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\begin{isabellebody}%
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\def\isabellecontext{logic}%
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\isadelimtheory
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\isanewline
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\isanewline
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\isanewline
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\endisadelimtheory
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ logic\ \isakeyword{imports}\ base\ \isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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\isadelimtheory
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\endisadelimtheory
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%
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\isamarkupchapter{Primitive logic \label{ch:logic}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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The logical foundations of Isabelle/Isar are that of the Pure logic,
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which has been introduced as a natural-deduction framework in
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\cite{paulson700}. This is essentially the same logic as ``\isa{{\isasymlambda}HOL}'' in the more abstract setting of Pure Type Systems (PTS)
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\cite{Barendregt-Geuvers:2001}, although there are some key
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differences in the specific treatment of simple types in
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Isabelle/Pure.
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Following type-theoretic parlance, the Pure logic consists of three
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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
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\isa{{\isasymLongrightarrow}} for implication (proofs depending on proofs).
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Pure derivations are relative to a logical theory, which declares
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type constructors, term constants, and axioms. Theory declarations
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support schematic polymorphism, which is strictly speaking outside
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the logic.\footnote{Incidently, this is the main logical reason, why
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the theory context \isa{{\isasymTheta}} is separate from the context \isa{{\isasymGamma}} of the core calculus.}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Types \label{sec:types}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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The language of types is an uninterpreted order-sorted first-order
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algebra; types are qualified by ordered type classes.
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\medskip A \emph{type class} is an abstract syntactic entity
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declared in the theory context. The \emph{subclass relation} \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}} is specified by stating an acyclic
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generating relation; the transitive closure is maintained
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internally. The resulting relation is an ordering: reflexive,
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transitive, and antisymmetric.
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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
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intersection. Notationally, the curly braces are omitted for
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singleton intersections, i.e.\ any class \isa{c} may be read as
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a sort \isa{{\isacharbraceleft}c{\isacharbraceright}}. The ordering on type classes is extended to
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sorts according to the meaning of intersections: \isa{{\isacharbraceleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}\ c\isactrlisub m{\isacharbraceright}\ {\isasymsubseteq}\ {\isacharbraceleft}d\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ d\isactrlisub n{\isacharbraceright}} iff
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\isa{{\isasymforall}j{\isachardot}\ {\isasymexists}i{\isachardot}\ c\isactrlisub i\ {\isasymsubseteq}\ d\isactrlisub j}. The empty intersection
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\isa{{\isacharbraceleft}{\isacharbraceright}} refers to the universal sort, which is the largest
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element wrt.\ the sort order. The intersections of all (finitely
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many) classes declared in the current theory are the minimal
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elements wrt.\ the sort order.
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\medskip A \emph{fixed type variable} is a pair of a basic name
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(starting with a \isa{{\isacharprime}} character) and a sort constraint. For
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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
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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}.
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Note that \emph{all} syntactic components contribute to the identity
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of type variables, including the sort constraint. The core logic
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handles type variables with the same name but different sorts as
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different, although some outer layers of the system make it hard to
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produce anything like this.
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A \emph{type constructor} \isa{{\isasymkappa}} is a \isa{k}-ary operator
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on types declared in the theory. Type constructor application is
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usually written postfix as \isa{{\isacharparenleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub k{\isacharparenright}{\isasymkappa}}.
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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
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parentheses are omitted, e.g.\ \isa{{\isasymalpha}\ list} instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharparenright}list}. Further notation is provided for specific constructors,
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notably the right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}} instead of
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\isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}.
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A \emph{type} is defined inductively over type variables and type
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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}.
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A \emph{type abbreviation} is a syntactic abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} of an arbitrary type expression \isa{{\isasymtau}} over
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variables \isa{\isactrlvec {\isasymalpha}}. Type abbreviations looks like type
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constructors at the surface, but are fully expanded before entering
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the logical core.
