theory Generic
imports Base Main
begin
chapter {* Generic tools and packages \label{ch:gen-tools} *}
section {* Configuration options \label{sec:config} *}
text {* Isabelle/Pure maintains a record of named configuration
options within the theory or proof context, with values of type
@{ML_type bool}, @{ML_type int}, @{ML_type real}, or @{ML_type
string}. Tools may declare options in ML, and then refer to these
values (relative to the context). Thus global reference variables
are easily avoided. The user may change the value of a
configuration option by means of an associated attribute of the same
name. This form of context declaration works particularly well with
commands such as @{command "declare"} or @{command "using"} like
this:
*}
declare [[show_main_goal = false]]
notepad
begin
note [[show_main_goal = true]]
end
text {* For historical reasons, some tools cannot take the full proof
context into account and merely refer to the background theory.
This is accommodated by configuration options being declared as
``global'', which may not be changed within a local context.
\begin{matharray}{rcll}
@{command_def "print_configs"} & : & @{text "context \<rightarrow>"} \\
\end{matharray}
@{rail "
@{syntax name} ('=' ('true' | 'false' | @{syntax int} | @{syntax float} | @{syntax name}))?
"}
\begin{description}
\item @{command "print_configs"} prints the available configuration
options, with names, types, and current values.
\item @{text "name = value"} as an attribute expression modifies the
named option, with the syntax of the value depending on the option's
type. For @{ML_type bool} the default value is @{text true}. Any
attempt to change a global option in a local context is ignored.
\end{description}
*}
section {* Basic proof tools *}
subsection {* Miscellaneous methods and attributes \label{sec:misc-meth-att} *}
text {*
\begin{matharray}{rcl}
@{method_def unfold} & : & @{text method} \\
@{method_def fold} & : & @{text method} \\
@{method_def insert} & : & @{text method} \\[0.5ex]
@{method_def erule}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def drule}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def frule}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def intro} & : & @{text method} \\
@{method_def elim} & : & @{text method} \\
@{method_def succeed} & : & @{text method} \\
@{method_def fail} & : & @{text method} \\
\end{matharray}
@{rail "
(@@{method fold} | @@{method unfold} | @@{method insert}) @{syntax thmrefs}
;
(@@{method erule} | @@{method drule} | @@{method frule})
('(' @{syntax nat} ')')? @{syntax thmrefs}
;
(@@{method intro} | @@{method elim}) @{syntax thmrefs}?
"}
\begin{description}
\item @{method unfold}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{method fold}~@{text
"a\<^sub>1 \<dots> a\<^sub>n"} expand (or fold back) the given definitions throughout
all goals; any chained facts provided are inserted into the goal and
subject to rewriting as well.
\item @{method insert}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} inserts theorems as facts
into all goals of the proof state. Note that current facts
indicated for forward chaining are ignored.
\item @{method erule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, @{method
drule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, and @{method frule}~@{text
"a\<^sub>1 \<dots> a\<^sub>n"} are similar to the basic @{method rule}
method (see \secref{sec:pure-meth-att}), but apply rules by
elim-resolution, destruct-resolution, and forward-resolution,
respectively \cite{isabelle-implementation}. The optional natural
number argument (default 0) specifies additional assumption steps to
be performed here.
Note that these methods are improper ones, mainly serving for
experimentation and tactic script emulation. Different modes of
basic rule application are usually expressed in Isar at the proof
language level, rather than via implicit proof state manipulations.
For example, a proper single-step elimination would be done using
the plain @{method rule} method, with forward chaining of current
facts.
\item @{method intro} and @{method elim} repeatedly refine some goal
by intro- or elim-resolution, after having inserted any chained
facts. Exactly the rules given as arguments are taken into account;
this allows fine-tuned decomposition of a proof problem, in contrast
to common automated tools.
\item @{method succeed} yields a single (unchanged) result; it is
the identity of the ``@{text ","}'' method combinator (cf.\
\secref{sec:proof-meth}).
\item @{method fail} yields an empty result sequence; it is the
identity of the ``@{text "|"}'' method combinator (cf.\
\secref{sec:proof-meth}).
\end{description}
\begin{matharray}{rcl}
@{attribute_def tagged} & : & @{text attribute} \\
@{attribute_def untagged} & : & @{text attribute} \\[0.5ex]
@{attribute_def THEN} & : & @{text attribute} \\
@{attribute_def unfolded} & : & @{text attribute} \\
@{attribute_def folded} & : & @{text attribute} \\
@{attribute_def abs_def} & : & @{text attribute} \\[0.5ex]
@{attribute_def rotated} & : & @{text attribute} \\
@{attribute_def (Pure) elim_format} & : & @{text attribute} \\
@{attribute_def standard}@{text "\<^sup>*"} & : & @{text attribute} \\
@{attribute_def no_vars}@{text "\<^sup>*"} & : & @{text attribute} \\
\end{matharray}
@{rail "
@@{attribute tagged} @{syntax name} @{syntax name}
;
@@{attribute untagged} @{syntax name}
;
@@{attribute THEN} ('[' @{syntax nat} ']')? @{syntax thmref}
;
(@@{attribute unfolded} | @@{attribute folded}) @{syntax thmrefs}
;
@@{attribute rotated} @{syntax int}?
"}
\begin{description}
\item @{attribute tagged}~@{text "name value"} and @{attribute
untagged}~@{text name} add and remove \emph{tags} of some theorem.
Tags may be any list of string pairs that serve as formal comment.
The first string is considered the tag name, the second its value.
Note that @{attribute untagged} removes any tags of the same name.
\item @{attribute THEN}~@{text a} composes rules by resolution; it
resolves with the first premise of @{text a} (an alternative
position may be also specified). See also @{ML_op "RS"} in
\cite{isabelle-implementation}.
\item @{attribute unfolded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{attribute
folded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} expand and fold back again the given
definitions throughout a rule.
\item @{attribute abs_def} turns an equation of the form @{prop "f x
y \<equiv> t"} into @{prop "f \<equiv> \<lambda>x y. t"}, which ensures that @{method
simp} or @{method unfold} steps always expand it. This also works
for object-logic equality.
\item @{attribute rotated}~@{text n} rotate the premises of a
theorem by @{text n} (default 1).
\item @{attribute (Pure) elim_format} turns a destruction rule into
elimination rule format, by resolving with the rule @{prop "PROP A \<Longrightarrow>
(PROP A \<Longrightarrow> PROP B) \<Longrightarrow> PROP B"}.
