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">
34 <entry id="grammar.id">&id;</entry>
36 <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>
41 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
46 <entry id="grammar.nat">&nat;</entry>
48 <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>
53 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
58 <entry id="grammar.char">&char;</entry>
60 <entry>[<emphasis role="bold">a</emphasis>-<emphasis role="bold">zA</emphasis>-<emphasis role="bold">Z0</emphasis>-<emphasis role="bold">9_-</emphasis>]</entry>
65 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
66 <title>uri-step</title>
70 <entry id="grammar.uri-step">&uri-step;</entry>
72 <entry>&char;[&char;]…</entry>
77 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
82 <entry id="grammar.uri">&uri;</entry>
84 <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>
93 <!-- ZACK: Sample EBNF snippet, see:
94 http://www.docbook.org/tdg/en/html/productionset.html -->
98 <production id="grammar.term">
101 <lineannotation></lineannotation>
107 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
112 <entry id="grammar.term">&term;</entry>
114 <entry>&sterm;</entry>
115 <entry>simple or delimited term</entry>
120 <entry>&term; &term;</entry>
121 <entry>application</entry>
126 <entry><emphasis role="bold">λ</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
127 <entry>λ-abstraction</entry>
132 <entry><emphasis role="bold">Π</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
133 <entry>dependent product meant to define a datatype</entry>
138 <entry><emphasis role="bold">∀</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
139 <entry>dependent product meant to define a proposition</entry>
144 <entry>&term; <emphasis role="bold">→</emphasis> &term;</entry>
145 <entry>non-dependent product (logical implication or function space)</entry>
150 <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>
151 <entry>local definition</entry>
157 <emphasis role="bold">let</emphasis>
158 [<emphasis role="bold">co</emphasis>]<emphasis role="bold">rec</emphasis>
161 <entry>(co)recursive definitions</entry>
167 [<emphasis role="bold">and</emphasis> &rec_def;]…
175 <emphasis role="bold">in</emphasis> &term;
183 <entry>user provided notation</entry>
186 <entry id="grammar.rec_def">&rec_def;</entry>
189 &id; [&id;|<emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&term;]… <emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis>]…
197 [<emphasis role="bold">on</emphasis> &nat;]
198 [<emphasis role="bold">:</emphasis> &term;]
199 <emphasis role="bold">≝</emphasis> &term;]
207 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
208 <title>Simple terms</title>
212 <entry id="grammar.sterm">&sterm;</entry>
214 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
220 <entry>&id;[<emphasis role="bold">\subst[</emphasis>
221 &id;<emphasis role="bold">≔</emphasis>&term;
222 [<emphasis role="bold">;</emphasis>&id;<emphasis role="bold">≔</emphasis>&term;]…
223 <emphasis role="bold">]</emphasis>]
225 <entry>identifier with optional explicit named substitution</entry>
231 <entry>a qualified reference</entry>
236 <entry><emphasis role="bold">Prop</emphasis></entry>
237 <entry>the impredicative sort of propositions</entry>
242 <entry><emphasis role="bold">Set</emphasis></entry>
243 <entry>the impredicate sort of datatypes</entry>
248 <entry><emphasis role="bold">CProp</emphasis></entry>
249 <entry>one fixed predicative sort of constructive propositions</entry>
254 <entry><emphasis role="bold">Type</emphasis></entry>
255 <entry>one predicative sort of datatypes</entry>
260 <entry><emphasis role="bold">?</emphasis></entry>
261 <entry>implicit argument</entry>
266 <entry><emphasis role="bold">?n</emphasis>
267 [<emphasis role="bold">[</emphasis>
268 [<emphasis role="bold">_</emphasis>|&term;]…
269 <emphasis role="bold">]</emphasis>]</entry>
270 <entry>metavariable</entry>
275 <entry><emphasis role="bold">match</emphasis> &term;
276 [ <emphasis role="bold">in</emphasis> &term; ]
277 [ <emphasis role="bold">return</emphasis> &term; ]
278 <emphasis role="bold">with</emphasis>
280 <entry>case analysis</entry>
286 <emphasis role="bold">[</emphasis>
287 &match_branch;[<emphasis role="bold">|</emphasis>&match_branch;]…
288 <emphasis role="bold">]</emphasis>
295 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
302 <entry>user provided notation at precedence 90</entry>
308 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
309 <title>Arguments</title>
313 <entry id="grammar.args">&args;</entry>
316 <emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]
318 <entry>ignored argument</entry>
324 <emphasis role="bold">(</emphasis><emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis>
326 <entry>ignored argument</entry>
