matita/help/C/sec_tactics.xml

   1
   2 <!-- ============ Tactics ====================== -->
   3 <chapter id="sec_tactics">
   4  <title>Tactics</title>
   5
   6   <sect1 id="tac_absurd">
   7     <title>absurd &lt;term&gt;</title>
   8     <titleabbrev>absurd</titleabbrev>
   9     <para><userinput>absurd P</userinput></para>
  10      <para>
  11       <variablelist>
  12         <varlistentry>
  13           <term>Pre-conditions:</term>
  14           <listitem>
  15             <para><command>P</command> must have type <command>Prop</command>.</para>
  16           </listitem>
  17         </varlistentry>
  18         <varlistentry>
  19           <term>Action:</term>
  20           <listitem>
  21             <para>it closes the current sequent by eliminating an
  22              absurd term.</para>
  23           </listitem>
  24         </varlistentry>
  25         <varlistentry>
  26           <term>New sequents to prove:</term>
  27           <listitem>
  28             <para>it opens two new sequents of conclusion <command>P</command>
  29              and <command>¬P</command>.</para>
  30           </listitem>
  31         </varlistentry>
  32       </variablelist>
  33      </para>
  34   </sect1>
  35   <sect1 id="tac_apply">
  36     <title>apply &lt;term&gt;</title>
  37     <titleabbrev>apply</titleabbrev>
  38     <para><userinput>apply t</userinput></para>
  39     <para>
  40       <variablelist>
  41         <varlistentry>
  42           <term>Pre-conditions:</term>
  43           <listitem>
  44             <para><command>t</command> must have type
  45              <command>T<subscript>1</subscript> → ... →
  46               T<subscript>n</subscript> → G</command>
  47              where <command>G</command> can be unified with the conclusion
  48              of the current sequent.</para>
  49           </listitem>
  50         </varlistentry>
  51         <varlistentry>
  52           <term>Action:</term>
  53           <listitem>
  54             <para>it closes the current sequent by applying <command>t</command> to <command>n</command> implicit arguments (that become new sequents).</para>
  55           </listitem>
  56         </varlistentry>
  57         <varlistentry>
  58           <term>New sequents to prove:</term>
  59           <listitem>
  60             <para>it opens a new sequent for each premise
  61              <command>T<subscript>i</subscript></command> that is not
  62              instantiated by unification. <command>T<subscript>i</subscript></command> is
  63              the conclusion of the <command>i</command>-th new sequent to
  64              prove.</para>
  65           </listitem>
  66         </varlistentry>
  67       </variablelist>
  68     </para>
  69   </sect1>
  70   <sect1 id="tac_assumption">
  71     <title>assumption</title>
  72     <titleabbrev>assumption</titleabbrev>
  73     <para><userinput>assumption </userinput></para>
  74     <para>
  75       <variablelist>
  76         <varlistentry>
  77           <term>Pre-conditions:</term>
  78           <listitem>
  79             <para>there must exist an hypothesis whose type can be unified with
  80              the conclusion of the current sequent.</para>
  81           </listitem>
  82         </varlistentry>
  83         <varlistentry>
  84           <term>Action:</term>
  85           <listitem>
  86             <para>it closes the current sequent exploiting an hypothesis.</para>
  87           </listitem>
  88         </varlistentry>
  89         <varlistentry>
  90           <term>New sequents to prove:</term>
  91           <listitem>
  92             <para>none</para>
  93           </listitem>
  94         </varlistentry>
  95       </variablelist>
  96     </para>
  97   </sect1>
  98   <sect1 id="tac_auto">
  99     <title>auto [depth=&lt;int&gt;] [width=&lt;int&gt;] [paramodulation] [full]</title>
 100     <titleabbrev>auto</titleabbrev>
 101     <para><userinput>auto depth=d width=w paramodulation full</userinput></para>
 102     <para>
 103       <variablelist>
 104         <varlistentry>
 105           <term>Pre-conditions:</term>
 106           <listitem>
 107             <para>none, but the tactic may fail finding a proof if every
 108              proof is in the search space that is pruned away. Pruning is
 109              controlled by <command>d</command> and <command>w</command>.
