--- a/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 07 15:16:51 2006 +0200
+++ b/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 07 20:12:08 2006 +0200
@@ -28,15 +28,18 @@
which has been introduced as a natural-deduction framework in
\cite{paulson700}. This is essentially the same logic as ``\isa{{\isasymlambda}HOL}'' in the more abstract framework of Pure Type Systems (PTS)
\cite{Barendregt-Geuvers:2001}, although there are some key
- differences in the practical treatment of simple types.
+ differences in the specific treatment of simple types in
+ Isabelle/Pure.
Following type-theoretic parlance, the Pure logic consists of three
levels of \isa{{\isasymlambda}}-calculus with corresponding arrows: \isa{{\isasymRightarrow}} for syntactic function space (terms depending on terms), \isa{{\isasymAnd}} for universal quantification (proofs depending on terms), and
\isa{{\isasymLongrightarrow}} for implication (proofs depending on proofs).
Pure derivations are relative to a logical theory, which declares
- type constructors, term constants, and axioms. Term constants and
- axioms support schematic polymorphism.%
+ type constructors, term constants, and axioms. Theory declarations
+ support schematic polymorphism, which is strictly speaking outside
+ the logic.\footnote{Incidently, this is the main logical reason, why
+ the theory context \isa{{\isasymTheta}} is separate from the context \isa{{\isasymGamma}} of the core calculus.}%
\end{isamarkuptext}%
\isamarkuptrue%
%
@@ -50,38 +53,42 @@
\medskip A \emph{type class} is an abstract syntactic entity
declared in the theory context. The \emph{subclass relation} \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}} is specified by stating an acyclic
- generating relation; the transitive closure maintained internally.
+ generating relation; the transitive closure is maintained
+ internally. The resulting relation is an ordering: reflexive,
+ transitive, and antisymmetric.
A \emph{sort} is a list of type classes written as \isa{{\isacharbraceleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlisub m{\isacharbraceright}}, which represents symbolic
intersection. Notationally, the curly braces are omitted for
singleton intersections, i.e.\ any class \isa{c} may be read as
a sort \isa{{\isacharbraceleft}c{\isacharbraceright}}. The ordering on type classes is extended to
- sorts in the canonical fashion: \isa{{\isacharbraceleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}\ c\isactrlisub m{\isacharbraceright}\ {\isasymsubseteq}\ {\isacharbraceleft}d\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ d\isactrlisub n{\isacharbraceright}} iff \isa{{\isasymforall}j{\isachardot}\ {\isasymexists}i{\isachardot}\ c\isactrlisub i\ {\isasymsubseteq}\ d\isactrlisub j}. The empty intersection \isa{{\isacharbraceleft}{\isacharbraceright}} refers to the
- universal sort, which is the largest element wrt.\ the sort order.
- The intersections of all (i.e.\ finitely many) classes declared in
- the current theory are the minimal elements wrt.\ sort order.
+ sorts according to the meaning of intersections: \isa{{\isacharbraceleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}\ c\isactrlisub m{\isacharbraceright}\ {\isasymsubseteq}\ {\isacharbraceleft}d\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ d\isactrlisub n{\isacharbraceright}} iff
+ \isa{{\isasymforall}j{\isachardot}\ {\isasymexists}i{\isachardot}\ c\isactrlisub i\ {\isasymsubseteq}\ d\isactrlisub j}. The empty intersection
+ \isa{{\isacharbraceleft}{\isacharbraceright}} refers to the universal sort, which is the largest
+ element wrt.\ the sort order. The intersections of all (finitely
+ many) classes declared in the current theory are the minimal
+ elements wrt.\ the sort order.
- \medskip A \emph{fixed type variable} is pair of a basic name
+ \medskip A \emph{fixed type variable} is a pair of a basic name
(starting with \isa{{\isacharprime}} character) and a sort constraint. For
example, \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ s{\isacharparenright}} which is usually printed as \isa{{\isasymalpha}\isactrlisub s}. A \emph{schematic type variable} is a pair of an
- indexname and a sort constraint. For example, \isa{{\isacharparenleft}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ s{\isacharparenright}} which is usually printed \isa{{\isacharquery}{\isasymalpha}\isactrlisub s}.