Note that the Classical Reasoner (\secref{sec:classical}) provides
its own version of this operation.
\item @{attribute standard} puts a theorem into the standard form of
object-rules at the outermost theory level. Note that this
operation violates the local proof context (including active
locales).
\item @{attribute no_vars} replaces schematic variables by free
ones; this is mainly for tuning output of pretty printed theorems.
\end{description}
*}
subsection {* Low-level equational reasoning *}
text {*
\begin{matharray}{rcl}
@{method_def subst} & : & @{text method} \\
@{method_def hypsubst} & : & @{text method} \\
@{method_def split} & : & @{text method} \\
\end{matharray}
@{rail "
@@{method subst} ('(' 'asm' ')')? \\ ('(' (@{syntax nat}+) ')')? @{syntax thmref}
;
@@{method split} @{syntax thmrefs}
"}
These methods provide low-level facilities for equational reasoning
that are intended for specialized applications only. Normally,
single step calculations would be performed in a structured text
(see also \secref{sec:calculation}), while the Simplifier methods
provide the canonical way for automated normalization (see
\secref{sec:simplifier}).
\begin{description}
\item @{method subst}~@{text eq} performs a single substitution step
using rule @{text eq}, which may be either a meta or object
equality.
\item @{method subst}~@{text "(asm) eq"} substitutes in an
assumption.
\item @{method subst}~@{text "(i \<dots> j) eq"} performs several
substitutions in the conclusion. The numbers @{text i} to @{text j}
indicate the positions to substitute at. Positions are ordered from
the top of the term tree moving down from left to right. For
example, in @{text "(a + b) + (c + d)"} there are three positions
where commutativity of @{text "+"} is applicable: 1 refers to @{text
"a + b"}, 2 to the whole term, and 3 to @{text "c + d"}.
If the positions in the list @{text "(i \<dots> j)"} are non-overlapping
(e.g.\ @{text "(2 3)"} in @{text "(a + b) + (c + d)"}) you may
assume all substitutions are performed simultaneously. Otherwise
the behaviour of @{text subst} is not specified.
\item @{method subst}~@{text "(asm) (i \<dots> j) eq"} performs the
substitutions in the assumptions. The positions refer to the
assumptions in order from left to right. For example, given in a
goal of the form @{text "P (a + b) \<Longrightarrow> P (c + d) \<Longrightarrow> \<dots>"}, position 1 of
commutativity of @{text "+"} is the subterm @{text "a + b"} and
position 2 is the subterm @{text "c + d"}.
\item @{method hypsubst} performs substitution using some
assumption; this only works for equations of the form @{text "x =
t"} where @{text x} is a free or bound variable.
\item @{method split}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} performs single-step case
splitting using the given rules. Splitting is performed in the
conclusion or some assumption of the subgoal, depending of the
structure of the rule.
Note that the @{method simp} method already involves repeated
application of split rules as declared in the current context, using
@{attribute split}, for example.
\end{description}
*}
subsection {* Further tactic emulations \label{sec:tactics} *}
text {*
The following improper proof methods emulate traditional tactics.
These admit direct access to the goal state, which is normally
considered harmful! In particular, this may involve both numbered
goal addressing (default 1), and dynamic instantiation within the
scope of some subgoal.
\begin{warn}
Dynamic instantiations refer to universally quantified parameters
of a subgoal (the dynamic context) rather than fixed variables and
term abbreviations of a (static) Isar context.
\end{warn}
Tactic emulation methods, unlike their ML counterparts, admit
simultaneous instantiation from both dynamic and static contexts.
If names occur in both contexts goal parameters hide locally fixed
variables. Likewise, schematic variables refer to term
abbreviations, if present in the static context. Otherwise the
schematic variable is interpreted as a schematic variable and left
to be solved by unification with certain parts of the subgoal.
Note that the tactic emulation proof methods in Isabelle/Isar are
consistently named @{text foo_tac}. Note also that variable names
occurring on left hand sides of instantiations must be preceded by a
question mark if they coincide with a keyword or contain dots. This
is consistent with the attribute @{attribute "where"} (see
\secref{sec:pure-meth-att}).
\begin{matharray}{rcl}
@{method_def rule_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def erule_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def drule_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def frule_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def cut_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def thin_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def subgoal_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def rename_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def rotate_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def tactic}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def raw_tactic}@{text "\<^sup>*"} & : & @{text method} \\
\end{matharray}
@{rail "
(@@{method rule_tac} | @@{method erule_tac} | @@{method drule_tac} |
@@{method frule_tac} | @@{method cut_tac} | @@{method thin_tac}) @{syntax goal_spec}? \\
( dynamic_insts @'in' @{syntax thmref} | @{syntax thmrefs} )
;
@@{method subgoal_tac} @{syntax goal_spec}? (@{syntax prop} +)
;
@@{method rename_tac} @{syntax goal_spec}? (@{syntax name} +)
;
@@{method rotate_tac} @{syntax goal_spec}? @{syntax int}?
;
(@@{method tactic} | @@{method raw_tactic}) @{syntax text}
;
dynamic_insts: ((@{syntax name} '=' @{syntax term}) + @'and')
"}
\begin{description}
\item @{method rule_tac} etc. do resolution of rules with explicit
instantiation. This works the same way as the ML tactics @{ML
res_inst_tac} etc. (see \cite{isabelle-implementation})
Multiple rules may be only given if there is no instantiation; then
@{method rule_tac} is the same as @{ML resolve_tac} in ML (see
\cite{isabelle-implementation}).
\item @{method cut_tac} inserts facts into the proof state as
assumption of a subgoal; instantiations may be given as well. Note
that the scope of schematic variables is spread over the main goal
statement and rule premises are turned into new subgoals. This is
in contrast to the regular method @{method insert} which inserts
closed rule statements.
\item @{method thin_tac}~@{text \<phi>} deletes the specified premise
from a subgoal. Note that @{text \<phi>} may contain schematic
variables, to abbreviate the intended proposition; the first
matching subgoal premise will be deleted. Removing useless premises
from a subgoal increases its readability and can make search tactics
run faster.
\item @{method subgoal_tac}~@{text "\<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} adds the propositions
@{text "\<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} as local premises to a subgoal, and poses the same
as new subgoals (in the original context).
