--- a/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 14 15:27:08 2006 +0200
+++ b/doc-src/IsarImplementation/Thy/document/logic.tex Thu Sep 14 15:51:20 2006 +0200
@@ -35,11 +35,12 @@
levels of \isa{{\isasymlambda}}-calculus with corresponding arrows: \isa{{\isasymRightarrow}} for syntactic function space (terms depending on terms), \isa{{\isasymAnd}} for universal quantification (proofs depending on terms), and
\isa{{\isasymLongrightarrow}} for implication (proofs depending on proofs).
- Pure derivations are relative to a logical theory, which declares
- type constructors, term constants, and axioms. Theory declarations
- support schematic polymorphism, which is strictly speaking outside
- the logic.\footnote{Incidently, this is the main logical reason, why
- the theory context \isa{{\isasymTheta}} is separate from the context \isa{{\isasymGamma}} of the core calculus.}%
+ Derivations are relative to a logical theory, which declares type
+ constructors, constants, and axioms. Theory declarations support
+ schematic polymorphism, which is strictly speaking outside the
+ logic.\footnote{This is the deeper logical reason, why the theory
+ context \isa{{\isasymTheta}} is separate from the proof context \isa{{\isasymGamma}}
+ of the core calculus.}%
\end{isamarkuptext}%
\isamarkuptrue%
%
@@ -57,7 +58,7 @@
internally. The resulting relation is an ordering: reflexive,
transitive, and antisymmetric.
- A \emph{sort} is a list of type classes written as \isa{{\isacharbraceleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlisub m{\isacharbraceright}}, which represents symbolic
+ A \emph{sort} is a list of type classes written as \isa{s\ {\isacharequal}\ {\isacharbraceleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlisub m{\isacharbraceright}}, which represents symbolic
intersection. Notationally, the curly braces are omitted for
singleton intersections, i.e.\ any class \isa{c} may be read as
a sort \isa{{\isacharbraceleft}c{\isacharbraceright}}. The ordering on type classes is extended to
@@ -69,9 +70,11 @@
elements wrt.\ the sort order.
\medskip A \emph{fixed type variable} is a pair of a basic name
- (starting with a \isa{{\isacharprime}} character) and a sort constraint. For
- example, \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ s{\isacharparenright}} which is usually printed as \isa{{\isasymalpha}\isactrlisub s}. A \emph{schematic type variable} is a pair of an
- indexname and a sort constraint. For example, \isa{{\isacharparenleft}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ s{\isacharparenright}} which is usually printed as \isa{{\isacharquery}{\isasymalpha}\isactrlisub s}.
+ (starting with a \isa{{\isacharprime}} character) and a sort constraint, e.g.\
+ \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ s{\isacharparenright}} which is usually printed as \isa{{\isasymalpha}\isactrlisub s}.
+ A \emph{schematic type variable} is a pair of an indexname and a
+ sort constraint, e.g.\ \isa{{\isacharparenleft}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ s{\isacharparenright}} which is usually
+ printed as \isa{{\isacharquery}{\isasymalpha}\isactrlisub s}.
Note that \emph{all} syntactic components contribute to the identity
of type variables, including the sort constraint. The core logic
@@ -81,19 +84,20 @@
A \emph{type constructor} \isa{{\isasymkappa}} is a \isa{k}-ary operator
on types declared in the theory. Type constructor application is
- usually written postfix as \isa{{\isacharparenleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub k{\isacharparenright}{\isasymkappa}}.
- For \isa{k\ {\isacharequal}\ {\isadigit{0}}} the argument tuple is omitted, e.g.\ \isa{prop} instead of \isa{{\isacharparenleft}{\isacharparenright}prop}. For \isa{k\ {\isacharequal}\ {\isadigit{1}}} the
- parentheses are omitted, e.g.\ \isa{{\isasymalpha}\ list} instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharparenright}list}. Further notation is provided for specific constructors,
- notably the right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}} instead of
- \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}.
+ written postfix as \isa{{\isacharparenleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub k{\isacharparenright}{\isasymkappa}}. For
+ \isa{k\ {\isacharequal}\ {\isadigit{0}}} the argument tuple is omitted, e.g.\ \isa{prop}
+ instead of \isa{{\isacharparenleft}{\isacharparenright}prop}. For \isa{k\ {\isacharequal}\ {\isadigit{1}}} the parentheses
+ are omitted, e.g.\ \isa{{\isasymalpha}\ list} instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharparenright}list}.
+ Further notation is provided for specific constructors, notably the
+ right-associative infix \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}} instead of \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}.
- A \emph{type} \isa{{\isasymtau}} is defined inductively over type variables
- and type constructors as follows: \isa{{\isasymtau}\ {\isacharequal}\ {\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharquery}{\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharparenleft}{\isasymtau}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlsub k{\isacharparenright}k}.
+ A \emph{type} is defined inductively over type variables and type
+ constructors as follows: \isa{{\isasymtau}\ {\isacharequal}\ {\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharquery}{\isasymalpha}\isactrlisub s\ {\isacharbar}\ {\isacharparenleft}{\isasymtau}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlsub k{\isacharparenright}k}.
A \emph{type abbreviation} is a syntactic definition \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} of an arbitrary type expression \isa{{\isasymtau}} over
- variables \isa{\isactrlvec {\isasymalpha}}. Type abbreviations looks like type
- constructors at the surface, but are fully expanded before entering
- the logical core.
+ variables \isa{\isactrlvec {\isasymalpha}}. Type abbreviations appear as type
+ constructors in the syntax, but are expanded before entering the
+ logical core.
A \emph{type arity} declares the image behavior of a type
constructor wrt.\ the algebra of sorts: \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub k{\isacharparenright}s} means that \isa{{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub k{\isacharparenright}{\isasymkappa}} is
@@ -103,16 +107,17 @@
\medskip The sort algebra is always maintained as \emph{coregular},
which means that type arities are consistent with the subclass
- relation: for each type constructor \isa{{\isasymkappa}} and classes \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}, any arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{1}}{\isacharparenright}c\isactrlisub {\isadigit{1}}} has a corresponding arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{2}}{\isacharparenright}c\isactrlisub {\isadigit{2}}} where \isa{\isactrlvec s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ \isactrlvec s\isactrlisub {\isadigit{2}}} holds component-wise.
+ relation: for any type constructor \isa{{\isasymkappa}}, and classes \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}, and arities \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{1}}{\isacharparenright}c\isactrlisub {\isadigit{1}}} and \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlisub {\isadigit{2}}{\isacharparenright}c\isactrlisub {\isadigit{2}}} holds \isa{\isactrlvec s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ \isactrlvec s\isactrlisub {\isadigit{2}}} component-wise.
