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-\begin{isabellebody}%
-\def\isabellecontext{logic}%
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ logic\ \isakeyword{imports}\ base\ \isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\isamarkupchapter{Primitive logic \label{ch:logic}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The logical foundations of Isabelle/Isar are that of the Pure logic,
- which has been introduced as a natural-deduction framework in
- \cite{paulson700}. This is essentially the same logic as ``\isa{{\isasymlambda}HOL}'' in the more abstract setting of Pure Type Systems (PTS)
- \cite{Barendregt-Geuvers:2001}, although there are some key
- differences in the specific treatment of simple types in
- Isabelle/Pure.
-
- Following type-theoretic parlance, the Pure logic consists of three
- levels of \isa{{\isasymlambda}}-calculus with corresponding arrows, \isa{{\isasymRightarrow}} for syntactic function space (terms depending on terms), \isa{{\isasymAnd}} for universal quantification (proofs depending on terms), and
- \isa{{\isasymLongrightarrow}} for implication (proofs depending on proofs).
-
- 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%
-%
-\isamarkupsection{Types \label{sec:types}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The language of types is an uninterpreted order-sorted first-order
- algebra; types are qualified by ordered type classes.
-
- \medskip A \emph{type class} is an abstract syntactic entity
- declared in the theory context. The \emph{subclass relation} \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}} is specified by stating an acyclic
- generating relation; the transitive closure is maintained
- internally. The resulting relation is an ordering: reflexive,
- transitive, and antisymmetric.
-
- A \emph{sort} is a list of type classes written as \isa{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
- sorts according to the meaning of intersections: \isa{{\isacharbraceleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}\ c\isactrlisub m{\isacharbraceright}\ {\isasymsubseteq}\ {\isacharbraceleft}d\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ d\isactrlisub n{\isacharbraceright}} iff
- \isa{{\isasymforall}j{\isachardot}\ {\isasymexists}i{\isachardot}\ c\isactrlisub i\ {\isasymsubseteq}\ d\isactrlisub j}. The empty intersection
- \isa{{\isacharbraceleft}{\isacharbraceright}} refers to the universal sort, which is the largest
- element wrt.\ the sort order. The intersections of all (finitely
- many) classes declared in the current theory are the minimal
- elements wrt.\ the sort order.
-
- \medskip A \emph{fixed type variable} is a pair of a basic name
- (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
- handles type variables with the same name but different sorts as
- different, although some outer layers of the system make it hard to
- produce anything like this.
-
- A \emph{type constructor} \isa{{\isasymkappa}} is a \isa{k}-ary operator
- on types declared in the theory. Type constructor application is
- 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} 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}{\isasymkappa}}.
-
- A \emph{type abbreviation} is a syntactic definition \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} of an arbitrary type expression \isa{{\isasymtau}} over
- variables \isa{\isactrlvec {\isasymalpha}}. Type abbreviations appear as type
- 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
- of sort \isa{s} if every argument type \isa{{\isasymtau}\isactrlisub i} is
- of sort \isa{s\isactrlisub i}. Arity declarations are implicitly
- 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}.
-
- \medskip The sort algebra is always maintained as \emph{coregular},
- which means that type arities are consistent with the subclass
- 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 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, type unification has most general solutions (modulo
- equivalence of sorts), so type-inference produces primary types as
- expected \cite{nipkow-prehofer}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
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-\isadelimmlref
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-\endisadelimmlref
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-\isatagmlref
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-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexmltype{class}\verb|type class| \\
- \indexmltype{sort}\verb|type sort| \\
- \indexmltype{arity}\verb|type arity| \\
- \indexmltype{typ}\verb|type typ| \\
- \indexml{map\_atyps}\verb|map_atyps: (typ -> typ) -> typ -> typ| \\
- \indexml{fold\_atyps}\verb|fold_atyps: (typ -> 'a -> 'a) -> typ -> 'a -> 'a| \\
- \end{mldecls}
- \begin{mldecls}
- \indexml{Sign.subsort}\verb|Sign.subsort: theory -> sort * sort -> bool| \\
- \indexml{Sign.of\_sort}\verb|Sign.of_sort: theory -> typ * sort -> bool| \\
- \indexml{Sign.add\_types}\verb|Sign.add_types: (string * int * mixfix) list -> theory -> theory| \\
- \indexml{Sign.add\_tyabbrs\_i}\verb|Sign.add_tyabbrs_i: |\isasep\isanewline%
-\verb| (string * string list * typ * mixfix) list -> theory -> theory| \\
- \indexml{Sign.primitive\_class}\verb|Sign.primitive_class: string * class list -> theory -> theory| \\
- \indexml{Sign.primitive\_classrel}\verb|Sign.primitive_classrel: class * class -> theory -> theory| \\
- \indexml{Sign.primitive\_arity}\verb|Sign.primitive_arity: arity -> theory -> theory| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|class| represents type classes; this is an alias for
- \verb|string|.
