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>Non terminal symbols are emphasized and have a link to their definition. E.g.: &term;</listitem>
9 <listitem>Terminal symbols are in bold. E.g.: <emphasis role="bold">theorem</emphasis></listitem>
10 <listitem>Optional sequences of elements are put in square brackets.
11 E.g.: [<emphasis role="bold">in</emphasis> &term;]</listitem>
12 <listitem>Alternatives are put in square brakets and they are separated
13 by vertical bars. E.g.: [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>]</listitem>
14 <listitem>Repetition of sequences of elements are given by putting the
15 first sequence in square brackets, that are followed by three dots.
16 E.g.: [<emphasis role="bold">and</emphasis> &term;]…</listitem>
18 <sect1 id="terms_and_co">
19 <title>Terms & co.</title>
21 <title>Lexical conventions</title>
27 <entry id="id">&id;</entry>
29 <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
39 <entry id="nat">&nat;</entry>
41 <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
51 <entry id="uri">&uri;</entry>
53 <entry><emphasis>〈〈&TODO;〉〉</emphasis></entry>
66 <entry id="term">&term;</entry>
68 <entry>&sterm;</entry>
69 <entry>simple or delimited term</entry>
74 <entry>&term; &term;</entry>
75 <entry>application</entry>
80 <entry><emphasis role="bold">λ</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
81 <entry>λ-abstraction</entry>
86 <entry><emphasis role="bold">Π</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
87 <entry>dependent product meant to define a datatype</entry>
92 <entry><emphasis role="bold">∀</emphasis>&args;<emphasis role="bold">.</emphasis>&term;</entry>
93 <entry>dependent product meant to define a proposition</entry>
98 <entry>&term; <emphasis role="bold">→</emphasis> &term;</entry>
99 <entry>non-dependent product (logical implication or function space)</entry>
104 <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>
105 <entry>local definition</entry>
110 <entry><emphasis role="bold">let</emphasis>
111 [<emphasis role="bold">co</emphasis>]<emphasis role="bold">rec</emphasis>
112 &id; [&id;|<emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&term;]… <emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis>]… [<emphasis role="bold">on</emphasis> &nat;]
113 [<emphasis role="bold">:</emphasis> &term;]
114 <emphasis role="bold">≝</emphasis> &term;
116 <entry>(co)recursive definitions</entry>
122 [<emphasis role="bold">and</emphasis>
123 [&id;|<emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&term;]… <emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis>]… [<emphasis role="bold">on</emphasis> &nat;]
124 [<emphasis role="bold">:</emphasis> &term;]
125 <emphasis role="bold">≝</emphasis> &term;]…
133 <emphasis role="bold">in</emphasis> &term;
141 <entry>user provided notation</entry>
152 <entry id="sterm">&sterm;</entry>
154 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
160 <entry>&id;[<emphasis role="bold">\subst[</emphasis>
161 &id;<emphasis role="bold">≔</emphasis>&term;
162 [<emphasis role="bold">;</emphasis>&id;<emphasis role="bold">≔</emphasis>&term;]…
163 <emphasis role="bold">]</emphasis>]
165 <entry>identifier with optional explicit named substitution</entry>
171 <entry>a qualified reference</entry>
176 <entry><emphasis role="bold">Prop</emphasis></entry>
177 <entry>the impredicative sort of propositions</entry>
182 <entry><emphasis role="bold">Set</emphasis></entry>
183 <entry>the impredicate sort of datatypes</entry>
188 <entry><emphasis role="bold">Type</emphasis></entry>
189 <entry>one predicative sort of datatypes</entry>
194 <entry><emphasis role="bold">?</emphasis></entry>
195 <entry>implicit argument</entry>
200 <entry><emphasis role="bold">?n</emphasis>
201 [<emphasis role="bold">[</emphasis>
202 [<emphasis role="bold">_</emphasis>|&term;]…
203 <emphasis role="bold">]</emphasis>]</entry>
204 <entry>metavariable</entry>
209 <entry><emphasis role="bold">match</emphasis> &term;
210 [ <emphasis role="bold">in</emphasis> &term; ]
211 [ <emphasis role="bold">return</emphasis> &term; ]
212 <emphasis role="bold">with</emphasis>
214 <entry>case analysis</entry>
220 <emphasis role="bold">[</emphasis>
221 &match_pattern; <emphasis role="bold"> ⇒ </emphasis> &term;
223 <emphasis role="bold">|</emphasis>
