role="bold">,</emphasis>&sterm;…]
1194              ]
1195              [<emphasis role="bold">as</emphasis> &id;]
1196             </para>
1197           </listitem>
1198         </varlistentry>
1199         <varlistentry>
1200           <term>Pre-conditions:</term>
1201           <listitem>
1202             <para>
1203              <command>t</command> must have at least <command>d</command>
1204              independent premises and <command>n</command> must not be
1205              greater than <command>d</command>.
1206             </para>
1207           </listitem>
1208         </varlistentry>
1209         <varlistentry>
1210           <term>Action:</term>
1211           <listitem>
1212             <para>
1213              Invokes <command>letin H ≝ (t ? ... ?)</command>
1214              with enough <command>?</command>'s to reach the
1215              <command>d</command>-th independent premise of
1216              <command>t</command>
1217              (<command>d</command> is maximum if unspecified).
1218              Then istantiates (by <command>apply</command>) with
1219              t<subscript>1</subscript>, ..., t<subscript>n</subscript>
1220              the <command>?</command>'s corresponding to the first
1221              <command>n</command> independent premises of
1222              <command>t</command>.
1223              Usually the other <command>?</command>'s preceding the
1224              <command>n</command>-th independent premise of
1225              <command>t</command> are istantiated as a consequence.
1226              If the <command>linear</command> flag is specified and if
1227              <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
1228              are (applications of) premises in the current context, they are
1229              <command>clear</command>ed.
1230             </para>
1231           </listitem>
1232         </varlistentry>
1233         <varlistentry>
1234           <term>New sequents to prove:</term>
1235           <listitem>
1236             <para>
1237              The ones opened by the tactics <command>lapply</command> invokes.
1238             </para>
1239           </listitem>
1240         </varlistentry>
1241       </variablelist>
1242     </para>
1243   </sect1>
1244   <sect1 id="tac_left">
1245     <title>left</title>
1246     <titleabbrev>left</titleabbrev>
1247     <para><userinput>left </userinput></para>
1248     <para>
1249       <variablelist>
1250         <varlistentry role="tactic.synopsis">
1251           <term>Synopsis:</term>
1252           <listitem>
1253             <para><emphasis role="bold">left</emphasis></para>
1254           </listitem>
1255         </varlistentry>
1256         <varlistentry>
1257           <term>Pre-conditions:</term>
1258           <listitem>
1259             <para>The conclusion of the current sequent must be
1260              an inductive type or the application of an inductive type
1261              with at least one constructor.</para>
1262           </listitem>
1263         </varlistentry>
1264         <varlistentry>
1265           <term>Action:</term>
1266           <listitem>
1267             <para>Equivalent to <command>constructor 1</command>.</para>
1268           </listitem>
1269         </varlistentry>
1270         <varlistentry>
1271           <term>New sequents to prove:</term>
1272           <listitem>
1273             <para>It opens a new sequent for each premise of the first
1274              constructor of the inductive type that is the conclusion of the
1275              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1276           </listitem>
1277         </varlistentry>
1278       </variablelist>
1279     </para>
1280   </sect1>
1281   <sect1 id="tac_letin">
1282     <title>letin</title>
1283     <titleabbrev>letin</titleabbrev>
1284     <para><userinput>letin x ≝ t</userinput></para>
1285     <para>
1286       <variablelist>
1287         <varlistentry role="tactic.synopsis">
1288           <term>Synopsis:</term>
1289           <listitem>
1290             <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
1291           </listitem>
