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

changeset 20537 | b6b49903db7e |

parent 20521 | 189811b39869 |

child 20542 | a54ca4e90874 |

--- a/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 14 15:27:08 2006 +0200 +++ b/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 14 15:51:20 2006 +0200 @@ -35,11 +35,12 @@ 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. 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.}% + Derivations are relative to a logical theory, which declares type + constructors, constants, and axioms. Theory declarations support + schematic polymorphism, which is strictly speaking outside the + logic.\footnote{This is the deeper logical reason, why the theory + context \isa{{\isasymTheta}} is separate from the proof context \isa{{\isasymGamma}} + of the core calculus.}% \end{isamarkuptext}% \isamarkuptrue% % @@ -57,7 +58,7 @@ 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 + A \emph{sort} is a list of type classes written as \isa{s\ {\isacharequal}\ {\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 @@ -69,9 +70,11 @@ elements wrt.\ the sort order. \medskip A \emph{fixed type variable} is a pair of a basic name - (starting with a \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 as \isa{{\isacharquery}{\isasymalpha}\isactrlisub s}. + (starting with a \isa{{\isacharprime}} character) and a sort constraint, e.g.\ + \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, e.g.\ \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 type variables, including the sort constraint. The core logic @@ -81,19 +84,20 @@ A \emph{type constructor} \isa{{\isasymkappa}} is a \isa{k}-ary operator on types declared in the theory. Type constructor application is - usually written postfix as \isa{{\isacharparenleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub k{\isacharparenright}{\isasymkappa}}. - 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 the right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}} instead of - \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}. + written postfix as \isa{{\isacharparenleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub k{\isacharparenright}{\isasymkappa}}. 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 the + right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}} instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}. - 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} 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 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. + variables \isa{\isactrlvec {\isasymalpha}}. Type abbreviations appear as type + constructors in the syntax, but are expanded before entering the + logical core. A \emph{type arity} declares the image behavior of a type constructor wrt.\ the algebra of sorts: \isa{{\isasymkappa}\ {\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}{\isasymkappa}} is @@ -103,16 +107,17 @@ \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{{\isasymkappa}} and classes \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}, any arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{1}}{\isacharparenright}c\isactrlisub {\isadigit{1}}} has a corresponding arity \isa{{\isasymkappa}\ {\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 component-wise. + relation: for any type constructor \isa{{\isasymkappa}}, and classes \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}, and arities \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{1}}{\isacharparenright}c\isactrlisub {\isadigit{1}}} and \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{2}}{\isacharparenright}c\isactrlisub {\isadigit{2}}} holds \isa{\isactrlvec s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ \isactrlvec s\isactrlisub {\isadigit{2}}} component-wise. The key property of a coregular order-sorted algebra is that sort - constraints may be always solved in a most general fashion: for each - type 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, 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}.% + constraints can be solved in a most general fashion: for each type + 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}.% \end{isamarkuptext}% \isamarkuptrue% % @@ -154,19 +159,19 @@ \item \verb|typ| represents types; this is a datatype with constructors \verb|TFree|, \verb|TVar|, \verb|Type|. - \item \verb|map_atyps|~\isa{f\ {\isasymtau}} applies mapping \isa{f} to - all atomic types (\verb|TFree|, \verb|TVar|) occurring in \isa{{\isasymtau}}. + \item \verb|map_atyps|~\isa{f\ {\isasymtau}} applies the mapping \isa{f} + to all atomic types (\verb|TFree|, \verb|TVar|) occurring in \isa{{\isasymtau}}. - \item \verb|fold_atyps|~\isa{f\ {\isasymtau}} iterates operation \isa{f} - over all occurrences of atoms (\verb|TFree|, \verb|TVar|) in \isa{{\isasymtau}}; the type structure is traversed from left to right. + \item \verb|fold_atyps|~\isa{f\ {\isasymtau}} iterates the operation \isa{f} over all occurrences of atomic types (\verb|TFree|, \verb|TVar|) + in \isa{{\isasymtau}}; the type structure is traversed from left to right. \item \verb|Sign.subsort|~\isa{thy\ {\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ s\isactrlisub {\isadigit{2}}{\isacharparenright}} 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 - is of a given sort. + \item \verb|Sign.of_sort|~\isa{thy\ {\isacharparenleft}{\isasymtau}{\isacharcomma}\ s{\isacharparenright}} tests whether type + \isa{{\isasymtau}} is of sort \isa{s}. - \item \verb|Sign.add_types|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ k{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares new + \item \verb|Sign.add_types|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ k{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares a new type constructors \isa{{\isasymkappa}} with \isa{k} arguments and optional mixfix syntax. @@ -174,13 +179,13 @@ defines a new type abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\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}, together with class + \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_arity|~\isa{{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec s{\isacharcomma}\ s{\isacharparenright}} declares - arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}. + the arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}. \end{description}% \end{isamarkuptext}% @@ -202,54 +207,56 @@ The language of terms is that of simply-typed \isa{{\isasymlambda}}-calculus with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72} - or \cite{paulson-ml2}), 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. + or \cite{paulson-ml2}), with the types being determined determined + by the corresponding binders. In contrast, free variables and + constants are have an explicit name and type in each occurrence. \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. A \emph{loose variables} is a bound variable that is - outside the current scope of local binders or the context. For + 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}}} - 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. + 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. + + A \emph{loose variables} 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. - 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}}. + 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 + \emph{schematic variable} is a pair of an indexname and a type, + e.g.\ \isa{{\isacharparenleft}{\isacharparenleft}x{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed as \isa{{\isacharquery}x\isactrlisub {\isasymtau}}. - \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}}. + \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. - 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}}. + 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 + the matcher \isa{{\isasymvartheta}\ {\isacharequal}\ {\isacharbraceleft}{\isacharquery}{\isasymalpha}\isactrlisub {\isadigit{1}}\ {\isasymmapsto}\ {\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isacharquery}{\isasymalpha}\isactrlisub n\ {\isasymmapsto}\ {\isasymtau}\isactrlisub n{\isacharbraceright}} presented in canonical order \isa{{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharparenright}}. Within a given theory context, + there is a one-to-one correspondence between any constant \isa{c\isactrlisub {\isasymtau}} and the application \isa{c{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharparenright}} of its type arguments. For example, with \isa{plus\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}}, the instance \isa{plus\isactrlbsub nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat\isactrlesub } corresponds to \isa{plus{\isacharparenleft}nat{\isacharparenright}}. 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. + polymorphic constants that the user-level type-checker would reject + due to violation of type class restrictions. - \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. + \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. - 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}}. + The inductive relation \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} assigns a (unique) type to a + term according to the structure of atomic terms, abstractions, and + applicatins: \[ \infer{\isa{a\isactrlisub {\isasymtau}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}}{} \qquad @@ -265,40 +272,40 @@ 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. + The identity of atomic terms consists both of the name and the type + 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. \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. + \isa{{\isasymsigma}}. This means that the term implicitly depends on type + arguments that are not accounted in 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. - \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. + \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. - \medskip Canonical operations on \isa{{\isasymlambda}}-terms include \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion. \isa{{\isasymalpha}}-conversion refers to capture-free + \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 - abstraction applied to