-
- \item \verb|sort| represents sorts; this is an alias for
- \verb|class list|.
-
- \item \verb|arity| represents type arities; this is an alias for
- 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.
-
- \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 the mapping \isa{f}
- to all atomic types (\verb|TFree|, \verb|TVar|) occurring in \isa{{\isasymtau}}.
-
- \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 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 a new
- type constructors \isa{{\isasymkappa}} with \isa{k} arguments and
- optional mixfix syntax.
-
- \item \verb|Sign.add_tyabbrs_i|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec {\isasymalpha}{\isacharcomma}\ {\isasymtau}{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}}
- 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 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 the class relation \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}.
-
- \item \verb|Sign.primitive_arity|~\isa{{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec s{\isacharcomma}\ s{\isacharparenright}} declares
- the arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
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-\endisatagmlref
-{\isafoldmlref}%
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-\isadelimmlref
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-\endisadelimmlref
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-\isamarkupsection{Terms \label{sec:terms}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\glossary{Term}{FIXME}
-
- 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}), 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 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}\isactrlbsub nat\isactrlesub {\isachardot}\ {\isasymlambda}\isactrlbsub nat\isactrlesub {\isachardot}\ {\isadigit{1}}\ {\isacharplus}\ {\isadigit{0}}} would
- correspond to \isa{{\isasymlambda}x\isactrlbsub nat\isactrlesub {\isachardot}\ {\isasymlambda}y\isactrlbsub nat\isactrlesub {\isachardot}\ x\ {\isacharplus}\ y} in a named
- representation. Note that a bound variable may be represented by
- different de-Bruijn indices at different occurrences, depending on
- the nesting of abstractions.
-
- A \emph{loose variable} is a bound variable that is outside the
- scope of local binders. The types (and names) for loose variables
- can be managed as a separate context, that is maintained as a stack
- of hypothetical binders. The core logic operates on closed terms,
- without any loose variables.
-
- A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
- \isa{{\isacharparenleft}x{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed \isa{x\isactrlisub {\isasymtau}}. A
- \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 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 all substitution instances
- \isa{c\isactrlisub {\isasymtau}} for \isa{{\isasymtau}\ {\isacharequal}\ {\isasymsigma}{\isasymvartheta}} are valid.
-
- The vector of \emph{type arguments} of constant \isa{c\isactrlisub {\isasymtau}}
- wrt.\ the declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} is defined as the codomain of
- 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
- due to violation of type class restrictions.
-
- \medskip An \emph{atomic} term is either a variable or constant. A
- \emph{term} is defined inductively over atomic terms, with
- abstraction and application as follows: \isa{t\ {\isacharequal}\ b\ {\isacharbar}\ x\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isacharquery}x\isactrlisub {\isasymtau}\ {\isacharbar}\ c\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ t\ {\isacharbar}\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}}.
- Parsing and printing takes care of converting between an external
- representation with named bound variables. Subsequently, we shall
- use the latter notation instead of internal de-Bruijn
- representation.
-
- The inductive relation \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} assigns a (unique) type to a
- term according to the structure of atomic terms, abstractions, and
- applicatins:
- \[
- \infer{\isa{a\isactrlisub {\isasymtau}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}}{}
- \qquad
- \infer{\isa{{\isacharparenleft}{\isasymlambda}x\isactrlsub {\isasymtau}{\isachardot}\ t{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymsigma}}}{\isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}}
- \qquad
- \infer{\isa{t\ u\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}}{\isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymsigma}} & \isa{u\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}}
- \]
- A \emph{well-typed term} is a term that can be typed according to these rules.
-
- Typing information can be omitted: type-inference is able to
- reconstruct the most general type of a raw term, while assigning
- most general types to all of its variables and constants.
- Type-inference depends on a context of type constraints for fixed
- variables, and declarations for polymorphic constants.
-
- The identity of atomic terms consists both of the name and the type
- 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
- contrast, mixed instances of polymorphic constants occur frequently.
-
- \medskip The \emph{hidden polymorphism} of a term \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}
- is the set of type variables occurring in \isa{t}, but not in
- \isa{{\isasymsigma}}. This means that the term implicitly depends on type
- arguments that are not accounted in the result type, i.e.\ there are
- different type instances \isa{t{\isasymvartheta}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} and \isa{t{\isasymvartheta}{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} with the same type. This slightly
- pathological situation notoriously demands additional care.