1292         </varlistentry>
1293         <varlistentry>
1294           <term>Pre-conditions:</term>
1295           <listitem>
1296             <para>None.</para>
1297           </listitem>
1298         </varlistentry>
1299         <varlistentry>
1300           <term>Action:</term>
1301           <listitem>
1302             <para>It adds to the context of the current sequent to prove a new
1303              definition <command>x ≝ t</command>.</para>
1304           </listitem>
1305         </varlistentry>
1306         <varlistentry>
1307           <term>New sequents to prove:</term>
1308           <listitem>
1309             <para>None.</para>
1310           </listitem>
1311         </varlistentry>
1312       </variablelist>
1313     </para>
1314   </sect1>
1315   <sect1 id="tac_normalize">
1316     <title>normalize</title>
1317     <titleabbrev>normalize</titleabbrev>
1318     <para><userinput>normalize patt</userinput></para>
1319     <para>
1320       <variablelist>
1321         <varlistentry role="tactic.synopsis">
1322           <term>Synopsis:</term>
1323           <listitem>
1324             <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
1325           </listitem>
1326         </varlistentry>
1327         <varlistentry>
1328           <term>Pre-conditions:</term>
1329           <listitem>
1330             <para>None.</para>
1331           </listitem>
1332         </varlistentry>
1333         <varlistentry>
1334           <term>Action:</term>
1335           <listitem>
1336             <para>It replaces all the terms matched by <command>patt</command>
1337              with their βδιζ-normal form.</para>
1338           </listitem>
1339         </varlistentry>
1340         <varlistentry>
1341           <term>New sequents to prove:</term>
1342           <listitem>
1343             <para>None.</para>
1344           </listitem>
1345         </varlistentry>
1346       </variablelist>
1347     </para>
1348   </sect1>
1349   <sect1 id="tac_reflexivity">
1350     <title>reflexivity</title>
1351     <titleabbrev>reflexivity</titleabbrev>
1352     <para><userinput>reflexivity </userinput></para>
1353     <para>
1354       <variablelist>
1355         <varlistentry role="tactic.synopsis">
1356           <term>Synopsis:</term>
1357           <listitem>
1358             <para><emphasis role="bold">reflexivity</emphasis></para>
1359           </listitem>
1360         </varlistentry>
1361         <varlistentry>
1362           <term>Pre-conditions:</term>
1363           <listitem>
1364             <para>The conclusion of the current sequent must be
1365              <command>t=t</command> for some term <command>t</command></para>
1366           </listitem>
1367         </varlistentry>
1368         <varlistentry>
1369           <term>Action:</term>
1370           <listitem>
1371             <para>It closes the current sequent by reflexivity
1372              of equality.</para>
1373           </listitem>
1374         </varlistentry>
1375         <varlistentry>
1376           <term>New sequents to prove:</term>
1377           <listitem>
1378             <para>None.</para>
1379           </listitem>
1380         </varlistentry>
1381       </variablelist>
1382     </para>
1383   </sect1>
1384   <sect1 id="tac_replace">
1385     <title>replace</title>
1386     <titleabbrev>change</titleabbrev>
1387     <para><userinput>change patt with t</userinput></para>
1388     <para>
1389       <variablelist>
1390         <varlistentry role="tactic.synopsis">
1391           <term>Synopsis:</term>
1392           <listitem>
1393             <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
1394           </listitem>
1395         </varlistentry>
1396         <varlistentry>
1397           <term>Pre-conditions:</term>
1398           <listitem>
1399             <para>None.</para>
1400           </listitem>
1401         </varlistentry>
1402         <varlistentry>
1403           <term>Action:</term>
1404           <listitem>
1405             <para>It replaces the subterms of the current sequent matched by
1406              <command>patt</command> with the new term <command>t</command>.
