--- a/doc-src/IsarRef/logics.tex Thu Mar 07 19:07:56 2002 +0100
+++ b/doc-src/IsarRef/logics.tex Thu Mar 07 22:52:07 2002 +0100
@@ -18,11 +18,13 @@
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 $prop$ that covers universal quantification $\Forall$ and
-implication $\Imp$). Common object-logics are sufficiently expressive to
-\emph{internalize} rule statements over $\Forall$ and $\Imp$ 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.\ $\All x x
-\in A \Imp P(x)$ versus $\forall x \in A. P(x)$).
+implication $\Imp$).
+
+Common object-logics are sufficiently expressive to internalize rule
+statements over $\Forall$ and $\Imp$ 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.\ $\All x x \in A \Imp P(x)$
+versus $\forall x \in A. P(x)$.
From the following language elements, only the $atomize$ method and
$rule_format$ attribute are occasionally required by end-users, the rest is
@@ -31,34 +33,36 @@
realistic examples.
Generic tools may refer to the information provided by object-logic
-declarations internally (e.g.\ locales \S\ref{sec:locale}, or the Classical
-Reasoner \S\ref{sec:classical}).
+declarations internally.
+
+\railalias{ruleformat}{rule\_format}
+\railterm{ruleformat}
\begin{rail}
'judgment' constdecl
;
- atomize ('(' 'full' ')')?
+ 'atomize' ('(' 'full' ')')?
;
ruleformat ('(' 'noasm' ')')?
;
\end{rail}
\begin{descr}
-
-\item [$\isarkeyword{judgment}~c::\sigma~~syn$] declares constant $c$ as the
+
+\item [$\isarkeyword{judgment}~c::\sigma~~(mx)$] declares constant $c$ as the
truth judgment of the current object-logic. Its type $\sigma$ should
specify a coercion of the category of object-level propositions to $prop$ of
- the Pure meta-logic; the mixfix annotation $syn$ would typically just link
+ the Pure meta-logic; the mixfix annotation $(mx)$ would typically just link
the object language (internally of syntactic category $logic$) with that of
$prop$. Only one $\isarkeyword{judgment}$ declaration may be given in any
theory development.
-
+
\item [$atomize$] (as a method) rewrites any non-atomic premises of a
sub-goal, using the meta-level equations declared via $atomize$ (as an
attribute) beforehand. As a result, heavily nested goals become amenable to
fundamental operations such as resolution (cf.\ the $rule$ method) and
proof-by-assumption (cf.\ $assumption$). Giving the ``$(full)$'' option
- here means to turn the subgoal into an object-statement (if possible),
+ 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 $atomize$ rules for a particular object-logic would
@@ -106,7 +110,7 @@
\item [$\isarkeyword{typedecl}~(\vec\alpha)t$] is similar to the original
$\isarkeyword{typedecl}$ of Isabelle/Pure (see \S\ref{sec:types-pure}), but
- also declares type arity $t :: (term, \dots, term) term$, making $t$ an
+ also declares type arity $t :: (type, \dots, type) type$, making $t$ an
actual HOL type constructor.
\item [$\isarkeyword{typedef}~(\vec\alpha)t = A$] sets up a goal stating
@@ -120,21 +124,22 @@
$Abs_t$ (this may be changed via an explicit $\isarkeyword{morphisms}$
declaration).
- Theorems $Rep_t$, $Rep_inverse$, and $Abs_inverse$ provide the most basic
- characterization as a corresponding injection/surjection pair (in both
+ Theorems $Rep_t$, $Rep_t_inverse$, and $Abs_t_inverse$ provide the most
+ basic characterization as a corresponding injection/surjection pair (in both
directions). Rules $Rep_t_inject$ and $Abs_t_inject$ provide a slightly
- more comfortable view on the injectivity part, suitable for automated proof
- tools (e.g.\ in $simp$ or $iff$ declarations). Rules $Rep_t_cases$,
- $Rep_t_induct$, and $Abs_t_cases$, $Abs_t_induct$ provide alternative views
- on surjectivity; these are already declared as type or set rules for the
- generic $cases$ and $induct$ methods.
+ more convenient view on the injectivity part, suitable for automated proof
+ tools (e.g.\ in $simp$ or $iff$ declarations). Rules
+ $Rep_t_cases/Rep_t_induct$, and $Abs_t_cases/Abs_t_induct$ provide
+ alternative views on surjectivity; these are already declared as set or type
+ rules for the generic $cases$ and $induct$ methods.