some argument term, substituting the argument + abstraction applied to an argument term, substituting the argument in the body: \isa{{\isacharparenleft}{\isasymlambda}x{\isachardot}\ b{\isacharparenright}a} becomes \isa{b{\isacharbrackleft}a{\isacharslash}x{\isacharbrackright}}; \isa{{\isasymeta}}-conversion contracts vacuous application-abstraction: \isa{{\isasymlambda}x{\isachardot}\ f\ x} becomes \isa{f}, provided that the bound variable - \isa{{\isadigit{0}}} does not occur in \isa{f}. + does not occur in \isa{f}. - Terms are almost always treated module \isa{{\isasymalpha}}-conversion, which - is implicit in the de-Bruijn representation. The names in - abstractions of bound variables are maintained only as a comment for - parsing and printing. Full \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence is usually - taken for granted higher rules (\secref{sec:rules}), anything - depending on higher-order unification or rewriting.% + Terms are normally treated modulo \isa{{\isasymalpha}}-conversion, which is + 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 + are based on higher-order unification and matching.% \end{isamarkuptext}% \isamarkuptrue% % @@ -328,38 +335,35 @@ \begin{description} - \item \verb|term| represents de-Bruijn terms with comments in - abstractions for bound variable names. This is a datatype with - constructors \verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|, \verb|Abs|, \verb|op $|. + \item \verb|term| represents de-Bruijn terms, with comments in + abstractions, and explicitly named free variables and constants; + this is a datatype with constructors \verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|, \verb|Abs|, \verb|op $|. \item \isa{t}~\verb|aconv|~\isa{u} checks \isa{{\isasymalpha}}-equivalence of two terms. This is the basic equality relation on type \verb|term|; raw datatype equality should only be used for operations related to parsing or printing! - \item \verb|map_term_types|~\isa{f\ t} applies mapping \isa{f} - to all types occurring in \isa{t}. + \item \verb|map_term_types|~\isa{f\ t} applies the mapping \isa{f} to all types occurring in \isa{t}. + + \item \verb|fold_types|~\isa{f\ t} iterates the operation \isa{f} over all occurrences of types in \isa{t}; the term + structure is traversed from left to right. - \item \verb|fold_types|~\isa{f\ t} iterates operation \isa{f} - over all occurrences of types in \isa{t}; the term structure is + \item \verb|map_aterms|~\isa{f\ t} applies the mapping \isa{f} + to all atomic terms (\verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|) occurring in \isa{t}. + + \item \verb|fold_aterms|~\isa{f\ t} iterates the operation \isa{f} over all occurrences of atomic terms (\verb|Bound|, \verb|Free|, + \verb|Var|, \verb|Const|) in \isa{t}; the term structure is traversed from left to right. - \item \verb|map_aterms|~\isa{f\ t} applies mapping \isa{f} to - all atomic terms (\verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|) - occurring in \isa{t}. - - \item \verb|fold_aterms|~\isa{f\ t} iterates operation \isa{f} - over all occurrences of atomic terms in (\verb|Bound|, \verb|Free|, - \verb|Var|, \verb|Const|) \isa{t}; the term structure is traversed - from left to right. + \item \verb|fastype_of|~\isa{t} determines the type of a + well-typed term. This operation is relatively slow, despite the + omission of any sanity checks. - \item \verb|fastype_of|~\isa{t} recomputes the type of a - well-formed term, while omitting any sanity checks. This operation - is relatively slow. + \item \verb|lambda|~\isa{a\ b} produces an abstraction \isa{{\isasymlambda}a{\isachardot}\ b}, where occurrences of the atomic term \isa{a} in the + body \isa{b} are replaced by bound variables. - \item \verb|lambda|~\isa{a\ b} produces an abstraction \isa{{\isasymlambda}a{\isachardot}\ b}, where occurrences of the original (atomic) term \isa{a} in the body \isa{b} are replaced by bound variables. - - \item \verb|betapply|~\isa{t\ u} produces an application \isa{t\ u}, with topmost \isa{{\isasymbeta}}-conversion if \isa{t} happens to - be an abstraction. + \item \verb|betapply|~\isa{{\isacharparenleft}t{\isacharcomma}\ u{\isacharparenright}} produces an application \isa{t\ u}, with topmost \isa{{\isasymbeta}}-conversion if \isa{t} is an + abstraction. \item \verb|Sign.add_consts_i|~\isa{{\isacharbrackleft}{\isacharparenleft}c{\isacharcomma}\ {\isasymsigma}{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares a new constant \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} with optional mixfix syntax. @@ -369,9 +373,9 @@ 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 two representations of constants, namely full - type instance vs.\ compact type arguments form (depending on the - most general declaration given in the context). + 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). \end{description}% \end{isamarkuptext}% @@ -427,27 +431,26 @@ A \emph{proposition} is a well-formed 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 are separate (derived) - rules for equality/equivalence \isa{{\isasymequiv}} and internal conjunction - \isa{{\isacharampersand}}.