\item @{method rename_tac}~@{text "x\<^sub>1 \<dots> x\<^sub>n"} renames parameters of a
goal according to the list @{text "x\<^sub>1, \<dots>, x\<^sub>n"}, which refers to the
\emph{suffix} of variables.
\item @{method rotate_tac}~@{text n} rotates the premises of a
subgoal by @{text n} positions: from right to left if @{text n} is
positive, and from left to right if @{text n} is negative; the
default value is 1.
\item @{method tactic}~@{text "text"} produces a proof method from
any ML text of type @{ML_type tactic}. Apart from the usual ML
environment and the current proof context, the ML code may refer to
the locally bound values @{ML_text facts}, which indicates any
current facts used for forward-chaining.
\item @{method raw_tactic} is similar to @{method tactic}, but
presents the goal state in its raw internal form, where simultaneous
subgoals appear as conjunction of the logical framework instead of
the usual split into several subgoals. While feature this is useful
for debugging of complex method definitions, it should not never
appear in production theories.
\end{description}
*}
section {* The Simplifier \label{sec:simplifier} *}
subsection {* Simplification methods *}
text {*
\begin{matharray}{rcl}
@{method_def simp} & : & @{text method} \\
@{method_def simp_all} & : & @{text method} \\
\end{matharray}
@{rail "
(@@{method simp} | @@{method simp_all}) opt? (@{syntax simpmod} * )
;
opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use' | 'asm_lr' ) ')'
;
@{syntax_def simpmod}: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') |
'split' (() | 'add' | 'del')) ':' @{syntax thmrefs}
"}
\begin{description}
\item @{method simp} invokes the Simplifier, after declaring
additional rules according to the arguments given. Note that the
@{text only} modifier first removes all other rewrite rules,
congruences, and looper tactics (including splits), and then behaves
like @{text add}.
\medskip The @{text cong} modifiers add or delete Simplifier
congruence rules (see also \secref{sec:simp-cong}), the default is
to add.
\medskip The @{text split} modifiers add or delete rules for the
Splitter (see also \cite{isabelle-ref}), the default is to add.
This works only if the Simplifier method has been properly setup to
include the Splitter (all major object logics such HOL, HOLCF, FOL,
ZF do this already).
\item @{method simp_all} is similar to @{method simp}, but acts on
all goals (backwards from the last to the first one).
\end{description}
By default the Simplifier methods take local assumptions fully into
account, using equational assumptions in the subsequent
normalization process, or simplifying assumptions themselves (cf.\
@{ML asm_full_simp_tac} in \cite{isabelle-ref}). In structured
proofs this is usually quite well behaved in practice: just the
local premises of the actual goal are involved, additional facts may
be inserted via explicit forward-chaining (via @{command "then"},
@{command "from"}, @{command "using"} etc.).
Additional Simplifier options may be specified to tune the behavior
further (mostly for unstructured scripts with many accidental local
facts): ``@{text "(no_asm)"}'' means assumptions are ignored
completely (cf.\ @{ML simp_tac}), ``@{text "(no_asm_simp)"}'' means
assumptions are used in the simplification of the conclusion but are
not themselves simplified (cf.\ @{ML asm_simp_tac}), and ``@{text
"(no_asm_use)"}'' means assumptions are simplified but are not used
in the simplification of each other or the conclusion (cf.\ @{ML
full_simp_tac}). For compatibility reasons, there is also an option
``@{text "(asm_lr)"}'', which means that an assumption is only used
for simplifying assumptions which are to the right of it (cf.\ @{ML
asm_lr_simp_tac}).
The configuration option @{text "depth_limit"} limits the number of
recursive invocations of the simplifier during conditional
rewriting.
\medskip The Splitter package is usually configured to work as part
of the Simplifier. The effect of repeatedly applying @{ML
split_tac} can be simulated by ``@{text "(simp only: split:
a\<^sub>1 \<dots> a\<^sub>n)"}''. There is also a separate @{text split}
method available for single-step case splitting.
*}
subsection {* Declaring rules *}
text {*
\begin{matharray}{rcl}
@{command_def "print_simpset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{attribute_def simp} & : & @{text attribute} \\
@{attribute_def split} & : & @{text attribute} \\
\end{matharray}
@{rail "
(@@{attribute simp} | @@{attribute split}) (() | 'add' | 'del')
"}
\begin{description}
\item @{command "print_simpset"} prints the collection of rules
declared to the Simplifier, which is also known as ``simpset''
internally \cite{isabelle-ref}.
\item @{attribute simp} declares simplification rules.
\item @{attribute split} declares case split rules.
\end{description}
*}
subsection {* Congruence rules\label{sec:simp-cong} *}
text {*
\begin{matharray}{rcl}
@{attribute_def cong} & : & @{text attribute} \\
\end{matharray}
@{rail "
@@{attribute cong} (() | 'add' | 'del')
"}
\begin{description}
\item @{attribute cong} declares congruence rules to the Simplifier
context.
\end{description}
Congruence rules are equalities of the form @{text [display]
"\<dots> \<Longrightarrow> f ?x\<^sub>1 \<dots> ?x\<^sub>n = f ?y\<^sub>1 \<dots> ?y\<^sub>n"}
This controls the simplification of the arguments of @{text f}. For
example, some arguments can be simplified under additional
assumptions: @{text [display] "?P\<^sub>1 \<longleftrightarrow> ?Q\<^sub>1 \<Longrightarrow> (?Q\<^sub>1 \<Longrightarrow> ?P\<^sub>2 \<longleftrightarrow> ?Q\<^sub>2) \<Longrightarrow>
(?P\<^sub>1 \<longrightarrow> ?P\<^sub>2) \<longleftrightarrow> (?Q\<^sub>1 \<longrightarrow> ?Q\<^sub>2)"}
Given this rule, the simplifier assumes @{text "?Q\<^sub>1"} and extracts
rewrite rules from it when simplifying @{text "?P\<^sub>2"}. Such local
assumptions are effective for rewriting formulae such as @{text "x =
0 \<longrightarrow> y + x = y"}.
%FIXME
%The local assumptions are also provided as theorems to the solver;
%see \secref{sec:simp-solver} below.