+ indexname and a sort constraint. For example, \isa{{\isacharparenleft}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ s{\isacharparenright}} which is usually printed as \isa{{\isacharquery}{\isasymalpha}\isactrlisub s}.
Note that \emph{all} syntactic components contribute to the identity
- of a type variables, including the literal sort constraint. The
- core logic handles type variables with the same name but different
- sorts as different, even though the outer layers of the system make
- it hard to produce anything like this.
+ of type variables, including the literal sort constraint. The core
+ logic handles type variables with the same name but different sorts
+ as different, although some outer layers of the system make it hard
+ to produce anything like this.
- A \emph{type constructor} is an \isa{k}-ary type operator
- declared in the theory.
+ A \emph{type constructor} is a \isa{k}-ary operator on types
+ declared in the theory. Type constructor application is usually
+ written postfix. For \isa{k\ {\isacharequal}\ {\isadigit{0}}} the argument tuple is omitted,
+ e.g.\ \isa{prop} instead of \isa{{\isacharparenleft}{\isacharparenright}prop}. For \isa{k\ {\isacharequal}\ {\isadigit{1}}} the parentheses are omitted, e.g.\ \isa{{\isasymalpha}\ list} instead of
+ \isa{{\isacharparenleft}{\isasymalpha}{\isacharparenright}list}. Further notation is provided for specific
+ constructors, notably right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}}
+ instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun} constructor.
A \emph{type} is defined inductively over type variables and type
- constructors: \isa{{\isasymtau}\ {\isacharequal}\ {\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharquery}{\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharparenleft}{\isasymtau}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlsub k{\isacharparenright}c}. Type constructor application is usually written
- postfix. For \isa{k\ {\isacharequal}\ {\isadigit{0}}} the argument tuple is omitted, e.g.\
- \isa{prop} instead of \isa{{\isacharparenleft}{\isacharparenright}prop}. For \isa{k\ {\isacharequal}\ {\isadigit{1}}} the
- parentheses are omitted, e.g.\ \isa{{\isasymtau}\ list} instead of \isa{{\isacharparenleft}{\isasymtau}{\isacharparenright}\ list}. Further notation is provided for specific
- constructors, notably right-associative infix \isa{{\isasymtau}\isactrlisub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymtau}\isactrlisub {\isadigit{2}}} instead of \isa{{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymtau}\isactrlisub {\isadigit{2}}{\isacharparenright}fun}
- constructor.
+ constructors as follows: \isa{{\isasymtau}\ {\isacharequal}\ {\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharquery}{\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharparenleft}{\isasymtau}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlsub k{\isacharparenright}c}.
A \emph{type abbreviation} is a syntactic abbreviation of an
arbitrary type expression of the theory. Type abbreviations looks
@@ -89,21 +96,23 @@
core logic encounters them.
A \emph{type arity} declares the image behavior of a type
- constructor wrt.\ the algebra of sorts: \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub n{\isacharparenright}s} means that \isa{{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub k{\isacharparenright}c} is
+ constructor wrt.\ the algebra of sorts: \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub k{\isacharparenright}s} means that \isa{{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub k{\isacharparenright}c} is
of sort \isa{s} if each argument type \isa{{\isasymtau}\isactrlisub i} is of
- sort \isa{s\isactrlisub i}. The sort algebra is always maintained as
- \emph{coregular}, which means that type arities are consistent with
- the subclass relation: for each type constructor \isa{c} and
- classes \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}, any arity \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{1}}{\isacharparenright}c\isactrlisub {\isadigit{1}}} has a corresponding arity \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{2}}{\isacharparenright}c\isactrlisub {\isadigit{2}}} where \isa{\isactrlvec s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ \isactrlvec s\isactrlisub {\isadigit{2}}} holds pointwise for all argument sorts.