-
- \medskip A \emph{term abbreviation} is a syntactic definition \isa{c\isactrlisub {\isasymsigma}\ {\isasymequiv}\ t} of a closed term \isa{t} of type \isa{{\isasymsigma}},
- without any hidden polymorphism. A term abbreviation looks like a
- constant in the syntax, but is expanded before entering the logical
- core. Abbreviations are usually reverted when printing terms, using
- \isa{t\ {\isasymrightarrow}\ c\isactrlisub {\isasymsigma}} as rules for higher-order rewriting.
-
- \medskip Canonical operations on \isa{{\isasymlambda}}-terms include \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion: \isa{{\isasymalpha}}-conversion refers to capture-free
- renaming of bound variables; \isa{{\isasymbeta}}-conversion contracts an
- 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
- does not occur in \isa{f}.
-
- 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 standard operations (\secref{sec:obj-rules})
- that are based on higher-order unification and matching.%
-\end{isamarkuptext}%
-\isamarkuptrue%
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-\isadelimmlref
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-\endisadelimmlref
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-\isatagmlref
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-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexmltype{term}\verb|type term| \\
- \indexml{op aconv}\verb|op aconv: term * term -> bool| \\
- \indexml{map\_types}\verb|map_types: (typ -> typ) -> term -> term| \\
- \indexml{fold\_types}\verb|fold_types: (typ -> 'a -> 'a) -> term -> 'a -> 'a| \\
- \indexml{map\_aterms}\verb|map_aterms: (term -> term) -> term -> term| \\
- \indexml{fold\_aterms}\verb|fold_aterms: (term -> 'a -> 'a) -> term -> 'a -> 'a| \\
- \end{mldecls}
- \begin{mldecls}
- \indexml{fastype\_of}\verb|fastype_of: term -> typ| \\
- \indexml{lambda}\verb|lambda: term -> term -> term| \\
- \indexml{betapply}\verb|betapply: term * term -> term| \\
- \indexml{Sign.declare\_const}\verb|Sign.declare_const: Properties.T -> (binding * typ) * mixfix ->|\isasep\isanewline%
-\verb| theory -> term * theory| \\
- \indexml{Sign.add\_abbrev}\verb|Sign.add_abbrev: string -> Properties.T -> binding * term ->|\isasep\isanewline%
-\verb| theory -> (term * term) * theory| \\
- \indexml{Sign.const\_typargs}\verb|Sign.const_typargs: theory -> string * typ -> typ list| \\
- \indexml{Sign.const\_instance}\verb|Sign.const_instance: theory -> string * typ list -> typ| \\
- \end{mldecls}
-
- \begin{description}
-
- \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_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|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|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|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|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.declare_const|~\isa{properties\ {\isacharparenleft}{\isacharparenleft}c{\isacharcomma}\ {\isasymsigma}{\isacharparenright}{\isacharcomma}\ mx{\isacharparenright}}
- declares a new constant \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} with optional mixfix
- syntax.
-
- \item \verb|Sign.add_abbrev|~\isa{print{\isacharunderscore}mode\ properties\ {\isacharparenleft}c{\isacharcomma}\ t{\isacharparenright}}
- introduces a new term abbreviation \isa{c\ {\isasymequiv}\ t}.
