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author | wenzelm |

Fri, 15 Sep 2006 16:49:41 +0200 | |

changeset 20543 | dc294418ff17 |

parent 20542 | a54ca4e90874 |

child 20544 | 893e7a9546ff |

tuned;

doc-src/IsarImplementation/Thy/document/logic.tex | file | annotate | diff | comparison | revisions | |

doc-src/IsarImplementation/Thy/logic.thy | file | annotate | diff | comparison | revisions |

--- a/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 14 22:48:37 2006 +0200 +++ b/doc-src/IsarImplementation/Thy/document/logic.tex Fri Sep 15 16:49:41 2006 +0200 @@ -32,7 +32,7 @@ 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 + 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). Derivations are relative to a logical theory, which declares type @@ -92,7 +92,7 @@ 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}. + 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}{\isasymkappa}}. 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 appear as type @@ -114,10 +114,9 @@ constructor \isa{{\isasymkappa}} 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 a type scheme \isa{{\isacharparenleft}{\isasymalpha}\isactrlbsub s\isactrlisub {\isadigit{1}}\isactrlesub {\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlbsub s\isactrlisub k\isactrlesub {\isacharparenright}{\isasymkappa}} is of sort \isa{s}. - Consequently, unification on the algebra of types has most general - solutions (modulo equivalence of sorts). This means that - type-inference will produce primary types as expected - \cite{nipkow-prehofer}.% + Consequently, type unification has most general solutions (modulo + equivalence of sorts), so type-inference produces primary types as + expected \cite{nipkow-prehofer}.% \end{isamarkuptext}% \isamarkuptrue% % @@ -182,7 +181,7 @@ \item \verb|Sign.primitive_class|~\isa{{\isacharparenleft}c{\isacharcomma}\ {\isacharbrackleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlisub n{\isacharbrackright}{\isacharparenright}} declares a new class \isa{c}, together with 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}}}. + \item \verb|Sign.primitive_classrel|~\isa{{\isacharparenleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ c\isactrlisub {\isadigit{2}}{\isacharparenright}} declares the class relation \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}. \item \verb|Sign.primitive_arity|~\isa{{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec s{\isacharcomma}\ s{\isacharparenright}} declares the arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}. @@ -214,17 +213,17 @@ \medskip A \emph{bound variable} is a natural number \isa{b}, which accounts for the number of intermediate binders between the variable occurrence in the body and its binding position. For - example, the de-Bruijn term \isa{{\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ {\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ {\isadigit{1}}\ {\isacharplus}\ {\isadigit{0}}} - would correspond to \isa{{\isasymlambda}x\isactrlisub {\isasymtau}{\isachardot}\ {\isasymlambda}y\isactrlisub {\isasymtau}{\isachardot}\ x\ {\isacharplus}\ y} in a - named representation. Note that a bound variable may be represented - by different de-Bruijn indices at different occurrences, depending - on the nesting of abstractions. + example, the de-Bruijn term \isa{{\isasymlambda}\isactrlbsub nat\isactrlesub {\isachardot}\ {\isasymlambda}\isactrlbsub nat\isactrlesub {\isachardot}\ {\isadigit{1}}\ {\isacharplus}\ {\isadigit{0}}} would + correspond to \isa{{\isasymlambda}x\isactrlbsub nat\isactrlesub {\isachardot}\ {\isasymlambda}y\isactrlbsub nat\isactrlesub {\isachardot}\ x\ {\isacharplus}\ y} in a named + representation. Note that a bound variable may be represented by + different de-Bruijn indices at different occurrences, depending on + the nesting of abstractions. - A \emph{loose variables} is a bound variable that is outside the + A \emph{loose variable} is a bound variable that is outside the scope of local binders. The types (and names) for loose variables - can be managed as a separate context, that is maintained inside-out - like a stack of hypothetical binders. The core logic only operates - on closed terms, without any loose variables. + can be managed as a separate context, that is maintained as a stack + of hypothetical binders. The core logic operates on closed terms, + without any loose variables. A \emph{fixed variable} is a pair of a basic name and a type, e.g.\ \isa{{\isacharparenleft}x{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed \isa{x\isactrlisub {\isasymtau}}. A @@ -233,8 +232,8 @@ \medskip A \emph{constant} is a pair of a basic name and a type, e.g.\ \isa{{\isacharparenleft}c{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed as \isa{c\isactrlisub {\isasymtau}}. Constants are declared in the context as polymorphic - families \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}, meaning that valid all substitution - instances \isa{c\isactrlisub {\isasymtau}} for \isa{{\isasymtau}\ {\isacharequal}\ {\isasymsigma}{\isasymvartheta}} are valid. + families \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}, meaning that all substitution instances + \isa{c\isactrlisub {\isasymtau}} for \isa{{\isasymtau}\ {\isacharequal}\ {\isasymsigma}{\isasymvartheta}} are valid. The vector of \emph{type arguments} of constant \isa{c\isactrlisub {\isasymtau}} wrt.