2 <!-- =========== Terms, declarations and definitions ============ -->
4 <chapter id="sec_terms">
6 <para>To describe syntax in this manual we use the following conventions:</para>
8 <listitem><para>Non terminal symbols are emphasized and have a link to their
9 definition. E.g.: &term;</para></listitem>
10 <listitem><para>Terminal symbols are in bold. E.g.:
11 <emphasis role="bold">theorem</emphasis></para></listitem>
12 <listitem><para>Optional sequences of elements are put in square brackets.
13 E.g.: [<emphasis role="bold">in</emphasis> &term;]</para></listitem>
14 <listitem><para>Alternatives are put in square brakets and they are
15 separated by vertical bars. E.g.: [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>]</para></listitem>
16 <listitem><para>Repetitions of a sequence of elements are given by putting the
17 sequence in square brackets, that are followed by three dots. The empty
18 sequence is a valid repetition.
19 E.g.: [<emphasis role="bold">and</emphasis> &term;]…</para></listitem>
20 <listitem><para>Characters belonging to a set of characters are given
21 by listing the set elements in square brackets. Hyphens are used to
22 specify ranges of characters in the set.
23 E.g.: [<emphasis role="bold">a</emphasis>-<emphasis role="bold">zA</emphasis>-<emphasis role="bold">Z0</emphasis>-<emphasis role="bold">9_-</emphasis>]</para></listitem>
25 <sect1 id="terms_and_co">
26 <title>Terms & co.</title>
28 <title>Lexical conventions</title>
29 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
30 <title>qstring</title>
34 <entry id="grammar.qstring">&qstring;</entry>
36 <entry><emphasis role="bold">"</emphasis><emphasis>〈〈any sequence of characters excluded "〉〉</emphasis><emphasis role="bold">"</emphasis></entry>
41 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
46 <entry id="grammar.id">&id;</entry>
48 <entry><emphasis>〈〈any sequence of letters, underscores or valid <ulink type="http" url="http://www.w3.org/TR/2004/REC-xml-20040204/#NT-Digit">XML digits</ulink> prefixed by a latin letter ([a-zA-Z]) and post-fixed by a possible empty sequence of decorators ([?'`])〉〉</emphasis></entry>
53 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
58 <entry id="grammar.nat">&nat;</entry>
60 <entry><emphasis>〈〈any sequence of valid <ulink type="http" url="http://www.w3.org/TR/2004/REC-xml-20040204/#NT-Digit">XML digits</ulink>〉〉</emphasis></entry>
65 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
70 <entry id="grammar.char">&char;</entry>
72 <entry>[<emphasis role="bold">a</emphasis>-<emphasis role="bold">zA</emphasis>-<emphasis role="bold">Z0</emphasis>-<emphasis role="bold">9_-</emphasis>]</entry>
77 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
78 <title>uri-step</title>
82 <entry id="grammar.uri-step">&uri-step;</entry>
84 <entry>&char;[&char;]…</entry>
89 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
94 <entry id="grammar.uri">&uri;</entry>
96 <entry>[<emphasis role="bold">cic:/</emphasis>|<emphasis role="bold">theory:/</emphasis>]&uri-step;[<emphasis role="bold">/</emphasis>&uri-step;]…<emphasis role="bold">.</emphasis>&id;[<emphasis role="bold">.</emphasis>&id;]…[<emphasis role="bold">#xpointer(</emphasis>&nat;<emphasis role="bold">/</emphasis>&nat;[<emphasis role="bold">/</emphasis>&nat;]…<emphasis role="bold">)</emphasis>]</entry>
101 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
102 <title>csymbol</title>
106 <entry id="grammar.csymbol">&csymbol;</entry>
108 <entry><emphasis role="bold">'</emphasis>&id;</entry>
117 <!-- ZACK: Sample EBNF snippet, see:
118 http://www.docbook.org/tdg/en/html/productionset.html -->
122 <production id="grammar.term">
125 <lineannotation></lineannotation>
131 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
136 <entry id="grammar.term">&term;</entry>
138 <entry>&sterm;</entry>
139 <entry>simple or delimited term</entry>
144 <entry>&term; &term;</entry>
145 <entry>application</entry>
150 <entry><emphasis role="bold">λ</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
151 <entry>λ-abstraction</entry>
156 <entry><emphasis role="bold">Π</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