331 <entry>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]</entry>
337 <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis></entry>
341 <entry id="grammar.args2">&args2;</entry>
349 <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…<emphasis role="bold">:</emphasis> &term;<emphasis role="bold">)</emphasis></entry>
356 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
357 <title>Pattern matching</title>
361 <entry id="grammar.match_branch">&match_branch;</entry>
363 <entry>&match_pattern; <emphasis role="bold">⇒</emphasis> &term;</entry>
367 <entry id="grammar.match_pattern">&match_pattern;</entry>
370 <entry>0-ary constructor</entry>
375 <entry><emphasis role="bold">(</emphasis>&id; &id; [&id;]…<emphasis role="bold">)</emphasis></entry>
376 <entry>n-ary constructor (binds the n arguments)</entry>
386 <sect1 id="axiom_definition_declaration">
387 <title>Definitions and declarations</title>
389 <title><emphasis role="bold">axiom</emphasis> &id;<emphasis role="bold">:</emphasis> &term;</title>
390 <titleabbrev>axiom</titleabbrev>
391 <para><userinput>axiom H: P</userinput></para>
392 <para><command>H</command> is declared as an axiom that states <command>P</command></para>
394 <sect2 id="definition">
395 <title><emphasis role="bold">definition</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
396 <titleabbrev>definition</titleabbrev>
397 <para><userinput>definition f: T ≝ t</userinput></para>
398 <para><command>f</command> is defined as <command>t</command>;
399 <command>T</command> is its type. An error is raised if the type of
400 <command>t</command> is not convertible to <command>T</command>.</para>
401 <para><command>T</command> is inferred from <command>t</command> if
403 <para><command>t</command> can be omitted only if <command>T</command> is
404 given. In this case Matita enters in interactive mode and
405 <command>f</command> must be defined by means of tactics.</para>
406 <para>Notice that the command is equivalent to <command>theorem f: T ≝ t</command>.</para>
408 <sect2 id="inductive">
409 <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;]…
410 [<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;]…]…
412 <titleabbrev>(co)inductive types declaration</titleabbrev>
413 <para><userinput>inductive i x y z: S ≝ k1:T1 | … | kn:Tn with i' : S' ≝ k1':T1' | … | km':Tm'</userinput></para>
414 <para>Declares a family of two mutually inductive types
415 <command>i</command> and <command>i'</command> whose types are
416 <command>S</command> and <command>S'</command>, which must be convertible
418 <para>The constructors <command>ki</command> of type <command>Ti</command>
419 and <command>ki'</command> of type <command>Ti'</command> are also
420 simultaneously declared. The declared types <command>i</command> and
421 <command>i'</command> may occur in the types of the constructors, but
422 only in strongly positive positions according to the rules of the
424 <para>The whole family is parameterized over the arguments <command>x,y,z</command>.</para>
425 <para>If the keyword <command>coinductive</command> is used, the declared
426 types are considered mutually coinductive.</para>
427 <para>Elimination principles for the record are automatically generated
428 by Matita, if allowed by the typing rules of the calculus according to
429 the sort <command>S</command>. If generated,
430 they are named <command>i_ind</command>, <command>i_rec</command> and
431 <command>i_rect</command> according to the sort of their induction
435 <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>
436 <titleabbrev>record</titleabbrev>
437 <para><userinput>record id x y z: S ≝ { f1: T1; …; fn:Tn }</userinput></para>
438 <para>Declares a new record family <command>id</command> parameterized over
439 <command>x,y,z</command>.</para>
440 <para><command>S</command> is the type of the record
441 and it must be convertible to a sort.</para>
442 <para>Each field <command>fi</command> is declared by giving its type
443 <command>Ti</command>. A record without any field is admitted.</para>
444 <para>Elimination principles for the record are automatically generated
445 by Matita, if allowed by the typing rules of the calculus according to
446 the sort <command>S</command>. If generated,
447 they are named <command>i_ind</command>, <command>i_rec</command> and
448 <command>i_rect</command> according to the sort of their induction
450 <para>For each field <command>fi</command> a record projection
451 <command>fi</command> is also automatically generated if projection
452 is allowed by the typing rules of the calculus according to the
453 sort <command>S</command>, the type <command>T1</command> and
454 the definability of depending record projections.</para>
455 <para>If the type of a field is declared with <command>:></command>,
456 the corresponding record projection becomes an implicit coercion.