\ the declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} is defined as the codomain of @@ -248,11 +247,13 @@ polymorphic constants that the user-level type-checker would reject due to violation of type class restrictions. - \medskip A \emph{term} 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. + \medskip An \emph{atomic} term is either a variable or constant. A + \emph{term} is defined inductively over atomic terms, 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 inductive relation \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} assigns a (unique) type to a term according to the structure of atomic terms, abstractions, and @@ -276,23 +277,20 @@ component. This means that different variables \isa{x\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{1}}\isactrlesub } and \isa{x\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{2}}\isactrlesub } may become the same after type instantiation. Some outer layers of the system make it hard to produce variables of the same name, but different types. In - particular, type-inference always demands ``consistent'' type - constraints for free variables. In contrast, mixed instances of - polymorphic constants occur frequently. + contrast, mixed instances of polymorphic constants occur frequently. \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 type - arguments that are not accounted in result type, i.e.\ there are + arguments that are not accounted in the result 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 demands special care. + pathological situation notoriously demands additional care. \medskip A \emph{term abbreviation} is a syntactic definition \isa{c\isactrlisub {\isasymsigma}\ {\isasymequiv}\ t} of a closed term \isa{t} of type \isa{{\isasymsigma}}, without any hidden polymorphism. A term abbreviation looks like a - constant in the syntax, but is fully expanded before entering the - logical core. Abbreviations are usually reverted when printing - terms, using the collective \isa{t\ {\isasymrightarrow}\ c\isactrlisub {\isasymsigma}} as rules for - higher-order rewriting. + constant in the syntax, but is expanded before entering the logical + core. Abbreviations are usually reverted when printing terms, using + \isa{t\ {\isasymrightarrow}\ c\isactrlisub {\isasymsigma}} as rules for higher-order rewriting. \medskip Canonical operations on \isa{{\isasymlambda}}-terms include \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion: \isa{{\isasymalpha}}-conversion refers to capture-free renaming of bound variables; \isa{{\isasymbeta}}-conversion contracts an @@ -304,7 +302,7 @@ implicit in the de-Bruijn representation. Names for bound variables in abstractions are maintained separately as (meaningless) comments, mostly for parsing and printing. Full \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion is - commonplace in various higher operations (\secref{sec:rules}) that + commonplace in various standard operations (\secref{sec:rules}) that are based on higher-order unification and matching.% \end{isamarkuptext}% \isamarkuptrue% @@ -373,9 +371,8 @@ mixfix syntax. \item \verb|Sign.const_typargs|~\isa{thy\ {\isacharparenleft}c{\isacharcomma}\ {\isasymtau}{\isacharparenright}} and \verb|Sign.const_instance|~\isa{thy\ {\isacharparenleft}c{\isacharcomma}\ {\isacharbrackleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharbrackright}{\isacharparenright}} - convert between the representations of polymorphic constants: the - full type instance vs.\ the compact type arguments form (depending - on the most general declaration given in the context). + convert between two representations of polymorphic constants: full + type instance vs.\ compact type arguments form. \end{description}% \end{isamarkuptext}% @@ -428,7 +425,7 @@ \seeglossary{type variable}. The distinguishing feature of different variables is their binding scope. FIXME} - A \emph{proposition} is a well-formed term of type \isa{prop}, a + A \emph{proposition} is a well-typed term of type \isa{prop}, a \emph{theorem} is a proven proposition (depending on a context of hypotheses and the background theory). Primitive inferences include plain natural deduction rules for the primary connectives \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} of the framework. There is also a builtin @@ -441,16 +438,16 @@ \isamarkuptrue% % \begin{isamarkuptext}% -The theory \isa{Pure} contains declarations for the standard - connectives \isa{{\isasymAnd}}, \isa{{\isasymLongrightarrow}}, and \isa{{\isasymequiv}} of the logical - framework, see \figref{fig:pure-connectives}. The derivability - judgment \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} is defined - inductively by the primitive inferences given in - \figref{fig:prim-rules}, with the global restriction that hypotheses - \isa{{\isasymGamma}} may \emph{not} contain schematic variables. The builtin - equality is conceptually axiomatized as shown in +The theory \isa{Pure} contains constant declarations for the + primitive connectives \isa{{\isasymAnd}}, \isa{{\isasymLongrightarrow}}, and \isa{{\isasymequiv}} of + the logical framework, see \figref{fig:pure-connectives}. The + derivability judgment \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} is + defined inductively by the primitive inferences given in + \figref{fig:prim-rules}, with the global restriction that the + hypotheses must \emph{not} contain any schematic variables. The + builtin equality is conceptually axiomatized as shown in \figref{fig:pure-equality}, although the implementation works - directly with derived inference rules. + directly with derived inferences. \begin{figure}[htb] \begin{center} @@ -498,8 +495,8 @@ \end{figure} The introduction and elimination rules for \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} are analogous to formation of dependently typed \isa{{\isasymlambda}}-terms representing the underlying proof objects. Proof terms - are irrelevant in the Pure logic, though, they may never occur - within propositions. The system provides a runtime option to record + are irrelevant in the Pure logic, though; they cannot occur within + propositions. The system provides a runtime option to record explicit proof terms for primitive inferences. Thus all three levels of \isa{{\isasymlambda}}-calculus become explicit: \isa{{\isasymRightarrow}} for terms, and \isa{{\isasymAnd}{\isacharslash}{\isasymLongrightarrow}} for proofs (cf.