% + plain natural deduction rules for the primary connectives \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} of the framework. There is also a builtin + notion of equality/equivalence \isa{{\isasymequiv}}.% \end{isamarkuptext}% \isamarkuptrue% % -\isamarkupsubsection{Standard connectives and rules% +\isamarkupsubsection{Primitive connectives and rules% } \isamarkuptrue% % \begin{isamarkuptext}% -The basic theory is called \isa{Pure}, it contains declarations - for the standard logical connectives \isa{{\isasymAnd}}, \isa{{\isasymLongrightarrow}}, and - \isa{{\isasymequiv}} of the 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 syntactic restriction that - hypotheses may never contain schematic variables. The builtin - equality is conceptually axiomatized shown in +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 \figref{fig:pure-equality}, although the implementation works - directly with (derived) inference rules. + directly with derived inference rules. \begin{figure}[htb] \begin{center} @@ -456,7 +459,7 @@ \isa{{\isasymLongrightarrow}\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} & implication (right associative infix) \\ \isa{{\isasymequiv}\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ prop} & equality relation (infix) \\ \end{tabular} - \caption{Standard connectives of Pure}\label{fig:pure-connectives} + \caption{Primitive connectives of Pure}\label{fig:pure-connectives} \end{center} \end{figure} @@ -468,9 +471,9 @@ \infer[\isa{{\isacharparenleft}assume{\isacharparenright}}]{\isa{A\ {\isasymturnstile}\ A}}{} \] \[ - \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b\ x}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ b\ x} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}} + \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ b{\isacharbrackleft}x{\isacharbrackright}} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}} \qquad - \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ b\ a}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b\ x}} + \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ b{\isacharbrackleft}a{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}}} \] \[ \infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isacharminus}\ A\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}} @@ -484,44 +487,39 @@ \begin{figure}[htb] \begin{center} \begin{tabular}{ll} - \isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ b\ x{\isacharparenright}\ a\ {\isasymequiv}\ b\ a} & \isa{{\isasymbeta}}-conversion \\ + \isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}{\isacharparenright}\ a\ {\isasymequiv}\ b{\isacharbrackleft}a{\isacharbrackright}} & \isa{{\isasymbeta}}-conversion \\ \isa{{\isasymturnstile}\ x\ {\isasymequiv}\ x} & reflexivity \\ \isa{{\isasymturnstile}\ x\ {\isasymequiv}\ y\ {\isasymLongrightarrow}\ P\ x\ {\isasymLongrightarrow}\ P\ y} & substitution \\ \isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ f\ x\ {\isasymequiv}\ g\ x{\isacharparenright}\ {\isasymLongrightarrow}\ f\ {\isasymequiv}\ g} & extensionality \\ - \isa{{\isasymturnstile}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymequiv}\ B} & coincidence with equivalence \\ + \isa{{\isasymturnstile}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymequiv}\ B} & logical equivalence \\ \end{tabular} - \caption{Conceptual axiomatization of builtin equality}\label{fig:pure-equality} + \caption{Conceptual axiomatization of \isa{{\isasymequiv}}}\label{fig:pure-equality} \end{center} \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 \emph{irrelevant} in the Pure logic, they may never occur within - propositions, i.e.\ the \isa{{\isasymLongrightarrow}} arrow is non-dependent. The - system provides a runtime option to record explicit proof terms for - primitive inferences, cf.\ \cite{Berghofer-Nipkow:2000:TPHOL}. Thus - the three-fold \isa{{\isasymlambda}}-structure can be made explicit. + 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 + 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.\ + \cite{Berghofer-Nipkow:2000:TPHOL}). - Observe that locally fixed parameters (as used in rule \isa{{\isasymAnd}{\isacharunderscore}intro}) need not be recorded in the hypotheses, because the - simple syntactic types of Pure are always inhabitable. The typing - ``assumption'' \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} is logically vacuous, it disappears - automatically whenever the statement body ceases to mention variable - \isa{x\isactrlisub {\isasymtau}}.\footnote{This greatly simplifies many basic - reasoning steps, and is the key difference to the formulation of - this logic as ``\isa{{\isasymlambda}HOL}'' in the PTS framework - \cite{Barendregt-Geuvers:2001}.} + 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.