157 <entry>dependent product meant to define a datatype</entry>
162 <entry><emphasis role="bold">∀</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
163 <entry>dependent product meant to define a proposition</entry>
168 <entry>&term; <emphasis role="bold">→</emphasis> &term;</entry>
169 <entry>non-dependent product (logical implication or function space)</entry>
174 <entry><emphasis role="bold">let</emphasis> [&id;|(&id;<emphasis role="bold">:</emphasis> &term;)] <emphasis role="bold">≝</emphasis> &term; <emphasis role="bold">in</emphasis> &term;</entry>
175 <entry>local definition</entry>
181 <emphasis role="bold">let</emphasis>
182 [<emphasis role="bold">co</emphasis>]<emphasis role="bold">rec</emphasis>
185 <entry>(co)recursive definitions</entry>
191 [<emphasis role="bold">and</emphasis> &rec_def;]…
199 <emphasis role="bold">in</emphasis> &term;
207 <entry>user provided notation</entry>
210 <entry id="grammar.rec_def">&rec_def;</entry>
213 &id; [&id;|<emphasis role="bold">_</emphasis>|<emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]… <emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis>]…
221 [<emphasis role="bold">on</emphasis> &id;]
222 [<emphasis role="bold">:</emphasis> &term;]
223 <emphasis role="bold">≝</emphasis> &term;]
231 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
232 <title>Simple terms</title>
236 <entry id="grammar.sterm">&sterm;</entry>
238 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
244 <entry>&id;[<emphasis role="bold">\subst[</emphasis>
245 &id;<emphasis role="bold">≔</emphasis>&term;
246 [<emphasis role="bold">;</emphasis>&id;<emphasis role="bold">≔</emphasis>&term;]…
247 <emphasis role="bold">]</emphasis>]
249 <entry>identifier with optional explicit named substitution</entry>
255 <entry>a qualified reference</entry>
260 <entry><emphasis role="bold">Prop</emphasis></entry>
261 <entry>the impredicative sort of propositions</entry>
266 <entry><emphasis role="bold">Set</emphasis></entry>
267 <entry>the impredicate sort of datatypes</entry>
272 <entry><emphasis role="bold">CProp</emphasis></entry>
273 <entry>one fixed predicative sort of constructive propositions</entry>
278 <entry><emphasis role="bold">Type</emphasis></entry>
279 <entry>one predicative sort of datatypes</entry>
284 <entry><emphasis role="bold">?</emphasis></entry>
285 <entry>implicit argument</entry>
290 <entry><emphasis role="bold">?n</emphasis>
291 [<emphasis role="bold">[</emphasis>
292 [<emphasis role="bold">_</emphasis>|&term;]…
293 <emphasis role="bold">]</emphasis>]</entry>
294 <entry>metavariable</entry>
299 <entry><emphasis role="bold">match</emphasis> &term;
300 [ <emphasis role="bold">in</emphasis> &id; ]
301 [ <emphasis role="bold">return</emphasis> &term; ]
302 <emphasis role="bold">with</emphasis>
304 <entry>case analysis</entry>
310 <emphasis role="bold">[</emphasis>
311 &match_branch;[<emphasis role="bold">|</emphasis>&match_branch;]…
312 <emphasis role="bold">]</emphasis>
319 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
326 <entry>user provided notation at precedence 90</entry>
332 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
333 <title>Arguments</title>
337 <entry id="grammar.args">&args;</entry>
340 <emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]
342 <entry>ignored argument</entry>
348 <emphasis role="bold">(</emphasis><emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis>
350 <entry>ignored argument</entry>
355 <entry>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]</entry>
361 <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis></entry>
365 <entry id="grammar.args2">&args2;</entry>
373 <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…<emphasis role="bold">:</emphasis> &term;<emphasis role="bold">)</emphasis></entry>
380 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
381 <title>Pattern matching</title>
385 <entry id="grammar.match_branch">&match_branch;</entry>
387 <entry>&match_pattern; <emphasis role="bold">⇒</emphasis> &term;</entry>
391 <entry id="grammar.match_pattern">&match_pattern;</entry>
394 <entry>0-ary constructor</entry>