1407              For each subterm matched by the pattern, <command>t</command> is
1408              disambiguated in the context of the subterm.</para>
1409           </listitem>
1410         </varlistentry>
1411         <varlistentry>
1412           <term>New sequents to prove:</term>
1413           <listitem>
1414             <para>For each matched term <command>t'</command> it opens
1415              a new sequent to prove whose conclusion is
1416              <command>t'=t</command>.</para>
1417           </listitem>
1418         </varlistentry>
1419       </variablelist>
1420     </para>
1421   </sect1>
1422   <sect1 id="tac_rewrite">
1423     <title>rewrite</title>
1424     <titleabbrev>rewrite</titleabbrev>
1425     <para><userinput>rewrite dir p patt</userinput></para>
1426     <para>
1427       <variablelist>
1428         <varlistentry role="tactic.synopsis">
1429           <term>Synopsis:</term>
1430           <listitem>
1431             <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold">&lt;</emphasis>|<emphasis role="bold">&gt;</emphasis>] &sterm; &pattern;</para>
1432           </listitem>
1433         </varlistentry>
1434         <varlistentry>
1435           <term>Pre-conditions:</term>
1436           <listitem>
1437             <para><command>p</command> must be the proof of an equality,
1438              possibly under some hypotheses.</para>
1439           </listitem>
1440         </varlistentry>
1441         <varlistentry>
1442           <term>Action:</term>
1443           <listitem>
1444             <para>It looks in every term matched by <command>patt</command>
1445              for all the occurrences of the
1446              left hand side of the equality that <command>p</command> proves
1447              (resp. the right hand side if <command>dir</command> is
1448              <command>&lt;</command>). Every occurence found is replaced with
1449              the opposite side of the equality.</para>
1450           </listitem>
1451         </varlistentry>
1452         <varlistentry>
1453           <term>New sequents to prove:</term>
1454           <listitem>
1455             <para>It opens one new sequent for each hypothesis of the
1456              equality proved by <command>p</command> that is not closed
1457              by unification.</para>
1458           </listitem>
1459         </varlistentry>
1460       </variablelist>
1461     </para>
1462   </sect1>
1463   <sect1 id="tac_right">
1464     <title>right</title>
1465     <titleabbrev>right</titleabbrev>
1466     <para><userinput>right </userinput></para>
1467     <para>
1468       <variablelist>
1469         <varlistentry role="tactic.synopsis">
1470           <term>Synopsis:</term>
1471           <listitem>
1472             <para><emphasis role="bold">right</emphasis></para>
1473           </listitem>
1474         </varlistentry>
1475         <varlistentry>
1476           <term>Pre-conditions:</term>
1477           <listitem>
1478             <para>The conclusion of the current sequent must be
1479              an inductive type or the application of an inductive type with
1480              at least two constructors.</para>
1481           </listitem>
1482         </varlistentry>
1483         <varlistentry>
1484           <term>Action:</term>
1485           <listitem>
1486             <para>Equivalent to <command>constructor 2</command>.</para>
1487           </listitem>
1488         </varlistentry>
1489         <varlistentry>
1490           <term>New sequents to prove:</term>
1491           <listitem>
1492             <para>It opens a new sequent for each premise of the second
1493              constructor of the inductive type that is the conclusion of the
1494              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1495           </listitem>
1496         </varlistentry>
1497       </variablelist>
1498     </para>
1499   </sect1>
1500   <sect1 id="tac_ring">
1501     <title>ring</title>
1502     <titleabbrev>ring</titleabbrev>
1503     <para><userinput>ring </userinput></para>
1504     <para>
1505       <variablelist>
1506         <varlistentry role="tactic.synopsis">
1507           <term>Synopsis:</term>
1508           <listitem>
1509             <para><emphasis role="bold">ring</emphasis></para>
1510           </listitem>
1511         </varlistentry>
1512         <varlistentry>
1513           <term>Pre-conditions:</term>
1514           <listitem>
1515             <para>The conclusion of the current sequent must be an
1516              equality over Coq's real numbers that can be proved using
1517              the ring properties of the real numbers only.</para>