} \medskip FIXME \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence and primitive definitions Since the basic representation of terms already accounts for \isa{{\isasymalpha}}-conversion, Pure equality essentially acts like \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence on terms, while coinciding with bi-implication. \medskip The axiomatization of a theory is implicitly closed by - forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymvartheta}} for - 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}. - Alternatively, the term substitution rules could be derived from - \isa{{\isasymAnd}{\isacharunderscore}intro{\isacharslash}elim}. The versions for types are genuine - admissible rules, due to the lack of true polymorphism in the logic. + 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}. \begin{figure}[htb] \begin{center} @@ -539,11 +537,14 @@ \end{center} \end{figure} - Since \isa{{\isasymGamma}} may never contain any schematic variables, the - \isa{instantiate} do not require an explicit side-condition. In - principle, variables could be substituted in hypotheses as well, but - this could disrupt monotonicity of the basic calculus: derivations - could leave the current proof context.% + Note that \isa{instantiate} does not require an explicit + side-condition, because \isa{{\isasymGamma}} may never contain schematic + 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.% \end{isamarkuptext}% \isamarkuptrue% % @@ -584,16 +585,16 @@ \isamarkuptrue% % \begin{isamarkuptext}% -Pure also provides various auxiliary connectives based on primitive - definitions, see \figref{fig:pure-aux}. These are normally not - exposed to the user, but appear in internal encodings only. +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. \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}}) \\ + \isa{prop\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop} & (prefix \isa{{\isacharhash}}, hidden) \\ \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] @@ -604,35 +605,33 @@ \end{center} \end{figure} - Conjunction as an explicit connective allows to treat both - simultaneous assumptions and conclusions uniformly. The definition - allows to derive the usual introduction \isa{{\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ A\ {\isacharampersand}\ B}, - and destructions \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ A} and \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ B}. For - example, several claims may be stated at the same time, which is - intermediately represented as an assumption, but the user only - encounters several sub-goals, and several resulting facts in the - very end (cf.\ \secref{sec:tactical-goals}). + 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.\ + \secref{sec:tactical-goals}). - The \isa{{\isacharhash}} marker allows complex propositions (nested \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}}) to appear formally 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{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. Despite its logically vacous meaning, this is - occasionally useful to treat syntactic terms and proven propositions - uniformly, as in a type-theoretic framework. + 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{TYPE} constructor (which is the canonical - representative of the unspecified type \isa{{\isasymalpha}\ itself}) injects - the language of types into that of terms. There is specific - notation \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} for \isa{TYPE\isactrlbsub {\isasymtau}\ itself\isactrlesub }. - Although being devoid of any particular meaning, the term \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} is able to carry the type \isa{{\isasymtau}} formally. \isa{TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}} may be used as an additional formal argument in primitive - definitions, in order to avoid hidden polymorphism (cf.\ - \secref{sec:terms}). For example, \isa{c\ TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}\ {\isasymequiv}\ A{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}} turns - out as a formally correct definition of some proposition \isa{A} - that depends on an additional type argument.% + The \isa{TYPE} constructor is the canonical representative of + the unspecified type \isa{{\isasymalpha}\ itself}; it essentially injects the + language of types into that of terms. There is specific notation + \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} for \isa{TYPE\isactrlbsub {\isasymtau}\ itself\isactrlesub }. + Although being devoid of any particular meaning, the \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} accounts for the type \isa{{\isasymtau}} within the term + language. In particular, \isa{TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}} may be used as formal + argument in primitive definitions, in order to circumvent hidden + polymorphism (cf.\ \secref{sec:terms}). For example, \isa{c\ TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}\ {\isasymequiv}\ A{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}} defines \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ itself\ {\isasymRightarrow}\ prop} in terms of + a proposition \isa{A} that depends on an additional type + argument, which is essentially a predicate on types.% \end{isamarkuptext}% \isamarkuptrue% %