399 <entry><emphasis role="bold">(</emphasis>&id; &id; [&id;]…<emphasis role="bold">)</emphasis></entry>
400 <entry>n-ary constructor (binds the n arguments)</entry>
405 <entry>&id; &id; [&id;]…</entry>
406 <entry>n-ary constructor (binds the n arguments)</entry>
411 <entry><emphasis role="bold">_</emphasis></entry>
412 <entry>any remaining constructor (ignoring its arguments)</entry>
422 <sect1 id="axiom_definition_declaration">
423 <title>Definitions and declarations</title>
425 <title><emphasis role="bold">axiom</emphasis> &id;<emphasis role="bold">:</emphasis> &term;</title>
426 <titleabbrev>axiom</titleabbrev>
427 <para><userinput>axiom H: P</userinput></para>
428 <para><command>H</command> is declared as an axiom that states <command>P</command></para>
430 <sect2 id="definition">
431 <title><emphasis role="bold">definition</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
432 <titleabbrev>definition</titleabbrev>
433 <para><userinput>definition f: T ≝ t</userinput></para>
434 <para><command>f</command> is defined as <command>t</command>;
435 <command>T</command> is its type. An error is raised if the type of
436 <command>t</command> is not convertible to <command>T</command>.</para>
437 <para><command>T</command> is inferred from <command>t</command> if
439 <para><command>t</command> can be omitted only if <command>T</command> is
440 given. In this case Matita enters in interactive mode and
441 <command>f</command> must be defined by means of tactics.</para>
442 <para>Notice that the command is equivalent to <command>theorem f: T ≝ t</command>.</para>
445 <title><emphasis role="bold">letrec</emphasis> &TODO;</title>
446 <titleabbrev>&TODO;</titleabbrev>
449 <sect2 id="inductive">
450 <title>[<emphasis role="bold">inductive</emphasis>|<emphasis role="bold">coinductive</emphasis>] &id; [&args2;]… <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis> [<emphasis role="bold">|</emphasis>] [&id;<emphasis role="bold">:</emphasis>&term;] [<emphasis role="bold">|</emphasis> &id;<emphasis role="bold">:</emphasis>&term;]…
451 [<emphasis role="bold">with</emphasis> &id; <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis> [<emphasis role="bold">|</emphasis>] [&id;<emphasis role="bold">:</emphasis>&term;] [<emphasis role="bold">|</emphasis> &id;<emphasis role="bold">:</emphasis>&term;]…]…
453 <titleabbrev>(co)inductive types declaration</titleabbrev>
454 <para><userinput>inductive i x y z: S ≝ k1:T1 | … | kn:Tn with i' : S' ≝ k1':T1' | … | km':Tm'</userinput></para>
455 <para>Declares a family of two mutually inductive types
456 <command>i</command> and <command>i'</command> whose types are
457 <command>S</command> and <command>S'</command>, which must be convertible
459 <para>The constructors <command>ki</command> of type <command>Ti</command>
460 and <command>ki'</command> of type <command>Ti'</command> are also
461 simultaneously declared. The declared types <command>i</command> and
462 <command>i'</command> may occur in the types of the constructors, but
463 only in strongly positive positions according to the rules of the
465 <para>The whole family is parameterized over the arguments <command>x,y,z</command>.</para>
466 <para>If the keyword <command>coinductive</command> is used, the declared
467 types are considered mutually coinductive.</para>
468 <para>Elimination principles for the record are automatically generated
469 by Matita, if allowed by the typing rules of the calculus according to
470 the sort <command>S</command>. If generated,
471 they are named <command>i_ind</command>, <command>i_rec</command> and
472 <command>i_rect</command> according to the sort of their induction
476 <title><emphasis role="bold">record</emphasis> &id; [&args2;]… <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis><emphasis role="bold">{</emphasis>[&id; [<emphasis role="bold">:</emphasis>|<emphasis role="bold">:></emphasis>] &term;] [<emphasis role="bold">;</emphasis>&id; [<emphasis role="bold">:</emphasis>|<emphasis role="bold">:></emphasis>] &term;]…<emphasis role="bold">}</emphasis></title>
477 <titleabbrev>record</titleabbrev>
478 <para><userinput>record id x y z: S ≝ { f1: T1; …; fn:Tn }</userinput></para>
