--- /dev/null
+<!-- ================= Tactics ========================= -->
+<chapter id="sec_declarative_tactics">
+ <title>Declarative Tactics</title>
+
+ <sect1
+ id="declarative_tactics_quickref">
+ <title>Quick reference card</title>
+ <para>
+ &declarativetacticref;
+ </para>
+ </sect1>
+
+
+ <sect1 id="tac_assume">
+ <title>assume</title>
+ <titleabbrev>assume</titleabbrev>
+ <para><userinput>assume x : t</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">assume</emphasis> &id; <emphasis role="bold"> : </emphasis> &sterm;</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>The conclusion of the current proof must be
+ <command>∀x:T.P</command> or
+ <command>T→P</command> where <command>T</command> is
+ a data type (i.e. <command>T</command> has type
+ <command>Set</command> or <command>Type</command>).</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It adds to the context of the current sequent to prove a new
+ declaration <command>x : T </command>. The new conclusion becomes
+ <command>P</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_byinduction">
+ <title>by induction hypothesis we know</title>
+ <titleabbrev>by induction hypothesis we know</titleabbrev>
+ <para><userinput>by induction hypothesis we know t (id)</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem><para><emphasis role="bold">by induction hypothesis we know</emphasis> &term; <emphasis role="bold"> (</emphasis> &id; <emphasis role="bold">)</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>To be used in a proof by induction to state the inductive
+ hypothesis.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para> Introduces the inductive hypothesis. </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_case">
+ <title>case</title>
+ <titleabbrev>case</titleabbrev>
+ <para><userinput>case id (id1:t1) … (idn:tn)</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">case</emphasis> &id; [<emphasis role="bold">(</emphasis> &id; <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">)</emphasis>] … </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>To be used in a proof by induction or by cases to start
+ a new case</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>Starts the new case <command>id</command> declaring
+ the local parameters <command>(id1:t1) … (idn:tn)</command></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_bydone">
+ <title>done</title>
+ <titleabbrev>done</titleabbrev>
+ <para><userinput>justification done</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para>&justification; <emphasis role="bold">done</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It closes the current sequent given the justification.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+
+ <sect1 id="tac_exitselim">
+ <title>let such that</title>
+ <titleabbrev>let such that</titleabbrev>
+ <para><userinput>justification let x:t such that p (id)</userinput>
+ </para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para>&justification; <emphasis role="bold">let</emphasis> &id;
+ <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">such that</emphasis> &term;
+ <emphasis role="bold">(</emphasis> &id; <emphasis role="bold">)</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It derives <command>∃x:t.p</command> using the
+ <command>justification</command> and then it introduces in the context
+ <command>x</command> and the hypothesis
+ <command>p</command> labelled with
+ <command>id</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_obtain">
+ <title>obtain</title>
+ <titleabbrev>obtain</titleabbrev>
+ <para><userinput>obtain H t1 = t2 justification</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para>[<emphasis role="bold">obtain</emphasis> &id; | <emphasis role="bold">conclude</emphasis> &term;] <emphasis role="bold">=</emphasis> &term; [&autoparams; | <emphasis role="bold">using</emphasis> &term; | <emphasis role="bold">using once</emphasis> &term; | <emphasis role="bold">proof</emphasis>] [<emphasis role="bold">done</emphasis>]</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para><command>conclude</command> can be used only if the current
+ sequent is stating an equality. The left hand side must be omitted
+ in an equality chain.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>Starts or continues an equality chain. If the chain starts
+ with <command>obtain H</command> a new subproof named
+ <command>H</command> is started.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>If the chain starts
+ with <command>obtain H</command> a nre sequent for
+ <command>t2 = ?</command> is opened.