+ sort \isa{s\isactrlisub i}. Arity declarations are implicitly
+ completed, i.e.\ \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}c} entails \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}c{\isacharprime}} for any \isa{c{\isacharprime}\ {\isasymsupseteq}\ c}.
- The key property of the order-sorted algebra of types is that sort
+ \medskip The sort algebra is always maintained as \emph{coregular},
+ which means that type arities are consistent with the subclass
+ relation: for each type constructor \isa{c} and classes \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}, any arity \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{1}}{\isacharparenright}c\isactrlisub {\isadigit{1}}} has a corresponding arity \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{2}}{\isacharparenright}c\isactrlisub {\isadigit{2}}} where \isa{\isactrlvec s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ \isactrlvec s\isactrlisub {\isadigit{2}}} holds pointwise for all argument sorts.
+
+ The key property of a coregular order-sorted algebra is that sort
constraints may be always fulfilled in a most general fashion: for
each type constructor \isa{c} and sort \isa{s} there is a
- most general vector of argument sorts \isa{{\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub k{\isacharparenright}} such that \isa{{\isacharparenleft}{\isasymtau}\isactrlbsub s\isactrlisub {\isadigit{1}}\isactrlesub {\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlbsub s\isactrlisub k\isactrlesub {\isacharparenright}} for arbitrary \isa{{\isasymtau}\isactrlisub i} of
- sort \isa{s\isactrlisub i}. This means the unification problem on
- the algebra of types has most general solutions (modulo renaming and
- equivalence of sorts). As a consequence, type-inference is able to
- produce primary types.%
+ most general vector of argument sorts \isa{{\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub k{\isacharparenright}} such that a type scheme \isa{{\isacharparenleft}{\isasymalpha}\isactrlbsub s\isactrlisub {\isadigit{1}}\isactrlesub {\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlbsub s\isactrlisub k\isactrlesub {\isacharparenright}c} is
+ of sort \isa{s}. Consequently, the unification problem on the
+ algebra of types has most general solutions (modulo renaming and
+ equivalence of sorts). Moreover, the usual type-inference algorithm
+ will produce primary types as expected \cite{nipkow-prehofer}.%
\end{isamarkuptext}%
\isamarkuptrue%
%
@@ -147,18 +156,18 @@
tests the subsort relation \isa{s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ s\isactrlisub {\isadigit{2}}}.
\item \verb|Sign.of_sort|~\isa{thy\ {\isacharparenleft}{\isasymtau}{\isacharcomma}\ s{\isacharparenright}} tests whether a type
- expression of of a given sort.
+ is of a given sort.
\item \verb|Sign.add_types|~\isa{{\isacharbrackleft}{\isacharparenleft}c{\isacharcomma}\ k{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares new
- type constructors \isa{c} with \isa{k} arguments, and
+ type constructors \isa{c} with \isa{k} arguments and
optional mixfix syntax.
\item \verb|Sign.add_tyabbrs_i|~\isa{{\isacharbrackleft}{\isacharparenleft}c{\isacharcomma}\ \isactrlvec {\isasymalpha}{\isacharcomma}\ {\isasymtau}{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}}
- defines type abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}c} (with optional
- mixfix syntax) as \isa{{\isasymtau}}.
+ defines a new type abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}c\ {\isacharequal}\ {\isasymtau}} with
+ optional mixfix syntax.
\item \verb|Sign.primitive_class|~\isa{{\isacharparenleft}c{\isacharcomma}\ {\isacharbrackleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlisub n{\isacharbrackright}{\isacharparenright}} declares new class \isa{c} derived together with
- class relations \isa{c\ {\isasymsubseteq}\ c\isactrlisub i}.
+ class relations \isa{c\ {\isasymsubseteq}\ c\isactrlisub i}, for \isa{i\ {\isacharequal}\ {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ n}.