-
- \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 two representations of polymorphic constants: full
- type instance vs.\ compact type arguments form.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagmlref
-{\isafoldmlref}%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isamarkupsection{Theorems \label{sec:thms}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\glossary{Proposition}{FIXME A \seeglossary{term} of
- \seeglossary{type} \isa{prop}. Internally, there is nothing
- special about propositions apart from their type, but the concrete
- syntax enforces a clear distinction. Propositions are structured
- via implication \isa{A\ {\isasymLongrightarrow}\ B} or universal quantification \isa{{\isasymAnd}x{\isachardot}\ B\ x} --- anything else is considered atomic. The canonical
- form for propositions is that of a \seeglossary{Hereditary Harrop
- Formula}. FIXME}
-
- \glossary{Theorem}{A proven proposition within a certain theory and
- proof context, formally \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}}; both contexts are
- rarely spelled out explicitly. Theorems are usually normalized
- according to the \seeglossary{HHF} format. FIXME}
-
- \glossary{Fact}{Sometimes used interchangeably for
- \seeglossary{theorem}. Strictly speaking, a list of theorems,
- essentially an extra-logical conjunction. Facts emerge either as
- local assumptions, or as results of local goal statements --- both
- may be simultaneous, hence the list representation. FIXME}
-
- \glossary{Schematic variable}{FIXME}
-
- \glossary{Fixed variable}{A variable that is bound within a certain
- proof context; an arbitrary-but-fixed entity within a portion of
- proof text. FIXME}
-
- \glossary{Free variable}{Synonymous for \seeglossary{fixed
- variable}. FIXME}
-
- \glossary{Bound variable}{FIXME}
-
- \glossary{Variable}{See \seeglossary{schematic variable},
- \seeglossary{fixed variable}, \seeglossary{bound variable}, or
- \seeglossary{type variable}. The distinguishing feature of
- different variables is their binding scope. FIXME}
-
- A \emph{proposition} is a well-typed term of type \isa{prop}, a
- \emph{theorem} is a proven proposition (depending on a context of
- hypotheses and the background theory). Primitive inferences include
- plain natural deduction rules for the primary connectives \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} of the framework. There is also a builtin
- notion of equality/equivalence \isa{{\isasymequiv}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Primitive connectives and rules \label{sec:prim-rules}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The theory \isa{Pure} contains constant declarations for the
- primitive connectives \isa{{\isasymAnd}}, \isa{{\isasymLongrightarrow}}, and \isa{{\isasymequiv}} of
- the logical framework, see \figref{fig:pure-connectives}. The
- derivability judgment \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} is
- defined inductively by the primitive inferences given in
- \figref{fig:prim-rules}, with the global restriction that the
- hypotheses must \emph{not} contain any schematic variables. The
- builtin equality is conceptually axiomatized as shown in
- \figref{fig:pure-equality}, although the implementation works
- directly with derived inferences.
-
- \begin{figure}[htb]
- \begin{center}
- \begin{tabular}{ll}
- \isa{all\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}{\isasymalpha}\ {\isasymRightarrow}\ prop{\isacharparenright}\ {\isasymRightarrow}\ prop} & universal quantification (binder \isa{{\isasymAnd}}) \\
- \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{Primitive connectives of Pure}\label{fig:pure-connectives}
- \end{center}
- \end{figure}
-
- \begin{figure}[htb]
- \begin{center}
- \[
- \infer[\isa{{\isacharparenleft}axiom{\isacharparenright}}]{\isa{{\isasymturnstile}\ A}}{\isa{A\ {\isasymin}\ {\isasymTheta}}}
- \qquad
- \infer[\isa{{\isacharparenleft}assume{\isacharparenright}}]{\isa{A\ {\isasymturnstile}\ A}}{}
- \]
- \[
- \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ b{\isacharbrackleft}x{\isacharbrackright}} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}}
- \qquad
- \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ b{\isacharbrackleft}a{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}}}
- \]
- \[
- \infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isacharminus}\ A\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}}
- \qquad
- \infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\isactrlsub {\isadigit{1}}\ {\isasymunion}\ {\isasymGamma}\isactrlsub {\isadigit{2}}\ {\isasymturnstile}\ B}}{\isa{{\isasymGamma}\isactrlsub {\isadigit{1}}\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B} & \isa{{\isasymGamma}\isactrlsub {\isadigit{2}}\ {\isasymturnstile}\ A}}
- \]
- \caption{Primitive inferences of Pure}\label{fig:prim-rules}
- \end{center}
- \end{figure}
-
- \begin{figure}[htb]
- \begin{center}
- \begin{tabular}{ll}
- \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} & logical equivalence \\
- \end{tabular}
- \caption{Conceptual axiomatization of Pure equality}\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 irrelevant in the Pure logic, though; they cannot occur within
- propositions. The system provides a runtime option to record
- explicit proof terms for primitive inferences. Thus all three
- levels of \isa{{\isasymlambda}}-calculus become explicit: \isa{{\isasymRightarrow}} for
- terms, and \isa{{\isasymAnd}{\isacharslash}{\isasymLongrightarrow}} for proofs (cf.\
- \cite{Berghofer-Nipkow:2000:TPHOL}).
-
- 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. ``Assumptions'' \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} for type-membership are only present as long as some \isa{x\isactrlisub {\isasymtau}} occurs in the statement body.\footnote{This is the key
- difference to ``\isa{{\isasymlambda}HOL}'' in the PTS framework
- \cite{Barendregt-Geuvers:2001}, where hypotheses \isa{x\ {\isacharcolon}\ A} are
- treated uniformly for propositions and types.}
-
- \medskip The axiomatization of a theory is implicitly closed by
- forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymvartheta}} holds for any substitution instance of an axiom
- \isa{{\isasymturnstile}\ A}. By pushing substitutions through derivations
- inductively, we also get admissible \isa{generalize} and \isa{instance} rules as shown in \figref{fig:subst-rules}.