The key property of a coregular order-sorted algebra is that sort
- constraints may be always solved in a most general fashion: for each
- type constructor \isa{{\isasymkappa}} and sort \isa{s} there is a most
- general vector of argument sorts \isa{{\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub k{\isacharparenright}} such that a type scheme \isa{{\isacharparenleft}{\isasymalpha}\isactrlbsub s\isactrlisub {\isadigit{1}}\isactrlesub {\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlbsub s\isactrlisub k\isactrlesub {\isacharparenright}{\isasymkappa}} is
- of sort \isa{s}. Consequently, the unification problem on the
- algebra of types has most general solutions (modulo renaming and
- equivalence of sorts). Moreover, the usual type-inference algorithm
- will produce primary types as expected \cite{nipkow-prehofer}.%
+ constraints can be solved in a most general fashion: for each type
+ constructor \isa{{\isasymkappa}} and sort \isa{s} there is a most general
+ vector of argument sorts \isa{{\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlisub k{\isacharparenright}} such
+ that a type scheme \isa{{\isacharparenleft}{\isasymalpha}\isactrlbsub s\isactrlisub {\isadigit{1}}\isactrlesub {\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlbsub s\isactrlisub k\isactrlesub {\isacharparenright}{\isasymkappa}} is of sort \isa{s}.
+ Consequently, unification on the algebra of types has most general
+ solutions (modulo equivalence of sorts). This means that
+ type-inference will produce primary types as expected
+ \cite{nipkow-prehofer}.%
\end{isamarkuptext}%
\isamarkuptrue%
%
@@ -154,19 +159,19 @@
\item \verb|typ| represents types; this is a datatype with
constructors \verb|TFree|, \verb|TVar|, \verb|Type|.
- \item \verb|map_atyps|~\isa{f\ {\isasymtau}} applies mapping \isa{f} to
- all atomic types (\verb|TFree|, \verb|TVar|) occurring in \isa{{\isasymtau}}.
+ \item \verb|map_atyps|~\isa{f\ {\isasymtau}} applies the mapping \isa{f}
+ to all atomic types (\verb|TFree|, \verb|TVar|) occurring in \isa{{\isasymtau}}.
- \item \verb|fold_atyps|~\isa{f\ {\isasymtau}} iterates operation \isa{f}
- over all occurrences of atoms (\verb|TFree|, \verb|TVar|) in \isa{{\isasymtau}}; the type structure is traversed from left to right.
+ \item \verb|fold_atyps|~\isa{f\ {\isasymtau}} iterates the operation \isa{f} over all occurrences of atomic types (\verb|TFree|, \verb|TVar|)
+ in \isa{{\isasymtau}}; the type structure is traversed from left to right.
\item \verb|Sign.subsort|~\isa{thy\ {\isacharparenleft}s\isactrlisub {\isadigit{1}}{\isacharcomma}\ s\isactrlisub {\isadigit{2}}{\isacharparenright}}
tests the subsort relation \isa{s\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ s\isactrlisub {\isadigit{2}}}.
- \item \verb|Sign.of_sort|~\isa{thy\ {\isacharparenleft}{\isasymtau}{\isacharcomma}\ s{\isacharparenright}} tests whether a type
- is of a given sort.
+ \item \verb|Sign.of_sort|~\isa{thy\ {\isacharparenleft}{\isasymtau}{\isacharcomma}\ s{\isacharparenright}} tests whether type
+ \isa{{\isasymtau}} is of sort \isa{s}.
- \item \verb|Sign.add_types|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ k{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares new
+ \item \verb|Sign.add_types|~\isa{{\isacharbrackleft}{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ k{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares a new
type constructors \isa{{\isasymkappa}} with \isa{k} arguments and
optional mixfix syntax.
@@ -174,13 +179,13 @@
defines a new type abbreviation \isa{{\isacharparenleft}\isactrlvec {\isasymalpha}{\isacharparenright}{\isasymkappa}\ {\isacharequal}\ {\isasymtau}} with
optional mixfix syntax.
- \item \verb|Sign.primitive_class|~\isa{{\isacharparenleft}c{\isacharcomma}\ {\isacharbrackleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlisub n{\isacharbrackright}{\isacharparenright}} declares new class \isa{c}, together with class
+ \item \verb|Sign.primitive_class|~\isa{{\isacharparenleft}c{\isacharcomma}\ {\isacharbrackleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlisub n{\isacharbrackright}{\isacharparenright}} declares a new class \isa{c}, together with class
relations \isa{c\ {\isasymsubseteq}\ c\isactrlisub i}, for \isa{i\ {\isacharequal}\ {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ n}.
\item \verb|Sign.primitive_classrel|~\isa{{\isacharparenleft}c\isactrlisub {\isadigit{1}}{\isacharcomma}\ c\isactrlisub {\isadigit{2}}{\isacharparenright}} declares class relation \isa{c\isactrlisub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlisub {\isadigit{2}}}.
\item \verb|Sign.primitive_arity|~\isa{{\isacharparenleft}{\isasymkappa}{\isacharcomma}\ \isactrlvec s{\isacharcomma}\ s{\isacharparenright}} declares
- arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}.
+ the arity \isa{{\isasymkappa}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}s}.
\end{description}%
\end{isamarkuptext}%
@@ -202,54 +207,56 @@
The language of terms is that of simply-typed \isa{{\isasymlambda}}-calculus
with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
- or \cite{paulson-ml2}), and named free variables and constants.
- Terms with loose bound variables are usually considered malformed.
- The types of variables and constants is stored explicitly at each
- occurrence in the term.
+ or \cite{paulson-ml2}), with the types being determined determined
+ by the corresponding binders. In contrast, free variables and
+ constants are have an explicit name and type in each occurrence.
\medskip A \emph{bound variable} is a natural number \isa{b},
- which refers to the next binder that is \isa{b} steps upwards
- from the occurrence of \isa{b} (counting from zero). Bindings
- may be introduced as abstractions within the term, or as a separate
- context (an inside-out list). This associates each bound variable
- with a type. A \emph{loose variables} is a bound variable that is
- outside the current scope of local binders or the context. For
+ which accounts for the number of intermediate binders between the
+ variable occurrence in the body and its binding position. For
example, the de-Bruijn term \isa{{\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ {\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ {\isadigit{1}}\ {\isacharplus}\ {\isadigit{0}}}
- corresponds to \isa{{\isasymlambda}x\isactrlisub {\isasymtau}{\isachardot}\ {\isasymlambda}y\isactrlisub {\isasymtau}{\isachardot}\ x\ {\isacharplus}\ y} in a named
- representation. Also note that the very same bound variable may get
- different numbers at different occurrences.
+ would correspond to \isa{{\isasymlambda}x\isactrlisub {\isasymtau}{\isachardot}\ {\isasymlambda}y\isactrlisub {\isasymtau}{\isachardot}\ x\ {\isacharplus}\ y} in a
+ named representation. Note that a bound variable may be represented
+ by different de-Bruijn indices at different occurrences, depending
+ on the nesting of abstractions.
+
+ A \emph{loose variables} is a bound variable that is outside the
+ scope of local binders. The types (and names) for loose variables
+ can be managed as a separate context, that is maintained inside-out
+ like a stack of hypothetical binders. The core logic only operates
+ on closed terms, without any loose variables.
- A \emph{fixed variable} is a pair of a basic name and a type. For
- example, \isa{{\isacharparenleft}x{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed \isa{x\isactrlisub {\isasymtau}}. A \emph{schematic variable} is a pair of an
- indexname and a type. For example, \isa{{\isacharparenleft}{\isacharparenleft}x{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is
- usually printed as \isa{{\isacharquery}x\isactrlisub {\isasymtau}}.