1518           </listitem>
1519         </varlistentry>
1520         <varlistentry>
1521           <term>Action:</term>
1522           <listitem>
1523             <para>It closes the current sequent veryfying the equality by
1524              means of computation (i.e. this is a reflexive tactic, implemented
1525              exploiting the &quot;two level reasoning&quot; technique).</para>
1526           </listitem>
1527         </varlistentry>
1528         <varlistentry>
1529           <term>New sequents to prove:</term>
1530           <listitem>
1531             <para>None.</para>
1532           </listitem>
1533         </varlistentry>
1534       </variablelist>
1535     </para>
1536   </sect1>
1537   <sect1 id="tac_simplify">
1538     <title>simplify</title>
1539     <titleabbrev>simplify</titleabbrev>
1540     <para><userinput>simplify patt</userinput></para>
1541     <para>
1542       <variablelist>
1543         <varlistentry role="tactic.synopsis">
1544           <term>Synopsis:</term>
1545           <listitem>
1546             <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
1547           </listitem>
1548         </varlistentry>
1549         <varlistentry>
1550           <term>Pre-conditions:</term>
1551           <listitem>
1552             <para>None.</para>
1553           </listitem>
1554         </varlistentry>
1555         <varlistentry>
1556           <term>Action:</term>
1557           <listitem>
1558             <para>It replaces all the terms matched by <command>patt</command>
1559              with other convertible terms that are supposed to be simpler.</para>
1560           </listitem>
1561         </varlistentry>
1562         <varlistentry>
1563           <term>New sequents to prove:</term>
1564           <listitem>
1565             <para>None.</para>
1566           </listitem>
1567         </varlistentry>
1568       </variablelist>
1569     </para>
1570   </sect1>
1571   <sect1 id="tac_split">
1572     <title>split</title>
1573     <titleabbrev>split</titleabbrev>
1574     <para><userinput>split </userinput></para>
1575     <para>
1576       <variablelist>
1577         <varlistentry role="tactic.synopsis">
1578           <term>Synopsis:</term>
1579           <listitem>
1580             <para><emphasis role="bold">split</emphasis></para>
1581           </listitem>
1582         </varlistentry>
1583         <varlistentry>
1584           <term>Pre-conditions:</term>
1585           <listitem>
1586             <para>The conclusion of the current sequent must be
1587              an inductive type or the application of an inductive type with
1588              at least one constructor.</para>
1589           </listitem>
1590         </varlistentry>
1591         <varlistentry>
1592           <term>Action:</term>
1593           <listitem>
1594             <para>Equivalent to <command>constructor 1</command>.</para>
1595           </listitem>
1596         </varlistentry>
1597         <varlistentry>
1598           <term>New sequents to prove:</term>
1599           <listitem>
1600             <para>It opens a new sequent for each premise of the first
1601              constructor of the inductive type that is the conclusion of the
1602              current sequent. For more details, see the <command>constructor</command> tactic.</para>
1603           </listitem>
1604         </varlistentry>
1605       </variablelist>
1606     </para>
1607   </sect1>
1608
1609   <sect1 id="tac_subst">
1610     <title>subst</title>
1611     <titleabbrev>subst</titleabbrev>
1612     <para><userinput>subst</userinput></para>
1613     <para>
1614       <variablelist>
1615         <varlistentry role="tactic.synopsis">
1616           <term>Synopsis:</term>
1617           <listitem>
1618             <para><emphasis role="bold">subst</emphasis></para>
1619           </listitem>
1620         </varlistentry>
1621         <varlistentry>
1622           <term>Pre-conditions:</term>
1623           <listitem><para>
1624             None.
1625           </para></listitem>
1626         </varlistentry>
1627         <varlistentry>
1628           <term>Action:</term>
1629           <listitem><para>
1630             For each premise of the form
1631             <command>H: x = t</command> or <command>H: t = x</command>
1632             where <command>x</command> is a local variable and
1633             <command>t</command> does not depend on <command>x</command>,
1634             the tactic rewrites <command>H</command> wherever
1635             <command>x</command> appears clearing <command>H</command> and
1636             <command>x</command> afterwards.