 110              Moreover, only lemmas whose type signature is a subset of the
 111              signature of the current sequent are considered. The signature of
 112              a sequent is ...TODO</para>
 113           </listitem>
 114         </varlistentry>
 115         <varlistentry>
 116           <term>Action:</term>
 117           <listitem>
 118             <para>it closes the current sequent by repeated application of
 119              rewriting steps (unless <command>paramodulation</command> is
 120              omitted), hypothesis and lemmas in the library.</para>
 121           </listitem>
 122         </varlistentry>
 123         <varlistentry>
 124           <term>New sequents to prove:</term>
 125           <listitem>
 126             <para>none</para>
 127           </listitem>
 128         </varlistentry>
 129       </variablelist>
 130     </para>
 131   </sect1>
 132   <sect1 id="tac_clear">
 133     <title>clear &lt;id&gt;</title>
 134     <titleabbrev>clear</titleabbrev>
 135     <para><userinput>clear H</userinput></para>
 136     <para>
 137       <variablelist>
 138         <varlistentry>
 139           <term>Pre-conditions:</term>
 140           <listitem>
 141             <para><command>H</command> must be an hypothesis of the
 142              current sequent to prove.</para>
 143           </listitem>
 144         </varlistentry>
 145         <varlistentry>
 146           <term>Action:</term>
 147           <listitem>
 148             <para>it hides the hypothesis <command>H</command> from the
 149              current sequent.</para>
 150           </listitem>
 151         </varlistentry>
 152         <varlistentry>
 153           <term>New sequents to prove:</term>
 154           <listitem>
 155             <para>none</para>
 156           </listitem>
 157         </varlistentry>
 158       </variablelist>
 159     </para>
 160   </sect1>
 161   <sect1 id="tac_clearbody">
 162     <title>clearbody &lt;id&gt;</title>
 163     <titleabbrev>clearbody</titleabbrev>
 164     <para><userinput>clearbody H</userinput></para>
 165     <para>
 166       <variablelist>
 167         <varlistentry>
 168           <term>Pre-conditions:</term>
 169           <listitem>
 170             <para><command>H</command> must be an hypothesis of the
 171              current sequent to prove.</para>
 172           </listitem>
 173         </varlistentry>
 174         <varlistentry>
 175           <term>Action:</term>
 176           <listitem>
 177             <para>it hides the definiens of a definition in the current
 178              sequent context. Thus the definition becomes an hypothesis.</para>
 179           </listitem>
 180         </varlistentry>
 181         <varlistentry>
 182           <term>New sequents to prove:</term>
 183           <listitem>
 184             <para>none.</para>
 185           </listitem>
 186         </varlistentry>
 187       </variablelist>
 188     </para>
 189   </sect1>
 190   <sect1 id="tac_change">
 191     <title>change &lt;pattern&gt; with &lt;term&gt;</title>
 192     <titleabbrev>change</titleabbrev>
 193     <para><userinput>change patt with t</userinput></para>
 194     <para>
 195       <variablelist>
 196         <varlistentry>
 197           <term>Pre-conditions:</term>
 198           <listitem>
 199             <para>each subterm matched by the pattern must be convertible
 200              with the term <command>t</command> disambiguated in the context
 201              of the matched subterm.</para>
 202           </listitem>
 203         </varlistentry>
 204         <varlistentry>
 205           <term>Action:</term>
 206           <listitem>
 207             <para>it replaces the subterms of the current sequent matched by
 208              <command>patt</command> with the new term <command>t</command>.
