</para>
</sect1>
- <sect1 id="tac_absurd">
- <title>absurd</title>
- <titleabbrev>absurd</titleabbrev>
- <para><userinput>absurd P</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">absurd</emphasis> &sterm;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para><command>P</command> must have type <command>Prop</command>.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It closes the current sequent by eliminating an
- absurd term.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>It opens two new sequents of conclusion <command>P</command>
- and <command>¬P</command>.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
<sect1 id="tac_apply">
- <title>apply</title>
- <titleabbrev>apply</titleabbrev>
- <para><userinput>apply t</userinput></para>
+ <title>@</title>
+ <titleabbrev>@</titleabbrev>
+ <para><userinput>@t</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">apply</emphasis> &sterm;</para>
+ <para><emphasis role="bold">@</emphasis> &sterm;</para>
</listitem>
</varlistentry>
<varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_applyS">
- <title>applyS</title>
- <titleabbrev>applyS</titleabbrev>
- <para><userinput>applyS t auto_params</userinput></para>
+ <sect1 id="tac_auto">
+ <title>//</title>
+ <titleabbrev>//</titleabbrev>
+ <para><userinput>/params/</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">applyS</emphasis> &sterm; &autoparams;</para>
+ <para><emphasis role="bold">/</emphasis>&autoparams;<emphasis role="bold">/</emphasis>. </para>
+ <!--<para><emphasis role="bold">autobatch</emphasis> &autoparams;</para>-->
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para><command>t</command> must have type
- <command>T<subscript>1</subscript> → ... →
- T<subscript>n</subscript> → G</command>.</para>
+ <para>None, but the tactic may fail finding a proof if every
+ proof is in the search space that is pruned away. Pruning is
+ controlled by the optional <command>params</command>.
+ Moreover, only lemmas whose type signature is a subset of the
+ signature of the current sequent are considered. The signature of
+ a sequent is essentially the set of constats appearing in it.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para><command>applyS</command> is useful when
- <command>apply</command> fails because the current goal
- and the conclusion of the applied theorems are extensionally
- equivalent up to instantiation of metavariables, but cannot
- be unified. E.g. the goal is <command>P(n*O+m)</command> and
- the theorem to be applied proves <command>∀m.P(m+O)</command>.
- </para>
- <para>
- It tries to automatically rewrite the current goal using
- <link linkend="tac_auto">auto paramodulation</link>
- to make it unifiable with <command>G</command>.
- Then it closes the current sequent by applying
- <command>t</command> to <command>n</command>
- implicit arguments (that become new sequents).
- The <command>auto_params</command> parameters are passed
- directly to <command>auto paramodulation</command>.
- </para>
+ <para>It closes the current sequent by repeated application of
+ rewriting steps (unless <command>paramodulation</command> is
+ omitted), hypothesis and lemmas in the library.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>It opens a new sequent for each premise
- <command>T<subscript>i</subscript></command> that is not
- instantiated by unification. <command>T<subscript>i</subscript></command> is
- the conclusion of the <command>i</command>-th new sequent to
- prove.</para>
+ <para>None</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_assumption">
- <title>assumption</title>
- <titleabbrev>assumption</titleabbrev>
- <para><userinput>assumption </userinput></para>
+ <sect1 id="tac_intro">
+ <title>#</title>
+ <titleabbrev>#</titleabbrev>
+ <para><userinput>#H</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">assumption</emphasis></para>
+ <para><emphasis role="bold">#</emphasis>&id;</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>There must exist an hypothesis whose type can be unified with
- the conclusion of the current sequent.</para>
+ <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 closes the current sequent exploiting an hypothesis.</para>
+ <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>None</para>
+ <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>.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_auto">
- <title>auto</title>
- <titleabbrev>auto</titleabbrev>
- <para><userinput>auto params</userinput></para>
+ <sect1 id="tac_intro_clear">
+ <title>#_</title>
+ <titleabbrev>#_</titleabbrev>
+ <para><userinput>#_</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">auto</emphasis> &autoparams;. </para>
- <para><emphasis role="bold">autobatch</emphasis> &autoparams;</para>
+ <para><emphasis role="bold">#_</emphasis></para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>None, but the tactic may fail finding a proof if every
- proof is in the search space that is pruned away. Pruning is
- controlled by the optional <command>params</command>.
- Moreover, only lemmas whose type signature is a subset of the
- signature of the current sequent are considered. The signature of
- a sequent is essentially the set of constats appearing in it.