+ A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
+ \isa{{\isacharparenleft}x{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed \isa{x\isactrlisub {\isasymtau}}. A
+ \emph{schematic variable} is a pair of an indexname and a type,
+ e.g.\ \isa{{\isacharparenleft}{\isacharparenleft}x{\isacharcomma}\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed as \isa{{\isacharquery}x\isactrlisub {\isasymtau}}.
- \medskip A \emph{constant} is a atomic terms consisting of a basic
- name and a type. Constants are declared in the context as
- polymorphic families \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}, meaning that any \isa{c\isactrlisub {\isasymtau}} is a valid constant for all substitution instances
- \isa{{\isasymtau}\ {\isasymle}\ {\isasymsigma}}.
+ \medskip A \emph{constant} is a pair of a basic name and a type,
+ e.g.\ \isa{{\isacharparenleft}c{\isacharcomma}\ {\isasymtau}{\isacharparenright}} which is usually printed as \isa{c\isactrlisub {\isasymtau}}. Constants are declared in the context as polymorphic
+ families \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}, meaning that valid all substitution
+ instances \isa{c\isactrlisub {\isasymtau}} for \isa{{\isasymtau}\ {\isacharequal}\ {\isasymsigma}{\isasymvartheta}} are valid.
- The list of \emph{type arguments} of \isa{c\isactrlisub {\isasymtau}} wrt.\ the
- declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} is the codomain of the type matcher
- presented in canonical order (according to the left-to-right
- occurrences of type variables in in \isa{{\isasymsigma}}). Thus \isa{c\isactrlisub {\isasymtau}} can be represented more compactly as \isa{c{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharparenright}}. For example, the instance \isa{plus\isactrlbsub nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat\isactrlesub } of some \isa{plus\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}} has the singleton list \isa{nat} as type arguments, the
- constant may be represented as \isa{plus{\isacharparenleft}nat{\isacharparenright}}.
+ The vector of \emph{type arguments} of constant \isa{c\isactrlisub {\isasymtau}}
+ wrt.\ the declaration \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} is defined as the codomain of
+ the matcher \isa{{\isasymvartheta}\ {\isacharequal}\ {\isacharbraceleft}{\isacharquery}{\isasymalpha}\isactrlisub {\isadigit{1}}\ {\isasymmapsto}\ {\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isacharquery}{\isasymalpha}\isactrlisub n\ {\isasymmapsto}\ {\isasymtau}\isactrlisub n{\isacharbraceright}} presented in canonical order \isa{{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharparenright}}. Within a given theory context,
+ there is a one-to-one correspondence between any constant \isa{c\isactrlisub {\isasymtau}} and the application \isa{c{\isacharparenleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharparenright}} of its type arguments. For example, with \isa{plus\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}}, the instance \isa{plus\isactrlbsub nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat\isactrlesub } corresponds to \isa{plus{\isacharparenleft}nat{\isacharparenright}}.
Constant declarations \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} may contain sort constraints
for type variables in \isa{{\isasymsigma}}. These are observed by
type-inference as expected, but \emph{ignored} by the core logic.
This means the primitive logic is able to reason with instances of
- polymorphic constants that the user-level type-checker would reject.
+ polymorphic constants that the user-level type-checker would reject
+ due to violation of type class restrictions.
- \medskip A \emph{term} \isa{t} is defined inductively over
- variables and constants, with abstraction and application as
- follows: \isa{t\ {\isacharequal}\ b\ {\isacharbar}\ x\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isacharquery}x\isactrlisub {\isasymtau}\ {\isacharbar}\ c\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ t\ {\isacharbar}\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}}. Parsing and printing takes
- care of converting between an external representation with named
- bound variables. Subsequently, we shall use the latter notation
- instead of internal de-Bruijn representation.
+ \medskip A \emph{term} is defined inductively over variables and
+ constants, with abstraction and application as follows: \isa{t\ {\isacharequal}\ b\ {\isacharbar}\ x\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isacharquery}x\isactrlisub {\isasymtau}\ {\isacharbar}\ c\isactrlisub {\isasymtau}\ {\isacharbar}\ {\isasymlambda}\isactrlisub {\isasymtau}{\isachardot}\ t\ {\isacharbar}\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}}. Parsing and printing takes care of
+ converting between an external representation with named bound
+ variables. Subsequently, we shall use the latter notation instead
+ of internal de-Bruijn representation.
- The subsequent inductive relation \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} assigns a
- (unique) type to a term, using the special type constructor \isa{{\isacharparenleft}{\isasymalpha}{\isacharcomma}\ {\isasymbeta}{\isacharparenright}fun}, which is written \isa{{\isasymalpha}\ {\isasymRightarrow}\ {\isasymbeta}}.
+ The inductive relation \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} assigns a (unique) type to a
+ term according to the structure of atomic terms, abstractions, and
+ applicatins:
\[
\infer{\isa{a\isactrlisub {\isasymtau}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}}{}
\qquad
@@ -265,40 +272,40 @@
Type-inference depends on a context of type constraints for fixed
variables, and declarations for polymorphic constants.
- The identity of atomic terms consists both of the name and the type.
- Thus different entities \isa{c\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{1}}\isactrlesub } and
- \isa{c\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{2}}\isactrlesub } may well identified by type
- instantiation, by mapping \isa{{\isasymtau}\isactrlisub {\isadigit{1}}} and \isa{{\isasymtau}\isactrlisub {\isadigit{2}}} to the same \isa{{\isasymtau}}. Although,
- different type instances of constants of the same basic name are
- commonplace, this rarely happens for variables: type-inference
- always demands ``consistent'' type constraints.
+ The identity of atomic terms consists both of the name and the type
+ component. This means that different variables \isa{x\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{1}}\isactrlesub } and \isa{x\isactrlbsub {\isasymtau}\isactrlisub {\isadigit{2}}\isactrlesub } may become the same after type
+ instantiation. Some outer layers of the system make it hard to
+ produce variables of the same name, but different types. In
+ particular, type-inference always demands ``consistent'' type
+ constraints for free variables. In contrast, mixed instances of
+ polymorphic constants occur frequently.
\medskip The \emph{hidden polymorphism} of a term \isa{t\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}}
is the set of type variables occurring in \isa{t}, but not in
- \isa{{\isasymsigma}}. This means that the term implicitly depends on the
- values of various type variables that are not visible in the overall
- type, i.e.\ there are different type instances \isa{t{\isasymvartheta}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} and \isa{t{\isasymvartheta}{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} with the same type. This
- slightly pathological situation is apt to cause strange effects.
+ \isa{{\isasymsigma}}. This means that the term implicitly depends on type
+ arguments that are not accounted in result type, i.e.\ there are
+ different type instances \isa{t{\isasymvartheta}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} and \isa{t{\isasymvartheta}{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} with the same type. This slightly
+ pathological situation demands special care.
- \medskip A \emph{term abbreviation} is a syntactic definition \isa{c\isactrlisub {\isasymsigma}\ {\isasymequiv}\ t} of an arbitrary closed term \isa{t} of type
- \isa{{\isasymsigma}} without any hidden polymorphism. A term abbreviation
- looks like a constant at the surface, but is fully expanded before
- entering the logical core. Abbreviations are usually reverted when
- printing terms, using rules \isa{t\ {\isasymrightarrow}\ c\isactrlisub {\isasymsigma}} has a
- higher-order term rewrite system.