\end{descr}
-Raw type declarations are rarely used in practice; the main application is
-with experimental (or even axiomatic!) theory fragments. Instead of primitive
-HOL type definitions, user-level theories usually refer to higher-level
-packages such as $\isarkeyword{record}$ (see \S\ref{sec:hol-record}) or
-$\isarkeyword{datatype}$ (see \S\ref{sec:hol-datatype}).
+Note that raw type declarations are rarely used in practice; the main
+application is with experimental (or even axiomatic!) theory fragments.
+Instead of primitive HOL type definitions, user-level theories usually refer
+to higher-level packages such as $\isarkeyword{record}$ (see
+\S\ref{sec:hol-record}) or $\isarkeyword{datatype}$ (see
+\S\ref{sec:hol-datatype}).
\subsection{Adhoc tuples}
@@ -153,13 +158,12 @@
\end{rail}
\begin{descr}
-
+
\item [$split_format~\vec p@1 \dots \vec p@n$] puts expressions of low-level
tuple types into canonical form as specified by the arguments given; $\vec
- p@i$ refers to occurrences in premise $i$ of the rule. The
- $split_format~(complete)$ form causes \emph{all} arguments in function
- applications to be represented canonically according to their tuple type
- structure.
+ p@i$ refers to occurrences in premise $i$ of the rule. The $(complete)$
+ option causes \emph{all} arguments in function applications to be
+ represented canonically according to their tuple type structure.
Note that these operations tend to invent funny names for new local
parameters to be introduced.
@@ -169,8 +173,8 @@
\subsection{Records}\label{sec:hol-record}
-In principle, records merely generalize the concept of tuples where components
-may be addressed by labels instead of just position. The logical
+In principle, records merely generalize the concept of tuples, where
+components may be addressed by labels instead of just position. The logical
infrastructure of records in Isabelle/HOL is slightly more advanced, though,
supporting truly extensible record schemes. This admits operations that are
polymorphic with respect to record extension, yielding ``object-oriented''
@@ -203,8 +207,8 @@
``$\more$'' notation (which is actually part of the syntax). The improper
field ``$\more$'' of a record scheme is called the \emph{more part}.
Logically it is just a free variable, which is occasionally referred to as
-\emph{row variable} in the literature. The more part of a record scheme may
-be instantiated by zero or more further components. For example, the above
+``row variable'' in the literature. The more part of a record scheme may be
+instantiated by zero or more further components. For example, the previous
scheme may get instantiated to $\record{x = a\fs y = b\fs z = c\fs \more =
m'}$, where $m'$ refers to a different more part. Fixed records are special
instances of record schemes, where ``$\more$'' is properly terminated by the
@@ -295,11 +299,11 @@
reverse than in the actual term. Since repeated updates are just function
applications, fields may be freely permuted in $\record{x \asn a\fs y \asn
b\fs z \asn c}$, as far as logical equality is concerned. Thus
-commutativity of updates can be proven within the logic for any two fields,
-but not as a general theorem: fields are not first-class values.
+commutativity of independent updates can be proven within the logic for any
+two fields, but not as a general theorem.
\medskip The \textbf{make} operation provides a cumulative record constructor
-functions:
+function:
\begin{matharray}{lll}
t{\dtt}make & \ty & \vec\sigma \To \record{\vec c \ty \vec \sigma} \\
\end{matharray}
@@ -336,25 +340,26 @@
\record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma} \\
\end{matharray}
-\noindent Note that $t{\dtt}make$ and $t{\dtt}fields$ are actually coincide for root records.
+\noindent Note that $t{\dtt}make$ and $t{\dtt}fields$ actually coincide for root records.
\subsubsection{Derived rules and proof tools}\label{sec:hol-record-thms}
The record package proves several results internally, declaring these facts to
appropriate proof tools. This enables users to reason about record structures
-quite comfortably. Assume that $t$ is a record type as specified above.
+quite conveniently. Assume that $t$ is a record type as specified above.
\begin{enumerate}
-
+
\item Standard conversions for selectors or updates applied to record
constructor terms are made part of the default Simplifier context; thus
proofs by reduction of basic operations merely require the $simp$ method
- without further arguments. These rules are available as $t{\dtt}simps$.
-
+ without further arguments. These rules are available as $t{\dtt}simps$,
+ too.
+
\item Selectors applied to updated records are automatically reduced by an
- internal simplification procedure, which is also part of the default
- Simplifier context.
+ internal simplification procedure, which is also part of the standard
+ Simplifier setup.