 209              For each subterm matched by the pattern, <command>t</command> is
 210              disambiguated in the context of the subterm.</para>
 211           </listitem>
 212         </varlistentry>
 213         <varlistentry>
 214           <term>New sequents to prove:</term>
 215           <listitem>
 216             <para>none.</para>
 217           </listitem>
 218         </varlistentry>
 219       </variablelist>
 220     </para>
 221   </sect1>
 222   <sect1 id="tac_constructor">
 223     <title>constructor &lt;int&gt;</title>
 224     <titleabbrev>constructor</titleabbrev>
 225     <para><userinput>constructor n</userinput></para>
 226     <para>
 227       <variablelist>
 228         <varlistentry>
 229           <term>Pre-conditions:</term>
 230           <listitem>
 231             <para>the conclusion of the current sequent must be
 232              an inductive type or the application of an inductive type.</para>
 233           </listitem>
 234         </varlistentry>
 235         <varlistentry>
 236           <term>Action:</term>
 237           <listitem>
 238             <para>it applies the <command>n</command>-th constructor of the
 239              inductive type of the conclusion of the current sequent.</para>
 240           </listitem>
 241         </varlistentry>
 242         <varlistentry>
 243           <term>New sequents to prove:</term>
 244           <listitem>
 245             <para>it opens a new sequent for each premise of the constructor
 246              that can not be inferred by unification. For more details,
 247              see the <command>apply</command> tactic.</para>
 248           </listitem>
 249         </varlistentry>
 250       </variablelist>
 251     </para>
 252   </sect1>
 253   <sect1 id="tac_contradiction">
 254     <title>contradiction</title>
 255     <titleabbrev>contradiction</titleabbrev>
 256     <para><userinput>contradiction </userinput></para>
 257     <para>
 258       <variablelist>
 259         <varlistentry>
 260           <term>Pre-conditions:</term>
 261           <listitem>
 262             <para>there must be in the current context an hypothesis of type
 263              <command>False</command>.</para>
 264           </listitem>
 265         </varlistentry>
 266         <varlistentry>
 267           <term>Action:</term>
 268           <listitem>
 269             <para>it closes the current sequent by applying an hypothesis of
 270              type <command>False</command>.</para>
 271           </listitem>
 272         </varlistentry>
 273         <varlistentry>
 274           <term>New sequents to prove:</term>
 275           <listitem>
 276             <para>none</para>
 277           </listitem>
 278         </varlistentry>
 279       </variablelist>
 280     </para>
 281   </sect1>
 282   <sect1 id="tac_cut">
 283     <title>cut &lt;term&gt; [as &lt;id&gt;]</title>
 284     <titleabbrev>cut</titleabbrev>
 285     <para><userinput>cut P as H</userinput></para>
 286     <para>
 287       <variablelist>
 288         <varlistentry>
 289           <term>Pre-conditions:</term>
 290           <listitem>
 291             <para><command>P</command> must have type <command>Prop</command>.</para>
 292           </listitem>
 293         </varlistentry>
 294         <varlistentry>
 295           <term>Action:</term>
 296           <listitem>
 297             <para>it closes the current sequent.</para>
 298           </listitem>
 299         </varlistentry>
 300         <varlistentry>
 301           <term>New sequents to prove:</term>
 302           <listitem>
 303             <para>it opens two new sequents. The first one has an extra
 304              hypothesis <command>H:P</command>. If <command>H</command> is
 305              omitted, the name of the hypothesis is automatically generated.