\ @@ -507,15 +504,15 @@ Observe that locally fixed parameters (as in \isa{{\isasymAnd}{\isacharunderscore}intro}) need not be recorded in the hypotheses, because the simple syntactic - types of Pure are always inhabitable. Typing ``assumptions'' \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} are (implicitly) present only with occurrences of \isa{x\isactrlisub {\isasymtau}} in the statement body.\footnote{This is the key - difference ``\isa{{\isasymlambda}HOL}'' in the PTS framework - \cite{Barendregt-Geuvers:2001}, where \isa{x\ {\isacharcolon}\ A} hypotheses are - treated explicitly for types, in the same way as propositions.} + types of Pure are always inhabitable. ``Assumptions'' \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} for type-membership are only present as long as some \isa{x\isactrlisub {\isasymtau}} occurs in the statement body.\footnote{This is the key + difference to ``\isa{{\isasymlambda}HOL}'' in the PTS framework + \cite{Barendregt-Geuvers:2001}, where hypotheses \isa{x\ {\isacharcolon}\ A} are + treated uniformly for propositions and types.} \medskip The axiomatization of a theory is implicitly closed by forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymvartheta}} holds for any substitution instance of an axiom - \isa{{\isasymturnstile}\ A}. By pushing substitution through derivations - inductively, we get admissible \isa{generalize} and \isa{instance} rules shown in \figref{fig:subst-rules}. + \isa{{\isasymturnstile}\ A}. By pushing substitutions through derivations + inductively, we also get admissible \isa{generalize} and \isa{instance} rules as shown in \figref{fig:subst-rules}. \begin{figure}[htb] \begin{center} @@ -538,34 +535,32 @@ variables. In principle, variables could be substituted in hypotheses as well, - but this would disrupt monotonicity reasoning: deriving \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymturnstile}\ B{\isasymvartheta}} from \isa{{\isasymGamma}\ {\isasymturnstile}\ B} is correct, but - \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymsupseteq}\ {\isasymGamma}} does not necessarily hold --- the result - belongs to a different proof context. + but this would disrupt the monotonicity of reasoning: deriving + \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymturnstile}\ B{\isasymvartheta}} from \isa{{\isasymGamma}\ {\isasymturnstile}\ B} is + correct, but \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymsupseteq}\ {\isasymGamma}} does not necessarily hold: + the result belongs to a different proof context. - \medskip The system allows axioms to be stated either as plain - propositions, or as arbitrary functions (``oracles'') that produce - propositions depending on given arguments. It is possible to trace - the used of axioms (and defined theorems) in derivations. - Invocations of oracles are recorded invariable. + \medskip An \emph{oracle} is a function that produces axioms on the + fly. Logically, this is an instance of the \isa{axiom} rule + (\figref{fig:prim-rules}), but there is an operational difference. + The system always records oracle invocations within derivations of + theorems. Tracing plain axioms (and named theorems) is optional. Axiomatizations should be limited to the bare minimum, typically as part of the initial logical basis of an object-logic formalization. - Normally, theories will be developed definitionally, by stating - restricted equalities over constants. + Later on, theories are usually developed in a strictly definitional + fashion, by stating only certain equalities over new constants. - A \emph{simple definition} consists of a constant declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} together with an axiom \isa{{\isasymturnstile}\ c\ {\isasymequiv}\ t}, where \isa{t} is a closed term without any hidden polymorphism. The RHS may - depend on further defined constants, but not \isa{c} itself. - Definitions of constants with function type may be presented - liberally as \isa{c\ \isactrlvec \ {\isasymequiv}\ t} instead of the puristic \isa{c\ {\isasymequiv}\ {\isasymlambda}\isactrlvec x{\isachardot}\ t}. + A \emph{simple definition} consists of a constant declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} together with an axiom \isa{{\isasymturnstile}\ c\ {\isasymequiv}\ t}, where \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} is a closed term without any hidden polymorphism. The RHS + may depend on further defined constants, but not \isa{c} itself. + Definitions of functions may be presented as \isa{c\ \isactrlvec x\ {\isasymequiv}\ t} instead of the puristic \isa{c\ {\isasymequiv}\ {\isasymlambda}\isactrlvec x{\isachardot}\ t}. - An \emph{overloaded definition} consists may give zero or one - equality axioms \isa{c{\isacharparenleft}{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}{\isacharparenright}\ {\isasymequiv}\ t} for each type - constructor \isa{{\isasymkappa}}, with distinct variables \isa{\isactrlvec {\isasymalpha}} - as formal arguments. The RHS may mention previously defined - constants as above, or arbitrary constants \isa{d{\isacharparenleft}{\isasymalpha}\isactrlisub i{\isacharparenright}} - for some \isa{{\isasymalpha}\isactrlisub i} projected from \isa{\isactrlvec {\isasymalpha}}. - Thus overloaded definitions essentially work by primitive recursion - over the syntactic structure of a single type argument.% + An \emph{overloaded definition} consists of a collection of axioms + for the same constant, with zero or one equations \isa{c{\isacharparenleft}{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}{\isacharparenright}\ {\isasymequiv}\ t} for each type constructor \isa{{\isasymkappa}} (for + distinct variables \isa{\isactrlvec {\isasymalpha}}). The RHS may mention + previously defined constants as above, or arbitrary constants \isa{d{\isacharparenleft}{\isasymalpha}\isactrlisub i{\isacharparenright}} for some \isa{{\isasymalpha}\isactrlisub i} projected from \isa{\isactrlvec {\isasymalpha}}. Thus overloaded definitions essentially work by + primitive recursion over the syntactic structure of a single type + argument.