224 &match_pattern; <emphasis role="bold"> ⇒ </emphasis> &term;
225 ]…<emphasis role="bold">]</emphasis> </entry>
231 <entry><emphasis role="bold">(</emphasis>&term;<emphasis role="bold">:</emphasis>&term;<emphasis role="bold">)</emphasis></entry>
238 <entry>user provided notation at precedence 90</entry>
249 <entry id="args">&args;</entry>
252 <emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]
254 <entry>ignored argument</entry>
260 <emphasis role="bold">(</emphasis><emphasis role="bold">_</emphasis>[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis>
262 <entry>ignored argument</entry>
267 <entry>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]</entry>
273 <entry><emphasis role="bold">(</emphasis>&id;[<emphasis role="bold">,</emphasis>&id;]…[<emphasis role="bold">:</emphasis> &term;]<emphasis role="bold">)</emphasis></entry>
285 <entry id="match_pattern">&match_pattern;</entry>
288 <entry>0-ary constructor</entry>
293 <entry><emphasis role="bold">(</emphasis>&id; &id; [&id;]…<emphasis role="bold">)</emphasis></entry>
294 <entry>n-ary constructor (binds the n arguments)</entry>
303 <sect1 id="axiom_definition_declaration">
304 <title>Definitions and declarations</title>
306 <title>axiom &id;: &term;</title>
307 <titleabbrev>axiom</titleabbrev>
308 <para><userinput>axiom H: P</userinput></para>
309 <para><command>H</command> is declared as an axiom that states <command>P</command></para>
311 <sect2 id="definition">
312 <title>definition &id;[: &term;] [≝ &term;]</title>
313 <titleabbrev>definition</titleabbrev>
314 <para><userinput>definition f: T ≝ t</userinput></para>
315 <para><command>f</command> is defined as <command>t</command>;
316 <command>T</command> is its type. An error is raised if the type of
317 <command>t</command> is not convertible to <command>T</command>.</para>
318 <para><command>T</command> is inferred from <command>t</command> if
320 <para><command>t</command> can be omitted only if <command>T</command> is
321 given. In this case Matita enters in interactive mode and
322 <command>f</command> must be defined by means of tactics.</para>
323 <para>Notice that the command is equivalent to <command>theorem f: T ≝ t</command>.</para>
325 <sect2 id="inductive">
326 <title>[co]inductive &id; (of inductive types)</title>
327 <titleabbrev>(co)inductive types declaration</titleabbrev>
328 <para> &TODO; </para>
333 <title>Proofs</title>
335 <title>theorem &id;[: &term;] [≝ &term;]</title>
336 <titleabbrev>theorem</titleabbrev>
337 <para><userinput>theorem f: P ≝ p</userinput></para>
338 <para>Proves a new theorem <command>f</command> whose thesis is
339 <command>P</command>.</para>
340 <para>If <command>p</command> is provided, it must be a proof term for
341 <command>P</command>. Otherwise an interactive proof is started.</para>
342 <para><command>P</command> can be omitted only if the proof is not
344 <para>Proving a theorem already proved in the library is an error.
345 To provide an alternative name and proof for the same theorem, use
346 <command>variant f: P ≝ p</command>.</para>
347 <para>A warning is raised if the name of the theorem cannot be obtained
348 by mangling the name of the constants in its thesis.</para>
349 <para>Notice that the command is equivalent to <command>definition f: T ≝ t</command>.</para>
352 <title>variant &id;[: &term;] [≝ &term;]</title>
353 <titleabbrev>variant</titleabbrev>
354 <para><userinput>variant f: T ≝ t</userinput></para>
355 <para>Same as <command>theorem f: T ≝ t</command>, but it does not
356 complain if the theorem has already been proved. To be used to give
357 an alternative name or proof to a theorem.</para>
360 <title>lemma &id;[: &term;] [≝ &term;]</title>
361 <titleabbrev>lemma</titleabbrev>
362 <para><userinput>lemma f: T ≝ t</userinput></para>
363 <para>Same as <command>theorem f: T ≝ t</command></para>
366 <title>fact &id;[: &term;] [≝ &term;]</title>
367 <titleabbrev>fact</titleabbrev>
368 <para><userinput>fact f: T ≝ t</userinput></para>
369 <para>Same as <command>theorem f: T ≝ t</command></para>
372 <title>remark &id;[: &term;] [≝ &term;]</title>
373 <titleabbrev>remark</titleabbrev>
374 <para><userinput>remark f: T ≝ t</userinput></para>
375 <para>Same as <command>theorem f: T ≝ t</command></para>