- </para>
+ <para>The conclusion of the sequent to prove must be an implication.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It closes the current sequent by repeated application of
- rewriting steps (unless <command>paramodulation</command> is
- omitted), hypothesis and lemmas in the library.</para>
+ <para>It applies the ``a fortiori'' rule for implication,
+ closing the current sequent.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>None</para>
+ <para>It opens a new sequent whose conclusion is the conclusion
+ of the implication of the original sequent.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_cases">
- <title>cases</title>
- <titleabbrev>cases</titleabbrev>
- <para><userinput>
- cases t pattern hyps
- </userinput></para>
+ <sect1 id="tac_intro">
+ <title>##</title>
+ <titleabbrev>##</titleabbrev>
+ <para><userinput>##</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para>
- <emphasis role="bold">cases</emphasis>
- &term; &pattern; [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]
- </para>
+ <para><emphasis role="bold">##</emphasis></para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>
- <command>t</command> must inhabit an inductive type
- </para>
+ <para>None.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>
- It proceed by cases on <command>t</command>. The new generated
- hypothesis in each branch are named according to
- <command>hyps</command>.
- The elimintation predicate is restricted by
- <command>pattern</command>. In particular, if some hypothesis
- is listed in <command>pattern</command>, the hypothesis is
- generalized and cleared before proceeding by cases on
- <command>t</command>. Currently, we only support patterns of the
- form <command>H<subscript>1</subscript> … H<subscript>n</subscript> ⊢ %</command>. This limitation will be lifted in the future.
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>One new sequent for each constructor of the type of
- <command>t</command>. Each sequent has a new hypothesis for
- each argument of the constructor.</para>
+ <para>This macro expands to the longest possible list of
+ <command>#H<subscript>i</subscript></command> tactics. The
+ names of the introduced hypotheses are automatically
+ generated.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
<sect1 id="tac_clear">
- <title>clear</title>
- <titleabbrev>clear</titleabbrev>
- <para><userinput>
- clear H<subscript>1</subscript> ... H<subscript>m</subscript>
- </userinput></para>
+ <title>-</title>
+ <titleabbrev>-</titleabbrev>
+ <para><userinput>-H</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
<para>
- <emphasis role="bold">clear</emphasis>
- &id; [&id;…]
+ <emphasis role="bold">-</emphasis>&id;
</para>
</listitem>
</varlistentry>
<term>Pre-conditions:</term>
<listitem>
<para>
- <command>
- H<subscript>1</subscript> ... H<subscript>m</subscript>
- </command> must be hypotheses of the
+ <command>H</command> must be an hypothesis of the
current sequent to prove.
</para>
</listitem>
<term>Action:</term>
<listitem>
<para>
- It hides the hypotheses
- <command>
- H<subscript>1</subscript> ... H<subscript>m</subscript>
- </command> from the current sequent.
+ It hides the hypothesis <command>H</command>
+ from the current sequent.
</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_clearbody">
- <title>clearbody</title>
- <titleabbrev>clearbody</titleabbrev>
- <para><userinput>clearbody H</userinput></para>
+ <sect1 id="tac_constructor">
+ <title>%</title>
+ <titleabbrev>%</titleabbrev>
+ <para><userinput>%n {args}</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">clearbody</emphasis> &id;</para>
+ <para><emphasis role="bold">%</emphasis> [&nat;] [<emphasis role="bol">{</emphasis>&sterm;…<emphasis role="bol">}</emphasis>]</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para><command>H</command> must be an hypothesis of the
- current sequent to prove.</para>
+ <para>The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type with
+ at least <command>n</command> constructors.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It hides the definiens of a definition in the current
- sequent context. Thus the definition becomes an hypothesis.</para>
+ <para>It applies the <command>n</command>-th constructor of the
+ inductive type of the conclusion of the current sequent to
+ the arguments <command>args</command>.
+ If <command>n</command> is omitted, it defaults to 1.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>None.</para>
+ <para>It opens a new sequent for each premise of the constructor
+ that can not be inferred by unification. For more details,
+ see the <command>apply</command> tactic.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_compose">
- <title>compose</title>
- <titleabbrev>compose</titleabbrev>
- <para><userinput>compose n t1 with t2 hyps</userinput></para>
+ <sect1 id="tac_decompose">
+ <title>*</title>
+ <titleabbrev>*</titleabbrev>
+ <para><userinput>
+ * as H
+ </userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">compose</emphasis> [&nat;] &sterm; [<emphasis role="bold">with</emphasis> &sterm;] [&intros-spec;]</para>
+ <para>
+ <emphasis role="bold">*</emphasis>
+ [<emphasis role="bold">as</emphasis> &id;]
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para></para>
+ <para>The current conclusion must be of the form
+ <command>T → G</command> where <command>I</command> is
+ an inductive type applied to its arguments, if any.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>Composes t1 with t2 in every possible way
- <command>n</command> times introducing generated terms
- as if <command>intros hyps</command> was issued.</para>
- <para>If <command>t1:∀x:A.B[x]</command> and
- <command>t2:∀x,y:A.B[x]→B[y]→C[x,y]</command> it generates:
- <itemizedlist>
- <listitem>
- <para><command>λx,y:A.t2 x y (t1 x) : ∀x,y:A.B[y]→C[x,y]</command></para>
- </listitem>
- <listitem>
- <para><command>λx,y:A.λH:B[x].t2 x y H (t1 y) : ∀x,y:A.B[x]→C[x,y]
- </command></para>
- </listitem>
- </itemizedlist>
- </para>
- <para>If <command>t2</command> is omitted it composes
- <command>t1</command>
- with every hypothesis that can be introduced.