 306              The second sequent has conclusion <command>P</command> and
 307              hypotheses the hypotheses of the current sequent to prove.</para>
 308           </listitem>
 309         </varlistentry>
 310       </variablelist>
 311     </para>
 312   </sect1>
 313   <sect1 id="tac_decompose">
 314     <title>decompose [&lt;ident list&gt;] &lt;ident&gt; [&lt;intros_spec&gt;]</title>
 315     <titleabbrev>decompose</titleabbrev>
 316     <para><userinput>decompose ???</userinput></para>
 317     <para>
 318       <variablelist>
 319         <varlistentry>
 320           <term>Pre-conditions:</term>
 321           <listitem>
 322             <para>TODO.</para>
 323           </listitem>
 324         </varlistentry>
 325         <varlistentry>
 326           <term>Action:</term>
 327           <listitem>
 328             <para>TODO.</para>
 329           </listitem>
 330         </varlistentry>
 331         <varlistentry>
 332           <term>New sequents to prove:</term>
 333           <listitem>
 334             <para>TODO.</para>
 335           </listitem>
 336         </varlistentry>
 337       </variablelist>
 338     </para>
 339   </sect1>
 340   <sect1 id="tac_discriminate">
 341     <title>discriminate &lt;term&gt;</title>
 342     <titleabbrev>discriminate</titleabbrev>
 343     <para><userinput>discriminate p</userinput></para>
 344     <para>
 345       <variablelist>
 346         <varlistentry>
 347           <term>Pre-conditions:</term>
 348           <listitem>
 349             <para><command>p</command> must have type <command>K<subscript>1</subscript> t<subscript>1</subscript> ... t<subscript>n</subscript> = K'<subscript>1</subscript> t'<subscript>1</subscript> ... t'<subscript>m</subscript></command> where <command>K</command> and <command>K'</command> must be different constructors of the same inductive type and each argument list can be empty if
 350 its constructor takes no arguments.</para>
 351           </listitem>
 352         </varlistentry>
 353         <varlistentry>
 354           <term>Action:</term>
 355           <listitem>
 356             <para>it closes the current sequent by proving the absurdity of
 357              <command>p</command>.</para>
 358           </listitem>
 359         </varlistentry>
 360         <varlistentry>
 361           <term>New sequents to prove:</term>
 362           <listitem>
 363             <para>none.</para>
 364           </listitem>
 365         </varlistentry>
 366       </variablelist>
 367     </para>
 368   </sect1>
 369   <sect1 id="tac_elim">
 370     <title>elim &lt;term&gt; [using &lt;term&gt;] [&lt;intros_spec&gt;]</title>
 371     <titleabbrev>elim</titleabbrev>
 372     <para><userinput>elim t using th hyps</userinput></para>
 373     <para>
 374       <variablelist>
 375         <varlistentry>
 376           <term>Pre-conditions:</term>
 377           <listitem>
 378             <para><command>t</command> must inhabit an inductive type and
 379              <command>th</command> must be an elimination principle for that
 380              inductive type. If <command>th</command> is omitted the appropriate
 381              standard elimination principle is chosen.</para>
 382           </listitem>
 383         </varlistentry>
 384         <varlistentry>
 385           <term>Action:</term>
 386           <listitem>
 387             <para>it proceeds by cases on the values of <command>t</command>,
 388              according to the elimination principle <command>th</command>.
 389             </para>
 390           </listitem>
 391         </varlistentry>
 392         <varlistentry>
 393           <term>New sequents to prove:</term>
 394           <listitem>
 395             <para>it opens one new sequent for each case. The names of
 396              the new hypotheses are picked by <command>hyps</command>, if
 397              provided.</para>
 398           </listitem>
 399         </varlistentry>
 400       </variablelist>
 401     </para>
 402   </sect1>