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A \emph{type arity} declares the image behavior of a type
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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
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of sort \isa{s} if every argument type \isa{{\isasymtau}\isactrlisub i} is
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of sort \isa{s\isactrlisub i}. Arity declarations are implicitly
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completed, i.e.\ \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}c} entails \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}c{\isacharprime}} for any \isa{c{\isacharprime}\ {\isasymsupseteq}\ c}.
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\medskip The sort algebra is always maintained as \emph{coregular},
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which means that type arities are consistent with the subclass
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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 componentwise.
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The key property of a coregular order-sorted algebra is that sort
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constraints may be always solved in a most general fashion: for each
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type constructor \isa{{\isasymkappa}} and sort \isa{s} there is a most
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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
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of sort \isa{s}. Consequently, the unification problem on the
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algebra of types has most general solutions (modulo renaming and
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equivalence of sorts). Moreover, the usual type-inference algorithm
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will produce primary types as expected \cite{nipkow-prehofer}.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isadelimmlref
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\endisadelimmlref
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\isatagmlref
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%
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\begin{isamarkuptext}%
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\begin{mldecls}
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\indexmltype{class}\verb|type class| \\
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\indexmltype{sort}\verb|type sort| \\
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\indexmltype{arity}\verb|type arity| \\
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\indexmltype{typ}\verb|type typ| \\
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\indexml{fold-atyps}\verb|fold_atyps: (typ -> 'a -> 'a) -> typ -> 'a -> 'a| \\
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\indexml{Sign.subsort}\verb|Sign.subsort: theory -> sort * sort -> bool| \\
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\indexml{Sign.of-sort}\verb|Sign.of_sort: theory -> typ * sort -> bool| \\
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\indexml{Sign.add-types}\verb|Sign.add_types: (bstring * int * mixfix) list -> theory -> theory| \\
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\indexml{Sign.add-tyabbrs-i}\verb|Sign.add_tyabbrs_i: |\isasep\isanewline%
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\verb| (bstring * string list * typ * mixfix) list -> theory -> theory| \\
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\indexml{Sign.primitive-class}\verb|Sign.primitive_class: string * class list -> theory -> theory| \\
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\indexml{Sign.primitive-classrel}\verb|Sign.primitive_classrel: class * class -> theory -> theory| \\
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\indexml{Sign.primitive-arity}\verb|Sign.primitive_arity: arity -> theory -> theory| \\
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\end{mldecls}
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\begin{description}
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\item \verb|class| represents type classes; this is an alias for
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\verb|string|.
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\item \verb|sort| represents sorts; this is an alias for
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\verb|class list|.
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\item \verb|arity| represents type arities; this is an alias for
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triples of the form \isa{{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec s{\isacharcomma}\ s{\isacharparenright}} for \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s} described above.
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\item \verb|typ| represents types; this is a datatype with
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constructors \verb|TFree|, \verb|TVar|, \verb|Type|.
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\item \verb|fold_atyps|~\isa{f\ {\isasymtau}} iterates function \isa{f}
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over all occurrences of atoms (\verb|TFree| or \verb|TVar|) of \isa{{\isasymtau}}; the type structure is traversed from left to right.
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\item \verb|Sign.subsort|~\isa{thy\ {\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ s\isactrlisub {\isadigit{2}}{\isacharparenright}}
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tests the subsort relation \isa{s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ s\isactrlisub {\isadigit{2}}}.
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\item \verb|Sign.of_sort|~\isa{thy\ {\isacharparenleft}{\isasymtau}{\isacharcomma}\ s{\isacharparenright}} tests whether a type
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is of a given sort.
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\item \verb|Sign.add_types|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ k{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares new
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type constructors \isa{{\isasymkappa}} with \isa{k} arguments and
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optional mixfix syntax.
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\item \verb|Sign.add_tyabbrs_i|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec {\isasymalpha}{\isacharcomma}\ {\isasymtau}{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}}
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defines a new type abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} with
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optional mixfix syntax.