- <command>n</command> iterates the process.</para>
+ <para>
+ It introduces a new hypothesis <command>H</command> of type
+ <command>T</command>. Then it proceeds by cases over
+ <command>H</command>. Finally, if the name <command>H</command>
+ is not specified, it clears the new hypothesis from all contexts.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>The same, but with more hypothesis eventually introduced
- by the &intros-spec;.</para>
+ <para>
+ The ones generated by case analysis.
+ </para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_change">
- <title>change</title>
- <titleabbrev>change</titleabbrev>
- <para><userinput>change patt with t</userinput></para>
+ <sect1 id="tac_rewrite">
+ <title>></title>
+ <titleabbrev>></titleabbrev>
+ <para><userinput>> p patt</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
+ <para>[<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>] &sterm; &pattern;</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>Each subterm matched by the pattern must be convertible
- with the term <command>t</command> disambiguated in the context
- of the matched subterm.</para>
+ <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 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>
+ <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><</command> is used).
+ Every occurence found is replaced with
+ the opposite side of the equality.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>None.</para>
+ <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_constructor">
- <title>constructor</title>
- <titleabbrev>constructor</titleabbrev>
- <para><userinput>constructor n</userinput></para>
+ <sect1 id="tac_applyS">
+ <title>applyS</title>
+ <titleabbrev>applyS</titleabbrev>
+ <para><userinput>applyS t auto_params</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">constructor</emphasis> &nat;</para>
+ <para><emphasis role="bold">applyS</emphasis> &sterm; &autoparams;</para>
</listitem>
</varlistentry>
<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 <command>n</command> constructors.</para>
+ <para><command>t</command> must have type
+ <command>T<subscript>1</subscript> → ... →
+ T<subscript>n</subscript> → G</command>.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It applies the <command>n</command>-th constructor of the
- inductive type of the conclusion of the current sequent.</para>
+ <para><command>applyS</command> is useful when
+ <command>apply</command> fails because the current goal
+ and the conclusion of the applied theorems are extensionally
+ equivalent up to instantiation of metavariables, but cannot
+ be unified. E.g. the goal is <command>P(n*O+m)</command> and
+ the theorem to be applied proves <command>∀m.P(m+O)</command>.
+ </para>
+ <para>
+ It tries to automatically rewrite the current goal using
+ <link linkend="tac_auto">auto paramodulation</link>
+ to make it unifiable with <command>G</command>.
+ Then it closes the current sequent by applying
+ <command>t</command> to <command>n</command>
+ implicit arguments (that become new sequents).
+ The <command>auto_params</command> parameters are passed
+ directly to <command>auto paramodulation</command>.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>It opens a new sequent for each premise of the constructor
- that can not be inferred by unification. For more details,
- see the <command>apply</command> tactic.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_contradiction">
- <title>contradiction</title>
- <titleabbrev>contradiction</titleabbrev>
- <para><userinput>contradiction </userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">contradiction</emphasis></para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>There must be in the current context an hypothesis of type
- <command>False</command>.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It closes the current sequent by applying an hypothesis of
- type <command>False</command>.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_cut">
- <title>cut</title>
- <titleabbrev>cut</titleabbrev>
- <para><userinput>cut P as H</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">cut</emphasis> &sterm; [<emphasis role="bold">as</emphasis> &id;]</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para><command>P</command> must have type <command>Prop</command>.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It closes the current sequent.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>It opens two new sequents. The first one has an extra
- hypothesis <command>H:P</command>. If <command>H</command> is
- omitted, the name of the hypothesis is automatically generated.