479 <para>Declares a new record family <command>id</command> parameterized over
480 <command>x,y,z</command>.</para>
481 <para><command>S</command> is the type of the record
482 and it must be convertible to a sort.</para>
483 <para>Each field <command>fi</command> is declared by giving its type
484 <command>Ti</command>. A record without any field is admitted.</para>
485 <para>Elimination principles for the record are automatically generated
486 by Matita, if allowed by the typing rules of the calculus according to
487 the sort <command>S</command>. If generated,
488 they are named <command>i_ind</command>, <command>i_rec</command> and
489 <command>i_rect</command> according to the sort of their induction
491 <para>For each field <command>fi</command> a record projection
492 <command>fi</command> is also automatically generated if projection
493 is allowed by the typing rules of the calculus according to the
494 sort <command>S</command>, the type <command>T1</command> and
495 the definability of depending record projections.</para>
496 <para>If the type of a field is declared with <command>:></command>,
497 the corresponding record projection becomes an implicit coercion.
498 This is just syntactic sugar and it has the same effect of declaring the
499 record projection as a coercion later on.</para>
504 <title>Proofs</title>
506 <title><emphasis role="bold">theorem</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
507 <titleabbrev>theorem</titleabbrev>
508 <para><userinput>theorem f: P ≝ p</userinput></para>
509 <para>Proves a new theorem <command>f</command> whose thesis is
510 <command>P</command>.</para>
511 <para>If <command>p</command> is provided, it must be a proof term for
512 <command>P</command>. Otherwise an interactive proof is started.</para>
513 <para><command>P</command> can be omitted only if the proof is not
515 <para>Proving a theorem already proved in the library is an error.
516 To provide an alternative name and proof for the same theorem, use
517 <command>variant f: P ≝ p</command>.</para>
518 <para>A warning is raised if the name of the theorem cannot be obtained
519 by mangling the name of the constants in its thesis.</para>
520 <para>Notice that the command is equivalent to <command>definition f: T ≝ t</command>.</para>
523 <title><emphasis role="bold">variant</emphasis> &id;<emphasis role="bold">:</emphasis> &term; <emphasis role="bold">≝</emphasis> &term;</title>
524 <titleabbrev>variant</titleabbrev>
525 <para><userinput>variant f: T ≝ t</userinput></para>
526 <para>Same as <command>theorem f: T ≝ t</command>, but it does not
527 complain if the theorem has already been proved. To be used to give
528 an alternative name or proof to a theorem.</para>
531 <title><emphasis role="bold">lemma</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
532 <titleabbrev>lemma</titleabbrev>
533 <para><userinput>lemma f: T ≝ t</userinput></para>
534 <para>Same as <command>theorem f: T ≝ t</command></para>
537 <title><emphasis role="bold">fact</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
538 <titleabbrev>fact</titleabbrev>
539 <para><userinput>fact f: T ≝ t</userinput></para>
540 <para>Same as <command>theorem f: T ≝ t</command></para>
543 <title><emphasis role="bold">remark</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
544 <titleabbrev>remark</titleabbrev>
545 <para><userinput>remark f: T ≝ t</userinput></para>
546 <para>Same as <command>theorem f: T ≝ t</command></para>
550 <sect1 id="tacticargs">
551 <title>Tactic arguments</title>
552 <para>This section documents the syntax of some recurring arguments for
555 <sect2 id="introsspec">
556 <title>intros-spec</title>
557 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
558 <title>intros-spec</title>
562 <entry id="grammar.intros-spec">&intros-spec;</entry>
564 <entry>[&nat;] [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]</entry>
569 <para>The natural number is the number of new hypotheses to be introduced. The list of identifiers gives the name for the first hypotheses.</para>
573 <title>pattern</title>
574 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
575 <title>pattern</title>