\medskip The following congruence rule for bounded quantifiers also
supplies contextual information --- about the bound variable:
@{text [display] "(?A = ?B) \<Longrightarrow> (\<And>x. x \<in> ?B \<Longrightarrow> ?P x \<longleftrightarrow> ?Q x) \<Longrightarrow>
(\<forall>x \<in> ?A. ?P x) \<longleftrightarrow> (\<forall>x \<in> ?B. ?Q x)"}
\medskip This congruence rule for conditional expressions can
supply contextual information for simplifying the arms:
@{text [display] "?p = ?q \<Longrightarrow> (?q \<Longrightarrow> ?a = ?c) \<Longrightarrow> (\<not> ?q \<Longrightarrow> ?b = ?d) \<Longrightarrow>
(if ?p then ?a else ?b) = (if ?q then ?c else ?d)"}
A congruence rule can also \emph{prevent} simplification of some
arguments. Here is an alternative congruence rule for conditional
expressions that conforms to non-strict functional evaluation:
@{text [display] "?p = ?q \<Longrightarrow> (if ?p then ?a else ?b) = (if ?q then ?a else ?b)"}
Only the first argument is simplified; the others remain unchanged.
This can make simplification much faster, but may require an extra
case split over the condition @{text "?q"} to prove the goal.
*}
subsection {* Simplification procedures *}
text {* Simplification procedures are ML functions that produce proven
rewrite rules on demand. They are associated with higher-order
patterns that approximate the left-hand sides of equations. The
Simplifier first matches the current redex against one of the LHS
patterns; if this succeeds, the corresponding ML function is
invoked, passing the Simplifier context and redex term. Thus rules
may be specifically fashioned for particular situations, resulting
in a more powerful mechanism than term rewriting by a fixed set of
rules.
Any successful result needs to be a (possibly conditional) rewrite
rule @{text "t \<equiv> u"} that is applicable to the current redex. The
rule will be applied just as any ordinary rewrite rule. It is
expected to be already in \emph{internal form}, bypassing the
automatic preprocessing of object-level equivalences.
\begin{matharray}{rcl}
@{command_def "simproc_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
simproc & : & @{text attribute} \\
\end{matharray}
@{rail "
@@{command simproc_setup} @{syntax name} '(' (@{syntax term} + '|') ')' '='
@{syntax text} \\ (@'identifier' (@{syntax nameref}+))?
;
@@{attribute simproc} (('add' ':')? | 'del' ':') (@{syntax name}+)
"}
\begin{description}
\item @{command "simproc_setup"} defines a named simplification
procedure that is invoked by the Simplifier whenever any of the
given term patterns match the current redex. The implementation,
which is provided as ML source text, needs to be of type @{ML_type
"morphism -> simpset -> cterm -> thm option"}, where the @{ML_type
cterm} represents the current redex @{text r} and the result is
supposed to be some proven rewrite rule @{text "r \<equiv> r'"} (or a
generalized version), or @{ML NONE} to indicate failure. The
@{ML_type simpset} argument holds the full context of the current
Simplifier invocation, including the actual Isar proof context. The
@{ML_type morphism} informs about the difference of the original
compilation context wrt.\ the one of the actual application later
on. The optional @{keyword "identifier"} specifies theorems that
represent the logical content of the abstract theory of this
simproc.
Morphisms and identifiers are only relevant for simprocs that are
defined within a local target context, e.g.\ in a locale.
\item @{text "simproc add: name"} and @{text "simproc del: name"}
add or delete named simprocs to the current Simplifier context. The
default is to add a simproc. Note that @{command "simproc_setup"}
already adds the new simproc to the subsequent context.
\end{description}
*}
subsubsection {* Example *}
text {* The following simplification procedure for @{thm
[source=false, show_types] unit_eq} in HOL performs fine-grained
control over rule application, beyond higher-order pattern matching.
Declaring @{thm unit_eq} as @{attribute simp} directly would make
the simplifier loop! Note that a version of this simplification
procedure is already active in Isabelle/HOL. *}
simproc_setup unit ("x::unit") = {*
fn _ => fn _ => fn ct =>
if HOLogic.is_unit (term_of ct) then NONE
else SOME (mk_meta_eq @{thm unit_eq})
*}
text {* Since the Simplifier applies simplification procedures
frequently, it is important to make the failure check in ML
reasonably fast. *}
subsection {* Forward simplification *}
text {*
\begin{matharray}{rcl}
@{attribute_def simplified} & : & @{text attribute} \\
\end{matharray}
@{rail "
@@{attribute simplified} opt? @{syntax thmrefs}?
;
opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use') ')'
"}
\begin{description}
\item @{attribute simplified}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} causes a theorem to
be simplified, either by exactly the specified rules @{text "a\<^sub>1, \<dots>,
a\<^sub>n"}, or the implicit Simplifier context if no arguments are given.
The result is fully simplified by default, including assumptions and
conclusion; the options @{text no_asm} etc.\ tune the Simplifier in
the same way as the for the @{text simp} method.
Note that forward simplification restricts the simplifier to its
most basic operation of term rewriting; solver and looper tactics
\cite{isabelle-ref} are \emph{not} involved here. The @{text
simplified} attribute should be only rarely required under normal
circumstances.
\end{description}
*}
section {* The Classical Reasoner \label{sec:classical} *}
subsection {* Basic concepts *}
text {* Although Isabelle is generic, many users will be working in
some extension of classical first-order logic. Isabelle/ZF is built
upon theory FOL, while Isabelle/HOL conceptually contains
first-order logic as a fragment. Theorem-proving in predicate logic
is undecidable, but many automated strategies have been developed to
assist in this task.
Isabelle's classical reasoner is a generic package that accepts
certain information about a logic and delivers a suite of automatic
proof tools, based on rules that are classified and declared in the
context. These proof procedures are slow and simplistic compared
with high-end automated theorem provers, but they can save
considerable time and effort in practice. They can prove theorems
such as Pelletier's \cite{pelletier86} problems 40 and 41 in a few
milliseconds (including full proof reconstruction): *}
lemma "(\<exists>y. \<forall>x. F x y \<longleftrightarrow> F x x) \<longrightarrow> \<not> (\<forall>x. \<exists>y. \<forall>z. F z y \<longleftrightarrow> \<not> F z x)"
by blast
lemma "(\<forall>z. \<exists>y. \<forall>x. f x y \<longleftrightarrow> f x z \<and> \<not> f x x) \<longrightarrow> \<not> (\<exists>z. \<forall>x. f x z)"
by blast
text {* The proof tools are generic. They are not restricted to
first-order logic, and have been heavily used in the development of
the Isabelle/HOL library and applications. The tactics can be
traced, and their components can be called directly; in this manner,
any proof can be viewed interactively. *}
subsubsection {* The sequent calculus *}
text {* Isabelle supports natural deduction, which is easy to use for
interactive proof. But natural deduction does not easily lend
itself to automation, and has a bias towards intuitionism. For
certain proofs in classical logic, it can not be called natural.