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\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
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relations \isa{c\ {\isasymsubseteq}\ c\isactrlisub i}, for \isa{i\ {\isacharequal}\ {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ n}.
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\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}}}.
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\item \verb|Sign.primitive_arity|~\isa{{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec s{\isacharcomma}\ s{\isacharparenright}} declares
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arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}.
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\end{description}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\endisatagmlref
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{\isafoldmlref}%
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%
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\isadelimmlref
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%
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\endisadelimmlref
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%
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\isamarkupsection{Terms \label{sec:terms}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\glossary{Term}{FIXME}
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The language of terms is that of simply-typed \isa{{\isasymlambda}}-calculus
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with de-Bruijn indices for bound variables, and named free
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variables, and constants. Terms with loose bound variables are
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usually considered malformed. The types of variables and constants
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is stored explicitly at each occurrence in the term (which is a
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known performance issue).
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FIXME de-Bruijn representation of lambda terms
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Term syntax provides explicit abstraction \isa{{\isasymlambda}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ b{\isacharparenleft}x{\isacharparenright}}
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and application \isa{t\ u}, while types are usually implicit
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thanks to type-inference.
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\[
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\infer{\isa{a\isactrlisub {\isasymtau}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}}{}
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\qquad
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\infer{\isa{{\isacharparenleft}{\isasymlambda}x\isactrlsub {\isasymtau}{\isachardot}\ t{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymsigma}}}{\isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}}
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\qquad
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\infer{\isa{t\ u\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}}{\isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymsigma}} & \isa{u\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}}
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\]%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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FIXME
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\glossary{Schematic polymorphism}{FIXME}
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\glossary{Type variable}{FIXME}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Theorems \label{sec:thms}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\glossary{Proposition}{A \seeglossary{term} of \seeglossary{type}
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\isa{prop}. Internally, there is nothing special about
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propositions apart from their type, but the concrete syntax enforces
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a clear distinction. Propositions are structured via implication
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\isa{A\ {\isasymLongrightarrow}\ B} or universal quantification \isa{{\isasymAnd}x{\isachardot}\ B\ x} ---
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anything else is considered atomic. The canonical form for
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propositions is that of a \seeglossary{Hereditary Harrop Formula}. FIXME}
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\glossary{Theorem}{A proven proposition within a certain theory and
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proof context, formally \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}}; both contexts are
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rarely spelled out explicitly. Theorems are usually normalized
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according to the \seeglossary{HHF} format. FIXME}
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\glossary{Fact}{Sometimes used interchangably for
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\seeglossary{theorem}. Strictly speaking, a list of theorems,
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essentially an extra-logical conjunction. Facts emerge either as
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local assumptions, or as results of local goal statements --- both
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may be simultaneous, hence the list representation. FIXME}
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\glossary{Schematic variable}{FIXME}
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\glossary{Fixed variable}{A variable that is bound within a certain
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proof context; an arbitrary-but-fixed entity within a portion of
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proof text. FIXME}
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\glossary{Free variable}{Synonymous for \seeglossary{fixed
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variable}. FIXME}
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\glossary{Bound variable}{FIXME}
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\glossary{Variable}{See \seeglossary{schematic variable},
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\seeglossary{fixed variable}, \seeglossary{bound variable}, or
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\seeglossary{type variable}. The distinguishing feature of
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different variables is their binding scope. FIXME}
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A \emph{proposition} is a well-formed term of type \isa{prop}.
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The connectives of minimal logic are declared as constants of the
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basic theory:
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\smallskip
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\begin{tabular}{ll}
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\isa{all\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}{\isasymalpha}\ {\isasymRightarrow}\ prop{\isacharparenright}\ {\isasymRightarrow}\ prop} & universal quantification (binder \isa{{\isasymAnd}}) \\
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\isa{{\isasymLongrightarrow}\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} & implication (right associative infix) \\
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\end{tabular}
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\medskip A \emph{theorem} is a proven proposition, depending on a
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collection of assumptions, and axioms from the theory context. The
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judgment \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} is defined
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inductively by the primitive inferences given in
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\figref{fig:prim-rules}; there is a global syntactic restriction
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that the hypotheses may not contain schematic variables.