-
- \begin{figure}[htb]
- \begin{center}
- \[
- \infer{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}{\isasymalpha}{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}} & \isa{{\isasymalpha}\ {\isasymnotin}\ {\isasymGamma}}}
- \quad
- \infer[\quad\isa{{\isacharparenleft}generalize{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}x{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}x{\isacharbrackright}} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}}
- \]
- \[
- \infer{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isasymtau}{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}{\isasymalpha}{\isacharbrackright}}}
- \quad
- \infer[\quad\isa{{\isacharparenleft}instantiate{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}t{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B{\isacharbrackleft}{\isacharquery}x{\isacharbrackright}}}
- \]
- \caption{Admissible substitution rules}\label{fig:subst-rules}
- \end{center}
- \end{figure}
-
- 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 the monotonicity of reasoning: deriving
- \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymturnstile}\ B{\isasymvartheta}} from \isa{{\isasymGamma}\ {\isasymturnstile}\ B} is
- correct, but \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymsupseteq}\ {\isasymGamma}} does not necessarily hold:
- the result belongs to a different proof context.
-
- \medskip An \emph{oracle} is a function that produces axioms on the
- fly. Logically, this is an instance of the \isa{axiom} rule
- (\figref{fig:prim-rules}), but there is an operational difference.
- The system always records oracle invocations within derivations of
- theorems. Tracing plain axioms (and named theorems) is optional.
-
- Axiomatizations should be limited to the bare minimum, typically as
- part of the initial logical basis of an object-logic formalization.
- Later on, theories are usually developed in a strictly definitional
- fashion, by stating only certain equalities over new constants.
-
- A \emph{simple definition} consists of a constant declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} together with an axiom \isa{{\isasymturnstile}\ c\ {\isasymequiv}\ t}, where \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} is a closed term without any hidden polymorphism. The RHS
- may depend on further defined constants, but not \isa{c} itself.
- Definitions of functions may be presented as \isa{c\ \isactrlvec x\ {\isasymequiv}\ t} instead of the puristic \isa{c\ {\isasymequiv}\ {\isasymlambda}\isactrlvec x{\isachardot}\ t}.
-
- An \emph{overloaded definition} consists of a collection of axioms
- for the same constant, with zero or one equations \isa{c{\isacharparenleft}{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}{\isacharparenright}\ {\isasymequiv}\ t} for each type constructor \isa{{\isasymkappa}} (for
- distinct variables \isa{\isactrlvec {\isasymalpha}}). The RHS may mention
- previously defined constants as above, or arbitrary constants \isa{d{\isacharparenleft}{\isasymalpha}\isactrlisub i{\isacharparenright}} for some \isa{{\isasymalpha}\isactrlisub i} projected from \isa{\isactrlvec {\isasymalpha}}. Thus overloaded definitions essentially work by
- primitive recursion over the syntactic structure of a single type
- argument.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isatagmlref
-%
-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexmltype{ctyp}\verb|type ctyp| \\
- \indexmltype{cterm}\verb|type cterm| \\
- \indexml{Thm.ctyp\_of}\verb|Thm.ctyp_of: theory -> typ -> ctyp| \\
- \indexml{Thm.cterm\_of}\verb|Thm.cterm_of: theory -> term -> cterm| \\
- \end{mldecls}
- \begin{mldecls}
- \indexmltype{thm}\verb|type thm| \\
- \indexml{proofs}\verb|proofs: int ref| \\
- \indexml{Thm.assume}\verb|Thm.assume: cterm -> thm| \\
- \indexml{Thm.forall\_intr}\verb|Thm.forall_intr: cterm -> thm -> thm| \\
- \indexml{Thm.forall\_elim}\verb|Thm.forall_elim: cterm -> thm -> thm| \\
- \indexml{Thm.implies\_intr}\verb|Thm.implies_intr: cterm -> thm -> thm| \\
- \indexml{Thm.implies\_elim}\verb|Thm.implies_elim: thm -> thm -> thm| \\
- \indexml{Thm.generalize}\verb|Thm.generalize: string list * string list -> int -> thm -> thm| \\
- \indexml{Thm.instantiate}\verb|Thm.instantiate: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm| \\
- \indexml{Thm.axiom}\verb|Thm.axiom: theory -> string -> thm| \\
- \indexml{Thm.add\_oracle}\verb|Thm.add_oracle: bstring * ('a -> cterm) -> theory|\isasep\isanewline%
-\verb| -> (string * ('a -> thm)) * theory| \\
- \end{mldecls}
- \begin{mldecls}
- \indexml{Theory.add\_axioms\_i}\verb|Theory.add_axioms_i: (binding * term) list -> theory -> theory| \\
- \indexml{Theory.add\_deps}\verb|Theory.add_deps: string -> string * typ -> (string * typ) list -> theory -> theory| \\
- \indexml{Theory.add\_defs\_i}\verb|Theory.add_defs_i: bool -> bool -> (binding * term) list -> theory -> theory| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|ctyp| and \verb|cterm| represent certified types
- and terms, respectively. These are abstract datatypes that
- guarantee that its values have passed the full well-formedness (and
- well-typedness) checks, relative to the declarations of type
- constructors, constants etc. in the theory.