% \end{isamarkuptext}% \isamarkuptrue% % @@ -613,18 +608,22 @@ \item \verb|thm| represents proven propositions. This is an abstract datatype that guarantees that its values have been constructed by basic principles of the \verb|Thm| module. + Every \verb|thm| value contains a sliding back-reference to the + enclosing theory, cf.\ \secref{sec:context-theory}. - \item \verb|proofs| determines the detail of proof recording: \verb|0| - records only oracles, \verb|1| records oracles, axioms and named - theorems, \verb|2| records full proof terms. + \item \verb|proofs| determines the detail of proof recording within + \verb|thm| values: \verb|0| records only oracles, \verb|1| records + oracles, axioms and named theorems, \verb|2| records full proof + terms. \item \verb|Thm.assume|, \verb|Thm.forall_intr|, \verb|Thm.forall_elim|, \verb|Thm.implies_intr|, and \verb|Thm.implies_elim| correspond to the primitive inferences of \figref{fig:prim-rules}. \item \verb|Thm.generalize|~\isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharcomma}\ \isactrlvec x{\isacharparenright}} corresponds to the \isa{generalize} rules of - \figref{fig:subst-rules}. A collection of type and term variables - is generalized simultaneously, according to the given basic names. + \figref{fig:subst-rules}. Here collections of type and term + variables are generalized simultaneously, specified by the given + basic names. \item \verb|Thm.instantiate|~\isa{{\isacharparenleft}\isactrlvec {\isasymalpha}\isactrlisub s{\isacharcomma}\ \isactrlvec x\isactrlisub {\isasymtau}{\isacharparenright}} corresponds to the \isa{instantiate} rules of \figref{fig:subst-rules}. Type variables are substituted before @@ -634,24 +633,22 @@ \item \verb|Thm.get_axiom_i|~\isa{thy\ name} retrieves a named axiom, cf.\ \isa{axiom} in \figref{fig:prim-rules}. - \item \verb|Thm.invoke_oracle_i|~\isa{thy\ name\ arg} invokes the - oracle function \isa{name} of the theory. Logically, this is - just another instance of \isa{axiom} in \figref{fig:prim-rules}, - but the system records an explicit trace of oracle invocations with - the \isa{thm} value. + \item \verb|Thm.invoke_oracle_i|~\isa{thy\ name\ arg} invokes a + named oracle function, cf.\ \isa{axiom} in + \figref{fig:prim-rules}. - \item \verb|Theory.add_axioms_i|~\isa{{\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ A{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} adds - arbitrary axioms, without any checking of the proposition. + \item \verb|Theory.add_axioms_i|~\isa{{\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ A{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares + arbitrary propositions as axioms. - \item \verb|Theory.add_oracle|~\isa{{\isacharparenleft}name{\isacharcomma}\ f{\isacharparenright}} declares an - oracle, i.e.\ a function for generating arbitrary axioms. + \item \verb|Theory.add_oracle|~\isa{{\isacharparenleft}name{\isacharcomma}\ f{\isacharparenright}} declares an oracle + function for generating arbitrary axioms on the fly. - \item \verb|Theory.add_deps|~\isa{name\ c\isactrlisub {\isasymtau}\ \isactrlvec d\isactrlisub {\isasymsigma}} declares dependencies of a new specification for - constant \isa{c\isactrlisub {\isasymtau}} from relative to existing ones for - constants \isa{\isactrlvec d\isactrlisub {\isasymsigma}}. + \item \verb|Theory.add_deps|~\isa{name\ c\isactrlisub {\isasymtau}\ \isactrlvec d\isactrlisub {\isasymsigma}} declares dependencies of a named specification + for constant \isa{c\isactrlisub {\isasymtau}}, relative to existing + specifications for constants \isa{\isactrlvec d\isactrlisub {\isasymsigma}}. - \item \verb|Theory.add_defs_i|~\isa{unchecked\ overloaded\ {\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ c\ \isactrlvec x\ {\isasymequiv}\ t{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} states a definitional axiom for an already - declared constant \isa{c}. Dependencies are recorded using \verb|Theory.add_deps|, unless the \isa{unchecked} option is set. + \item \verb|Theory.add_defs_i|~\isa{unchecked\ overloaded\ {\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ c\ \isactrlvec x\ {\isasymequiv}\ t{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} states a definitional axiom for an existing + constant \isa{c}. Dependencies are recorded (cf.\ \verb|Theory.add_deps|), unless the \isa{unchecked} option is set. \end{description}% \end{isamarkuptext}% @@ -664,21 +661,21 @@ % \endisadelimmlref % -\isamarkupsubsection{Auxiliary connectives% +\isamarkupsubsection{Auxiliary definitions% } \isamarkuptrue% % \begin{isamarkuptext}% -Theory \isa{Pure} also defines a few auxiliary connectives, see - \figref{fig:pure-aux}. These are normally not exposed to the user, - but appear in internal encodings only. +Theory \isa{Pure} provides a few auxiliary definitions, see + \figref{fig:pure-aux}. These special constants are normally not + exposed to the user, but appear in internal encodings. \begin{figure}[htb] \begin{center} \begin{tabular}{ll} \isa{conjunction\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} & (infix \isa{{\isacharampersand}}) \\ \isa{{\isasymturnstile}\ A\ {\isacharampersand}\ B\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}C{\isachardot}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isacharparenright}} \\[1ex] - \isa{prop\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop} & (prefix \isa{{\isacharhash}}, hidden) \\ + \isa{prop\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop} & (prefix \isa{{\isacharhash}}, suppressed) \\ \isa{{\isacharhash}A\ {\isasymequiv}\ A} \\[1ex] \isa{term\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ prop} & (prefix \isa{TERM}) \\ \isa{term\ x\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}A{\isachardot}\ A\ {\isasymLongrightarrow}\ A{\isacharparenright}} \\[1ex] @@ -692,19 +689,19 @@ Derived conjunction rules include introduction \isa{A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ A\ {\isacharampersand}\ B}, and destructions \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ A} and \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ B}. Conjunction allows to treat simultaneous assumptions and conclusions uniformly. For example, multiple claims are intermediately - represented as explicit conjunction, but this is usually refined - into separate sub-goals before the user continues the proof; the - final result is projected into a list of theorems (cf.