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\begin{figure}[htb]
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\begin{center}
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\[
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\infer[\isa{{\isacharparenleft}axiom{\isacharparenright}}]{\isa{{\isasymturnstile}\ A}}{\isa{A\ {\isasymin}\ {\isasymTheta}}}
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\qquad
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\infer[\isa{{\isacharparenleft}assume{\isacharparenright}}]{\isa{A\ {\isasymturnstile}\ A}}{}
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\]
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\[
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\infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b\ x}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ b\ x} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}}
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\qquad
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\infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ b\ a}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b\ x}}
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\]
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\[
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\infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isacharminus}\ A\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}}
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\qquad
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\infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\isactrlsub {\isadigit{1}}\ {\isasymunion}\ {\isasymGamma}\isactrlsub {\isadigit{2}}\ {\isasymturnstile}\ B}}{\isa{{\isasymGamma}\isactrlsub {\isadigit{1}}\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B} & \isa{{\isasymGamma}\isactrlsub {\isadigit{2}}\ {\isasymturnstile}\ A}}
|
|
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\]
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|
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\caption{Primitive inferences of the Pure logic}\label{fig:prim-rules}
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\end{center}
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|
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\end{figure}
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|
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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
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are \emph{irrelevant} in the Pure logic, they may never occur within
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propositions, i.e.\ the \isa{{\isasymLongrightarrow}} arrow of the framework is a
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non-dependent one.
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|
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|
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Also note that fixed parameters as in \isa{{\isasymAnd}{\isacharunderscore}intro} need not be
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|
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recorded in the context \isa{{\isasymGamma}}, since syntactic types are
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|
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always inhabitable. An ``assumption'' \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} is logically
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vacuous, because \isa{{\isasymtau}} is always non-empty. This is the deeper
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|
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reason why \isa{{\isasymGamma}} only consists of hypothetical proofs, but no
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|
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hypothetical terms.
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|
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The corresponding proof terms are left implicit in the classic
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|
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``LCF-approach'', although they could be exploited separately
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|
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\cite{Berghofer-Nipkow:2000}. The implementation provides a runtime
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|
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option to control the generation of full proof terms.
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|
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|
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\medskip The axiomatization of a theory is implicitly closed by
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|
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forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymtheta}} for
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|
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any substirution instance of axiom \isa{{\isasymturnstile}\ A}. By pushing
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|
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substitution through derivations inductively, we get admissible
|
|
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substitution rules for theorems shown in \figref{fig:subst-rules}.
|
|
332 |
|
|
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\begin{figure}[htb]
|
|
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\begin{center}
|
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|
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\[
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20502
|
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\infer{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}{\isasymalpha}{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}} & \isa{{\isasymalpha}\ {\isasymnotin}\ {\isasymGamma}}}
|
|
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\quad
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|
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\infer[\quad\isa{{\isacharparenleft}generalize{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}x{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}x{\isacharbrackright}} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}}
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|
339 |
\]
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|
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\[
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|
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\infer{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isasymtau}{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}{\isasymalpha}{\isacharbrackright}}}
|
|
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\quad
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|
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\infer[\quad\isa{{\isacharparenleft}instantiate{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}t{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}x{\isacharbrackright}}}
|
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|
344 |
\]
|
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|
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\caption{Admissible substitution rules}\label{fig:subst-rules}
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|
346 |
\end{center}
|
|
347 |
\end{figure}
|
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|
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|
|
349 |
Note that \isa{instantiate{\isacharunderscore}term} could be derived using \isa{{\isasymAnd}{\isacharunderscore}intro{\isacharslash}elim}, but this is not how it is implemented. The type
|
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|
350 |
instantiation rule is a genuine admissible one, due to the lack of
|
|
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true polymorphism in the logic.
