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doc-src/IsarImplementation/Thy/document/logic.tex

changeset 20514 | 5ede702cd2ca |

parent 20502 | 08d227db6c74 |

child 20519 | d7ad1217c24a |

--- a/doc-src/IsarImplementation/Thy/document/logic.tex Tue Sep 12 12:16:17 2006 +0200 +++ b/doc-src/IsarImplementation/Thy/document/logic.tex Tue Sep 12 14:50:11 2006 +0200 @@ -87,10 +87,10 @@ notably the right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}} instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}. - A \emph{type} is defined inductively over type variables and type - 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}k}. + A \emph{type} \isa{{\isasymtau}} is defined inductively over type variables + and type 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}k}. - A \emph{type abbreviation} is a syntactic abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} of an arbitrary type expression \isa{{\isasymtau}} over + A \emph{type abbreviation} is a syntactic definition \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} of an arbitrary type expression \isa{{\isasymtau}} over variables \isa{\isactrlvec {\isasymalpha}}. Type abbreviations looks like type constructors at the surface, but are fully expanded before entering the logical core. @@ -197,38 +197,124 @@ \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). + 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. + + \medskip A \emph{bound variable} is a natural number \isa{b}, + which refers to the next binder that is \isa{b} steps upwards + from the occurrence of \isa{b} (counting from zero). Bindings + may be introduced as abstractions within the term, or as a separate + context (an inside-out list). This associates each bound variable + with a type, and a name that is maintained as a comment for parsing + and printing. A \emph{loose variables} is a bound variable that is + outside the current scope of local binders or the context. For + example, the de-Bruijn term \isa{{\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ {\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ {\isadigit{1}}\ {\isacharplus}\ {\isadigit{0}}} + corresponds to \isa{{\isasymlambda}x\isactrlisub {\isasymtau}{\isachardot}\ {\isasymlambda}y\isactrlisub {\isasymtau}{\isachardot}\ x\ {\isacharplus}\ y} in a named + representation. Also note that the very same bound variable may get + different numbers at different occurrences. + + A \emph{fixed variable} is a pair of a basic name and a type. For + example, \isa{{\isacharparenleft}x{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed \isa{x\isactrlisub {\isasymtau}}. A \emph{schematic variable} is a pair of an + indexname and a type. For example, \isa{{\isacharparenleft}{\isacharparenleft}x{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is + usually printed as \isa{{\isacharquery}x\isactrlisub {\isasymtau}}. - FIXME de-Bruijn representation of lambda terms + \medskip A \emph{constant} is a atomic terms consisting of a basic + name and a type. Constants are declared in the context as + polymorphic families \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}, meaning that any \isa{c\isactrlisub {\isasymtau}} is a valid constant for all substitution instances + \isa{{\isasymtau}\ {\isasymle}\ {\isasymsigma}}. + + The list of \emph{type arguments} of \isa{c\isactrlisub {\isasymtau}} wrt.\ the + declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} is the codomain of the type matcher + presented in canonical order (according to the left-to-right + occurrences of type variables in in \isa{{\isasymsigma}}). Thus \isa{c\isactrlisub {\isasymtau}} can be represented more compactly as \isa{c{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharparenright}}. For example, the instance \isa{plus\isactrlbsub nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat\isactrlesub } of some \isa{plus\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}} has the singleton list \isa{nat} as type arguments, the + constant may be represented as \isa{plus{\isacharparenleft}nat{\isacharparenright}}. - Term syntax provides explicit abstraction \isa{{\isasymlambda}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ b{\isacharparenleft}x{\isacharparenright}} - and application \isa{t\ u}, while types are usually implicit - thanks to type-inference. + Constant declarations \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} may contain sort constraints + for type variables in \isa{{\isasymsigma}}. These are observed by + type-inference as expected, but \emph{ignored} by the core logic. + This means the primitive logic is able to reason with instances of + polymorphic constants that the user-level type-checker would reject. + \medskip A \emph{term} \isa{t} is defined inductively over + variables and