+ \medskip A \emph{term abbreviation} is a syntactic definition \isa{c\isactrlisub {\isasymsigma}\ {\isasymequiv}\ t} of a closed term \isa{t} of type \isa{{\isasymsigma}},
+ without any hidden polymorphism. A term abbreviation looks like a
+ constant in the syntax, but is fully expanded before entering the
+ logical core. Abbreviations are usually reverted when printing
+ terms, using the collective \isa{t\ {\isasymrightarrow}\ c\isactrlisub {\isasymsigma}} as rules for
+ higher-order rewriting.
- \medskip Canonical operations on \isa{{\isasymlambda}}-terms include \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion. \isa{{\isasymalpha}}-conversion refers to capture-free
+ \medskip Canonical operations on \isa{{\isasymlambda}}-terms include \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion: \isa{{\isasymalpha}}-conversion refers to capture-free
renaming of bound variables; \isa{{\isasymbeta}}-conversion contracts an
- abstraction applied to some argument term, substituting the argument
+ abstraction applied to an argument term, substituting the argument
in the body: \isa{{\isacharparenleft}{\isasymlambda}x{\isachardot}\ b{\isacharparenright}a} becomes \isa{b{\isacharbrackleft}a{\isacharslash}x{\isacharbrackright}}; \isa{{\isasymeta}}-conversion contracts vacuous application-abstraction: \isa{{\isasymlambda}x{\isachardot}\ f\ x} becomes \isa{f}, provided that the bound variable
- \isa{{\isadigit{0}}} does not occur in \isa{f}.
+ does not occur in \isa{f}.
- Terms are almost always treated module \isa{{\isasymalpha}}-conversion, which
- is implicit in the de-Bruijn representation. The names in
- abstractions of bound variables are maintained only as a comment for
- parsing and printing. Full \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence is usually
- taken for granted higher rules (\secref{sec:rules}), anything
- depending on higher-order unification or rewriting.%
+ Terms are normally treated modulo \isa{{\isasymalpha}}-conversion, which is
+ implicit in the de-Bruijn representation. Names for bound variables
+ in abstractions are maintained separately as (meaningless) comments,
+ mostly for parsing and printing. Full \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-conversion is
+ commonplace in various higher operations (\secref{sec:rules}) that
+ are based on higher-order unification and matching.%
\end{isamarkuptext}%
\isamarkuptrue%
%
@@ -328,38 +335,35 @@
\begin{description}
- \item \verb|term| represents de-Bruijn terms with comments in
- abstractions for bound variable names. This is a datatype with
- constructors \verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|, \verb|Abs|, \verb|op $|.
+ \item \verb|term| represents de-Bruijn terms, with comments in
+ abstractions, and explicitly named free variables and constants;
+ this is a datatype with constructors \verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|, \verb|Abs|, \verb|op $|.
\item \isa{t}~\verb|aconv|~\isa{u} checks \isa{{\isasymalpha}}-equivalence of two terms. This is the basic equality relation
on type \verb|term|; raw datatype equality should only be used
for operations related to parsing or printing!
- \item \verb|map_term_types|~\isa{f\ t} applies mapping \isa{f}
- to all types occurring in \isa{t}.
+ \item \verb|map_term_types|~\isa{f\ t} applies the mapping \isa{f} to all types occurring in \isa{t}.
+
+ \item \verb|fold_types|~\isa{f\ t} iterates the operation \isa{f} over all occurrences of types in \isa{t}; the term
+ structure is traversed from left to right.
- \item \verb|fold_types|~\isa{f\ t} iterates operation \isa{f}
- over all occurrences of types in \isa{t}; the term structure is
+ \item \verb|map_aterms|~\isa{f\ t} applies the mapping \isa{f}
+ to all atomic terms (\verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|) occurring in \isa{t}.
+
+ \item \verb|fold_aterms|~\isa{f\ t} iterates the operation \isa{f} over all occurrences of atomic terms (\verb|Bound|, \verb|Free|,
+ \verb|Var|, \verb|Const|) in \isa{t}; the term structure is
traversed from left to right.
- \item \verb|map_aterms|~\isa{f\ t} applies mapping \isa{f} to
- all atomic terms (\verb|Bound|, \verb|Free|, \verb|Var|, \verb|Const|)
- occurring in \isa{t}.
-
- \item \verb|fold_aterms|~\isa{f\ t} iterates operation \isa{f}
- over all occurrences of atomic terms in (\verb|Bound|, \verb|Free|,
- \verb|Var|, \verb|Const|) \isa{t}; the term structure is traversed
- from left to right.
+ \item \verb|fastype_of|~\isa{t} determines the type of a
+ well-typed term. This operation is relatively slow, despite the
+ omission of any sanity checks.
- \item \verb|fastype_of|~\isa{t} recomputes the type of a
- well-formed term, while omitting any sanity checks. This operation
- is relatively slow.
+ \item \verb|lambda|~\isa{a\ b} produces an abstraction \isa{{\isasymlambda}a{\isachardot}\ b}, where occurrences of the atomic term \isa{a} in the
+ body \isa{b} are replaced by bound variables.
- \item \verb|lambda|~\isa{a\ b} produces an abstraction \isa{{\isasymlambda}a{\isachardot}\ b}, where occurrences of the original (atomic) term \isa{a} in the body \isa{b} are replaced by bound variables.
-
- \item \verb|betapply|~\isa{t\ u} produces an application \isa{t\ u}, with topmost \isa{{\isasymbeta}}-conversion if \isa{t} happens to
- be an abstraction.
+ \item \verb|betapply|~\isa{{\isacharparenleft}t{\isacharcomma}\ u{\isacharparenright}} produces an application \isa{t\ u}, with topmost \isa{{\isasymbeta}}-conversion if \isa{t} is an
+ abstraction.
\item \verb|Sign.add_consts_i|~\isa{{\isacharbrackleft}{\isacharparenleft}c{\isacharcomma}\ {\isasymsigma}{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharbrackright}} declares a
new constant \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}} with optional mixfix syntax.
@@ -369,9 +373,9 @@
mixfix syntax.
\item \verb|Sign.const_typargs|~\isa{thy\ {\isacharparenleft}c{\isacharcomma}\ {\isasymtau}{\isacharparenright}} and \verb|Sign.const_instance|~\isa{thy\ {\isacharparenleft}c{\isacharcomma}\ {\isacharbrackleft}{\isasymtau}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymtau}\isactrlisub n{\isacharbrackright}{\isacharparenright}}
- convert between the two representations of constants, namely full
- type instance vs.\ compact type arguments form (depending on the
- most general declaration given in the context).
+ convert between the representations of polymorphic constants: the
+ full type instance vs.\ the compact type arguments form (depending
+ on the most general declaration given in the context).