|
20499
|
352 |
|
20502
|
353 |
Since \isa{{\isasymGamma}} may never contain any schematic variables, the
|
|
354 |
\isa{instantiate} do not require an explicit side-condition. In
|
|
355 |
principle, variables could be substituted in hypotheses as well, but
|
|
356 |
this could disrupt monotonicity of the basic calculus: derivations
|
|
357 |
could leave the current proof context.
|
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|
358 |
|
20502
|
359 |
\medskip The framework also provides builtin equality \isa{{\isasymequiv}},
|
|
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which is conceptually axiomatized shown in \figref{fig:equality},
|
|
361 |
although the implementation provides derived rules directly:
|
|
362 |
|
|
363 |
\begin{figure}[htb]
|
|
364 |
\begin{center}
|
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|
365 |
\begin{tabular}{ll}
|
|
366 |
\isa{{\isasymequiv}\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ prop} & equality relation (infix) \\
|
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|
367 |
\isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ b\ x{\isacharparenright}\ a\ {\isasymequiv}\ b\ a} & \isa{{\isasymbeta}}-conversion \\
|
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|
368 |
\isa{{\isasymturnstile}\ x\ {\isasymequiv}\ x} & reflexivity law \\
|
|
369 |
\isa{{\isasymturnstile}\ x\ {\isasymequiv}\ y\ {\isasymLongrightarrow}\ P\ x\ {\isasymLongrightarrow}\ P\ y} & substitution law \\
|
|
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\isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ f\ x\ {\isasymequiv}\ g\ x{\isacharparenright}\ {\isasymLongrightarrow}\ f\ {\isasymequiv}\ g} & extensionality \\
|
|
371 |
\isa{{\isasymturnstile}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymequiv}\ B} & coincidence with equivalence \\
|
|
372 |
\end{tabular}
|
20502
|
373 |
\caption{Conceptual axiomatization of equality.}\label{fig:equality}
|
|
374 |
\end{center}
|
|
375 |
\end{figure}
|
|
376 |
|
|
377 |
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.
|
|
378 |
|
|
379 |
|
|
380 |
\medskip Conjunction is defined in Pure as a derived connective, see
|
|
381 |
\figref{fig:conjunction}. This is occasionally useful to represent
|
|
382 |
simultaneous statements behind the scenes --- framework conjunction
|
|
383 |
is usually not exposed to the user.
|
|
384 |
|
|
385 |
\begin{figure}[htb]
|
|
386 |
\begin{center}
|
|
387 |
\begin{tabular}{ll}
|
|
388 |
\isa{{\isacharampersand}\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} & conjunction (hidden) \\
|
|
389 |
\isa{{\isasymturnstile}\ A\ {\isacharampersand}\ B\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}C{\isachardot}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isacharparenright}} \\
|
|
390 |
\end{tabular}
|
|
391 |
\caption{Definition of conjunction.}\label{fig:equality}
|
|
392 |
\end{center}
|
|
393 |
\end{figure}
|
|
394 |
|
|
395 |
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}.%
|
18537
|
396 |
\end{isamarkuptext}%
|
|
397 |
\isamarkuptrue%
|
|
398 |
%
|
20491
|
399 |
\isamarkupsection{Rules \label{sec:rules}%
|
18537
|
400 |
}
|
|
401 |
\isamarkuptrue%
|
|
402 |
%
|
|
403 |
\begin{isamarkuptext}%
|
|
404 |
FIXME
|
|
405 |
|
20491
|
406 |
A \emph{rule} is any Pure theorem in HHF normal form; there is a
|
|
407 |
separate calculus for rule composition, which is modeled after
|
|
408 |
Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
|
|
409 |
rules to be nested arbitrarily, similar to \cite{extensions91}.