 403   <sect1 id="tac_elimType">
 404     <title>elimType &lt;term&gt; [using &lt;term&gt;]</title>
 405     <titleabbrev>elimType</titleabbrev>
 406     <para><userinput>elimType T using th</userinput></para>
 407     <para>
 408       <variablelist>
 409         <varlistentry>
 410           <term>Pre-conditions:</term>
 411           <listitem>
 412             <para><command>T</command> must be an inductive type.</para>
 413           </listitem>
 414         </varlistentry>
 415         <varlistentry>
 416           <term>Action:</term>
 417           <listitem>
 418             <para>TODO (severely bugged now).</para>
 419           </listitem>
 420         </varlistentry>
 421         <varlistentry>
 422           <term>New sequents to prove:</term>
 423           <listitem>
 424             <para>TODO</para>
 425           </listitem>
 426         </varlistentry>
 427       </variablelist>
 428     </para>
 429   </sect1>
 430   <sect1 id="tac_exact">
 431     <title>exact &lt;term&gt;</title>
 432     <titleabbrev>exact</titleabbrev>
 433     <para><userinput>exact p</userinput></para>
 434     <para>
 435       <variablelist>
 436         <varlistentry>
 437           <term>Pre-conditions:</term>
 438           <listitem>
 439             <para>the type of <command>p</command> must be convertible
 440              with the conclusion of the current sequent.</para>
 441           </listitem>
 442         </varlistentry>
 443         <varlistentry>
 444           <term>Action:</term>
 445           <listitem>
 446             <para>it closes the current sequent using <command>p</command>.</para>
 447           </listitem>
 448         </varlistentry>
 449         <varlistentry>
 450           <term>New sequents to prove:</term>
 451           <listitem>
 452             <para>none.</para>
 453           </listitem>
 454         </varlistentry>
 455       </variablelist>
 456     </para>
 457   </sect1>
 458   <sect1 id="tac_exists">
 459     <title>exists</title>
 460     <titleabbrev>exists</titleabbrev>
 461     <para><userinput>exists </userinput></para>
 462     <para>
 463       <variablelist>
 464         <varlistentry>
 465           <term>Pre-conditions:</term>
 466           <listitem>
 467             <para>the conclusion of the current sequent must be
 468              an inductive type or the application of an inductive type.</para>
 469           </listitem>
 470         </varlistentry>
 471         <varlistentry>
 472           <term>Action:</term>
 473           <listitem>
 474             <para>equivalent to <command>constructor 1</command>.</para>
 475           </listitem>
 476         </varlistentry>
 477         <varlistentry>
 478           <term>New sequents to prove:</term>
 479           <listitem>
 480             <para>it opens a new sequent for each premise of the first
 481              constructor of the inductive type that is the conclusion of the
 482              current sequent. For more details, see the <command>constructor</command> tactic.</para>
 483           </listitem>
 484         </varlistentry>
 485       </variablelist>
 486     </para>
 487   </sect1>
 488   <sect1 id="tac_fail">
 489     <title>fail </title>
 490     <titleabbrev>failt</titleabbrev>
 491     <para><userinput>fail</userinput></para>
 492     <para>
 493       <variablelist>
 494         <varlistentry>
 495           <term>Pre-conditions:</term>
 496           <listitem>
 497             <para>none.</para>
 498           </listitem>
 499         </varlistentry>
 500         <varlistentry>
 501           <term>Action:</term>
 502           <listitem>
 503             <para>this tactic always fail.</para>
 504           </listitem>
 505         </varlistentry>
 506         <varlistentry>
 507           <term>New sequents to prove:</term>
 508           <listitem>
 509             <para>N.A.</para>
 510           </listitem>
 511         </varlistentry>
 512       </variablelist>
 513     </para>
 514   </sect1>