\end{description}%
\end{isamarkuptext}%
@@ -427,27 +431,26 @@
A \emph{proposition} is a well-formed term of type \isa{prop}, a
\emph{theorem} is a proven proposition (depending on a context of
hypotheses and the background theory). Primitive inferences include
- plain natural deduction rules for the primary connectives \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} of the framework. There are separate (derived)
- rules for equality/equivalence \isa{{\isasymequiv}} and internal conjunction
- \isa{{\isacharampersand}}.%
+ plain natural deduction rules for the primary connectives \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} of the framework. There is also a builtin
+ notion of equality/equivalence \isa{{\isasymequiv}}.%
\end{isamarkuptext}%
\isamarkuptrue%
%
-\isamarkupsubsection{Standard connectives and rules%
+\isamarkupsubsection{Primitive connectives and rules%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
-The basic theory is called \isa{Pure}, it contains declarations
- for the standard logical connectives \isa{{\isasymAnd}}, \isa{{\isasymLongrightarrow}}, and
- \isa{{\isasymequiv}} of the framework, see \figref{fig:pure-connectives}.
- The derivability judgment \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} is
- defined inductively by the primitive inferences given in
- \figref{fig:prim-rules}, with the global syntactic restriction that
- hypotheses may never contain schematic variables. The builtin
- equality is conceptually axiomatized shown in
+The theory \isa{Pure} contains declarations for the standard
+ connectives \isa{{\isasymAnd}}, \isa{{\isasymLongrightarrow}}, and \isa{{\isasymequiv}} of the logical
+ framework, see \figref{fig:pure-connectives}. The derivability
+ judgment \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} is defined
+ inductively by the primitive inferences given in
+ \figref{fig:prim-rules}, with the global restriction that hypotheses
+ \isa{{\isasymGamma}} may \emph{not} contain schematic variables. The builtin
+ equality is conceptually axiomatized as shown in
\figref{fig:pure-equality}, although the implementation works
- directly with (derived) inference rules.
+ directly with derived inference rules.
\begin{figure}[htb]
\begin{center}
@@ -456,7 +459,7 @@
\isa{{\isasymLongrightarrow}\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} & implication (right associative infix) \\
\isa{{\isasymequiv}\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ prop} & equality relation (infix) \\
\end{tabular}
- \caption{Standard connectives of Pure}\label{fig:pure-connectives}
+ \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
\end{center}
\end{figure}
@@ -468,9 +471,9 @@
\infer[\isa{{\isacharparenleft}assume{\isacharparenright}}]{\isa{A\ {\isasymturnstile}\ A}}{}
\]
\[
- \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b\ x}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ b\ x} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}}
+ \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ b{\isacharbrackleft}x{\isacharbrackright}} & \isa{x\ {\isasymnotin}\ {\isasymGamma}}}
\qquad
- \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ b\ a}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b\ x}}
+ \infer[\isa{{\isacharparenleft}{\isasymAnd}{\isacharunderscore}elim{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isasymturnstile}\ b{\isacharbrackleft}a{\isacharbrackright}}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}}}
\]
\[
\infer[\isa{{\isacharparenleft}{\isasymLongrightarrow}{\isacharunderscore}intro{\isacharparenright}}]{\isa{{\isasymGamma}\ {\isacharminus}\ A\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}}
@@ -484,44 +487,39 @@
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
- \isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ b\ x{\isacharparenright}\ a\ {\isasymequiv}\ b\ a} & \isa{{\isasymbeta}}-conversion \\
+ \isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ b{\isacharbrackleft}x{\isacharbrackright}{\isacharparenright}\ a\ {\isasymequiv}\ b{\isacharbrackleft}a{\isacharbrackright}} & \isa{{\isasymbeta}}-conversion \\
\isa{{\isasymturnstile}\ x\ {\isasymequiv}\ x} & reflexivity \\
\isa{{\isasymturnstile}\ x\ {\isasymequiv}\ y\ {\isasymLongrightarrow}\ P\ x\ {\isasymLongrightarrow}\ P\ y} & substitution \\
\isa{{\isasymturnstile}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ f\ x\ {\isasymequiv}\ g\ x{\isacharparenright}\ {\isasymLongrightarrow}\ f\ {\isasymequiv}\ g} & extensionality \\
- \isa{{\isasymturnstile}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymequiv}\ B} & coincidence with equivalence \\
+ \isa{{\isasymturnstile}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymequiv}\ B} & logical equivalence \\
\end{tabular}
- \caption{Conceptual axiomatization of builtin equality}\label{fig:pure-equality}
+ \caption{Conceptual axiomatization of \isa{{\isasymequiv}}}\label{fig:pure-equality}
\end{center}
\end{figure}
- The introduction and elimination rules for \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} are analogous to formation of (dependently typed) \isa{{\isasymlambda}}-terms representing the underlying proof objects. Proof terms
- are \emph{irrelevant} in the Pure logic, they may never occur within
- propositions, i.e.\ the \isa{{\isasymLongrightarrow}} arrow is non-dependent. The
- system provides a runtime option to record explicit proof terms for
- primitive inferences, cf.\ \cite{Berghofer-Nipkow:2000:TPHOL}. Thus
- the three-fold \isa{{\isasymlambda}}-structure can be made explicit.
+ The introduction and elimination rules for \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}} are analogous to formation of dependently typed \isa{{\isasymlambda}}-terms representing the underlying proof objects. Proof terms
+ are irrelevant in the Pure logic, though, they may never occur
+ within propositions. The system provides a runtime option to record
+ explicit proof terms for primitive inferences. Thus all three
+ levels of \isa{{\isasymlambda}}-calculus become explicit: \isa{{\isasymRightarrow}} for
+ terms, and \isa{{\isasymAnd}{\isacharslash}{\isasymLongrightarrow}} for proofs (cf.\
+ \cite{Berghofer-Nipkow:2000:TPHOL}).
- Observe that locally fixed parameters (as used in rule \isa{{\isasymAnd}{\isacharunderscore}intro}) need not be recorded in the hypotheses, because the
- simple syntactic types of Pure are always inhabitable. The typing
- ``assumption'' \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} is logically vacuous, it disappears
- automatically whenever the statement body ceases to mention variable
- \isa{x\isactrlisub {\isasymtau}}.\footnote{This greatly simplifies many basic
- reasoning steps, and is the key difference to the formulation of
- this logic as ``\isa{{\isasymlambda}HOL}'' in the PTS framework
- \cite{Barendregt-Geuvers:2001}.}
+ Observe that locally fixed parameters (as in \isa{{\isasymAnd}{\isacharunderscore}intro}) need
+ not be recorded in the hypotheses, because the simple syntactic
+ types of Pure are always inhabitable. Typing ``assumptions'' \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} are (implicitly) present only with occurrences of \isa{x\isactrlisub {\isasymtau}} in the statement body.\footnote{This is the key
+ difference ``\isa{{\isasymlambda}HOL}'' in the PTS framework
+ \cite{Barendregt-Geuvers:2001}, where \isa{x\ {\isacharcolon}\ A} hypotheses are
+ treated explicitly for types, in the same way as propositions.}
\medskip FIXME \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence and primitive definitions
Since the basic representation of terms already accounts for \isa{{\isasymalpha}}-conversion, Pure equality essentially acts like \isa{{\isasymalpha}{\isasymbeta}{\isasymeta}}-equivalence on terms, while coinciding with bi-implication.