-
- \item \verb|ctyp_of|~\isa{thy\ {\isasymtau}} and \verb|cterm_of|~\isa{thy\ t} explicitly checks types and terms, respectively. This also
- involves some basic normalizations, such expansion of type and term
- abbreviations from the theory context.
-
- Re-certification is relatively slow and should be avoided in tight
- reasoning loops. There are separate operations to decompose
- certified entities (including actual theorems).
-
- \item \verb|thm| represents proven propositions. This is an
- abstract datatype that guarantees that its values have been
- constructed by basic principles of the \verb|Thm| module.
- Every \verb|thm| value contains a sliding back-reference to the
- enclosing theory, cf.\ \secref{sec:context-theory}.
-
- \item \verb|proofs| determines the detail of proof recording within
- \verb|thm| values: \verb|0| records only oracles, \verb|1| records
- oracles, axioms and named theorems, \verb|2| records full proof
- terms.
-
- \item \verb|Thm.assume|, \verb|Thm.forall_intr|, \verb|Thm.forall_elim|, \verb|Thm.implies_intr|, and \verb|Thm.implies_elim|
- correspond to the primitive inferences of \figref{fig:prim-rules}.
-
- \item \verb|Thm.generalize|~\isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharcomma}\ \isactrlvec x{\isacharparenright}}
- corresponds to the \isa{generalize} rules of
- \figref{fig:subst-rules}. Here collections of type and term
- variables are generalized simultaneously, specified by the given
- basic names.
-
- \item \verb|Thm.instantiate|~\isa{{\isacharparenleft}\isactrlvec {\isasymalpha}\isactrlisub s{\isacharcomma}\ \isactrlvec x\isactrlisub {\isasymtau}{\isacharparenright}} corresponds to the \isa{instantiate} rules
- of \figref{fig:subst-rules}. Type variables are substituted before
- term variables. Note that the types in \isa{\isactrlvec x\isactrlisub {\isasymtau}}
- refer to the instantiated versions.
-
- \item \verb|Thm.axiom|~\isa{thy\ name} retrieves a named
- axiom, cf.\ \isa{axiom} in \figref{fig:prim-rules}.
-
- \item \verb|Thm.add_oracle|~\isa{{\isacharparenleft}name{\isacharcomma}\ oracle{\isacharparenright}} produces a named
- oracle rule, essentially generating arbitrary axioms on the fly,
- cf.\ \isa{axiom} in \figref{fig:prim-rules}.
-
- \item \verb|Theory.add_axioms_i|~\isa{{\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ A{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares
- arbitrary propositions as axioms.
-
- \item \verb|Theory.add_deps|~\isa{name\ c\isactrlisub {\isasymtau}\ \isactrlvec d\isactrlisub {\isasymsigma}} declares dependencies of a named specification
- for constant \isa{c\isactrlisub {\isasymtau}}, relative to existing
- specifications for constants \isa{\isactrlvec d\isactrlisub {\isasymsigma}}.
-
- \item \verb|Theory.add_defs_i|~\isa{unchecked\ overloaded\ {\isacharbrackleft}{\isacharparenleft}name{\isacharcomma}\ c\ \isactrlvec x\ {\isasymequiv}\ t{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} states a definitional axiom for an existing
- constant \isa{c}. Dependencies are recorded (cf.\ \verb|Theory.add_deps|), unless the \isa{unchecked} option is set.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagmlref
-{\isafoldmlref}%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isamarkupsubsection{Auxiliary definitions%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Theory \isa{Pure} provides a few auxiliary definitions, see
- \figref{fig:pure-aux}. These special constants are normally not
- exposed to the user, but appear in internal encodings.
