- </sect2>
- <sect2 id="tac_generalize">
- <title>generalize <pattern> [as <id>]</title>
- <para>The tactic <command>generalize</command> </para>
- </sect2>
- <sect2 id="tac_id">
- <title>id</title>
- <para>The tactic <command>id</command> </para>
- </sect2>
- <sect2 id="tac_injection">
- <title>injection <term></title>
- <para>The tactic <command>injection</command> </para>
- </sect2>
- <sect2 id="tac_intro">
- <title>intro [<ident>]</title>
- <para>The tactic <command>intro</command> </para>
- </sect2>
- <sect2 id="tac_intros">
- <title>intros <intros_spec></title>
- <para>The tactic <command>intros</command> </para>
- </sect2>
- <sect2 id="tac_inversion">
- <title>intros <term></title>
- <para>The tactic <command>intros</command> </para>
- </sect2>
- <sect2 id="tac_lapply">
- <title>lapply [depth=<int>] <term> [to <term list] [using <ident>]</title>
- <para>The tactic <command>lapply</command> </para>
- </sect2>
- <sect2 id="tac_left">
- <title>left</title>
- <para>The tactic <command>left</command> </para>
- </sect2>
- <sect2 id="tac_letin">
- <title>letin <ident> ≝ <term></title>
- <para>The tactic <command>letin</command> </para>
- </sect2>
- <sect2 id="tac_normalize">
- <title>normalize <pattern></title>
- <para>The tactic <command>normalize</command> </para>
- </sect2>
- <sect2 id="tac_paramodulation">
- <title>paramodulation <pattern></title>
- <para>The tactic <command>paramodulation</command> </para>
- </sect2>
- <sect2 id="tac_reduce">
- <title>reduce <pattern></title>
- <para>The tactic <command>reduce</command> </para>
- </sect2>
- <sect2 id="tac_reflexivity">
- <title>reflexivity</title>
- <para>The tactic <command>reflexivity</command> </para>
- </sect2>
- <sect2 id="tac_replace">
- <title>replace <pattern> with <term></title>
- <para>The tactic <command>replace</command> </para>
- </sect2>
- <sect2 id="tac_rewrite">
- <title>rewrite {<|>} <term> <pattern></title>
- <para>The tactic <command>rewrite</command> </para>
- </sect2>
- <sect2 id="tac_right">
- <title>right</title>
- <para>The tactic <command>right</command> </para>
- </sect2>
- <sect2 id="tac_ring">
- <title>ring</title>
- <para>The tactic <command>ring</command> </para>
- </sect2>
- <sect2 id="tac_simplify">
- <title>simplify <pattern></title>
- <para>The tactic <command>simplify</command> </para>
- </sect2>
- <sect2 id="tac_split">
- <title>split</title>
- <para>The tactic <command>split</command> </para>
- </sect2>
- <sect2 id="tac_symmetry">
- <title>symmetry</title>
+ </sect1>
+ <sect1 id="tac_generalize">
+ <title><emphasis role="bold">generalize</emphasis> <pattern> [<emphasis role="bold">as</emphasis> &id;]</title>
+ <titleabbrev>generalize</titleabbrev>
+ <para><userinput>generalize patt as H</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>All the terms matched by <command>patt</command> must be
+ convertible and close in the context of the current sequent.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It closes the current sequent by applying a stronger
+ lemma that is proved using the new generated sequent.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens a new sequent where the current sequent conclusion
+ <command>G</command> is generalized to
+ <command>∀x.G{x/t}</command> where <command>{x/t}</command>
+ is a notation for the replacement with <command>x</command> of all
+ the occurrences of the term <command>t</command> matched by
+ <command>patt</command>. If <command>patt</command> matches no
+ subterm then <command>t</command> is defined as the
+ <command>wanted</command> part of the pattern.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_id">
+ <title><emphasis role="bold">id</emphasis></title>
+ <titleabbrev>id</titleabbrev>
+ <para><userinput>id </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>This identity tactic does nothing without failing.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_injection">
+ <title><emphasis role="bold">injection</emphasis> &sterm;</title>
+ <titleabbrev><emphasis role="bold">injection</emphasis></titleabbrev>
+ <para><userinput>injection p</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para><command>p</command> must have type <command>K t<subscript>1</subscript> ... t<subscript>n</subscript> = K t'<subscript>1</subscript> ... t'<subscript>n</subscript></command> where both argument lists are empty if
+<command>K</command> takes no arguments.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It derives new hypotheses by injectivity of
+ <command>K</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>The new sequent to prove is equal to the current sequent
+ with the additional hypotheses
+ <command>t<subscript>1</subscript>=t'<subscript>1</subscript></command> ... <command>t<subscript>n</subscript>=t'<subscript>n</subscript></command>.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_intro">
+ <title><emphasis role="bold">intro</emphasis> [&id;]</title>