 515   <sect1 id="tac_fold">
 516     <title>fold &lt;reduction_kind&gt; &lt;term&gt; &lt;pattern&gt;</title>
 517     <titleabbrev>fold</titleabbrev>
 518     <para><userinput>fold red t patt</userinput></para>
 519     <para>
 520       <variablelist>
 521         <varlistentry>
 522           <term>Pre-conditions:</term>
 523           <listitem>
 524             <para>the pattern must not specify the wanted term.</para>
 525           </listitem>
 526         </varlistentry>
 527         <varlistentry>
 528           <term>Action:</term>
 529           <listitem>
 530             <para>first of all it locates all the subterms matched by
 531              <command>patt</command>. In the context of each matched subterm
 532              it disambiguates the term <command>t</command> and reduces it
 533              to its <command>red</command> normal form; then it replaces with
 534              <command>t</command> every occurrence of the normal form in the
 535              matched subterm.</para>
 536           </listitem>
 537         </varlistentry>
 538         <varlistentry>
 539           <term>New sequents to prove:</term>
 540           <listitem>
 541             <para>none.</para>
 542           </listitem>
 543         </varlistentry>
 544       </variablelist>
 545     </para>
 546   </sect1>
 547   <sect1 id="tac_fourier">
 548     <title>fourier</title>
 549     <titleabbrev>fourier</titleabbrev>
 550     <para><userinput>fourier </userinput></para>
 551     <para>
 552       <variablelist>
 553         <varlistentry>
 554           <term>Pre-conditions:</term>
 555           <listitem>
 556             <para>the conclusion of the current sequent must be a linear
 557              inequation over real numbers taken from standard library of
 558              Coq. Moreover the inequations in the hypotheses must imply the
 559              inequation in the conclusion of the current sequent.</para>
 560           </listitem>
 561         </varlistentry>
 562         <varlistentry>
 563           <term>Action:</term>
 564           <listitem>
 565             <para>it closes the current sequent by applying the Fourier method.</para>
 566           </listitem>
 567         </varlistentry>
 568         <varlistentry>
 569           <term>New sequents to prove:</term>
 570           <listitem>
 571             <para>none.</para>
 572           </listitem>
 573         </varlistentry>
 574       </variablelist>
 575     </para>
 576   </sect1>
 577   <sect1 id="tac_fwd">
 578     <title>fwd &lt;ident&gt; [&lt;ident list&gt;]</title>
 579     <titleabbrev>fwd</titleabbrev>
 580     <para><userinput>fwd ...TODO</userinput></para>
 581     <para>
 582       <variablelist>
 583         <varlistentry>
 584           <term>Pre-conditions:</term>
 585           <listitem>
 586             <para>TODO.</para>
 587           </listitem>
 588         </varlistentry>
 589         <varlistentry>
 590           <term>Action:</term>
 591           <listitem>
 592             <para>TODO.</para>
 593           </listitem>
 594         </varlistentry>
 595         <varlistentry>
 596           <term>New sequents to prove:</term>
 597           <listitem>
 598             <para>TODO.</para>
 599           </listitem>
 600         </varlistentry>
 601       </variablelist>
 602     </para>
 603   </sect1>
 604   <sect1 id="tac_generalize">
 605     <title>generalize &lt;pattern&gt; [as &lt;id&gt;]</title>
 606     <titleabbrev>generalize</titleabbrev>
 607     <para>The tactic <command>generalize</command> </para>
 608   </sect1>
 609   <sect1 id="tac_id">
 610     <title>id</title>
 611     <titleabbrev>id</titleabbrev>
 612     <para>The tactic <command>id</command> </para>
 613   </sect1>
 614   <sect1 id="tac_injection">
 615     <title>injection &lt;term&gt;</title>
 616     <titleabbrev>injection</titleabbrev>
 617     <para>The tactic <command>injection</command> </para>
 618   </sect1>
 619   <sect1 id="tac_intro">
 620     <title>intro [&lt;ident&gt;]</title>