The \emph{sequent calculus}, a generalization of natural deduction,
is easier to automate.
A \textbf{sequent} has the form @{text "\<Gamma> \<turnstile> \<Delta>"}, where @{text "\<Gamma>"}
and @{text "\<Delta>"} are sets of formulae.\footnote{For first-order
logic, sequents can equivalently be made from lists or multisets of
formulae.} The sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} is
\textbf{valid} if @{text "P\<^sub>1 \<and> \<dots> \<and> P\<^sub>m"} implies @{text "Q\<^sub>1 \<or> \<dots> \<or>
Q\<^sub>n"}. Thus @{text "P\<^sub>1, \<dots>, P\<^sub>m"} represent assumptions, each of which
is true, while @{text "Q\<^sub>1, \<dots>, Q\<^sub>n"} represent alternative goals. A
sequent is \textbf{basic} if its left and right sides have a common
formula, as in @{text "P, Q \<turnstile> Q, R"}; basic sequents are trivially
valid.
Sequent rules are classified as \textbf{right} or \textbf{left},
indicating which side of the @{text "\<turnstile>"} symbol they operate on.
Rules that operate on the right side are analogous to natural
deduction's introduction rules, and left rules are analogous to
elimination rules. The sequent calculus analogue of @{text "(\<longrightarrow>I)"}
is the rule
\[
\infer[@{text "(\<longrightarrow>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<longrightarrow> Q"}}{@{text "P, \<Gamma> \<turnstile> \<Delta>, Q"}}
\]
Applying the rule backwards, this breaks down some implication on
the right side of a sequent; @{text "\<Gamma>"} and @{text "\<Delta>"} stand for
the sets of formulae that are unaffected by the inference. The
analogue of the pair @{text "(\<or>I1)"} and @{text "(\<or>I2)"} is the
single rule
\[
\infer[@{text "(\<or>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<or> Q"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P, Q"}}
\]
This breaks down some disjunction on the right side, replacing it by
both disjuncts. Thus, the sequent calculus is a kind of
multiple-conclusion logic.
To illustrate the use of multiple formulae on the right, let us
prove the classical theorem @{text "(P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}. Working
backwards, we reduce this formula to a basic sequent:
\[
\infer[@{text "(\<or>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}}
{\infer[@{text "(\<longrightarrow>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q), (Q \<longrightarrow> P)"}}
{\infer[@{text "(\<longrightarrow>R)"}]{@{text "P \<turnstile> Q, (Q \<longrightarrow> P)"}}
{@{text "P, Q \<turnstile> Q, P"}}}}
\]
This example is typical of the sequent calculus: start with the
desired theorem and apply rules backwards in a fairly arbitrary
manner. This yields a surprisingly effective proof procedure.
Quantifiers add only few complications, since Isabelle handles
parameters and schematic variables. See \cite[Chapter
10]{paulson-ml2} for further discussion. *}
subsubsection {* Simulating sequents by natural deduction *}
text {* Isabelle can represent sequents directly, as in the
object-logic LK. But natural deduction is easier to work with, and
most object-logics employ it. Fortunately, we can simulate the
sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} by the Isabelle formula
@{text "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>2 \<Longrightarrow> ... \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> Q\<^sub>1"} where the order of
the assumptions and the choice of @{text "Q\<^sub>1"} are arbitrary.
Elim-resolution plays a key role in simulating sequent proofs.
We can easily handle reasoning on the left. Elim-resolution with
the rules @{text "(\<or>E)"}, @{text "(\<bottom>E)"} and @{text "(\<exists>E)"} achieves
a similar effect as the corresponding sequent rules. For the other
connectives, we use sequent-style elimination rules instead of
destruction rules such as @{text "(\<and>E1, 2)"} and @{text "(\<forall>E)"}.
But note that the rule @{text "(\<not>L)"} has no effect under our
representation of sequents!
\[
\infer[@{text "(\<not>L)"}]{@{text "\<not> P, \<Gamma> \<turnstile> \<Delta>"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P"}}
\]
What about reasoning on the right? Introduction rules can only
affect the formula in the conclusion, namely @{text "Q\<^sub>1"}. The
other right-side formulae are represented as negated assumptions,
@{text "\<not> Q\<^sub>2, \<dots>, \<not> Q\<^sub>n"}. In order to operate on one of these, it
must first be exchanged with @{text "Q\<^sub>1"}. Elim-resolution with the
@{text swap} rule has this effect: @{text "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"}
To ensure that swaps occur only when necessary, each introduction
rule is converted into a swapped form: it is resolved with the
second premise of @{text "(swap)"}. The swapped form of @{text
"(\<and>I)"}, which might be called @{text "(\<not>\<and>E)"}, is
@{text [display] "\<not> (P \<and> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (\<not> R \<Longrightarrow> Q) \<Longrightarrow> R"}
Similarly, the swapped form of @{text "(\<longrightarrow>I)"} is
@{text [display] "\<not> (P \<longrightarrow> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P \<Longrightarrow> Q) \<Longrightarrow> R"}
Swapped introduction rules are applied using elim-resolution, which
deletes the negated formula. Our representation of sequents also
requires the use of ordinary introduction rules. If we had no
regard for readability of intermediate goal states, we could treat
the right side more uniformly by representing sequents as @{text
[display] "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> \<bottom>"}
*}
subsubsection {* Extra rules for the sequent calculus *}
text {* As mentioned, destruction rules such as @{text "(\<and>E1, 2)"} and
@{text "(\<forall>E)"} must be replaced by sequent-style elimination rules.