+ <titleabbrev>intro</titleabbrev>
+ <para><userinput>intro H</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The conclusion of the sequent to prove must be an implication
+ or a universal quantification.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It applies the right introduction rule for implication,
+ closing the current sequent.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens a new sequent to prove adding to the hypothesis
+ the antecedent of the implication and setting the conclusion
+ to the consequent of the implicaiton. The name of the new
+ hypothesis is <command>H</command> if provided; otherwise it
+ is automatically generated.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_intros">
+ <title><emphasis role="bold">intros</emphasis> <intros_spec></title>
+ <titleabbrev>intros</titleabbrev>
+ <para><userinput>intros hyps</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>If <command>hyps</command> specifies a number of hypotheses
+ to introduce, then the conclusion of the current sequent must
+ be formed by at least that number of imbricated implications
+ or universal quantifications.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It applies several times the right introduction rule for
+ implication, closing the current sequent.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens a new sequent to prove adding a number of new
+ hypotheses equal to the number of new hypotheses requested.
+ If the user does not request a precise number of new hypotheses,
+ it adds as many hypotheses as possible.
+ The name of each new hypothesis is either popped from the
+ user provided list of names, or it is automatically generated when
+ the list is (or becomes) empty.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_inversion">
+ <title><emphasis role="bold">inversion</emphasis> &sterm;</title>
+ <titleabbrev>inversion</titleabbrev>
+ <para><userinput>inversion t</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The type of the term <command>t</command> must be an inductive
+ type or the application of an inductive type.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It proceeds by cases on <command>t</command> paying attention
+ to the constraints imposed by the actual "right arguments"
+ of the inductive type.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens one new sequent to prove for each case in the
+ definition of the type of <command>t</command>. With respect to
+ a simple elimination, each new sequent has additional hypotheses
+ that states the equalities of the "right parameters"
+ of the inductive type with terms originally present in the
+ sequent to prove.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_lapply">
+ <title><emphasis role="bold">lapply</emphasis> [<emphasis role="bold">depth=</emphasis>&nat;] &sterm; [<emphasis role="bold">to</emphasis> <term list>] [<emphasis role="bold">using</emphasis> &id;]</title>
+ <titleabbrev>lapply</titleabbrev>
+ <para><userinput>lapply ???</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>TODO.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>TODO.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>TODO.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_left">
+ <title><emphasis role="bold">left</emphasis></title>
+ <titleabbrev>left</titleabbrev>
+ <para><userinput>left </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type
+ with at least one constructor.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>Equivalent to <command>constructor 1</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens a new sequent for each premise of the first
+ constructor of the inductive type that is the conclusion of the
+ current sequent. For more details, see the <command>constructor</command> tactic.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_letin">
+ <title><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</title>
+ <titleabbrev>letin</titleabbrev>
+ <para><userinput>letin x ≝ t</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It adds to the context of the current sequent to prove a new
+ definition <command>x ≝ t</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_normalize">
+ <title><emphasis role="bold">normalize</emphasis> <pattern></title>
+ <titleabbrev>normalize</titleabbrev>
+ <para><userinput>normalize patt</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It replaces all the terms matched by <command>patt</command>
+ with their βδιζ-normal form.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_paramodulation">
+ <title><emphasis role="bold">paramodulation</emphasis> <pattern></title>
+ <titleabbrev>paramodulation</titleabbrev>
+ <para><userinput>paramodulation patt</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>TODO.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>TODO.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>TODO.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_reduce">