579 <entry id="grammar.pattern">&pattern;</entry>
581 <entry><emphasis role="bold">in</emphasis>
582 [&id;[<emphasis role="bold">:</emphasis> &path;]]…
583 [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
584 <entry>simple pattern</entry>
589 <entry><emphasis role="bold">in match</emphasis> &path;
590 [<emphasis role="bold">in</emphasis>
591 [&id;[<emphasis role="bold">:</emphasis> &path;]]…
592 [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
593 <entry>full pattern</entry>
598 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
603 <entry id="grammar.path">&path;</entry>
605 <entry><emphasis>〈〈any &sterm; without occurrences of <emphasis role="bold">Set</emphasis>, <emphasis role="bold">Prop</emphasis>, <emphasis role="bold">CProp</emphasis>, <emphasis role="bold">Type</emphasis>, &id;, &uri; and user provided notation; however, <emphasis role="bold">%</emphasis> is now an additional production for &sterm;〉〉</emphasis></entry>
610 <para>A <emphasis>path</emphasis> locates zero or more subterms of a given term by mimicking the term structure up to:</para>
612 <listitem><para>Occurrences of the subterms to locate that are
613 represented by <emphasis role="bold">%</emphasis>.</para></listitem>
614 <listitem><para>Subterms without any occurrence of subterms to locate
615 that can be represented by <emphasis role="bold">?</emphasis>.
618 <para>Warning: the format for a path for a <emphasis role="bold">match</emphasis> … <emphasis role="bold">with</emphasis>
619 expression is restricted to: <emphasis role="bold">match</emphasis> &path;
620 <emphasis role="bold">with</emphasis>
621 <emphasis role="bold">[</emphasis>
622 <emphasis role="bold">_</emphasis>
623 <emphasis role="bold">⇒</emphasis>
625 <emphasis role="bold">|</emphasis> …
626 <emphasis role="bold">|</emphasis>
627 <emphasis role="bold">_</emphasis>
628 <emphasis role="bold">⇒</emphasis>
630 <emphasis role="bold">]</emphasis>
631 Its semantics is the following: the n-th
632 "<emphasis role="bold">_</emphasis>
633 <emphasis role="bold">⇒</emphasis>
634 &path;" branch is matched against the n-th constructor of the
635 inductive data type. The head λ-abstractions of &path; are matched
636 against the corresponding constructor arguments.
638 <para>For instance, the path
639 <userinput>∀_,_:?.(? ? % ?)→(? ? ? %)</userinput>
640 locates at once the subterms
641 <userinput>x+y</userinput> and <userinput>x*y</userinput> in the
642 term <userinput>∀x,y:nat.x+y=1→0=x*y</userinput>
643 (where the notation <userinput>A=B</userinput> hides the term
644 <userinput>(eq T A B)</userinput> for some type <userinput>T</userinput>).
646 <para>A <emphasis>simple pattern</emphasis> extends paths to locate
647 subterms in a whole sequent. In particular, the pattern
648 <userinput>in H: p K: q ⊢ r</userinput> locates at once all the subterms
649 located by the pattern <userinput>r</userinput> in the conclusion of the
650 sequent and by the patterns <userinput>p</userinput> and
651 <userinput>q</userinput> in the hypotheses <userinput>H</userinput>
652 and <userinput>K</userinput> of the sequent.
654 <para>If no list of hypotheses is provided in a simple pattern, no subterm
655 is selected in the hypothesis. If the <userinput>⊢ p</userinput>
656 part of the pattern is not provided, no subterm will be matched in the
657 conclusion if at least one hypothesis is provided; otherwise the whole
658 conclusion is selected.
660 <para>Finally, a <emphasis>full pattern</emphasis> is interpreted in three
661 steps. In the first step the <userinput>match T in</userinput>
662 part is ignored and a set <emphasis>S</emphasis> of subterms is
663 located as for the case of
664 simple patterns. In the second step the term <userinput>T</userinput>
665 is parsed and interpreted in the context of each subterm
666 <emphasis>s ∈ S</emphasis>. In the last term for each
667 <emphasis>s ∈ S</emphasis> the interpreted term <userinput>T</userinput>
668 computed in the previous step is looked for. The final set of subterms
669 located by the full pattern is the set of occurrences of
670 the interpreted <userinput>T</userinput> in the subterms <emphasis>s</emphasis>.