|
|
410 |
|
|
411 |
Normally, all theorems accessible to the user are proper rules.
|
|
412 |
Low-level inferences are occasional required internally, but the
|
|
413 |
result should be always presented in canonical form. The higher
|
|
414 |
interfaces of Isabelle/Isar will always produce proper rules. It is
|
|
415 |
important to maintain this invariant in add-on applications!
|
|
416 |
|
|
417 |
There are two main principles of rule composition: \isa{resolution} (i.e.\ backchaining of rules) and \isa{by{\isacharminus}assumption} (i.e.\ closing a branch); both principles are
|
|
418 |
combined in the variants of \isa{elim{\isacharminus}resosultion} and \isa{dest{\isacharminus}resolution}. Raw \isa{composition} is occasionally
|
|
419 |
useful as well, also it is strictly speaking outside of the proper
|
|
420 |
rule calculus.
|
|
421 |
|
|
422 |
Rules are treated modulo general higher-order unification, which is
|
|
423 |
unification modulo the equational theory of \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion
|
|
424 |
on \isa{{\isasymlambda}}-terms. Moreover, propositions are understood modulo
|
|
425 |
the (derived) equivalence \isa{{\isacharparenleft}A\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x{\isacharparenright}{\isacharparenright}\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ A\ {\isasymLongrightarrow}\ B\ x{\isacharparenright}}.
|
|
426 |
|
|
427 |
This means that any operations within the rule calculus may be
|
|
428 |
subject to spontaneous \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-HHF conversions. It is common
|
|
429 |
practice not to contract or expand unnecessarily. Some mechanisms
|
|
430 |
prefer an one form, others the opposite, so there is a potential
|
|
431 |
danger to produce some oscillation!
|
|
432 |
|
|
433 |
Only few operations really work \emph{modulo} HHF conversion, but
|
|
434 |
expect a normal form: quantifiers \isa{{\isasymAnd}} before implications
|
|
435 |
\isa{{\isasymLongrightarrow}} at each level of nesting.
|
|
436 |
|
18537
|
437 |
\glossary{Hereditary Harrop Formula}{The set of propositions in HHF
|
|
438 |
format is defined inductively as \isa{H\ {\isacharequal}\ {\isacharparenleft}{\isasymAnd}x\isactrlsup {\isacharasterisk}{\isachardot}\ H\isactrlsup {\isacharasterisk}\ {\isasymLongrightarrow}\ A{\isacharparenright}}, for variables \isa{x} and atomic propositions \isa{A}.
|
|
439 |
Any proposition may be put into HHF form by normalizing with the rule
|
|
440 |
\isa{{\isacharparenleft}A\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x{\isacharparenright}{\isacharparenright}\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ A\ {\isasymLongrightarrow}\ B\ x{\isacharparenright}}. In Isabelle, the outermost
|
|
441 |
quantifier prefix is represented via \seeglossary{schematic
|
|
442 |
variables}, such that the top-level structure is merely that of a
|
|
443 |
\seeglossary{Horn Clause}}.