In addition, we need rules to embody the classical equivalence
between @{text "P \<longrightarrow> Q"} and @{text "\<not> P \<or> Q"}. The introduction
rules @{text "(\<or>I1, 2)"} are replaced by a rule that simulates
@{text "(\<or>R)"}: @{text [display] "(\<not> Q \<Longrightarrow> P) \<Longrightarrow> P \<or> Q"}
The destruction rule @{text "(\<longrightarrow>E)"} is replaced by @{text [display]
"(P \<longrightarrow> Q) \<Longrightarrow> (\<not> P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"}
Quantifier replication also requires special rules. In classical
logic, @{text "\<exists>x. P x"} is equivalent to @{text "\<not> (\<forall>x. \<not> P x)"};
the rules @{text "(\<exists>R)"} and @{text "(\<forall>L)"} are dual:
\[
\infer[@{text "(\<exists>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x"}}{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x, P t"}}
\qquad
\infer[@{text "(\<forall>L)"}]{@{text "\<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}}{@{text "P t, \<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}}
\]
Thus both kinds of quantifier may be replicated. Theorems requiring
multiple uses of a universal formula are easy to invent; consider
@{text [display] "(\<forall>x. P x \<longrightarrow> P (f x)) \<and> P a \<longrightarrow> P (f\<^sup>n a)"} for any
@{text "n > 1"}. Natural examples of the multiple use of an
existential formula are rare; a standard one is @{text "\<exists>x. \<forall>y. P x
\<longrightarrow> P y"}.
Forgoing quantifier replication loses completeness, but gains
decidability, since the search space becomes finite. Many useful
theorems can be proved without replication, and the search generally
delivers its verdict in a reasonable time. To adopt this approach,
represent the sequent rules @{text "(\<exists>R)"}, @{text "(\<exists>L)"} and
@{text "(\<forall>R)"} by @{text "(\<exists>I)"}, @{text "(\<exists>E)"} and @{text "(\<forall>I)"},
respectively, and put @{text "(\<forall>E)"} into elimination form: @{text
[display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"}
Elim-resolution with this rule will delete the universal formula
after a single use. To replicate universal quantifiers, replace the
rule by @{text [display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"}
To replicate existential quantifiers, replace @{text "(\<exists>I)"} by
@{text [display] "(\<not> (\<exists>x. P x) \<Longrightarrow> P t) \<Longrightarrow> \<exists>x. P x"}
All introduction rules mentioned above are also useful in swapped
form.
Replication makes the search space infinite; we must apply the rules
with care. The classical reasoner distinguishes between safe and
unsafe rules, applying the latter only when there is no alternative.
Depth-first search may well go down a blind alley; best-first search
is better behaved in an infinite search space. However, quantifier
replication is too expensive to prove any but the simplest theorems.
*}
subsection {* Rule declarations *}
text {* The proof tools of the Classical Reasoner depend on
collections of rules declared in the context, which are classified
as introduction, elimination or destruction and as \emph{safe} or
\emph{unsafe}. In general, safe rules can be attempted blindly,
while unsafe rules must be used with care. A safe rule must never
reduce a provable goal to an unprovable set of subgoals.
The rule @{text "P \<Longrightarrow> P \<or> Q"} is unsafe because it reduces @{text "P
\<or> Q"} to @{text "P"}, which might turn out as premature choice of an
unprovable subgoal. Any rule is unsafe whose premises contain new
unknowns. The elimination rule @{text "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"} is
unsafe, since it is applied via elim-resolution, which discards the
assumption @{text "\<forall>x. P x"} and replaces it by the weaker
assumption @{text "P t"}. The rule @{text "P t \<Longrightarrow> \<exists>x. P x"} is
unsafe for similar reasons. The quantifier duplication rule @{text
"\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"} is unsafe in a different sense:
since it keeps the assumption @{text "\<forall>x. P x"}, it is prone to
looping. In classical first-order logic, all rules are safe except
those mentioned above.
The safe~/ unsafe distinction is vague, and may be regarded merely
as a way of giving some rules priority over others. One could argue
that @{text "(\<or>E)"} is unsafe, because repeated application of it
could generate exponentially many subgoals. Induction rules are
unsafe because inductive proofs are difficult to set up
automatically. Any inference is unsafe that instantiates an unknown
in the proof state --- thus matching must be used, rather than
unification. Even proof by assumption is unsafe if it instantiates
unknowns shared with other subgoals.
\begin{matharray}{rcl}
@{command_def "print_claset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{attribute_def intro} & : & @{text attribute} \\
@{attribute_def elim} & : & @{text attribute} \\
@{attribute_def dest} & : & @{text attribute} \\
@{attribute_def rule} & : & @{text attribute} \\
@{attribute_def iff} & : & @{text attribute} \\
@{attribute_def swapped} & : & @{text attribute} \\
\end{matharray}
@{rail "
(@@{attribute intro} | @@{attribute elim} | @@{attribute dest}) ('!' | () | '?') @{syntax nat}?
;
@@{attribute rule} 'del'
;
@@{attribute iff} (((() | 'add') '?'?) | 'del')
"}
\begin{description}
\item @{command "print_claset"} prints the collection of rules
declared to the Classical Reasoner, i.e.\ the @{ML_type claset}
within the context.
\item @{attribute intro}, @{attribute elim}, and @{attribute dest}
declare introduction, elimination, and destruction rules,
respectively. By default, rules are considered as \emph{unsafe}
(i.e.\ not applied blindly without backtracking), while ``@{text
"!"}'' classifies as \emph{safe}. Rule declarations marked by
``@{text "?"}'' coincide with those of Isabelle/Pure, cf.\
\secref{sec:pure-meth-att} (i.e.\ are only applied in single steps
of the @{method rule} method). The optional natural number
specifies an explicit weight argument, which is ignored by the
automated reasoning tools, but determines the search order of single
rule steps.
Introduction rules are those that can be applied using ordinary
resolution. Their swapped forms are generated internally, which
will be applied using elim-resolution. Elimination rules are
applied using elim-resolution. Rules are sorted by the number of
new subgoals they will yield; rules that generate the fewest
subgoals will be tried first. Otherwise, later declarations take
precedence over earlier ones.
Rules already present in the context with the same classification
are ignored. A warning is printed if the rule has already been
added with some other classification, but the rule is added anyway
as requested.
\item @{attribute rule}~@{text del} deletes all occurrences of a
rule from the classical context, regardless of its classification as
introduction~/ elimination~/ destruction and safe~/ unsafe.
\item @{attribute iff} declares logical equivalences to the
Simplifier and the Classical reasoner at the same time.
Non-conditional rules result in a safe introduction and elimination
pair; conditional ones are considered unsafe. Rules with negative
conclusion are automatically inverted (using @{text "\<not>"}-elimination
internally).