-
- \begin{figure}[htb]
- \begin{center}
- \begin{tabular}{ll}
- \isa{conjunction\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} & (infix \isa{{\isacharampersand}}) \\
- \isa{{\isasymturnstile}\ A\ {\isacharampersand}\ B\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}C{\isachardot}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isacharparenright}} \\[1ex]
- \isa{prop\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop} & (prefix \isa{{\isacharhash}}, suppressed) \\
- \isa{{\isacharhash}A\ {\isasymequiv}\ A} \\[1ex]
- \isa{term\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ prop} & (prefix \isa{TERM}) \\
- \isa{term\ x\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}A{\isachardot}\ A\ {\isasymLongrightarrow}\ A{\isacharparenright}} \\[1ex]
- \isa{TYPE\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ itself} & (prefix \isa{TYPE}) \\
- \isa{{\isacharparenleft}unspecified{\isacharparenright}} \\
- \end{tabular}
- \caption{Definitions of auxiliary connectives}\label{fig:pure-aux}
- \end{center}
- \end{figure}
-
- 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 refined into
- separate sub-goals before the user continues the proof; the final
- result is projected into a list of theorems (cf.\
- \secref{sec:tactical-goals}).
-
- The \isa{prop} marker (\isa{{\isacharhash}}) makes arbitrarily complex
- propositions appear as atomic, without changing the meaning: \isa{{\isasymGamma}\ {\isasymturnstile}\ A} and \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isacharhash}A} are interchangeable. See
- \secref{sec:tactical-goals} for specific operations.
-
- The \isa{term} marker turns any well-typed term into a derivable
- proposition: \isa{{\isasymturnstile}\ TERM\ t} holds unconditionally. Although
- this is logically vacuous, it allows to treat terms and proofs
- uniformly, similar to a type-theoretic framework.
-
- The \isa{TYPE} constructor is the canonical representative of
- the unspecified type \isa{{\isasymalpha}\ itself}; it essentially injects the
- 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}%
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-\begin{mldecls}
- \indexml{Conjunction.intr}\verb|Conjunction.intr: thm -> thm -> thm| \\
- \indexml{Conjunction.elim}\verb|Conjunction.elim: thm -> thm * thm| \\
- \indexml{Drule.mk\_term}\verb|Drule.mk_term: cterm -> thm| \\
- \indexml{Drule.dest\_term}\verb|Drule.dest_term: thm -> cterm| \\
- \indexml{Logic.mk\_type}\verb|Logic.mk_type: typ -> term| \\
- \indexml{Logic.dest\_type}\verb|Logic.dest_type: term -> typ| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Conjunction.intr| derives \isa{A\ {\isacharampersand}\ B} from \isa{A} and \isa{B}.
-
- \item \verb|Conjunction.elim| derives \isa{A} and \isa{B}
- from \isa{A\ {\isacharampersand}\ B}.
-
- \item \verb|Drule.mk_term| derives \isa{TERM\ t}.
-
- \item \verb|Drule.dest_term| recovers term \isa{t} from \isa{TERM\ t}.
-
- \item \verb|Logic.mk_type|~\isa{{\isasymtau}} produces the term \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}}.
-
- \item \verb|Logic.dest_type|~\isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} recovers the type
- \isa{{\isasymtau}}.
-
- \end{description}%
-\end{isamarkuptext}%
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-\isamarkupsection{Object-level rules \label{sec:obj-rules}%
-}
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-\begin{isamarkuptext}%
-FIXME
-
- A \emph{rule} is any Pure theorem in HHF normal form; there is a
- separate calculus for rule composition, which is modeled after
- Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
- rules to be nested arbitrarily, similar to \cite{extensions91}.
-
- Normally, all theorems accessible to the user are proper rules.
- Low-level inferences are occasional required internally, but the
- result should be always presented in canonical form. The higher
- interfaces of Isabelle/Isar will always produce proper rules. It is
- important to maintain this invariant in add-on applications!
-
- There are two main principles of rule composition: \isa{resolution} (i.e.\ backchaining of rules) and \isa{by{\isacharminus}assumption} (i.e.\ closing a branch); both principles are
- combined in the variants of \isa{elim{\isacharminus}resolution} and \isa{dest{\isacharminus}resolution}. Raw \isa{composition} is occasionally
- useful as well, also it is strictly speaking outside of the proper
- rule calculus.
-
- Rules are treated modulo general higher-order unification, which is
- unification modulo the equational theory of \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion
- on \isa{{\isasymlambda}}-terms. Moreover, propositions are understood modulo
- the (derived) equivalence \isa{{\isacharparenleft}A\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x{\isacharparenright}{\isacharparenright}\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ A\ {\isasymLongrightarrow}\ B\ x{\isacharparenright}}.
