diff -r e502690952be -r 98ba42f19995 doc-src/IsarImplementation/Thy/document/logic.tex --- 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