The ``@{text "?"}'' version of @{attribute iff} declares rules to
the Isabelle/Pure context only, and omits the Simplifier
declaration.
\item @{attribute swapped} turns an introduction rule into an
elimination, by resolving with the classical swap principle @{text
"\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"} in the second position. This is mainly for
illustrative purposes: the Classical Reasoner already swaps rules
internally as explained above.
\end{description}
*}
subsection {* Structured methods *}
text {*
\begin{matharray}{rcl}
@{method_def rule} & : & @{text method} \\
@{method_def contradiction} & : & @{text method} \\
\end{matharray}
@{rail "
@@{method rule} @{syntax thmrefs}?
"}
\begin{description}
\item @{method rule} as offered by the Classical Reasoner is a
refinement over the Pure one (see \secref{sec:pure-meth-att}). Both
versions work the same, but the classical version observes the
classical rule context in addition to that of Isabelle/Pure.
Common object logics (HOL, ZF, etc.) declare a rich collection of
classical rules (even if these would qualify as intuitionistic
ones), but only few declarations to the rule context of
Isabelle/Pure (\secref{sec:pure-meth-att}).
\item @{method contradiction} solves some goal by contradiction,
deriving any result from both @{text "\<not> A"} and @{text A}. Chained
facts, which are guaranteed to participate, may appear in either
order.
\end{description}
*}
subsection {* Automated methods *}
text {*
\begin{matharray}{rcl}
@{method_def blast} & : & @{text method} \\
@{method_def auto} & : & @{text method} \\
@{method_def force} & : & @{text method} \\
@{method_def fast} & : & @{text method} \\
@{method_def slow} & : & @{text method} \\
@{method_def best} & : & @{text method} \\
@{method_def fastforce} & : & @{text method} \\
@{method_def slowsimp} & : & @{text method} \\
@{method_def bestsimp} & : & @{text method} \\
@{method_def deepen} & : & @{text method} \\
\end{matharray}
@{rail "
@@{method blast} @{syntax nat}? (@{syntax clamod} * )
;
@@{method auto} (@{syntax nat} @{syntax nat})? (@{syntax clasimpmod} * )
;
@@{method force} (@{syntax clasimpmod} * )
;
(@@{method fast} | @@{method slow} | @@{method best}) (@{syntax clamod} * )
;
(@@{method fastforce} | @@{method slowsimp} | @@{method bestsimp})
(@{syntax clasimpmod} * )
;
@@{method deepen} (@{syntax nat} ?) (@{syntax clamod} * )
;
@{syntax_def clamod}:
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' @{syntax thmrefs}
;
@{syntax_def clasimpmod}: ('simp' (() | 'add' | 'del' | 'only') |
('cong' | 'split') (() | 'add' | 'del') |
'iff' (((() | 'add') '?'?) | 'del') |
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' @{syntax thmrefs}
"}
\begin{description}
\item @{method blast} is a separate classical tableau prover that
uses the same classical rule declarations as explained before.
Proof search is coded directly in ML using special data structures.
A successful proof is then reconstructed using regular Isabelle
inferences. It is faster and more powerful than the other classical
reasoning tools, but has major limitations too.
\begin{itemize}
\item It does not use the classical wrapper tacticals, such as the
integration with the Simplifier of @{method fastforce}.
\item It does not perform higher-order unification, as needed by the
rule @{thm [source=false] rangeI} in HOL. There are often
alternatives to such rules, for example @{thm [source=false]
range_eqI}.
\item Function variables may only be applied to parameters of the
subgoal. (This restriction arises because the prover does not use
higher-order unification.) If other function variables are present
then the prover will fail with the message \texttt{Function Var's
argument not a bound variable}.
\item Its proof strategy is more general than @{method fast} but can
be slower. If @{method blast} fails or seems to be running forever,
try @{method fast} and the other proof tools described below.
\end{itemize}
The optional integer argument specifies a bound for the number of
unsafe steps used in a proof. By default, @{method blast} starts
with a bound of 0 and increases it successively to 20. In contrast,
@{text "(blast lim)"} tries to prove the goal using a search bound
of @{text "lim"}. Sometimes a slow proof using @{method blast} can
be made much faster by supplying the successful search bound to this
proof method instead.
\item @{method auto} combines classical reasoning with
simplification. It is intended for situations where there are a lot
of mostly trivial subgoals; it proves all the easy ones, leaving the
ones it cannot prove. Occasionally, attempting to prove the hard
ones may take a long time.
The optional depth arguments in @{text "(auto m n)"} refer to its
builtin classical reasoning procedures: @{text m} (default 4) is for
@{method blast}, which is tried first, and @{text n} (default 2) is
for a slower but more general alternative that also takes wrappers
into account.
\item @{method force} is intended to prove the first subgoal
completely, using many fancy proof tools and performing a rather
exhaustive search. As a result, proof attempts may take rather long
or diverge easily.
\item @{method fast}, @{method best}, @{method slow} attempt to
prove the first subgoal using sequent-style reasoning as explained
before. Unlike @{method blast}, they construct proofs directly in
Isabelle.
There is a difference in search strategy and back-tracking: @{method
fast} uses depth-first search and @{method best} uses best-first
search (guided by a heuristic function: normally the total size of
the proof state).
Method @{method slow} is like @{method fast}, but conducts a broader
search: it may, when backtracking from a failed proof attempt, undo
even the step of proving a subgoal by assumption.
\item @{method fastforce}, @{method slowsimp}, @{method bestsimp}
are like @{method fast}, @{method slow}, @{method best},
respectively, but use the Simplifier as additional wrapper. The name
@{method fastforce}, reflects the behaviour of this popular method
better without requiring an understanding of its implementation.
\item @{method deepen} works by exhaustive search up to a certain
depth. The start depth is 4 (unless specified explicitly), and the
depth is increased iteratively up to 10. Unsafe rules are modified
to preserve the formula they act on, so that it be used repeatedly.
This method can prove more goals than @{method fast}, but is much
slower, for example if the assumptions have many universal
quantifiers.
\end{description}
Any of the above methods support additional modifiers of the context
of classical (and simplifier) rules, but the ones related to the
Simplifier are explicitly prefixed by @{text simp} here. The
semantics of these ad-hoc rule declarations is analogous to the
attributes given before. Facts provided by forward chaining are
inserted into the goal before commencing proof search.