\item Inject equations of a form analogous to $((x, y) = (x', y')) \equiv x=x'
\conj y=y'$ are declared to the Simplifier and Classical Reasoner as $iff$
@@ -368,10 +373,10 @@
terms are provided both in $cases$ and $induct$ format (cf.\ the generic
proof methods of the same name, \S\ref{sec:cases-induct}). Several
variations are available, for fixed records, record schemes, more parts etc.
-
+
The generic proof methods are sufficiently smart to pick the most sensible
rule according to the type of the indicated record expression: users just
- need to apply something like ``$(cases r)$'' to a certain proof problem.
+ need to apply something like ``$(cases~r)$'' to a certain proof problem.
\item The derived record operations $t{\dtt}make$, $t{\dtt}fields$,
$t{\dtt}extend$, $t{\dtt}truncate$ are \emph{not} treated automatically, but
@@ -471,7 +476,7 @@
\item [$\isarkeyword{recdef}$] defines general well-founded recursive
functions (using the TFL package), see also \cite{isabelle-HOL}. The
- $(permissive)$ option tells TFL to recover from failed proof attempts,
+ ``$(permissive)$'' option tells TFL to recover from failed proof attempts,
returning unfinished results. The $recdef_simp$, $recdef_cong$, and
$recdef_wf$ hints refer to auxiliary rules to be used in the internal
automated proof process of TFL. Additional $clasimpmod$ declarations (cf.\
@@ -496,8 +501,8 @@
$\isarkeyword{recdef}$ are numbered (starting from $1$).
The equations provided by these packages may be referred later as theorem list
-$f\mathord.simps$, where $f$ is the (collective) name of the functions
-defined. Individual equations may be named explicitly as well; note that for
+$f{\dtt}simps$, where $f$ is the (collective) name of the functions defined.
+Individual equations may be named explicitly as well; note that for
$\isarkeyword{recdef}$ each specification given by the user may result in
several theorems.
@@ -631,10 +636,10 @@
both goal addressing and dynamic instantiation. Note that named rule cases
are \emph{not} provided as would be by the proper $induct$ and $cases$ proof
methods (see \S\ref{sec:cases-induct}).
-
+
\item [$ind_cases$ and $\isarkeyword{inductive_cases}$] provide an interface
- to the \texttt{mk_cases} operation. Rules are simplified in an unrestricted
- forward manner.
+ to the internal \texttt{mk_cases} operation. Rules are simplified in an
+ unrestricted forward manner.
While $ind_cases$ is a proof method to apply the result immediately as
elimination rules, $\isarkeyword{inductive_cases}$ provides case split
@@ -648,7 +653,7 @@
from executable specifications, both functional and relational programs.
Isabelle/HOL instantiates these mechanisms in a way that is amenable to
end-user applications. See \cite{isabelle-HOL} for further information (this
-actually covers the new-style theory format).
+actually covers the new-style theory format as well).
\indexisarcmd{generate-code}\indexisarcmd{consts-code}\indexisarcmd{types-code}
\indexisaratt{code}
@@ -727,10 +732,10 @@
dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
\end{rail}
-Recursive domains in HOLCF are analogous to datatypes in classical HOL (cf.\
-\S\ref{sec:hol-datatype}). Mutual recursive is supported, but no nesting nor
+Recursive domains in HOLCF are analogous to datatypes in classical HOL (cf.\
+\S\ref{sec:hol-datatype}). Mutual recursion is supported, but no nesting nor
arbitrary branching. Domain constructors may be strict (default) or lazy, the
-latter admits to introduce infinitary objects in the typical LCF manner (e.g.\
+latter admits to introduce infinitary objects in the typical LCF manner (e.g.\
lazy lists). See also \cite{MuellerNvOS99} for a general discussion of HOLCF
domains.
@@ -742,7 +747,7 @@
The ZF logic is essentially untyped, so the concept of ``type checking'' is
performed as logical reasoning about set-membership statements. A special
method assists users in this task; a version of this is already declared as a
-``solver'' in the default Simplifier context.
+``solver'' in the standard Simplifier setup.
\indexisarcmd{print-tcset}\indexisaratt{typecheck}\indexisaratt{TC}
@@ -779,7 +784,7 @@
In ZF everything is a set. The generic inductive package also provides a
specific view for ``datatype'' specifications. Coinductive definitions are
-available as well.
+available in both cases, too.
\indexisarcmdof{ZF}{inductive}\indexisarcmdof{ZF}{coinductive}
\indexisarcmdof{ZF}{datatype}\indexisarcmdof{ZF}{codatatype}