457 This is just syntactic sugar and it has the same effect of declaring the
458 record projection as a coercion later on.</para>
463 <title>Proofs</title>
465 <title><emphasis role="bold">theorem</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
466 <titleabbrev>theorem</titleabbrev>
467 <para><userinput>theorem f: P ≝ p</userinput></para>
468 <para>Proves a new theorem <command>f</command> whose thesis is
469 <command>P</command>.</para>
470 <para>If <command>p</command> is provided, it must be a proof term for
471 <command>P</command>. Otherwise an interactive proof is started.</para>
472 <para><command>P</command> can be omitted only if the proof is not
474 <para>Proving a theorem already proved in the library is an error.
475 To provide an alternative name and proof for the same theorem, use
476 <command>variant f: P ≝ p</command>.</para>
477 <para>A warning is raised if the name of the theorem cannot be obtained
478 by mangling the name of the constants in its thesis.</para>
479 <para>Notice that the command is equivalent to <command>definition f: T ≝ t</command>.</para>
482 <title><emphasis role="bold">variant</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
483 <titleabbrev>variant</titleabbrev>
484 <para><userinput>variant f: T ≝ t</userinput></para>
485 <para>Same as <command>theorem f: T ≝ t</command>, but it does not
486 complain if the theorem has already been proved. To be used to give
487 an alternative name or proof to a theorem.</para>
490 <title><emphasis role="bold">lemma</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
491 <titleabbrev>lemma</titleabbrev>
492 <para><userinput>lemma f: T ≝ t</userinput></para>
493 <para>Same as <command>theorem f: T ≝ t</command></para>
496 <title><emphasis role="bold">fact</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
497 <titleabbrev>fact</titleabbrev>
498 <para><userinput>fact f: T ≝ t</userinput></para>
499 <para>Same as <command>theorem f: T ≝ t</command></para>
502 <title><emphasis role="bold">remark</emphasis> &id;[<emphasis role="bold">:</emphasis> &term;] [<emphasis role="bold">≝</emphasis> &term;]</title>
503 <titleabbrev>remark</titleabbrev>
504 <para><userinput>remark f: T ≝ t</userinput></para>
505 <para>Same as <command>theorem f: T ≝ t</command></para>
509 <sect1 id="tacticargs">
510 <title>Tactic arguments</title>
511 <para>This section documents the syntax of some recurring arguments for
514 <sect2 id="introsspec">
515 <title>intros-spec</title>
516 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
517 <title>intros-spec</title>
521 <entry id="grammar.intros-spec">&intros-spec;</entry>
523 <entry>[&nat;] [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]</entry>
528 <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>
532 <title>pattern</title>
533 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
534 <title>pattern</title>
538 <entry id="grammar.pattern">&pattern;</entry>
540 <entry><emphasis role="bold">in</emphasis>
541 [&id;[<emphasis role="bold">:</emphasis> &path;]]…
542 [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
543 <entry>simple pattern</entry>
548 <entry><emphasis role="bold">in match</emphasis> &term;
549 [<emphasis role="bold">in</emphasis>
550 [&id;[<emphasis role="bold">:</emphasis> &path;]]…
551 [<emphasis role="bold">⊢</emphasis> &path;]]</entry>
552 <entry>full pattern</entry>
557 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
562 <entry id="grammar.path">&path;</entry>
564 <entry><emphasis>〈〈any &sterm; whithout 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>
569 <para>A <emphasis>path</emphasis> locates zero or more subterms of a given term by mimicking the term structure up to:</para>
571 <listitem><para>Occurrences of the subterms to locate that are
572 represented by <emphasis role="bold">%</emphasis>.</para></listitem>
573 <listitem><para>Subterms without any occurrence of subterms to locate
574 that can be represented by <emphasis role="bold">?</emphasis>.