|
|
444 |
|
20499
|
445 |
\glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
|
|
446 |
|
|
447 |
|
|
448 |
\[
|
|
449 |
\infer[\isa{{\isacharparenleft}assumption{\isacharparenright}}]{\isa{C{\isasymvartheta}}}
|
|
450 |
{\isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec H\ \isactrlvec x\ {\isasymLongrightarrow}\ A\ \isactrlvec x{\isacharparenright}\ {\isasymLongrightarrow}\ C} & \isa{A{\isasymvartheta}\ {\isacharequal}\ H\isactrlsub i{\isasymvartheta}}~~\text{(for some~\isa{i})}}
|
|
451 |
\]
|
|
452 |
|
|
453 |
|
|
454 |
\[
|
|
455 |
\infer[\isa{{\isacharparenleft}compose{\isacharparenright}}]{\isa{\isactrlvec A{\isasymvartheta}\ {\isasymLongrightarrow}\ C{\isasymvartheta}}}
|
|
456 |
{\isa{\isactrlvec A\ {\isasymLongrightarrow}\ B} & \isa{B{\isacharprime}\ {\isasymLongrightarrow}\ C} & \isa{B{\isasymvartheta}\ {\isacharequal}\ B{\isacharprime}{\isasymvartheta}}}
|
|
457 |
\]
|
|
458 |
|
|
459 |
|
|
460 |
\[
|
|
461 |
\infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}lift{\isacharparenright}}]{\isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec A\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ B\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}}}{\isa{\isactrlvec A\ {\isacharquery}\isactrlvec a\ {\isasymLongrightarrow}\ B\ {\isacharquery}\isactrlvec a}}
|
|
462 |
\]
|
|
463 |
\[
|
|
464 |
\infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}lift{\isacharparenright}}]{\isa{{\isacharparenleft}\isactrlvec H\ {\isasymLongrightarrow}\ \isactrlvec A{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}\isactrlvec H\ {\isasymLongrightarrow}\ B{\isacharparenright}}}{\isa{\isactrlvec A\ {\isasymLongrightarrow}\ B}}
|
|
465 |
\]
|
|
466 |
|
|
467 |
The \isa{resolve} scheme is now acquired from \isa{{\isasymAnd}{\isacharunderscore}lift},
|
|
468 |
\isa{{\isasymLongrightarrow}{\isacharunderscore}lift}, and \isa{compose}.
|
|
469 |
|
|
470 |
\[
|
|
471 |
\infer[\isa{{\isacharparenleft}resolution{\isacharparenright}}]
|
|
472 |
{\isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec H\ \isactrlvec x\ {\isasymLongrightarrow}\ \isactrlvec A\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}{\isasymvartheta}\ {\isasymLongrightarrow}\ C{\isasymvartheta}}}
|
|
473 |
{\begin{tabular}{l}
|
|
474 |
\isa{\isactrlvec A\ {\isacharquery}\isactrlvec a\ {\isasymLongrightarrow}\ B\ {\isacharquery}\isactrlvec a} \\
|
|
475 |
\isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec H\ \isactrlvec x\ {\isasymLongrightarrow}\ B{\isacharprime}\ \isactrlvec x{\isacharparenright}\ {\isasymLongrightarrow}\ C} \\
|
|
476 |
\isa{{\isacharparenleft}{\isasymlambda}\isactrlvec x{\isachardot}\ B\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}{\isasymvartheta}\ {\isacharequal}\ B{\isacharprime}{\isasymvartheta}} \\
|
|
477 |
\end{tabular}}
|
|
478 |
\]
|
|
479 |
|
|
480 |
|
|
481 |
FIXME \isa{elim{\isacharunderscore}resolution}, \isa{dest{\isacharunderscore}resolution}%
|
18537
|
482 |
\end{isamarkuptext}%
|
|
483 |
\isamarkuptrue%
|
|
484 |
%
|
|
485 |
\isadelimtheory
|
|
486 |
%
|
|
487 |
\endisadelimtheory
|
|
488 |
%
|
|
489 |
\isatagtheory
|
|
490 |
\isacommand{end}\isamarkupfalse%
|
|
491 |
%
|
|
492 |
\endisatagtheory
|
|
493 |
{\isafoldtheory}%
|
|
494 |
%
|
|
495 |
\isadelimtheory
|
|
496 |
%
|
|
497 |
\endisadelimtheory
|
|
498 |
\isanewline
|
|
499 |
\end{isabellebody}%
|
|
500 |
%%% Local Variables:
|
|
501 |
%%% mode: latex
|
|
502 |
%%% TeX-master: "root"
|
|
503 |
%%% End:
|