</variablelist>
</para>
</sect1>
+ <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">applyS</emphasis> &sterm; &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>
+ </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>
+ </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>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
<sect1 id="tac_assumption">
<title>assumption</title>
<titleabbrev>assumption</titleabbrev>
<sect1 id="tac_auto">
<title>auto</title>
<titleabbrev>auto</titleabbrev>
- <para><userinput>auto depth=d width=w paramodulation full</userinput></para>
+ <para><userinput>auto params</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">auto</emphasis> [<emphasis role="bold">depth=</emphasis>&nat;] [<emphasis role="bold">width=</emphasis>&nat;] [<emphasis role="bold">paramodulation</emphasis>] [<emphasis role="bold">full</emphasis>]</para>
+ <para><emphasis role="bold">auto</emphasis> &autoparams;</para>
</listitem>
</varlistentry>
<varlistentry>
<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 <command>d</command> and <command>w</command>.
+ 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 ...TODO</para>
+ a sequent is ...&TODO;</para>
</listitem>
</varlistentry>
<varlistentry>
</variablelist>
</para>
</sect1>
+ <sect1 id="tac_cases">
+ <title>cases</title>
+ <titleabbrev>cases</titleabbrev>
+ <para><userinput>
+ cases t hyps
+ </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para>
+ <emphasis role="bold">cases</emphasis>
+ &term; [<emphasis role="bold">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para>
+ <command>t</command> must inhabit an inductive type
+ </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>.
+ </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>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
<sect1 id="tac_clear">
<title>clear</title>
<titleabbrev>clear</titleabbrev>
- <para><userinput>clear H</userinput></para>
+ <para><userinput>
+ clear H<subscript>1</subscript> ... H<subscript>m</subscript>
+ </userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">clear</emphasis> &id;</para>
+ <para>
+ <emphasis role="bold">clear</emphasis>
+ &id; [&id;…]
+ </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>
+ <command>
+ H<subscript>1</subscript> ... H<subscript>m</subscript>
+ </command> must be hypotheses of the
+ current sequent to prove.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It hides the hypothesis <command>H</command> from the
- current sequent.</para>
+ <para>
+ It hides the hypotheses
+ <command>
+ H<subscript>1</subscript> ... H<subscript>m</subscript>
+ </command> from the current sequent.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
</variablelist>
</para>
</sect1>
+ <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">compose</emphasis> [&nat;] &sterm; [<emphasis role="bold">with</emphasis> &sterm;] [&intros-spec;]</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem>
+ <para></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>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>The same, but with more hypothesis eventually introduced
+ by the &intros-spec;.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
<sect1 id="tac_change">
<title>change</title>
<titleabbrev>change</titleabbrev>
<title>decompose</title>
<titleabbrev>decompose</titleabbrev>
<para><userinput>
- decompose (T<subscript>1</subscript> ... T<subscript>n</subscript>) H hips
+ 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> &id; [&id;]… &intros-spec;</para>
+ <para>
+ <emphasis role="bold">decompose</emphasis>
+ [<emphasis role="bold">as</emphasis> &id;…]
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>
- <command>H</command> must inhabit one inductive type among
- <command>
- T<subscript>1</subscript> ... T<subscript>n</subscript>
- </command>
- and the types of a predefined list.
- </para>
+ <para>None.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
<para>
- Runs <command>elim H hyps</command>, clears H and tries to run
- itself recursively on each new identifier introduced by
- <command>elim</command> in the opened sequents.
+ 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>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_demodulation">
- <title>demodulation</title>
- <titleabbrev>demodulation</titleabbrev>
- <para><userinput>demodulation patt</userinput></para>
+ <sect1 id="tac_demodulate">
+ <title>demodulate</title>
+ <titleabbrev>demodulate</titleabbrev>
+ <para><userinput>demodulate</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">demodulation</emphasis> &pattern;</para>
+ <para><emphasis role="bold">demodulate</emphasis></para>
</listitem>
</varlistentry>
<varlistentry>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_discriminate">
- <title>discriminate</title>
- <titleabbrev>discriminate</titleabbrev>
- <para><userinput>discriminate p</userinput></para>
+ <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">discriminate</emphasis> &sterm;</para>
+ <para><emphasis role="bold">destruct</emphasis> &sterm;</para>
</listitem>
</varlistentry>
<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>m</subscript></command> where <command>K</command> and <command>K'</command> must be different constructors of the same inductive type and each argument list can be empty if
-its constructor takes no arguments.</para>
+ <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>It closes the current sequent by proving the absurdity of
- <command>p</command>.</para>
+ <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>
<sect1 id="tac_fwd">
<title>fwd</title>
<titleabbrev>fwd</titleabbrev>
- <para><userinput>fwd H</userinput></para>
+ <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">(</emphasis>[&id;]…<emphasis role="bold">)</emphasis>]</para>
+ <para><emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">as</emphasis> &id; [&id;]…]</para>
</listitem>
</varlistentry>
<varlistentry>
This tactic is under development.
It simplifies the current context by removing
<command>H</command> using the following methods:
- forward application of a suitable simplification theorem (chosen
- automatically) of which the type of <command>H</command> is a
- premise, decomposition, rewriting.
+ 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>None.</para>
+ <para>
+ The ones opened by the tactics <command>fwd</command> invokes.
+ </para>
</listitem>
</varlistentry>
</variablelist>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_injection">
- <title>injection</title>
- <titleabbrev>injection</titleabbrev>
- <para><userinput>injection p</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">injection</emphasis> &sterm;</para>
- </listitem>
- </varlistentry>
- <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>intro</title>
<titleabbrev>intro</titleabbrev>
<title>lapply</title>
<titleabbrev>lapply</titleabbrev>
<para><userinput>
- lapply depth=d t
+ lapply linear depth=d t
to t<subscript>1</subscript>, ..., t<subscript>n</subscript> as H
</userinput></para>
<para>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">lapply</emphasis> [<emphasis role="bold">depth=</emphasis>&nat;] &sterm; [<emphasis role="bold">to</emphasis> &sterm; [&sterm;]…] [<emphasis role="bold">as</emphasis> &id;]</para>
+ <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;]
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Pre-conditions:</term>
<listitem>
- <para>TODO.</para>
+ <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>.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>TODO.</para>
+ <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.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>New sequents to prove:</term>
<listitem>
- <para>TODO.</para>
+ <para>
+ The ones opened by the tactics <command>lapply</command> invokes.
+ </para>
</listitem>
</varlistentry>
</variablelist>
</variablelist>
</para>
</sect1>
- <sect1 id="tac_paramodulation">
- <title>paramodulation</title>
- <titleabbrev>paramodulation</titleabbrev>
- <para><userinput>paramodulation patt</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">paramodulation</emphasis> &pattern;</para>
- </listitem>
- </varlistentry>
- <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>reduce</title>
<titleabbrev>reduce</titleabbrev>
</variablelist>
</para>
</sect1>
+
+ <sect1 id="tac_subst">
+ <title>subst</title>
+ <titleabbrev>subst</titleabbrev>
+ <para><userinput>subst</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">subst</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-conditions:</term>
+ <listitem><para>
+ None.
+ </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>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem><para>
+ The one opened by the applied tactics.
+ </para></listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
<sect1 id="tac_symmetry">
<title>symmetry</title>
<titleabbrev>symmetry</titleabbrev>
projects / helm.git / blobdiff