+ <title><emphasis role="bold">reduce</emphasis> <pattern></title>
+ <titleabbrev>reduce</titleabbrev>
+ <para><userinput>reduce patt</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It replaces all the terms matched by <command>patt</command>
+ with their βδιζ-normal form.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_reflexivity">
+ <title><emphasis role="bold">reflexivity</emphasis></title>
+ <titleabbrev>reflexivity</titleabbrev>
+ <para><userinput>reflexivity </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The conclusion of the current sequent must be
+ <command>t=t</command> for some term <command>t</command></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It closes the current sequent by reflexivity
+ of equality.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_replace">
+ <title><emphasis role="bold">replace</emphasis> <pattern> <emphasis role="bold">with</emphasis> &sterm;</title>
+ <titleabbrev>change</titleabbrev>
+ <para><userinput>change patt with t</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It replaces the subterms of the current sequent matched by
+ <command>patt</command> with the new term <command>t</command>.
+ For each subterm matched by the pattern, <command>t</command> is
+ disambiguated in the context of the subterm.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>For each matched term <command>t'</command> it opens
+ a new sequent to prove whose conclusion is
+ <command>t'=t</command>.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_rewrite">
+ <title><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>] &sterm; <pattern></title>
+ <titleabbrev>rewrite</titleabbrev>
+ <para><userinput>rewrite dir p patt</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para><command>p</command> must be the proof of an equality,
+ possibly under some hypotheses.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It looks in every term matched by <command>patt</command>
+ for all the occurrences of the
+ left hand side of the equality that <command>p</command> proves
+ (resp. the right hand side if <command>dir</command> is
+ <command><</command>). Every occurence found is replaced with
+ the opposite side of the equality.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens one new sequent for each hypothesis of the
+ equality proved by <command>p</command> that is not closed
+ by unification.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_right">
+ <title><emphasis role="bold">right</emphasis></title>
+ <titleabbrev>right</titleabbrev>
+ <para><userinput>right </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type with
+ at least two constructors.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>Equivalent to <command>constructor 2</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens a new sequent for each premise of the second
+ constructor of the inductive type that is the conclusion of the
+ current sequent. For more details, see the <command>constructor</command> tactic.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_ring">
+ <title><emphasis role="bold">ring</emphasis></title>
+ <titleabbrev>ring</titleabbrev>
+ <para><userinput>ring </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The conclusion of the current sequent must be an
+ equality over Coq's real numbers that can be proved using
+ the ring properties of the real numbers only.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It closes the current sequent veryfying the equality by
+ means of computation (i.e. this is a reflexive tactic, implemented
+ exploiting the "two level reasoning" technique).</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_simplify">
+ <title><emphasis role="bold">simplify</emphasis> <pattern></title>
+ <titleabbrev>simplify</titleabbrev>
+ <para><userinput>simplify patt</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It replaces all the terms matched by <command>patt</command>
+ with other convertible terms that are supposed to be simpler.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_split">
+ <title><emphasis role="bold">split</emphasis></title>
+ <titleabbrev>split</titleabbrev>
+ <para><userinput>split </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type with
+ at least one constructor.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>Equivalent to <command>constructor 1</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens a new sequent for each premise of the first
+ constructor of the inductive type that is the conclusion of the
+ current sequent. For more details, see the <command>constructor</command> tactic.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+ <sect1 id="tac_symmetry">
+ <title><emphasis role="bold">symmetry</emphasis></title>
+ <titleabbrev>symmetry</titleabbrev>