- The second sequent has conclusion <command>P</command> and
- hypotheses the hypotheses of the current sequent to prove.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_decompose">
- <title>decompose</title>
- <titleabbrev>decompose</titleabbrev>
- <para><userinput>
- decompose as H<subscript>1</subscript> ... H<subscript>m</subscript>
- </userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para>
- <emphasis role="bold">decompose</emphasis>
- [<emphasis role="bold">as</emphasis> &id;…]
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>
- For each each premise <command>H</command> of type
- <command>T</command> in the current context where
- <command>T</command> is a non-recursive inductive type without
- right parameters and of sort Prop or CProp, the tactic runs
- <command>
- elim H as H<subscript>1</subscript> ... H<subscript>m</subscript>
- </command>, clears <command>H</command> and runs itself
- recursively on each new premise introduced by
- <command>elim</command> in the opened sequents.
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>
- The ones generated by all the <command>elim</command> tactics run.
- </para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_demodulate">
- <title>demodulate</title>
- <titleabbrev>demodulate</titleabbrev>
- <para><userinput>demodulate auto_params</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">demodulate</emphasis> &autoparams;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>&TODO;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_destruct">
- <title>destruct</title>
- <titleabbrev>destruct</titleabbrev>
- <para><userinput>destruct p</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para><command>p</command> must have type <command>E<subscript>1</subscript> = E<subscript>2</subscript></command> where the two sides of the equality are possibly applied constructors of an inductive type.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>The tactic recursively compare the two sides of the equality
- looking for different constructors in corresponding position.
- If two of them are found, the tactic closes the current sequent
- by proving the absurdity of <command>p</command>. Otherwise
- it adds a new hypothesis for each leaf of the formula that
- states the equality of the subformulae in the corresponding
- positions on the two sides of the equality.
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_elim">
- <title>elim</title>
- <titleabbrev>elim</titleabbrev>
- <para><userinput>elim t pattern using th hyps</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">elim</emphasis> &sterm; &pattern; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para><command>t</command> must inhabit an inductive type and
- <command>th</command> must be an elimination principle for that
- inductive type. If <command>th</command> is omitted the appropriate
- standard elimination principle is chosen.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It proceeds by cases on the values of <command>t</command>,
- according to the elimination principle <command>th</command>.
- The induction predicate is restricted by
- <command>pattern</command>. In particular, if some hypothesis
- is listed in <command>pattern</command>, the hypothesis is
- generalized and cleared before eliminating <command>t</command>
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>It opens one new sequent for each case. The names of
- the new hypotheses are picked by <command>hyps</command>, if
- provided. If hyps specifies also a number of hypotheses that
- is less than the number of new hypotheses for a new sequent,
- then the exceeding hypothesis will be kept as implications in
- the conclusion of the sequent.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_elimType">
- <title>elimType</title>
- <titleabbrev>elimType</titleabbrev>
- <para><userinput>elimType T using th hyps</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para><command>T</command> must be an inductive type.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>TODO (severely bugged now).</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>TODO</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_exact">
- <title>exact</title>
- <titleabbrev>exact</titleabbrev>
- <para><userinput>exact p</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">exact</emphasis> &sterm;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>The type of <command>p</command> must be convertible
- with the conclusion of the current sequent.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It closes the current sequent using <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_exists">
- <title>exists</title>
- <titleabbrev>exists</titleabbrev>
- <para><userinput>exists </userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">exists</emphasis></para>
- </listitem>
- </varlistentry>
- <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_fail">
- <title>fail</title>
- <titleabbrev>fail</titleabbrev>
- <para><userinput>fail</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">fail</emphasis></para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>This tactic always fail.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>N.A.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_fold">
- <title>fold</title>
- <titleabbrev>fold</titleabbrev>
- <para><userinput>fold red t patt</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>The pattern must not specify the wanted term.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>First of all it locates all the subterms matched by
- <command>patt</command>. In the context of each matched subterm
- it disambiguates the term <command>t</command> and reduces it
- to its <command>red</command> normal form; then it replaces with
- <command>t</command> every occurrence of the normal form in the
- matched subterm.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_fourier">
- <title>fourier</title>
- <titleabbrev>fourier</titleabbrev>
- <para><userinput>fourier </userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">fourier</emphasis></para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>The conclusion of the current sequent must be a linear
- inequation over real numbers taken from standard library of
- Coq. Moreover the inequations in the hypotheses must imply the
- inequation in the conclusion of the current sequent.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It closes the current sequent by applying the Fourier method.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_fwd">
- <title>fwd</title>
- <titleabbrev>fwd</titleabbrev>
- <para><userinput>fwd H as H<subscript>0</subscript> ... H<subscript>n</subscript></userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-conditions:</term>
- <listitem>
- <para>
- The type of <command>H</command> must be the premise of a
- forward simplification theorem.
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>
- This tactic is under development.