\medskip The axiomatization of a theory is implicitly closed by
- forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymvartheta}} for
- any substitution instance of axiom \isa{{\isasymturnstile}\ A}. By pushing
- substitution through derivations inductively, we get admissible
- substitution rules for theorems shown in \figref{fig:subst-rules}.
- Alternatively, the term substitution rules could be derived from
- \isa{{\isasymAnd}{\isacharunderscore}intro{\isacharslash}elim}. The versions for types are genuine
- admissible rules, due to the lack of true polymorphism in the logic.
+ forming all instances of type and term variables: \isa{{\isasymturnstile}\ A{\isasymvartheta}} holds for any substitution instance of an axiom
+ \isa{{\isasymturnstile}\ A}. By pushing substitution through derivations
+ inductively, we get admissible \isa{generalize} and \isa{instance} rules shown in \figref{fig:subst-rules}.
\begin{figure}[htb]
\begin{center}
@@ -539,11 +537,14 @@
\end{center}
\end{figure}
- Since \isa{{\isasymGamma}} may never contain any schematic variables, the
- \isa{instantiate} do not require an explicit side-condition. In
- principle, variables could be substituted in hypotheses as well, but
- this could disrupt monotonicity of the basic calculus: derivations
- could leave the current proof context.%
+ Note that \isa{instantiate} does not require an explicit
+ side-condition, because \isa{{\isasymGamma}} may never contain schematic
+ variables.
+
+ In principle, variables could be substituted in hypotheses as well,
+ but this would disrupt monotonicity reasoning: deriving \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymturnstile}\ B{\isasymvartheta}} from \isa{{\isasymGamma}\ {\isasymturnstile}\ B} is correct, but
+ \isa{{\isasymGamma}{\isasymvartheta}\ {\isasymsupseteq}\ {\isasymGamma}} does not necessarily hold --- the result
+ belongs to a different proof context.%
\end{isamarkuptext}%
\isamarkuptrue%
%
@@ -584,16 +585,16 @@
\isamarkuptrue%
%
\begin{isamarkuptext}%
-Pure also provides various auxiliary connectives based on primitive
- definitions, see \figref{fig:pure-aux}. These are normally not
- exposed to the user, but appear in internal encodings only.
+Theory \isa{Pure} also defines a few auxiliary connectives, see
+ \figref{fig:pure-aux}. These are normally not exposed to the user,
+ but appear in internal encodings only.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
\isa{conjunction\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop\ {\isasymRightarrow}\ prop} & (infix \isa{{\isacharampersand}}) \\
\isa{{\isasymturnstile}\ A\ {\isacharampersand}\ B\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}C{\isachardot}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isacharparenright}} \\[1ex]
- \isa{prop\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop} & (prefix \isa{{\isacharhash}}) \\
+ \isa{prop\ {\isacharcolon}{\isacharcolon}\ prop\ {\isasymRightarrow}\ prop} & (prefix \isa{{\isacharhash}}, hidden) \\
\isa{{\isacharhash}A\ {\isasymequiv}\ A} \\[1ex]
\isa{term\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymRightarrow}\ prop} & (prefix \isa{TERM}) \\
\isa{term\ x\ {\isasymequiv}\ {\isacharparenleft}{\isasymAnd}A{\isachardot}\ A\ {\isasymLongrightarrow}\ A{\isacharparenright}} \\[1ex]
@@ -604,35 +605,33 @@
\end{center}
\end{figure}
- Conjunction as an explicit connective allows to treat both
- simultaneous assumptions and conclusions uniformly. The definition
- allows to derive the usual introduction \isa{{\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ A\ {\isacharampersand}\ B},
- and destructions \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ A} and \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ B}. For
- example, several claims may be stated at the same time, which is
- intermediately represented as an assumption, but the user only
- encounters several sub-goals, and several resulting facts in the
- very end (cf.\ \secref{sec:tactical-goals}).
+ Derived conjunction rules include introduction \isa{A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ A\ {\isacharampersand}\ B}, and destructions \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ A} and \isa{A\ {\isacharampersand}\ B\ {\isasymLongrightarrow}\ B}.
+ Conjunction allows to treat simultaneous assumptions and conclusions
+ uniformly. For example, multiple claims are intermediately
+ represented as explicit conjunction, but this is usually refined
+ into separate sub-goals before the user continues the proof; the
+ final result is projected into a list of theorems (cf.\
+ \secref{sec:tactical-goals}).
- The \isa{{\isacharhash}} marker allows complex propositions (nested \isa{{\isasymAnd}} and \isa{{\isasymLongrightarrow}}) to appear formally as atomic, without changing
- the meaning: \isa{{\isasymGamma}\ {\isasymturnstile}\ A} and \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isacharhash}A} are
- interchangeable. See \secref{sec:tactical-goals} for specific
- operations.
+ The \isa{prop} marker (\isa{{\isacharhash}}) makes arbitrarily complex
+ propositions appear as atomic, without changing the meaning: \isa{{\isasymGamma}\ {\isasymturnstile}\ A} and \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isacharhash}A} are interchangeable. See
+ \secref{sec:tactical-goals} for specific operations.
- The \isa{TERM} marker turns any well-formed term into a
- derivable proposition: \isa{{\isasymturnstile}\ TERM\ t} holds
- unconditionally. Despite its logically vacous meaning, this is
- occasionally useful to treat syntactic terms and proven propositions
- uniformly, as in a type-theoretic framework.
+ The \isa{term} marker turns any well-formed term into a
+ derivable proposition: \isa{{\isasymturnstile}\ TERM\ t} holds unconditionally.
+ Although this is logically vacuous, it allows to treat terms and
+ proofs uniformly, similar to a type-theoretic framework.
- The \isa{TYPE} constructor (which is the canonical
- representative of the unspecified type \isa{{\isasymalpha}\ itself}) injects
- the language of types into that of terms. There is specific
- notation \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} for \isa{TYPE\isactrlbsub {\isasymtau}\ itself\isactrlesub }.
- Although being devoid of any particular meaning, the term \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} is able to carry the type \isa{{\isasymtau}} formally. \isa{TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}} may be used as an additional formal argument in primitive
- definitions, in order to avoid hidden polymorphism (cf.\
- \secref{sec:terms}). For example, \isa{c\ TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}\ {\isasymequiv}\ A{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}} turns
- out as a formally correct definition of some proposition \isa{A}
- that depends on an additional type argument.%
+ The \isa{TYPE} constructor is the canonical representative of
+ the unspecified type \isa{{\isasymalpha}\ itself}; it essentially injects the
+ language of types into that of terms. There is specific notation
+ \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} for \isa{TYPE\isactrlbsub {\isasymtau}\ itself\isactrlesub }.