-
- This means that any operations within the rule calculus may be
- subject to spontaneous \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-HHF conversions. It is common
- practice not to contract or expand unnecessarily. Some mechanisms
- prefer an one form, others the opposite, so there is a potential
- danger to produce some oscillation!
-
- Only few operations really work \emph{modulo} HHF conversion, but
- expect a normal form: quantifiers \isa{{\isasymAnd}} before implications
- \isa{{\isasymLongrightarrow}} at each level of nesting.
-
-\glossary{Hereditary Harrop Formula}{The set of propositions in HHF
-format is defined inductively as \isa{H\ {\isacharequal}\ {\isacharparenleft}{\isasymAnd}x\isactrlsup {\isacharasterisk}{\isachardot}\ H\isactrlsup {\isacharasterisk}\ {\isasymLongrightarrow}\ A{\isacharparenright}}, for variables \isa{x} and atomic propositions \isa{A}.
-Any proposition may be put into HHF form by normalizing with the rule
-\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
-quantifier prefix is represented via \seeglossary{schematic
-variables}, such that the top-level structure is merely that of a
-\seeglossary{Horn Clause}}.
-
-\glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
-
-
- \[
- \infer[\isa{{\isacharparenleft}assumption{\isacharparenright}}]{\isa{C{\isasymvartheta}}}
- {\isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec H\ \isactrlvec x\ {\isasymLongrightarrow}\ A\ \isactrlvec x{\isacharparenright}\ {\isasymLongrightarrow}\ C} & \isa{A{\isasymvartheta}\ {\isacharequal}\ H\isactrlsub i{\isasymvartheta}}~~\text{(for some~\isa{i})}}
- \]
-
-
- \[
- \infer[\isa{{\isacharparenleft}compose{\isacharparenright}}]{\isa{\isactrlvec A{\isasymvartheta}\ {\isasymLongrightarrow}\ C{\isasymvartheta}}}
- {\isa{\isactrlvec A\ {\isasymLongrightarrow}\ B} & \isa{B{\isacharprime}\ {\isasymLongrightarrow}\ C} & \isa{B{\isasymvartheta}\ {\isacharequal}\ B{\isacharprime}{\isasymvartheta}}}
- \]
-
-
- \[
- \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}lift{\isacharparenright}}]{\isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec A\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ B\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}}}{\isa{\isactrlvec A\ {\isacharquery}\isactrlvec a\ {\isasymLongrightarrow}\ B\ {\isacharquery}\isactrlvec a}}
- \]
- \[
- \infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}lift{\isacharparenright}}]{\isa{{\isacharparenleft}\isactrlvec H\ {\isasymLongrightarrow}\ \isactrlvec A{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}\isactrlvec H\ {\isasymLongrightarrow}\ B{\isacharparenright}}}{\isa{\isactrlvec A\ {\isasymLongrightarrow}\ B}}
- \]
-
- The \isa{resolve} scheme is now acquired from \isa{{\isasymAnd}{\isacharunderscore}lift},
- \isa{{\isasymLongrightarrow}{\isacharunderscore}lift}, and \isa{compose}.
-
- \[
- \infer[\isa{{\isacharparenleft}resolution{\isacharparenright}}]
- {\isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec H\ \isactrlvec x\ {\isasymLongrightarrow}\ \isactrlvec A\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}{\isasymvartheta}\ {\isasymLongrightarrow}\ C{\isasymvartheta}}}
- {\begin{tabular}{l}
- \isa{\isactrlvec A\ {\isacharquery}\isactrlvec a\ {\isasymLongrightarrow}\ B\ {\isacharquery}\isactrlvec a} \\
- \isa{{\isacharparenleft}{\isasymAnd}\isactrlvec x{\isachardot}\ \isactrlvec H\ \isactrlvec x\ {\isasymLongrightarrow}\ B{\isacharprime}\ \isactrlvec x{\isacharparenright}\ {\isasymLongrightarrow}\ C} \\
- \isa{{\isacharparenleft}{\isasymlambda}\isactrlvec x{\isachardot}\ B\ {\isacharparenleft}{\isacharquery}\isactrlvec a\ \isactrlvec x{\isacharparenright}{\isacharparenright}{\isasymvartheta}\ {\isacharequal}\ B{\isacharprime}{\isasymvartheta}} \\
- \end{tabular}}
- \]
-
-
- FIXME \isa{elim{\isacharunderscore}resolution}, \isa{dest{\isacharunderscore}resolution}%
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