doc-src/IsarImplementation/Thy/document/logic.tex
 author wenzelm Fri, 08 Sep 2006 19:44:43 +0200 changeset 20494 99ad217b6974 parent 20493 48fea5e99505 child 20499 18845f9dbd09 permissions -rw-r--r--
tuned;

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\isamarkupchapter{Primitive logic \label{ch:logic}%
}
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\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).

Pure derivations are relative to a logical theory, which declares
type constructors, term constants, and axioms.  Theory declarations
support schematic polymorphism, which is strictly speaking outside
the logic.\footnote{Incidently, this is the main logical reason, why
the theory context \isa{{\isasymTheta}} is separate from the context \isa{{\isasymGamma}} of the core calculus.}%
\end{isamarkuptext}%
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\isamarkupsection{Types \label{sec:types}%
}
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\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{{\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.  For
example, \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ s{\isacharparenright}} which is usually printed as \isa{{\isasymalpha}\isactrlisub s}.  A \emph{schematic type variable} is a pair of an
indexname and a sort constraint.  For example, \isa{{\isacharparenleft}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ s{\isacharparenright}} which is usually printed as \isa{{\isacharquery}{\isasymalpha}\isactrlisub s}.

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
usually written postfix as \isa{{\isacharparenleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub k{\isacharparenright}{\isasymkappa}}.
For \isa{k\ {\isacharequal}\ {\isadigit{0}}} the argument tuple is omitted, e.g.\ \isa{prop} instead of \isa{{\isacharparenleft}{\isacharparenright}prop}.  For \isa{k\ {\isacharequal}\ {\isadigit{1}}} the
parentheses are omitted, e.g.\ \isa{{\isasymalpha}\ list} instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharparenright}list}.  Further notation is provided for specific constructors,
notably the right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}} instead of
\isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}.

A \emph{type} is defined inductively over type variables and type
constructors as follows: \isa{{\isasymtau}\ {\isacharequal}\ {\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharquery}{\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharparenleft}{\isasymtau}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlsub k{\isacharparenright}k}.

A \emph{type abbreviation} is a syntactic abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} of an arbitrary type expression \isa{{\isasymtau}} over
variables \isa{\isactrlvec {\isasymalpha}}.  Type abbreviations looks like type
constructors at the surface, but are fully expanded before entering
the logical core.

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 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.

The key property of a coregular order-sorted algebra is that sort
constraints may be always solved in a most general fashion: for each
type constructor \isa{{\isasymkappa}} and sort \isa{s} there is a most
general vector of argument sorts \isa{{\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub k{\isacharparenright}} such that a type scheme \isa{{\isacharparenleft}{\isasymalpha}\isactrlbsub s\isactrlisub {\isadigit{1}}\isactrlesub {\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlbsub s\isactrlisub k\isactrlesub {\isacharparenright}{\isasymkappa}} is
of sort \isa{s}.  Consequently, the unification problem on the
algebra of types has most general solutions (modulo renaming and
equivalence of sorts).  Moreover, the usual type-inference algorithm
will produce primary types as expected \cite{nipkow-prehofer}.%
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\begin{mldecls}
\indexmltype{class}\verb|type class| \\
\indexmltype{sort}\verb|type sort| \\
\indexmltype{arity}\verb|type arity| \\
\indexmltype{typ}\verb|type typ| \\
\indexml{fold-atyps}\verb|fold_atyps: (typ -> 'a -> 'a) -> typ -> 'a -> 'a| \\
\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: (bstring * int * mixfix) list -> theory -> theory| \\
\verb|  (bstring * 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|fold_atyps|~\isa{f\ {\isasymtau}} iterates function \isa{f}
over all occurrences of atoms (\verb|TFree| or \verb|TVar|) of \isa{{\isasymtau}}; the type structure is traversed from left to right.

tests the subsort relation \isa{s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ s\isactrlisub {\isadigit{2}}}.

\item \verb|Sign.of_sort|~\isa{thy\ {\isacharparenleft}{\isasymtau}{\isacharcomma}\ s{\isacharparenright}} tests whether a type
is of a given sort.

\item \verb|Sign.add_types|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ k{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares 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 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_arity|~\isa{{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec s{\isacharcomma}\ s{\isacharparenright}} declares
arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}.

\end{description}%
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\isamarkupsection{Terms \label{sec:terms}%
}
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\glossary{Term}{FIXME}

The language of terms is that of simply-typed \isa{{\isasymlambda}}-calculus
with de-Bruijn indices for bound variables, and named free
variables, and constants.  Terms with loose bound variables are
usually considered malformed.  The types of variables and constants
is stored explicitly at each occurrence in the term (which is a
known performance issue).

FIXME de-Bruijn representation of lambda terms

Term syntax provides explicit abstraction \isa{{\isasymlambda}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ b{\isacharparenleft}x{\isacharparenright}}
and application \isa{t\ u}, while types are usually implicit
thanks to type-inference.

Terms of type \isa{prop} are called
propositions.  Logical statements are composed via \isa{{\isasymAnd}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} and \isa{A\ {\isasymLongrightarrow}\ B}.%
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FIXME

\glossary{Schematic polymorphism}{FIXME}

\glossary{Type variable}{FIXME}%
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\isamarkupsection{Theorems \label{sec:thms}%
}
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\begin{isamarkuptext}%
Primitive reasoning operates on judgments of the form \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}}, with standard introduction and elimination rules for \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} that refer to fixed parameters \isa{x} and
hypotheses \isa{A} from the context \isa{{\isasymGamma}}.  The
corresponding proof terms are left implicit in the classic
LCF-approach'', although they could be exploited separately
\cite{Berghofer-Nipkow:2000}.

The framework also provides definitional equality \isa{{\isasymequiv}\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ prop}, with \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion rules.  The internal
conjunction \isa{{\isacharampersand}\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} enables the view of
assumptions and conclusions emerging uniformly as simultaneous
statements.

FIXME

\glossary{Proposition}{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}.}

\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.}

\glossary{Fact}{Sometimes used interchangably 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.}

\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.}

\glossary{Free variable}{Synonymous for \seeglossary{fixed variable}.}

\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.}%
\end{isamarkuptext}%
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\isamarkupsection{Proof terms%
}
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\begin{isamarkuptext}%
FIXME !?%
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\isamarkupsection{Rules \label{sec: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}resosultion} 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}.}%
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