+ Although being devoid of any particular meaning, the \isa{TYPE{\isacharparenleft}{\isasymtau}{\isacharparenright}} accounts for the type \isa{{\isasymtau}} within the term
+ language. In particular, \isa{TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}} may be used as formal
+ argument in primitive definitions, in order to circumvent hidden
+ polymorphism (cf.\ \secref{sec:terms}). For example, \isa{c\ TYPE{\isacharparenleft}{\isasymalpha}{\isacharparenright}\ {\isasymequiv}\ A{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}} defines \isa{c\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ itself\ {\isasymRightarrow}\ prop} in terms of
+ a proposition \isa{A} that depends on an additional type
+ argument, which is essentially a predicate on types.%
\end{isamarkuptext}%
\isamarkuptrue%
%
--- a/doc-src/IsarImplementation/Thy/logic.thy Thu Sep 14 15:27:08 2006 +0200
+++ b/doc-src/IsarImplementation/Thy/logic.thy Thu Sep 14 15:51:20 2006 +0200
@@ -20,12 +20,12 @@
"\<And>"} for universal quantification (proofs depending on terms), and
@{text "\<Longrightarrow>"} 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 @{text "\<Theta>"} is separate from the context @{text
- "\<Gamma>"} of the core calculus.}
+ Derivations are relative to a logical theory, which declares type
+ constructors, constants, and axioms. Theory declarations support
+ schematic polymorphism, which is strictly speaking outside the
+ logic.\footnote{This is the deeper logical reason, why the theory
+ context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"}
+ of the core calculus.}
*}
@@ -42,8 +42,8 @@
internally. The resulting relation is an ordering: reflexive,
transitive, and antisymmetric.
- A \emph{sort} is a list of type classes written as @{text
- "{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
+ A \emph{sort} is a list of type classes written as @{text "s =
+ {c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
intersection. Notationally, the curly braces are omitted for
singleton intersections, i.e.\ any class @{text "c"} may be read as
a sort @{text "{c}"}. The ordering on type classes is extended to
@@ -56,11 +56,11 @@
elements wrt.\ the sort order.
\medskip A \emph{fixed type variable} is a pair of a basic name
- (starting with a @{text "'"} character) and a sort constraint. For
- example, @{text "('a, s)"} which is usually printed as @{text
- "\<alpha>\<^isub>s"}. A \emph{schematic type variable} is a pair of an
- indexname and a sort constraint. For example, @{text "(('a, 0),
- s)"} which is usually printed as @{text "?\<alpha>\<^isub>s"}.
+ (starting with a @{text "'"} character) and a sort constraint, e.g.\
+ @{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}.
+ A \emph{schematic type variable} is a pair of an indexname and a
+ sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually
+ printed as @{text "?\<alpha>\<^isub>s"}.
Note that \emph{all} syntactic components contribute to the identity
of type variables, including the sort constraint. The core logic
@@ -70,23 +70,23 @@
A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
on types declared in the theory. Type constructor application is
- usually written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.
- For @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text
- "prop"} instead of @{text "()prop"}. For @{text "k = 1"} the
- parentheses are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text
- "(\<alpha>)list"}. Further notation is provided for specific constructors,
- notably the right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of
- @{text "(\<alpha>, \<beta>)fun"}.
+ written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}. For
+ @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"}
+ instead of @{text "()prop"}. For @{text "k = 1"} the parentheses
+ are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}.
+ Further notation is provided for specific constructors, notably the
+ right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>,
+ \<beta>)fun"}.
- A \emph{type} @{text "\<tau>"} is defined inductively over type variables
- and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s |
- ?\<alpha>\<^isub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
+ A \emph{type} is defined inductively over type variables and type
+ constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
+ (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
A \emph{type abbreviation} is a syntactic definition @{text
"(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
- variables @{text "\<^vec>\<alpha>"}. Type abbreviations looks like type
- constructors at the surface, but are fully expanded before entering
- the logical core.
+ variables @{text "\<^vec>\<alpha>"}. 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: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
@@ -98,22 +98,22 @@
\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 @{text "\<kappa>"} and classes @{text
- "c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "\<kappa> ::
- (\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "\<kappa>
- :: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq>
- \<^vec>s\<^isub>2"} holds component-wise.
+ relation: for any type constructor @{text "\<kappa>"}, and classes @{text
+ "c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> ::
+ (\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> ::
+ (\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq>
+ \<^vec>s\<^isub>2"} component-wise.
The key property of a coregular order-sorted algebra is that sort
- constraints may be always solved in a most general fashion: for each
- type constructor @{text "\<kappa>"} and sort @{text "s"} there is a most
- general vector of argument sorts @{text "(s\<^isub>1, \<dots>,
- s\<^isub>k)"} such that a type scheme @{text
- "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is
- of sort @{text "s"}. Consequently, the unification problem on the
- algebra of types has most general solutions (modulo renaming and
- equivalence of sorts). Moreover, the usual type-inference algorithm
- will produce primary types as expected \cite{nipkow-prehofer}.
+ constraints can be solved in a most general fashion: for each type
+ constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general
+ vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such
+ that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>,
+ \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}.
+ Consequently, unification on the algebra of types has most general
+ solutions (modulo equivalence of sorts). This means that
+ type-inference will produce primary types as expected
+ \cite{nipkow-prehofer}.
*}
text %mlref {*
@@ -149,20 +149,21 @@
\item @{ML_type typ} represents types; this is a datatype with
constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
- \item @{ML map_atyps}~@{text "f \<tau>"} applies mapping @{text "f"} to
- all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text "\<tau>"}.
+ \item @{ML map_atyps}~@{text "f \<tau>"} applies the mapping @{text "f"}
+ to all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text
+ "\<tau>"}.
- \item @{ML fold_atyps}~@{text "f \<tau>"} iterates operation @{text "f"}
- over all occurrences of atoms (@{ML TFree}, @{ML TVar}) in @{text
- "\<tau>"}; the type structure is traversed from left to right.
+ \item @{ML fold_atyps}~@{text "f \<tau>"} iterates the operation @{text
+ "f"} over all occurrences of atomic types (@{ML TFree}, @{ML TVar})
+ in @{text "\<tau>"}; the type structure is traversed from left to right.
\item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
- \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether a type
- is of a given sort.
+ \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether type
+ @{text "\<tau>"} is of sort @{text "s"}.
- \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares new
+ \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares a new
type constructors @{text "\<kappa>"} with @{text "k"} arguments and
optional mixfix syntax.
@@ -171,7 +172,7 @@
optional mixfix syntax.
\item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
- c\<^isub>n])"} declares new class @{text "c"}, together with class
+ c\<^isub>n])"} declares a new class @{text "c"}, together with class
relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
\item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
@@ -179,7 +180,7 @@
c\<^isub>2"}.
\item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
- arity @{text "\<kappa> :: (\<^vec>s)s"}.
+ the arity @{text "\<kappa> :: (\<^vec>s)s"}.
\end{description}
*}
@@ -193,62 +194,66 @@
The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
- or \cite{paulson-ml2}), and named free variables and constants.
- Terms with loose bound variables are usually considered malformed.
- The types of variables and constants is stored explicitly at each
- occurrence in the term.
+ or \cite{paulson-ml2}), with the types being determined determined
+ by the corresponding binders. In contrast, free variables and
+ constants are have an explicit name and type in each occurrence.
\medskip A \emph{bound variable} is a natural number @{text "b"},
- which refers to the next binder that is @{text "b"} steps upwards
- from the occurrence of @{text "b"} (counting from zero). Bindings
- may be introduced as abstractions within the term, or as a separate
- context (an inside-out list). This associates each bound variable
- with a type. A \emph{loose variables} is a bound variable that is
- outside the current scope of local binders or the context. For
+ which accounts for the number of intermediate binders between the
+ variable occurrence in the body and its binding position. For
example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"}
- corresponds to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a named
- representation. Also note that the very same bound variable may get
- different numbers at different occurrences.