- It simplifies the current context by removing
- <command>H</command> using the following methods:
- forward application (by <command>lapply</command>) of a suitable
- simplification theorem, chosen automatically, of which the type
- of <command>H</command> is a premise,
- decomposition (by <command>decompose</command>),
- rewriting (by <command>rewrite</command>).
- <command>H<subscript>0</subscript> ... H<subscript>n</subscript></command>
- are passed to the tactics <command>fwd</command> invokes, as
- names for the premise they introduce.
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>
- The ones opened by the tactics <command>fwd</command> invokes.
- </para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
- <sect1 id="tac_generalize">
- <title>generalize</title>
- <titleabbrev>generalize</titleabbrev>
- <para><userinput>generalize patt as H</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]</para>
- </listitem>
- </varlistentry>
- <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>id</title>
- <titleabbrev>id</titleabbrev>
- <para><userinput>id </userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">id</emphasis></para>
- </listitem>
- </varlistentry>
- <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_intro">
- <title>intro</title>
- <titleabbrev>intro</titleabbrev>
- <para><userinput>intro H</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">intro</emphasis> [&id;]</para>
- </listitem>
- </varlistentry>
- <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>intros</title>
- <titleabbrev>intros</titleabbrev>
- <para><userinput>intros hyps</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">intros</emphasis> &intros-spec;</para>
- </listitem>
- </varlistentry>
- <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>
+ <para>It opens a new sequent for each premise
+ <command>T<subscript>i</subscript></command> that is not
+ instantiated by unification. <command>T<subscript>i</subscript></command> is
+ the conclusion of the <command>i</command>-th new sequent to
+ prove.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_inversion">
- <title>inversion</title>
- <titleabbrev>inversion</titleabbrev>
- <para><userinput>inversion t</userinput></para>
+ <sect1 id="tac_assumption">
+ <title>assumption</title>
+ <titleabbrev>assumption</titleabbrev>
+ <para><userinput>assumption </userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
+ <para><emphasis role="bold">assumption</emphasis></para>
</listitem>
</varlistentry>
<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>
+ <para>There must exist an hypothesis whose type can be unified with
+ the conclusion of the current sequent.</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>
+ <para>It closes the current sequent exploiting an hypothesis.</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>
+ <para>None</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_lapply">
- <title>lapply</title>
- <titleabbrev>lapply</titleabbrev>
+ <sect1 id="tac_cases">
+ <title>cases</title>
+ <titleabbrev>cases</titleabbrev>
<para><userinput>
- lapply linear depth=d t
- to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
+ cases t pattern
</userinput></para>
<para>
<variablelist>
<term>Synopsis:</term>
<listitem>
<para>
- <emphasis role="bold">lapply</emphasis>
- [<emphasis role="bold">linear</emphasis>]
- [<emphasis role="bold">depth=</emphasis>&nat;]
- &sterm;
- [<emphasis role="bold">to</emphasis>
- &sterm;
- [<emphasis role="bold">,</emphasis>&sterm;…]
- ]
- [<emphasis role="bold">as</emphasis> &id;]
+ <emphasis role="bold">cases</emphasis>
+ &term; &pattern;
</para>
</listitem>
</varlistentry>
<term>Pre-conditions:</term>
<listitem>
<para>
- <command>t</command> must have at least <command>d</command>
- independent premises and <command>n</command> must not be
- greater than <command>d</command>.
+ <command>t</command> must inhabit an inductive type
</para>
</listitem>
</varlistentry>
<term>Action:</term>
<listitem>
<para>
- Invokes <command>letin H ≝ (t ? ... ?)</command>
- with enough <command>?</command>'s to reach the
- <command>d</command>-th independent premise of
- <command>t</command>
- (<command>d</command> is maximum if unspecified).
- Then istantiates (by <command>apply</command>) with
- t<subscript>1</subscript>, ..., t<subscript>n</subscript>
- the <command>?</command>'s corresponding to the first
- <command>n</command> independent premises of
- <command>t</command>.
- Usually the other <command>?</command>'s preceding the
- <command>n</command>-th independent premise of
- <command>t</command> are istantiated as a consequence.
- If the <command>linear</command> flag is specified and if
- <command>t, t<subscript>1</subscript>, ..., t<subscript>n</subscript></command>
- are (applications of) premises in the current context, they are
- <command>clear</command>ed.
+ It proceed by cases on <command>t</command>. The new generated
+ hypothesis in each branch are named according to
+ <command>hyps</command>.