577 <para>For instance, the path
578 <userinput>∀_,_:?.(? ? % ?)→(? ? ? %)</userinput>
579 locates at once the subterms
580 <userinput>x+y</userinput> and <userinput>x*y</userinput> in the
581 term <userinput>∀x,y:nat.x+y=1→0=x*y</userinput>
582 (where the notation <userinput>A=B</userinput> hides the term
583 <userinput>(eq T A B)</userinput> for some type <userinput>T</userinput>).
585 <para>A <emphasis>simple pattern</emphasis> extends paths to locate
586 subterms in a whole sequent. In particular, the pattern
587 <userinput>in H: p K: q ⊢ r</userinput> locates at once all the subterms
588 located by the pattern <userinput>r</userinput> in the conclusion of the
589 sequent and by the patterns <userinput>p</userinput> and
590 <userinput>q</userinput> in the hypotheses <userinput>H</userinput>
591 and <userinput>K</userinput> of the sequent.
593 <para>If no list of hypotheses is provided in a simple pattern, no subterm
594 is selected in the hypothesis. If the <userinput>⊢ p</userinput>
595 part of the pattern is not provided, no subterm will be matched in the
596 conclusion if at least one hypothesis is provided; otherwise the whole
597 conclusion is selected.
599 <para>Finally, a <emphasis>full pattern</emphasis> is interpreted in three
600 steps. In the first step the <userinput>match T in</userinput>
601 part is ignored and a set <emphasis>S</emphasis> of subterms is
602 located as for the case of
603 simple patterns. In the second step the term <userinput>T</userinput>
604 is parsed and interpreted in the context of each subterm
605 <emphasis>s ∈ S</emphasis>. In the last term for each
606 <emphasis>s ∈ S</emphasis> the interpreted term <userinput>T</userinput>
607 computed in the previous step is looked for. The final set of subterms
608 located by the full pattern is the set of occurrences of
609 the interpreted <userinput>T</userinput> in the subterms <emphasis>s</emphasis>.
611 <para>A full pattern can always be replaced by a simple pattern,
612 often at the cost of increased verbosity or decreased readability.</para>
613 <para>Example: the pattern
614 <userinput>⊢ in match x+y in ∀_,_:?.(? ? % ?)</userinput>
615 locates only the first occurrence of <userinput>x+y</userinput>
616 in the sequent <userinput>x,y: nat ⊢ ∀z,w:nat. (x+y) * (z+w) =
617 z * (x+y) + w * (x+y)</userinput>. The corresponding simple pattern
618 is <userinput>⊢ ∀_,_:?.(? ? (? % ?) ?)</userinput>.
620 <para>Every tactic that acts on subterms of the selected sequents have
621 a pattern argument for uniformity. To automatically generate a simple
624 <listitem><para>Select in the current goal the subterms to pass to the
625 tactic by using the mouse. In order to perform a multiple selection of
626 subterms, hold the Ctrl key while selecting every subterm after the
627 first one.</para></listitem>
628 <listitem><para>From the contextual menu select "Copy".</para></listitem>
629 <listitem><para>From the "Edit" or the contextual menu select
630 "Paste as pattern"</para></listitem>
634 <sect2 id="reduction-kind">
635 <title>reduction-kind</title>
636 <para>Reduction kinds are normalization functions that transform a term
637 to a convertible but simpler one. Each reduction kind can be used both
638 as a tactic argument and as a stand-alone tactic.</para>
639 <table frame="topbot" rowsep="0" colsep="0" role="grammar">
640 <title>reduction-kind</title>
644 <entry id="grammar.reduction-kind">&reduction-kind;</entry>
646 <entry><emphasis role="bold">demodulate</emphasis></entry>
651 <entry><emphasis role="bold">normalize</emphasis></entry>
652 <entry>Computes the βδιζ-normal form</entry>
657 <entry><emphasis role="bold">reduce</emphasis></entry>
658 <entry>Computes the βδιζ-normal form</entry>
663 <entry><emphasis role="bold">simplify</emphasis></entry>
664 <entry>Computes a form supposed to be simpler</entry>
669 <entry><emphasis role="bold">unfold</emphasis> [&sterm;]</entry>
670 <entry>δ-reduces the constant or variable if specified, or that
671 in head position</entry>
676 <entry><emphasis role="bold">whd</emphasis></entry>
677 <entry>Computes the βδιζ-weak-head normal form</entry>