 621     <titleabbrev>intro</titleabbrev>
 622     <para>The tactic <command>intro</command> </para>
 623   </sect1>
 624   <sect1 id="tac_intros">
 625     <title>intros &lt;intros_spec&gt;</title>
 626     <titleabbrev>intros</titleabbrev>
 627     <para>The tactic <command>intros</command> </para>
 628   </sect1>
 629   <sect1 id="tac_inversion">
 630     <title>inversion &lt;term&gt;</title>
 631     <titleabbrev>inversion</titleabbrev>
 632     <para>The tactic <command>inversion</command> </para>
 633   </sect1>
 634   <sect1 id="tac_lapply">
 635     <title>lapply [depth=&lt;int&gt;] &lt;term&gt; [to &lt;term list] [using &lt;ident&gt;]</title>
 636     <titleabbrev>lapply</titleabbrev>
 637     <para>The tactic <command>lapply</command> </para>
 638   </sect1>
 639   <sect1 id="tac_left">
 640     <title>left</title>
 641     <titleabbrev>left</titleabbrev>
 642     <para>The tactic <command>left</command> </para>
 643   </sect1>
 644   <sect1 id="tac_letin">
 645     <title>letin &lt;ident&gt; ≝ &lt;term&gt;</title>
 646     <titleabbrev>letin</titleabbrev>
 647     <para>The tactic <command>letin</command> </para>
 648   </sect1>
 649   <sect1 id="tac_normalize">
 650     <title>normalize &lt;pattern&gt;</title>
 651     <titleabbrev>normalize</titleabbrev>
 652     <para>The tactic <command>normalize</command> </para>
 653   </sect1>
 654   <sect1 id="tac_paramodulation">
 655     <title>paramodulation &lt;pattern&gt;</title>
 656     <titleabbrev>paramodulation</titleabbrev>
 657     <para>The tactic <command>paramodulation</command> </para>
 658   </sect1>
 659   <sect1 id="tac_reduce">
 660     <title>reduce &lt;pattern&gt;</title>
 661     <titleabbrev>reduce</titleabbrev>
 662     <para>The tactic <command>reduce</command> </para>
 663   </sect1>
 664   <sect1 id="tac_reflexivity">
 665     <title>reflexivity</title>
 666     <titleabbrev>reflexivity</titleabbrev>
 667     <para>The tactic <command>reflexivity</command> </para>
 668   </sect1>
 669   <sect1 id="tac_replace">
 670     <title>replace &lt;pattern&gt; with &lt;term&gt;</title>
 671     <titleabbrev>replace</titleabbrev>
 672     <para>The tactic <command>replace</command> </para>
 673   </sect1>
 674   <sect1 id="tac_rewrite">
 675     <title>rewrite {&lt;|&gt;} &lt;term&gt; &lt;pattern&gt;</title>
 676     <titleabbrev>rewrite</titleabbrev>
 677     <para>The tactic <command>rewrite</command> </para>
 678   </sect1>
 679   <sect1 id="tac_right">
 680     <title>right</title>
 681     <titleabbrev>right</titleabbrev>
 682     <para>The tactic <command>right</command> </para>
 683   </sect1>
 684   <sect1 id="tac_ring">
 685     <title>ring</title>
 686     <titleabbrev>ring</titleabbrev>
 687     <para>The tactic <command>ring</command> </para>
 688   </sect1>
 689   <sect1 id="tac_simplify">
 690     <title>simplify &lt;pattern&gt;</title>
 691     <titleabbrev>simplify</titleabbrev>
 692     <para>The tactic <command>simplify</command> </para>
 693   </sect1>
 694   <sect1 id="tac_split">
 695     <title>split</title>
 696     <titleabbrev>split</titleabbrev>
 697     <para>The tactic <command>split</command> </para>
 698   </sect1>
 699   <sect1 id="tac_symmetry">
 700     <title>symmetry</title>
 701     <titleabbrev>symmetry</titleabbrev>
 702     <para>The tactic <command>symmetry</command> </para>
 703   </sect1>
 704   <sect1 id="tac_transitivity">
 705     <title>transitivity &lt;term&gt;</title>
 706     <titleabbrev>transitivity</titleabbrev>
 707     <para>The tactic <command>transitivity</command> </para>
 708   </sect1>
 709   <sect1 id="tac_unfold">
 710     <title>unfold [&lt;term&gt;] &lt;pattern&gt;</title>
 711     <titleabbrev>unfold</titleabbrev>
 712     <para>The tactic <command>unfold</command> </para>
 713   </sect1>
 714   <sect1 id="tac_whd">
 715     <title>whd &lt;pattern&gt;</title>
 716     <titleabbrev>whd</titleabbrev>
 717     <para>The tactic <command>whd</command> </para>
 718   </sect1>
 719
 720 </chapter>
 721