672 <para>A full pattern can always be replaced by a simple pattern,
673 often at the cost of increased verbosity or decreased readability.</para>
674 <para>Example: the pattern
675 <userinput>⊢ in match x+y in ∀_,_:?.(? ? % ?)</userinput>
676 locates only the first occurrence of <userinput>x+y</userinput>
677 in the sequent <userinput>x,y: nat ⊢ ∀z,w:nat. (x+y) * (z+w) =
678 z * (x+y) + w * (x+y)</userinput>. The corresponding simple pattern
679 is <userinput>⊢ ∀_,_:?.(? ? (? % ?) ?)</userinput>.
681 <para>Every tactic that acts on subterms of the selected sequents have
682 a pattern argument for uniformity. To automatically generate a simple
685 <listitem><para>Select in the current goal the subterms to pass to the
686 tactic by using the mouse. In order to perform a multiple selection of
687 subterms, hold the Ctrl key while selecting every subterm after the
688 first one.</para></listitem>
689 <listitem><para>From the contextual menu select "Copy".</para></listitem>
690 <listitem><para>From the "Edit" or the contextual menu select
691 "Paste as pattern"</para></listitem>
695 <sect2 id="reduction-kind">
696 <title>reduction-kind</title>
697 <para>Reduction kinds are normalization functions that transform a term
698 to a convertible but simpler one. Each reduction kind can be used both
699 as a tactic argument and as a stand-alone tactic.</para>
700 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
701 <title>reduction-kind</title>
705 <entry id="grammar.reduction-kind">&reduction-kind;</entry>
707 <entry><emphasis role="bold">normalize</emphasis></entry>
708 <entry>Computes the βδιζ-normal form</entry>
713 <entry><emphasis role="bold">simplify</emphasis></entry>
714 <entry>Computes a form supposed to be simpler</entry>
719 <entry><emphasis role="bold">unfold</emphasis> [&sterm;]</entry>
720 <entry>δ-reduces the constant or variable if specified, or that
721 in head position</entry>
726 <entry><emphasis role="bold">whd</emphasis></entry>
727 <entry>Computes the βδιζ-weak-head normal form</entry>
734 <sect2 id="auto-params">
735 <title>auto-params</title>
736 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
737 <title>auto-params</title>
741 <entry id="grammar.autoparams">&autoparams;</entry>
743 <entry>[&simpleautoparam;]…
744 [<emphasis role="bold">by</emphasis>
751 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
752 <title>simple-auto-param</title>
756 <entry id="grammar.simpleautoparam">&simpleautoparam;</entry>
758 <entry><emphasis role="bold">depth=&nat;</emphasis></entry>
759 <entry>Give a bound to the depth of the search tree</entry>
764 <entry><emphasis role="bold">width=&nat;</emphasis></entry>
765 <entry>The maximal width of the search tree</entry>
770 <entry><emphasis role="bold">library</emphasis></entry>
771 <entry>Search everywhere (not only in included files)</entry>
776 <entry><emphasis role="bold">type</emphasis></entry>
777 <entry>Try to close also goals of sort Type, otherwise only goals
778 living in sort Prop are attacked.
784 <entry><emphasis role="bold">paramodulation</emphasis></entry>
785 <entry>Try to close the goal performing unit-equality paramodulation
791 <entry><emphasis role="bold">timeout=&nat;</emphasis></entry>
792 <entry>Timeout in seconds
800 <sect2 id="justification">
801 <title>justification</title>
802 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
803 <title>justification</title>
807 <entry id="grammar.justification">&justification;</entry>
809 <entry><emphasis role="bold">using</emphasis> &term;</entry>
810 <entry>Proof term manually provided</entry>
815 <entry>&autoparams;</entry>
816 <entry>Call automation</entry>