constants, with abstraction and application as + follows: \isa{t\ {\isacharequal}\ b\ {\isacharbar}\ x\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isacharquery}x\isactrlisub {\isasymtau}\ {\isacharbar}\ c\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ t\ {\isacharbar}\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}}. Parsing and printing takes + care of converting between an external representation with named + bound variables. Subsequently, we shall use the latter notation + instead of internal de-Bruijn representation. + The subsequent inductive relation \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} assigns a + (unique) type to a term, using the special type constructor \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}, which is written \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}}. \[ \infer{\isa{a\isactrlisub {\isasymtau}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}}{} \qquad \infer{\isa{{\isacharparenleft}{\isasymlambda}x\isactrlsub {\isasymtau}{\isachardot}\ t{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymsigma}}}{\isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}} \qquad \infer{\isa{t\ u\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}}{\isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymsigma}} & \isa{u\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}} - \]% + \] + A \emph{well-typed term} is a term that can be typed according to these rules. + + Typing information can be omitted: type-inference is able to + reconstruct the most general type of a raw term, while assigning + most general types to all of its variables and constants. + Type-inference depends on a context of type constraints for fixed + variables, and declarations for polymorphic constants. + + The identity of atomic terms consists both of the name and the type. + Thus different entities \isa{c\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{1}}\isactrlesub } and + \isa{c\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{2}}\isactrlesub } may well identified by type + instantiation, by mapping \isa{{\isasymtau}\isactrlisub {\isadigit{1}}} and \isa{{\isasymtau}\isactrlisub {\isadigit{2}}} to the same \isa{{\isasymtau}}. Although, + different type instances of constants of the same basic name are + commonplace, this rarely happens for variables: type-inference + always demands ``consistent'' type constraints. + + \medskip The \emph{hidden polymorphism} of a term \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} + is the set of type variables occurring in \isa{t}, but not in + \isa{{\isasymsigma}}. This means that the term implicitly depends on the + values of various type variables that are not visible in the overall + type, i.e.\ there are different type instances \isa{t{\isasymvartheta}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} and \isa{t{\isasymvartheta}{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} with the same type. This + slightly pathological situation is apt to cause strange effects. + + \medskip A \emph{term abbreviation} is a syntactic definition \isa{c\isactrlisub {\isasymsigma}\ {\isasymequiv}\ t} of an arbitrary closed term \isa{t} of type + \isa{{\isasymsigma}} without any hidden polymorphism. A term abbreviation + looks like a constant at the surface, but is fully expanded before + entering the logical core. Abbreviations are usually reverted when + printing terms, using rules \isa{t\ {\isasymrightarrow}\ c\isactrlisub {\isasymsigma}} has a + higher-order term rewrite system.% \end{isamarkuptext}% \isamarkuptrue% % +\isadelimmlref +% +\endisadelimmlref +% +\isatagmlref +% \begin{isamarkuptext}% -FIXME +\begin{mldecls} + \indexmltype{term}\verb|type term| \\ + \indexml{map-aterms}\verb|map_aterms: (term -> term) -> term -> term| \\ + \indexml{fold-aterms}\verb|fold_aterms: (term -> 'a -> 'a) -> term -> 'a -> 'a| \\ + \indexml{fastype-of}\verb|fastype_of: term -> typ| \\ + \indexml{fold-types}\verb|fold_types: (typ -> 'a -> 'a) -> term -> 'a -> 'a| \\ + \end{mldecls} -\glossary{Schematic polymorphism}{FIXME} + \begin{description} -\glossary{Type variable}{FIXME}% + \item \verb|term| FIXME + + \end{description}% \end{isamarkuptext}% \isamarkuptrue% % +\endisatagmlref +{\isafoldmlref}% +% +\isadelimmlref +% +\endisadelimmlref +% \isamarkupsection{Theorems \label{sec:thms}% } \isamarkuptrue% @@ -326,7 +412,7 @@ \medskip The axiomatization of a theory is implicitly closed by forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymtheta}} for - any substirution instance of axiom \isa{{\isasymturnstile}\ A}. By pushing + any substitution instance of axiom \isa{{\isasymturnstile}\ A}. By pushing substitution through derivations inductively, we get admissible substitution rules for theorems shown in \figref{fig:subst-rules}.