+ </para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_suppose">
+ <title>suppose</title>
+ <titleabbrev>suppose</titleabbrev>
+ <para><userinput>suppose t1 (x) that is equivalent to t2</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">suppose</emphasis> &term; <emphasis role="bold"> (</emphasis> &id;
+ <emphasis role="bold">) </emphasis> [ <emphasis role="bold">that is equivalent to</emphasis> &term; ]</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>The conclusion of the current proof must be
+ <command>∀x:T.P</command> or
+ <command>T→P</command> where <command>T</command> is
+ a proposition (i.e. <command>T</command> has type
+ <command>Prop</command> or <command>CProp</command>).</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It adds to the context of the current sequent to prove a new
+ declaration <command>x : T </command>. The new conclusion becomes
+ <command>P</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_thesisbecomes">
+ <title>the thesis becomes</title>
+ <titleabbrev>the thesis becomes</titleabbrev>
+ <para><userinput>the thesis becomes t</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role ="bold">the thesis becomes</emphasis> &term; </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>The provided term <command>t</command> must be convertible with
+ current sequent.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It changes the current goal to the one provided.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_weneedtoprove">
+ <title>we need to prove</title>
+ <titleabbrev>we need to prove</titleabbrev>
+ <para><userinput>we need to prove t1 (id) or equivalently t2</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">we need to prove</emphasis> &term;
+ [<emphasis role="bold">(</emphasis>&id;
+ <emphasis role="bold">)</emphasis>]
+ [ <emphasis role="bold">or equivalently</emphasis> &term;]</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>If <command>id</command> is provided, starts a subproof that once concluded
+ will be named <command>id</command>. Otherwise states what needs to be proved.
+ If <command>t2</command> is provided, the new goal is
+ immediately changed to <command>t2</command> wich must
+ be equivalent to <command>t1</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>The stated one if <command>id</command> is provided</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+
+ <sect1 id="tac_andelim">
+ <title>we have</title>
+ <titleabbrev>we have</titleabbrev>
+ <para><userinput>justification we have t1 (id1) and t2 (id2)</userinput>
+ </para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic_synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para>&justification; <emphasis role="bold">we have</emphasis> &term;
+ <emphasis role="bold">( </emphasis> &id; <emphasis role="bold"> ) and </emphasis> &term;
+ <emphasis role="bold"> ( </emphasis> &id; <emphasis role="bold">)</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It derives <command>t1∧t2</command> using the
+ <command>justification</command> then it introduces in the context
+ <command>t1</command> labelled with <command>id1</command> and
+ <command>t2</command> labelled with <command>id2</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_weproceedbycases">
+ <title>we proceed by cases on</title>
+ <titleabbrev>we proceed by cases on</titleabbrev>
+ <para><userinput>we proceed by cases on t to prove th</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">we proceed by cases on</emphasis> &term; <emphasis role="bold">to prove</emphasis> &term; </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para><command>t</command> must inhabitant of an inductive type and
+ <command>th</command> must be the conclusion to be proved by
+ cases.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para> It proceeds by cases on <command>t</command> </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens one new sequent for each constructor of the
+ type of <command>t</command>.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_weproceedbyinduction">
+ <title>we proceed by induction on</title>
+ <titleabbrev>we proceed by induction on</titleabbrev>
+ <para><userinput>we proceed by induction on t to prove th</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">we proceed by induction on</emphasis> &term; <emphasis role="bold"> to prove </emphasis> &term; </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para><command>t</command> must inhabitant of an inductive type and
+ <command>th</command> must be the conclusion to be proved by induction.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It proceed by induction on <command>t</command>.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens one new sequent for each constructor of the
+ type of <command>t</command>.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+
+ <sect1 id="tac_bytermweproved">
+ <title>we proved</title>
+ <titleabbrev>we proved</titleabbrev>
+ <para><userinput>justification we proved t (id)</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para>&justification; <emphasis role="bold">we proved</emphasis> &term;
+ <emphasis role="bold">(</emphasis> &id;
+ <emphasis role="bold">)</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para><command>t</command>must have type <command>Prop</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It derives <command>t</command>
+ using the justification and labels the conclusion with
+ <command>id</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+</chapter>
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