\ + represented as explicit conjunction, but this is refined into + separate sub-goals before the user continues the proof; the final + result is projected into a list of theorems (cf.\ \secref{sec:tactical-goals}). The \isa{prop} marker (\isa{{\isacharhash}}) makes arbitrarily complex propositions appear as atomic, without changing the meaning: \isa{{\isasymGamma}\ {\isasymturnstile}\ A} and \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isacharhash}A} are interchangeable. See \secref{sec:tactical-goals} for specific operations. - The \isa{term} marker turns any well-formed term into a - derivable proposition: \isa{{\isasymturnstile}\ TERM\ t} holds unconditionally. - Although this is logically vacuous, it allows to treat terms and - proofs uniformly, similar to a type-theoretic framework. + The \isa{term} marker turns any well-typed term into a derivable + proposition: \isa{{\isasymturnstile}\ TERM\ t} holds unconditionally. Although + this is logically vacuous, it allows to treat terms and proofs + uniformly, similar to a type-theoretic framework. The \isa{TYPE} constructor is the canonical representative of the unspecified type \isa{{\isasymalpha}\ itself}; it essentially injects the @@ -739,12 +736,12 @@ \item \verb|Conjunction.intr| derives \isa{A\ {\isacharampersand}\ B} from \isa{A} and \isa{B}. - \item \verb|Conjunction.intr| derives \isa{A} and \isa{B} + \item \verb|Conjunction.elim| derives \isa{A} and \isa{B} from \isa{A\ {\isacharampersand}\ B}. - \item \verb|Drule.mk_term|~\isa{t} derives \isa{TERM\ t}. + \item \verb|Drule.mk_term| derives \isa{TERM\ t}. - \item \verb|Drule.dest_term|~\isa{TERM\ t} recovers term \isa{t}. + \item \verb|Drule.dest_term| recovers term \isa{t} from \isa{TERM\ t}. \item \verb|Logic.mk_type|~\isa{{\isasymtau}} produces the term \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}}.

--- a/doc-src/IsarImplementation/Thy/logic.thy Thu Sep 14 22:48:37 2006 +0200 +++ b/doc-src/IsarImplementation/Thy/logic.thy Fri Sep 15 16:49:41 2006 +0200 @@ -15,7 +15,7 @@ Isabelle/Pure. Following type-theoretic parlance, the Pure logic consists of three - levels of @{text "\<lambda>"}-calculus with corresponding arrows: @{text + levels of @{text "\<lambda>"}-calculus with corresponding arrows, @{text "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text "\<And>"} for universal quantification (proofs depending on terms), and @{text "\<Longrightarrow>"} for implication (proofs depending on proofs). @@ -80,7 +80,7 @@ A \emph{type} is defined inductively over type variables and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s | - (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}. + (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>"}. A \emph{type abbreviation} is a syntactic definition @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over @@ -110,10 +110,9 @@ vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}. - Consequently, unification on the algebra of types has most general - solutions (modulo equivalence of sorts). This means that - type-inference will produce primary types as expected - \cite{nipkow-prehofer}. + Consequently, type unification has most general solutions (modulo + equivalence of sorts), so type-inference produces primary types as + expected \cite{nipkow-prehofer}. *} text %mlref {* @@ -176,7 +175,7 @@ relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}. \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1, - c\<^isub>2)"} declares class relation @{text "c\<^isub>1 \<subseteq> + c\<^isub>2)"} declares the class relation @{text "c\<^isub>1 \<subseteq> c\<^isub>2"}. \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares @@ -201,17 +200,19 @@ \medskip A \emph{bound variable} is a natural number @{text "b"}, which accounts for the number of intermediate binders between the variable occurrence in the body and its binding position. For - example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"} - would correspond to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a - named representation. Note that a bound variable may be represented - by different de-Bruijn indices at different occurrences, depending - on the nesting of abstractions. + example, the de-Bruijn term @{text + "\<lambda>\<^bsub>nat\<^esub>. \<lambda>\<^bsub>nat\<^esub>. 1 + 0"} would + correspond to @{text + "\<lambda>x\<^bsub>nat\<^esub>. \<lambda>y\<^bsub>nat\<^esub>. x + y"} in a named + representation. Note that a bound variable may be represented by + different de-Bruijn indices at different occurrences, depending on + the nesting of abstractions. - A \emph{loose variables} is a bound variable that is outside the + A \emph{loose variable} is a bound variable that is outside the scope of local binders. The types (and names) for loose variables - can be managed as a separate context, that is maintained inside-out - like a stack of hypothetical binders. The core logic only operates - on closed terms, without any loose variables. + can be managed as a separate context, that is maintained as a stack + of hypothetical binders. The core logic operates on closed terms, + without any loose variables. A \emph{fixed variable} is a pair of a basic name and a type, e.g.