\item \verb|Sign.primitive_classrel|~\isa{{\isacharparenleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ c\isactrlisub {\isadigit{2}}{\isacharparenright}} declares class relation \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}.
@@ -183,6 +192,13 @@
\begin{isamarkuptext}%
\glossary{Term}{FIXME}
+ The language of terms is that of simply-typed \isa{{\isasymlambda}}-calculus
+ with de-Bruijn indices for bound variables, and named free
+ variables, and constants. Terms with loose bound variables are
+ usually considered malformed. The types of variables and constants
+ is stored explicitly at each occurrence in the term (which is a
+ known performance issue).
+
FIXME de-Bruijn representation of lambda terms
Term syntax provides explicit abstraction \isa{{\isasymlambda}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ b{\isacharparenleft}x{\isacharparenright}}
@@ -203,15 +219,6 @@
\end{isamarkuptext}%
\isamarkuptrue%
%
-\isamarkupsection{Proof terms%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-FIXME%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
\isamarkupsection{Theorems \label{sec:thms}%
}
\isamarkuptrue%
@@ -267,22 +274,53 @@
\end{isamarkuptext}%
\isamarkuptrue%
%
-\isamarkupsubsection{Primitive inferences%
+\isamarkupsection{Proof terms%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
-FIXME%
+FIXME !?%
\end{isamarkuptext}%
\isamarkuptrue%
%
-\isamarkupsubsection{Higher-order resolution%
+\isamarkupsection{Rules \label{sec:rules}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
FIXME
+ A \emph{rule} is any Pure theorem in HHF normal form; there is a
+ separate calculus for rule composition, which is modeled after
+ Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
+ rules to be nested arbitrarily, similar to \cite{extensions91}.
+
+ Normally, all theorems accessible to the user are proper rules.
+ Low-level inferences are occasional required internally, but the
+ result should be always presented in canonical form. The higher
+ interfaces of Isabelle/Isar will always produce proper rules. It is
+ important to maintain this invariant in add-on applications!
+
+ There are two main principles of rule composition: \isa{resolution} (i.e.\ backchaining of rules) and \isa{by{\isacharminus}assumption} (i.e.\ closing a branch); both principles are
+ combined in the variants of \isa{elim{\isacharminus}resosultion} and \isa{dest{\isacharminus}resolution}. Raw \isa{composition} is occasionally
+ useful as well, also it is strictly speaking outside of the proper
+ rule calculus.
+
+ Rules are treated modulo general higher-order unification, which is
+ unification modulo the equational theory of \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion
+ on \isa{{\isasymlambda}}-terms. Moreover, propositions are understood modulo
+ the (derived) equivalence \isa{{\isacharparenleft}A\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x{\isacharparenright}{\isacharparenright}\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ A\ {\isasymLongrightarrow}\ B\ x{\isacharparenright}}.
+
+ This means that any operations within the rule calculus may be
+ subject to spontaneous \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-HHF conversions. It is common
+ practice not to contract or expand unnecessarily. Some mechanisms
+ prefer an one form, others the opposite, so there is a potential
+ danger to produce some oscillation!
+
+ Only few operations really work \emph{modulo} HHF conversion, but
+ expect a normal form: quantifiers \isa{{\isasymAnd}} before implications
+ \isa{{\isasymLongrightarrow}} at each level of nesting.
+
\glossary{Hereditary Harrop Formula}{The set of propositions in HHF
format is defined inductively as \isa{H\ {\isacharequal}\ {\isacharparenleft}{\isasymAnd}x\isactrlsup {\isacharasterisk}{\isachardot}\ H\isactrlsup {\isacharasterisk}\ {\isasymLongrightarrow}\ A{\isacharparenright}}, for variables \isa{x} and atomic propositions \isa{A}.
Any proposition may be put into HHF form by normalizing with the rule
@@ -295,15 +333,6 @@
\end{isamarkuptext}%
\isamarkuptrue%
%
-\isamarkupsubsection{Equational reasoning%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-FIXME%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
\isadelimtheory
%
\endisadelimtheory