+ The elimintation predicate is restricted by
+ <command>pattern</command>. In particular, if some hypothesis
+ is listed in <command>pattern</command>, the hypothesis is
+ generalized and cleared before proceeding by cases on
+ <command>t</command>. Currently, we only support patterns of the
+ form <command>H<subscript>1</subscript> … H<subscript>n</subscript> ⊢ %</command>. This limitation will be lifted in the future.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>
- The ones opened by the tactics <command>lapply</command> invokes.
- </para>
+ <para>One new sequent for each constructor of the type of
+ <command>t</command>. Each sequent has a new hypothesis for
+ each argument of the constructor.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_left">
- <title>left</title>
- <titleabbrev>left</titleabbrev>
- <para><userinput>left </userinput></para>
+ <!--
+ <sect1 id="tac_clearbody">
+ <title>clearbody</title>
+ <titleabbrev>clearbody</titleabbrev>
+ <para><userinput>clearbody H</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">left</emphasis></para>
+ <para><emphasis role="bold">clearbody</emphasis> &id;</para>
</listitem>
</varlistentry>
<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>
+ <para><command>H</command> must be an hypothesis of the
+ current sequent to prove.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>Equivalent to <command>constructor 1</command>.</para>
+ <para>It hides the definiens of a definition in the current
+ sequent context. Thus the definition becomes an hypothesis.</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>
+ <para>None.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_letin">
- <title>letin</title>
- <titleabbrev>letin</titleabbrev>
- <para><userinput>letin x ≝ t</userinput></para>
+ -->
+ <!--
+ <sect1 id="tac_compose">
+ <title>compose</title>
+ <titleabbrev>compose</titleabbrev>
+ <para><userinput>compose n t1 with t2 hyps</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">≝</emphasis> &sterm;</para>
+ <para><emphasis role="bold">compose</emphasis> [&nat;] &sterm; [<emphasis role="bold">with</emphasis> &sterm;] [&intros-spec;]</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>None.</para>
+ <para></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>
+ <para>Composes t1 with t2 in every possible way
+ <command>n</command> times introducing generated terms
+ as if <command>intros hyps</command> was issued.</para>
+ <para>If <command>t1:∀x:A.B[x]</command> and
+ <command>t2:∀x,y:A.B[x]→B[y]→C[x,y]</command> it generates:
+ <itemizedlist>
+ <listitem>
+ <para><command>λx,y:A.t2 x y (t1 x) : ∀x,y:A.B[y]→C[x,y]</command></para>
+ </listitem>
+ <listitem>
+ <para><command>λx,y:A.λH:B[x].t2 x y H (t1 y) : ∀x,y:A.B[x]→C[x,y]
+ </command></para>
+ </listitem>
+ </itemizedlist>
+ </para>
+ <para>If <command>t2</command> is omitted it composes
+ <command>t1</command>
+ with every hypothesis that can be introduced.
+ <command>n</command> iterates the process.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>None.</para>
+ <para>The same, but with more hypothesis eventually introduced
+ by the &intros-spec;.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_normalize">
- <title>normalize</title>
- <titleabbrev>normalize</titleabbrev>
- <para><userinput>normalize patt</userinput></para>
+ -->
+ <sect1 id="tac_change">
+ <title>change</title>
+ <titleabbrev>change</titleabbrev>
+ <para><userinput>change patt with t</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">normalize</emphasis> &pattern;</para>
+ <para><emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>None.</para>
+ <para>Each subterm matched by the pattern must be convertible
+ with the term <command>t</command> disambiguated in the context
+ of the matched subterm.</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>
+ <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>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_reflexivity">
- <title>reflexivity</title>
- <titleabbrev>reflexivity</titleabbrev>
- <para><userinput>reflexivity </userinput></para>
+ <sect1 id="tac_cut">
+ <title>cut</title>
+ <titleabbrev>cut</titleabbrev>
+ <para><userinput>cut P</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">reflexivity</emphasis></para>
+ <para><emphasis role="bold">cut</emphasis> &sterm;</para>
</listitem>
</varlistentry>
<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>
+ <para><command>P</command> must be a type.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It closes the current sequent by reflexivity
- of equality.</para>
+ <para>It closes the current sequent.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>None.</para>
+ <para>It opens two new sequents. The first one has conclusion
+ <command>P → G</command> where <command>G</command> is the
+ old conclusion.