*}
subsection {* Semi-automated methods *}
text {* These proof methods may help in situations when the
fully-automated tools fail. The result is a simpler subgoal that
can be tackled by other means, such as by manual instantiation of
quantifiers.
\begin{matharray}{rcl}
@{method_def safe} & : & @{text method} \\
@{method_def clarify} & : & @{text method} \\
@{method_def clarsimp} & : & @{text method} \\
\end{matharray}
@{rail "
(@@{method safe} | @@{method clarify}) (@{syntax clamod} * )
;
@@{method clarsimp} (@{syntax clasimpmod} * )
"}
\begin{description}
\item @{method safe} repeatedly performs safe steps on all subgoals.
It is deterministic, with at most one outcome.
\item @{method clarify} performs a series of safe steps without
splitting subgoals; see also @{ML clarify_step_tac}.
\item @{method clarsimp} acts like @{method clarify}, but also does
simplification. Note that if the Simplifier context includes a
splitter for the premises, the subgoal may still be split.
\end{description}
*}
subsection {* Single-step tactics *}
text {*
\begin{matharray}{rcl}
@{index_ML safe_step_tac: "Proof.context -> int -> tactic"} \\
@{index_ML inst_step_tac: "Proof.context -> int -> tactic"} \\
@{index_ML step_tac: "Proof.context -> int -> tactic"} \\
@{index_ML slow_step_tac: "Proof.context -> int -> tactic"} \\
@{index_ML clarify_step_tac: "Proof.context -> int -> tactic"} \\
\end{matharray}
These are the primitive tactics behind the (semi)automated proof
methods of the Classical Reasoner. By calling them yourself, you
can execute these procedures one step at a time.
\begin{description}
\item @{ML safe_step_tac}~@{text "ctxt i"} performs a safe step on
subgoal @{text i}. The safe wrapper tacticals are applied to a
tactic that may include proof by assumption or Modus Ponens (taking
care not to instantiate unknowns), or substitution.
\item @{ML inst_step_tac} is like @{ML safe_step_tac}, but allows
unknowns to be instantiated.
\item @{ML step_tac}~@{text "ctxt i"} is the basic step of the proof
procedure. The unsafe wrapper tacticals are applied to a tactic
that tries @{ML safe_tac}, @{ML inst_step_tac}, or applies an unsafe
rule from the context.
\item @{ML slow_step_tac} resembles @{ML step_tac}, but allows
backtracking between using safe rules with instantiation (@{ML
inst_step_tac}) and using unsafe rules. The resulting search space
is larger.
\item @{ML clarify_step_tac}~@{text "ctxt i"} performs a safe step
on subgoal @{text i}. No splitting step is applied; for example,
the subgoal @{text "A \<and> B"} is left as a conjunction. Proof by
assumption, Modus Ponens, etc., may be performed provided they do
not instantiate unknowns. Assumptions of the form @{text "x = t"}
may be eliminated. The safe wrapper tactical is applied.
\end{description}
*}
section {* Object-logic setup \label{sec:object-logic} *}
text {*
\begin{matharray}{rcl}
@{command_def "judgment"} & : & @{text "theory \<rightarrow> theory"} \\
@{method_def atomize} & : & @{text method} \\
@{attribute_def atomize} & : & @{text attribute} \\
@{attribute_def rule_format} & : & @{text attribute} \\
@{attribute_def rulify} & : & @{text attribute} \\
\end{matharray}
The very starting point for any Isabelle object-logic is a ``truth
judgment'' that links object-level statements to the meta-logic
(with its minimal language of @{text prop} that covers universal
quantification @{text "\<And>"} and implication @{text "\<Longrightarrow>"}).
Common object-logics are sufficiently expressive to internalize rule
statements over @{text "\<And>"} and @{text "\<Longrightarrow>"} within their own
language. This is useful in certain situations where a rule needs
to be viewed as an atomic statement from the meta-level perspective,
e.g.\ @{text "\<And>x. x \<in> A \<Longrightarrow> P x"} versus @{text "\<forall>x \<in> A. P x"}.
From the following language elements, only the @{method atomize}
method and @{attribute rule_format} attribute are occasionally
required by end-users, the rest is for those who need to setup their
own object-logic. In the latter case existing formulations of
Isabelle/FOL or Isabelle/HOL may be taken as realistic examples.
Generic tools may refer to the information provided by object-logic
declarations internally.
@{rail "
@@{command judgment} @{syntax name} '::' @{syntax type} @{syntax mixfix}?
;
@@{attribute atomize} ('(' 'full' ')')?
;
@@{attribute rule_format} ('(' 'noasm' ')')?
"}
\begin{description}
\item @{command "judgment"}~@{text "c :: \<sigma> (mx)"} declares constant
@{text c} as the truth judgment of the current object-logic. Its
type @{text \<sigma>} should specify a coercion of the category of
object-level propositions to @{text prop} of the Pure meta-logic;
the mixfix annotation @{text "(mx)"} would typically just link the
object language (internally of syntactic category @{text logic})
with that of @{text prop}. Only one @{command "judgment"}
declaration may be given in any theory development.
\item @{method atomize} (as a method) rewrites any non-atomic
premises of a sub-goal, using the meta-level equations declared via
@{attribute atomize} (as an attribute) beforehand. As a result,
heavily nested goals become amenable to fundamental operations such
as resolution (cf.\ the @{method (Pure) rule} method). Giving the ``@{text
"(full)"}'' option here means to turn the whole subgoal into an
object-statement (if possible), including the outermost parameters
and assumptions as well.
A typical collection of @{attribute atomize} rules for a particular
object-logic would provide an internalization for each of the
connectives of @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"}.
Meta-level conjunction should be covered as well (this is
particularly important for locales, see \secref{sec:locale}).
\item @{attribute rule_format} rewrites a theorem by the equalities
declared as @{attribute rulify} rules in the current object-logic.
By default, the result is fully normalized, including assumptions
and conclusions at any depth. The @{text "(no_asm)"} option
restricts the transformation to the conclusion of a rule.
In common object-logics (HOL, FOL, ZF), the effect of @{attribute
rule_format} is to replace (bounded) universal quantification
(@{text "\<forall>"}) and implication (@{text "\<longrightarrow>"}) by the corresponding
rule statements over @{text "\<And>"} and @{text "\<Longrightarrow>"}.
\end{description}
*}
end