\ @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}. A @@ -222,8 +223,8 @@ \medskip A \emph{constant} is a pair of a basic name and a type, e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text "c\<^isub>\<tau>"}. Constants are declared in the context as polymorphic - families @{text "c :: \<sigma>"}, meaning that valid all substitution - instances @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid. + families @{text "c :: \<sigma>"}, meaning that all substitution instances + @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid. The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"} wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of @@ -243,13 +244,14 @@ polymorphic constants that the user-level type-checker would reject due to violation of type class restrictions. - \medskip A \emph{term} is defined inductively over variables and - constants, with abstraction and application as follows: @{text "t = - b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t | - t\<^isub>1 t\<^isub>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. + \medskip An \emph{atomic} term is either a variable or constant. A + \emph{term} is defined inductively over atomic terms, with + abstraction and application as follows: @{text "t = b | x\<^isub>\<tau> | + ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>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 inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a term according to the structure of atomic terms, abstractions, and @@ -275,25 +277,22 @@ "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type instantiation. Some outer layers of the system make it hard to produce variables of the same name, but different types. In - particular, type-inference always demands ``consistent'' type - constraints for free variables. In contrast, mixed instances of - polymorphic constants occur frequently. + contrast, mixed instances of polymorphic constants occur frequently. \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"} is the set of type variables occurring in @{text "t"}, but not in @{text "\<sigma>"}. This means that the term implicitly depends on type - arguments that are not accounted in result type, i.e.\ there are + arguments that are not accounted in the result type, i.e.\ there are different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text "t\<vartheta>' :: \<sigma>"} with the same type. This slightly - pathological situation demands special care. + pathological situation notoriously demands additional care. \medskip A \emph{term abbreviation} is a syntactic definition @{text "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"}, without any hidden polymorphism. A term abbreviation looks like a - constant in the syntax, but is fully expanded before entering the - logical core. Abbreviations are usually reverted when printing - terms, using the collective @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for - higher-order rewriting. + constant in the syntax, but is expanded before entering the logical + core. Abbreviations are usually reverted when printing terms, using + @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for higher-order rewriting. \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free @@ -308,7 +307,7 @@ implicit in the de-Bruijn representation. Names for bound variables in abstractions are maintained separately as (meaningless) comments, mostly for parsing and printing. Full @{text "\<alpha>\<beta>\<eta>"}-conversion is - commonplace in various higher operations (\secref{sec:rules}) that + commonplace in various standard operations (\secref{sec:rules}) that are based on higher-order unification and matching. *} @@ -379,9 +378,8 @@ \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"} - convert between the representations of polymorphic constants: the - full type instance vs.\ the compact type arguments form (depending - on the most general declaration given in the context). + convert between two representations of polymorphic constants: full + type instance vs.\ compact type arguments form. \end{description} *} @@ -426,7 +424,7 @@ \seeglossary{type variable}. The distinguishing feature of different variables is their binding scope. FIXME} - A \emph{proposition} is a well-formed term of type @{text "prop"}, a + A \emph{proposition} is a well-typed term of type @{text "prop"}, a \emph{theorem} is a proven proposition (depending on a context of hypotheses and the background theory). Primitive inferences include plain natural deduction rules for the primary connectives @{text @@ -437,16 +435,16 @@ subsection {* Primitive connectives and rules *} text {* - The theory @{text "Pure"} contains declarations for the standard - connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of the logical - framework, see \figref{fig:pure-connectives}. The derivability - judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined - inductively by the primitive inferences given in - \figref{fig:prim-rules}, with the global restriction that hypotheses - @{text "\<Gamma>"} may \emph{not} contain schematic variables. The builtin - equality is conceptually axiomatized as shown in + The theory @{text "Pure"} contains constant declarations for the + primitive connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of + the logical framework, see \figref{fig:pure-connectives}. The + derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is + defined inductively by the primitive inferences given in + \figref{fig:prim-rules}, with the global restriction that the + hypotheses must \emph{not} contain any schematic variables. The + builtin equality is conceptually axiomatized as shown in \figref{fig:pure-equality}, although the implementation works - directly with derived inference rules. + directly with derived inferences. \begin{figure}[htb] \begin{center} @@ -496,8 +494,8 @@ The introduction and elimination rules for @{text "\<And>"} and @{text "\<Longrightarrow>"} are analogous to formation of dependently typed @{text "\<lambda>"}-terms representing the underlying proof objects. Proof terms - are irrelevant in the Pure logic, though, they may never occur - within propositions. The system provides a runtime option to record + are irrelevant in the Pure logic, though; they cannot occur within + propositions. The system provides a runtime option to record explicit proof terms for primitive inferences. Thus all three levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\ @@ -505,19 +503,19 @@ Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need not be recorded in the hypotheses, because the simple syntactic - types of Pure are always inhabitable. Typing ``assumptions'' @{text - "x :: \<tau>"} are (implicitly) present only with occurrences of @{text - "x\<^isub>\<tau>"} in the statement body.