1637           </para></listitem>
1638         </varlistentry>
1639         <varlistentry>
1640           <term>New sequents to prove:</term>
1641           <listitem><para>
1642             The one opened by the applied tactics.
1643           </para></listitem>
1644         </varlistentry>
1645       </variablelist>
1646     </para>
1647   </sect1>
1648   <sect1 id="tac_symmetry">
1649     <title>symmetry</title>
1650     <titleabbrev>symmetry</titleabbrev>
1651     <para>The tactic <command>symmetry</command> </para>
1652     <para><userinput>symmetry </userinput></para>
1653     <para>
1654       <variablelist>
1655         <varlistentry role="tactic.synopsis">
1656           <term>Synopsis:</term>
1657           <listitem>
1658             <para><emphasis role="bold">symmetry</emphasis></para>
1659           </listitem>
1660         </varlistentry>
1661         <varlistentry>
1662           <term>Pre-conditions:</term>
1663           <listitem>
1664             <para>The conclusion of the current proof must be an equality.</para>
1665           </listitem>
1666         </varlistentry>
1667         <varlistentry>
1668           <term>Action:</term>
1669           <listitem>
1670             <para>It swaps the two sides of the equalityusing the symmetric
1671              property.</para>
1672           </listitem>
1673         </varlistentry>
1674         <varlistentry>
1675           <term>New sequents to prove:</term>
1676           <listitem>
1677             <para>None.</para>
1678           </listitem>
1679         </varlistentry>
1680       </variablelist>
1681     </para>
1682   </sect1>
1683   <sect1 id="tac_transitivity">
1684     <title>transitivity</title>
1685     <titleabbrev>transitivity</titleabbrev>
1686     <para><userinput>transitivity t</userinput></para>
1687     <para>
1688       <variablelist>
1689         <varlistentry role="tactic.synopsis">
1690           <term>Synopsis:</term>
1691           <listitem>
1692             <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
1693           </listitem>
1694         </varlistentry>
1695         <varlistentry>
1696           <term>Pre-conditions:</term>
1697           <listitem>
1698             <para>The conclusion of the current proof must be an equality.</para>
1699           </listitem>
1700         </varlistentry>
1701         <varlistentry>
1702           <term>Action:</term>
1703           <listitem>
1704             <para>It closes the current sequent by transitivity of the equality.</para>
1705           </listitem>
1706         </varlistentry>
1707         <varlistentry>
1708           <term>New sequents to prove:</term>
1709           <listitem>
1710             <para>It opens two new sequents <command>l=t</command> and
1711              <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
1712 the current sequent to prove.</para>
1713           </listitem>
1714         </varlistentry>
1715       </variablelist>
1716     </para>
1717   </sect1>
1718   <sect1 id="tac_unfold">
1719     <title>unfold</title>
1720     <titleabbrev>unfold</titleabbrev>
1721     <para><userinput>unfold t patt</userinput></para>
1722     <para>
1723       <variablelist>
1724         <varlistentry role="tactic.synopsis">
1725           <term>Synopsis:</term>
1726           <listitem>
1727             <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
1728           </listitem>
1729         </varlistentry>
1730         <varlistentry>
1731           <term>Pre-conditions:</term>
1732           <listitem>
1733             <para>None.</para>
1734           </listitem>
1735         </varlistentry>
1736         <varlistentry>
1737           <term>Action:</term>
1738           <listitem>
1739             <para>It finds all the occurrences of <command>t</command>
1740              (possibly applied to arguments) in the subterms matched by
1741              <command>patt</command>. Then it δ-expands each occurrence,
1742              also performing β-reduction of the obtained term. If
1743              <command>t</command> is omitted it defaults to each
1744              subterm matched by <command>patt</command>.</para>
1745           </listitem>
1746         </varlistentry>
1747         <varlistentry>
1748           <term>New sequents to prove:</term>
1749           <listitem>
1750             <para>None.</para>
1751           </listitem>
1752         </varlistentry>
1753       </variablelist>
1754     </para>
1755   </sect1>
1756   <sect1 id="tac_whd">
1757     <title>whd</title>
1758     <titleabbrev>whd</titleabbrev>
1759     <para><userinput>whd patt</userinput></para>
1760     <para>
1761       <variablelist>
1762         <varlistentry role="tactic.synopsis">
1763           <term>Synopsis:</term>
1764           <listitem>
1765             <para><emphasis role="bold">whd</emphasis> &pattern;</para>
1766           </listitem>
1767         </varlistentry>
1768         <varlistentry>
1769           <term>Pre-conditions:</term>
1770           <listitem>
1771             <para>None.</para>
1772           </listitem>
1773         </varlistentry>
1774         <varlistentry>
1775           <term>Action:</term>
1776           <listitem>
1777             <para>It replaces all the terms matched by <command>patt</command>
1778              with their βδιζ-weak-head normal form.</para>
1779           </listitem>
1780         </varlistentry>
1781         <varlistentry>
1782           <term>New sequents to prove:</term>
1783           <listitem>
1784             <para>None.</para>
1785           </listitem>
1786         </varlistentry>
1787       </variablelist>
1788     </para>
1789   </sect1>
1790
1791 </chapter>
1792