+ The second sequent has conclusion <command>P</command> and
+ hypotheses the hypotheses of the current sequent to prove.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_replace">
- <title>replace</title>
- <titleabbrev>change</titleabbrev>
- <para><userinput>change patt with t</userinput></para>
+ <!--
+ <sect1 id="tac_demodulate">
+ <title>demodulate</title>
+ <titleabbrev>demodulate</titleabbrev>
+ <para><userinput>demodulate auto_params</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;</para>
+ <para><emphasis role="bold">demodulate</emphasis> &autoparams;</para>
</listitem>
</varlistentry>
<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>
+ <para>&TODO;</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>
+ <para>None.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_rewrite">
- <title>rewrite</title>
- <titleabbrev>rewrite</titleabbrev>
- <para><userinput>rewrite dir p patt</userinput></para>
+ -->
+ <sect1 id="tac_destruct">
+ <title>destruct</title>
+ <titleabbrev>destruct</titleabbrev>
+ <para><userinput>destruct (H<subscript>0</subscript> ... H<subscript>n</subscript>) skip (K<subscript>0</subscript> ... K<subscript>m</subscript>)</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">rewrite</emphasis> [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>] &sterm; &pattern;</para>
+ <para><emphasis role="bold">destruct</emphasis>
+ [<emphasis role="bold">(</emphasis>&id;…<emphasis role="bold">)</emphasis>] [<emphasis role="bold">skip</emphasis> <emphasis role="bold">(</emphasis>&id;…<emphasis role="bold">)</emphasis>]</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para><command>p</command> must be the proof of an equality,
- possibly under some hypotheses.</para>
+ <para>Each hypothesis <command>H<subscript>i</subscript></command> must be either a Leibniz or a John Major equality where the two sides of the equality are possibly applied constructors of an inductive type.</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>
+ <para>The tactic recursively compare the two sides of each equality
+ looking for different constructors in corresponding position.
+ If two of them are found, the tactic closes the current sequent
+ by proving the absurdity of <command>p</command>. Otherwise
+ it adds a new hypothesis for each leaf of the formula that
+ states the equality of the subformulae in the corresponding
+ positions on the two sides of the equality. If the newly
+ added hypothesis is an equality between a variable and a term,
+ the variable is substituted for the term everywhere in the
+ sequent, except for the hypotheses <command>K<subscript>j</subscript></command>, and it is then cleared from the list of hypotheses.
+ </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>
+ <para>None.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_right">
- <title>right</title>
- <titleabbrev>right</titleabbrev>
- <para><userinput>right </userinput></para>
+ <sect1 id="tac_elim">
+ <title>elim</title>
+ <titleabbrev>elim</titleabbrev>
+ <para><userinput>elim t pattern</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">right</emphasis></para>
+ <para><emphasis role="bold">elim</emphasis> &sterm; &pattern;</para>
</listitem>
</varlistentry>
<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>
+ <para><command>t</command> must inhabit an inductive type.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>Equivalent to <command>constructor 2</command>.</para>
+ <para>It proceeds by cases on the values of <command>t</command>,
+ according to the most appropriate elimination principle for
+ the current goal.
+ The induction predicate is restricted by
+ <command>pattern</command>. In particular, if some hypothesis
+ is listed in <command>pattern</command>, the hypothesis is
+ generalized and cleared before eliminating <command>t</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>
+ <para>It opens one new sequent for each case.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_ring">
- <title>ring</title>
- <titleabbrev>ring</titleabbrev>
- <para><userinput>ring </userinput></para>
+ <!--
+ <sect1 id="tac_fail">
+ <title>fail</title>
+ <titleabbrev>fail</titleabbrev>
+ <para><userinput>fail</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">ring</emphasis></para>
+ <para><emphasis role="bold">fail</emphasis></para>
</listitem>
</varlistentry>
<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>
+ <para>None.</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>
+ <para>This tactic always fail.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>None.</para>
+ <para>N.A.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_simplify">
- <title>simplify</title>
- <titleabbrev>simplify</titleabbrev>
- <para><userinput>simplify patt</userinput></para>
+ -->
+ <sect1 id="tac_generalize">
+ <title>generalize</title>
+ <titleabbrev>generalize</titleabbrev>
+ <para><userinput>generalize patt</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">simplify</emphasis> &pattern;</para>
+ <para><emphasis role="bold">generalize</emphasis> &pattern;</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>None.</para>
+ <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 replaces all the terms matched by <command>patt</command>
- with other convertible terms that are supposed to be simpler.</para>
+ <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>None.</para>
+ <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_split">
- <title>split</title>
- <titleabbrev>split</titleabbrev>
- <para><userinput>split </userinput></para>
+ <!--
+ <sect1 id="tac_id">
+ <title>id</title>
+ <titleabbrev>id</titleabbrev>
+ <para><userinput>id </userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">split</emphasis></para>
+ <para><emphasis role="bold">id</emphasis></para>
</listitem>
</varlistentry>
<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>
+ <para>None.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>Equivalent to <command>constructor 1</command>.</para>
+ <para>This identity tactic does nothing without failing.</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>
+ <para>None.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
-
- <sect1 id="tac_subst">
- <title>subst</title>
- <titleabbrev>subst</titleabbrev>
- <para><userinput>subst</userinput></para>
+ -->
+ <sect1 id="tac_inversion">
+ <title>inversion</title>
+ <titleabbrev>inversion</titleabbrev>
+ <para><userinput>inversion t</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">subst</emphasis></para>
+ <para><emphasis role="bold">inversion</emphasis> &sterm;</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
- <listitem><para>
- None.