\footnote{This is the key - difference ``@{text "\<lambda>HOL"}'' in the PTS framework - \cite{Barendregt-Geuvers:2001}, where @{text "x : A"} hypotheses are - treated explicitly for types, in the same way as propositions.} + types of Pure are always inhabitable. ``Assumptions'' @{text "x :: + \<tau>"} for type-membership are only present as long as some @{text + "x\<^isub>\<tau>"} occurs in the statement body.\footnote{This is the key + difference to ``@{text "\<lambda>HOL"}'' in the PTS framework + \cite{Barendregt-Geuvers:2001}, where hypotheses @{text "x : A"} are + treated uniformly for propositions and types.} \medskip The axiomatization of a theory is implicitly closed by forming all instances of type and term variables: @{text "\<turnstile> A\<vartheta>"} holds for any substitution instance of an axiom - @{text "\<turnstile> A"}. By pushing substitution through derivations - inductively, we get admissible @{text "generalize"} and @{text - "instance"} rules shown in \figref{fig:subst-rules}. + @{text "\<turnstile> A"}. By pushing substitutions through derivations + inductively, we also get admissible @{text "generalize"} and @{text + "instance"} rules as shown in \figref{fig:subst-rules}. \begin{figure}[htb] \begin{center} @@ -540,38 +538,38 @@ variables. In principle, variables could be substituted in hypotheses as well, - but this would disrupt monotonicity reasoning: deriving @{text - "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is correct, but - @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold --- the result - belongs to a different proof context. + but this would disrupt the monotonicity of reasoning: deriving + @{text "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is + correct, but @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold: + the result belongs to a different proof context. - \medskip The system allows axioms to be stated either as plain - propositions, or as arbitrary functions (``oracles'') that produce - propositions depending on given arguments. It is possible to trace - the used of axioms (and defined theorems) in derivations. - Invocations of oracles are recorded invariable. + \medskip An \emph{oracle} is a function that produces axioms on the + fly. Logically, this is an instance of the @{text "axiom"} rule + (\figref{fig:prim-rules}), but there is an operational difference. + The system always records oracle invocations within derivations of + theorems. Tracing plain axioms (and named theorems) is optional. Axiomatizations should be limited to the bare minimum, typically as part of the initial logical basis of an object-logic formalization. - Normally, theories will be developed definitionally, by stating - restricted equalities over constants. + Later on, theories are usually developed in a strictly definitional + fashion, by stating only certain equalities over new constants. A \emph{simple definition} consists of a constant declaration @{text - "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text - "t"} is a closed term without any hidden polymorphism. The RHS may - depend on further defined constants, but not @{text "c"} itself. - Definitions of constants with function type may be presented - liberally as @{text "c \<^vec> \<equiv> t"} instead of the puristic @{text - "c \<equiv> \<lambda>\<^vec>x. t"}. + "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text "t + :: \<sigma>"} is a closed term without any hidden polymorphism. The RHS + may depend on further defined constants, but not @{text "c"} itself. + Definitions of functions may be presented as @{text "c \<^vec>x \<equiv> + t"} instead of the puristic @{text "c \<equiv> \<lambda>\<^vec>x. t"}. - An \emph{overloaded definition} consists may give zero or one - equality axioms @{text "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type - constructor @{text "\<kappa>"}, with distinct variables @{text "\<^vec>\<alpha>"} - as formal arguments. The RHS may mention previously defined - constants as above, or arbitrary constants @{text "d(\<alpha>\<^isub>i)"} - for some @{text "\<alpha>\<^isub>i"} projected from @{text "\<^vec>\<alpha>"}. - Thus overloaded definitions essentially work by primitive recursion - over the syntactic structure of a single type argument. + An \emph{overloaded definition} consists of a collection of axioms + for the same constant, with zero or one equations @{text + "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type constructor @{text "\<kappa>"} (for + distinct variables @{text "\<^vec>\<alpha>"}). The RHS may mention + previously defined constants as above, or arbitrary constants @{text + "d(\<alpha>\<^isub>i)"} for some @{text "\<alpha>\<^isub>i"} projected from @{text + "\<^vec>\<alpha>"}. Thus overloaded definitions essentially work by + primitive recursion over the syntactic structure of a single type + argument. *} text %mlref {* @@ -612,10 +610,13 @@ \item @{ML_type thm} represents proven propositions. This is an abstract datatype that guarantees that its values have been constructed by basic principles of the @{ML_struct Thm} module. + Every @{ML thm} value contains a sliding back-reference to the + enclosing theory, cf.