- </para></listitem>
+ <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>
- For each premise of the form
- <command>H: x = t</command> or <command>H: t = x</command>
- where <command>x</command> is a local variable and
- <command>t</command> does not depend on <command>x</command>,
- the tactic rewrites <command>H</command> wherever
- <command>x</command> appears clearing <command>H</command> and
- <command>x</command> afterwards.
- </para></listitem>
+ <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>
- The one opened by the applied tactics.
- </para></listitem>
+ <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. It uses either Leibniz or John Major equality
+ for the new hypotheses, according to the included files.</para>
+ </listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_symmetry">
- <title>symmetry</title>
- <titleabbrev>symmetry</titleabbrev>
- <para>The tactic <command>symmetry</command> </para>
- <para><userinput>symmetry </userinput></para>
+ <sect1 id="tac_lapply">
+ <title>lapply</title>
+ <titleabbrev>lapply</titleabbrev>
+ <para><userinput>
+ lapply t
+ </userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">symmetry</emphasis></para>
+ <para>
+ <emphasis role="bold">lapply</emphasis>
+ &sterm;
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>The conclusion of the current proof must be an equality.</para>
+ <para>None.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It swaps the two sides of the equalityusing the symmetric
- property.</para>
+ <para>
+ It generalizes the conclusion of the current goal
+ adding as a premise the type of <command>t</command>,
+ closing the current goal.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>None.</para>
+ <para>
+ The new sequent has conclusion <command>T → G</command> where
+ <command>T</command> is the type of <command>t</command>
+ and <command>G</command> the old conclusion.
+ </para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_transitivity">
- <title>transitivity</title>
- <titleabbrev>transitivity</titleabbrev>
- <para><userinput>transitivity t</userinput></para>
+ <sect1 id="tac_letin">
+ <title>letin</title>
+ <titleabbrev>letin</titleabbrev>
+ <para><userinput>letin x ≝ t</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">transitivity</emphasis> &sterm;</para>
+ <para><emphasis role="bold">letin</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 an equality.</para>
+ <para>None.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It closes the current sequent by transitivity of the equality.</para>
+ <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>It opens two new sequents <command>l=t</command> and
- <command>t=r</command> where <command>l</command> and <command>r</command> are the left and right hand side of the equality in the conclusion of
-the current sequent to prove.</para>
+ <para>None.</para>
</listitem>
</varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_unfold">
- <title>unfold</title>
- <titleabbrev>unfold</titleabbrev>
- <para><userinput>unfold t patt</userinput></para>
+ <sect1 id="tac_normalize">
+ <title>normalize</title>
+ <titleabbrev>normalize</titleabbrev>
+ <para><userinput>normalize patt nodelta</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;</para>
+ <para><emphasis role="bold">normalize</emphasis> &pattern;
+ [<emphasis role="bold">nodelta</emphasis>]</para>
</listitem>
</varlistentry>
<varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It finds all the occurrences of <command>t</command>
- (possibly applied to arguments) in the subterms matched by
- <command>patt</command>. Then it δ-expands each occurrence,
- also performing β-reduction of the obtained term. If
- <command>t</command> is omitted it defaults to each
- subterm matched by <command>patt</command>.</para>
+ <para>It replaces all the terms matched by <command>patt</command>
+ with their βδιζ-normal form. If <command>nodelta</command> is
+ specified, δ-expansions are not performed.</para>
</listitem>
</varlistentry>
<varlistentry>
<sect1 id="tac_whd">
<title>whd</title>
<titleabbrev>whd</titleabbrev>
- <para><userinput>whd patt</userinput></para>
+ <para><userinput>whd patt nodelta</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">whd</emphasis> &pattern;</para>
+ <para><emphasis role="bold">whd</emphasis> &pattern; [<emphasis role="bold">nodelta</emphasis>]</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
<para>It replaces all the terms matched by <command>patt</command>
- with their βδιζ-weak-head normal form.</para>
+ with their βδιζ-weak-head normal form. If <command>nodelta</command> is specified, δ-expansions are not performed.</para>
</listitem>
</varlistentry>
<varlistentry>
projects / helm.git / commitdiff