\ \secref{sec:context-theory}. - \item @{ML proofs} determines the detail of proof recording: @{ML 0} - records only oracles, @{ML 1} records oracles, axioms and named - theorems, @{ML 2} records full proof terms. + \item @{ML proofs} determines the detail of proof recording within + @{ML_type thm} values: @{ML 0} records only oracles, @{ML 1} records + oracles, axioms and named theorems, @{ML 2} records full proof + terms. \item @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim} @@ -623,8 +624,9 @@ \item @{ML Thm.generalize}~@{text "(\<^vec>\<alpha>, \<^vec>x)"} corresponds to the @{text "generalize"} rules of - \figref{fig:subst-rules}. A collection of type and term variables - is generalized simultaneously, according to the given basic names. + \figref{fig:subst-rules}. Here collections of type and term + variables are generalized simultaneously, specified by the given + basic names. \item @{ML Thm.instantiate}~@{text "(\<^vec>\<alpha>\<^isub>s, \<^vec>x\<^isub>\<tau>)"} corresponds to the @{text "instantiate"} rules @@ -635,45 +637,43 @@ \item @{ML Thm.get_axiom_i}~@{text "thy name"} retrieves a named axiom, cf.\ @{text "axiom"} in \figref{fig:prim-rules}. - \item @{ML Thm.invoke_oracle_i}~@{text "thy name arg"} invokes the - oracle function @{text "name"} of the theory. Logically, this is - just another instance of @{text "axiom"} in \figref{fig:prim-rules}, - but the system records an explicit trace of oracle invocations with - the @{text "thm"} value. + \item @{ML Thm.invoke_oracle_i}~@{text "thy name arg"} invokes a + named oracle function, cf.\ @{text "axiom"} in + \figref{fig:prim-rules}. - \item @{ML Theory.add_axioms_i}~@{text "[(name, A), \<dots>]"} adds - arbitrary axioms, without any checking of the proposition. + \item @{ML Theory.add_axioms_i}~@{text "[(name, A), \<dots>]"} declares + arbitrary propositions as axioms. - \item @{ML Theory.add_oracle}~@{text "(name, f)"} declares an - oracle, i.e.\ a function for generating arbitrary axioms. + \item @{ML Theory.add_oracle}~@{text "(name, f)"} declares an oracle + function for generating arbitrary axioms on the fly. \item @{ML Theory.add_deps}~@{text "name c\<^isub>\<tau> - \<^vec>d\<^isub>\<sigma>"} declares dependencies of a new specification for - constant @{text "c\<^isub>\<tau>"} from relative to existing ones for - constants @{text "\<^vec>d\<^isub>\<sigma>"}. + \<^vec>d\<^isub>\<sigma>"} declares dependencies of a named specification + for constant @{text "c\<^isub>\<tau>"}, relative to existing + specifications for constants @{text "\<^vec>d\<^isub>\<sigma>"}. \item @{ML Theory.add_defs_i}~@{text "unchecked overloaded [(name, c - \<^vec>x \<equiv> t), \<dots>]"} states a definitional axiom for an already - declared constant @{text "c"}. Dependencies are recorded using @{ML - Theory.add_deps}, unless the @{text "unchecked"} option is set. + \<^vec>x \<equiv> t), \<dots>]"} states a definitional axiom for an existing + constant @{text "c"}. Dependencies are recorded (cf.\ @{ML + Theory.add_deps}), unless the @{text "unchecked"} option is set. \end{description} *} -subsection {* Auxiliary connectives *} +subsection {* Auxiliary definitions *} text {* - Theory @{text "Pure"} also defines a few auxiliary connectives, see - \figref{fig:pure-aux}. These are normally not exposed to the user, - but appear in internal encodings only. + Theory @{text "Pure"} provides a few auxiliary definitions, see + \figref{fig:pure-aux}. These special constants are normally not + exposed to the user, but appear in internal encodings. \begin{figure}[htb] \begin{center} \begin{tabular}{ll} @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\ @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex] - @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, hidden) \\ + @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, suppressed) \\ @{text "#A \<equiv> A"} \\[1ex] @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\ @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex] @@ -688,9 +688,9 @@ B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}. Conjunction allows to treat simultaneous assumptions and conclusions uniformly. For example, multiple claims are intermediately - represented as explicit conjunction, but this is usually refined - into separate sub-goals before the user continues the proof; the - final result is projected into a list of theorems (cf.\ + represented as explicit conjunction, but this is refined into + separate sub-goals before the user continues the proof; the final + result is projected into a list of theorems (cf.\ \secref{sec:tactical-goals}). The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex @@ -698,10 +698,10 @@ "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable. See \secref{sec:tactical-goals} for specific operations. - The @{text "term"} marker turns any well-formed term into a - derivable proposition: @{text "\<turnstile> TERM t"} holds unconditionally. - Although this is logically vacuous, it allows to treat terms and - proofs uniformly, similar to a type-theoretic framework. + The @{text "term"} marker turns any well-typed term into a derivable + proposition: @{text "\<turnstile> TERM t"} holds unconditionally. Although + this is logically vacuous, it allows to treat terms and proofs + uniformly, similar to a type-theoretic framework. The @{text "TYPE"} constructor is the canonical representative of the unspecified type @{text "\<alpha> itself"}; it essentially injects the @@ -733,13 +733,13 @@ \item @{ML Conjunction.intr} derives @{text "A & B"} from @{text "A"} and @{text "B"}. - \item @{ML Conjunction.intr} derives @{text "A"} and @{text "B"} + \item @{ML Conjunction.elim} derives @{text "A"} and @{text "B"} from @{text "A & B"}. - \item @{ML Drule.mk_term}~@{text "t"} derives @{text "TERM t"}. + \item @{ML Drule.mk_term} derives @{text "TERM t"}. - \item @{ML Drule.dest_term}~@{text "TERM t"} recovers term @{text - "t"}. + \item @{ML Drule.dest_term} recovers term @{text "t"} from @{text + "TERM t"}. \item @{ML Logic.mk_type}~@{text "\<tau>"} produces the term @{text "TYPE(\<tau>)"}.