+ would correspond to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a
+ named representation. Note that a bound variable may be represented
+ by different de-Bruijn indices at different occurrences, depending
+ on the nesting of abstractions.
+
+ A \emph{loose variables} is a bound variable that is outside the
+ scope of local binders. The types (and names) for loose variables
+ can be managed as a separate context, that is maintained inside-out
+ like a stack of hypothetical binders. The core logic only operates
+ on closed terms, without any loose variables.
- A \emph{fixed variable} is a pair of a basic name and a type. For
- example, @{text "(x, \<tau>)"} which is usually printed @{text
- "x\<^isub>\<tau>"}. A \emph{schematic variable} is a pair of an
- indexname and a type. For example, @{text "((x, 0), \<tau>)"} which is
- usually printed as @{text "?x\<^isub>\<tau>"}.
+ A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
+ @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}. A
+ \emph{schematic variable} is a pair of an indexname and a type,
+ e.g.\ @{text "((x, 0), \<tau>)"} which is usually printed as @{text
+ "?x\<^isub>\<tau>"}.
- \medskip A \emph{constant} is a atomic terms consisting of a basic
- name and a type. Constants are declared in the context as
- polymorphic families @{text "c :: \<sigma>"}, meaning that any @{text
- "c\<^isub>\<tau>"} is a valid constant for all substitution instances
- @{text "\<tau> \<le> \<sigma>"}.
+ \medskip A \emph{constant} is a pair of a basic name and a type,
+ e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text
+ "c\<^isub>\<tau>"}. Constants are declared in the context as polymorphic
+ families @{text "c :: \<sigma>"}, meaning that valid all substitution
+ instances @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
- The list of \emph{type arguments} of @{text "c\<^isub>\<tau>"} wrt.\ the
- declaration @{text "c :: \<sigma>"} is the codomain of the type matcher
- presented in canonical order (according to the left-to-right
- occurrences of type variables in in @{text "\<sigma>"}). Thus @{text
- "c\<^isub>\<tau>"} can be represented more compactly as @{text
- "c(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}. For example, the instance @{text
- "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} of some @{text "plus :: \<alpha> \<Rightarrow> \<alpha>
- \<Rightarrow> \<alpha>"} has the singleton list @{text "nat"} as type arguments, the
- constant may be represented as @{text "plus(nat)"}.
+ The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"}
+ wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of
+ the matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>,
+ ?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in canonical order @{text
+ "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}. Within a given theory context,
+ there is a one-to-one correspondence between any constant @{text
+ "c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1, \<dots>,
+ \<tau>\<^isub>n)"} of its type arguments. For example, with @{text "plus
+ :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow>
+ nat\<^esub>"} corresponds to @{text "plus(nat)"}.
Constant declarations @{text "c :: \<sigma>"} may contain sort constraints
for type variables in @{text "\<sigma>"}. These are observed by
type-inference as expected, but \emph{ignored} by the core logic.
This means the primitive logic is able to reason with instances of
- polymorphic constants that the user-level type-checker would reject.
+ polymorphic constants that the user-level type-checker would reject
+ due to violation of type class restrictions.
- \medskip A \emph{term} @{text "t"} is defined inductively over
- variables and constants, with abstraction and application as
- follows: @{text "t = b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> |
- \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}. Parsing and printing takes
- care of converting between an external representation with named
- bound variables. Subsequently, we shall use the latter notation
- instead of internal de-Bruijn representation.
+ \medskip A \emph{term} is defined inductively over variables and
+ constants, with abstraction and application as follows: @{text "t =
+ b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t |
+ t\<^isub>1 t\<^isub>2"}. Parsing and printing takes care of
+ converting between an external representation with named bound
+ variables. Subsequently, we shall use the latter notation instead
+ of internal de-Bruijn representation.
- The subsequent inductive relation @{text "t :: \<tau>"} assigns a
- (unique) type to a term, using the special type constructor @{text
- "(\<alpha>, \<beta>)fun"}, which is written @{text "\<alpha> \<Rightarrow> \<beta>"}.
+ The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a
+ term according to the structure of atomic terms, abstractions, and
+ applicatins:
\[
\infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{}
\qquad
@@ -264,46 +269,47 @@
Type-inference depends on a context of type constraints for fixed
variables, and declarations for polymorphic constants.
- The identity of atomic terms consists both of the name and the type.
- Thus different entities @{text "c\<^bsub>\<tau>\<^isub>1\<^esub>"} and
- @{text "c\<^bsub>\<tau>\<^isub>2\<^esub>"} may well identified by type
- instantiation, by mapping @{text "\<tau>\<^isub>1"} and @{text
- "\<tau>\<^isub>2"} to the same @{text "\<tau>"}. Although,
- different type instances of constants of the same basic name are
- commonplace, this rarely happens for variables: type-inference
- always demands ``consistent'' type constraints.
+ The identity of atomic terms consists both of the name and the type
+ component. This means that different variables @{text
+ "x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text
+ "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type
+ instantiation. Some outer layers of the system make it hard to
+ produce variables of the same name, but different types. In
+ particular, type-inference always demands ``consistent'' type
+ constraints for free variables. In contrast, mixed instances of
+ polymorphic constants occur frequently.
\medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
is the set of type variables occurring in @{text "t"}, but not in
- @{text "\<sigma>"}. This means that the term implicitly depends on the
- values of various type variables that are not visible in the overall
- type, i.e.\ there are different type instances @{text "t\<vartheta>
- :: \<sigma>"} and @{text "t\<vartheta>' :: \<sigma>"} with the same type. This
- slightly pathological situation is apt to cause strange effects.
+ @{text "\<sigma>"}. This means that the term implicitly depends on type
+ arguments that are not accounted in result type, i.e.\ there are
+ different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text
+ "t\<vartheta>' :: \<sigma>"} with the same type. This slightly
+ pathological situation demands special care.
\medskip A \emph{term abbreviation} is a syntactic definition @{text
- "c\<^isub>\<sigma> \<equiv> t"} of an arbitrary closed term @{text "t"} of type
- @{text "\<sigma>"} without any hidden polymorphism. A term abbreviation
- looks like a constant at the surface, but is fully expanded before
- entering the logical core. Abbreviations are usually reverted when
- printing terms, using rules @{text "t \<rightarrow> c\<^isub>\<sigma>"} has a
- higher-order term rewrite system.
+ "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"},
+ without any hidden polymorphism. A term abbreviation looks like a
+ constant in the syntax, but is fully expanded before entering the
+ logical core. Abbreviations are usually reverted when printing
+ terms, using the collective @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for
+ higher-order rewriting.
\medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
- "\<alpha>\<beta>\<eta>"}-conversion. @{text "\<alpha>"}-conversion refers to capture-free
+ "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free
renaming of bound variables; @{text "\<beta>"}-conversion contracts an
- abstraction applied to some argument term, substituting the argument
+ abstraction applied to an argument term, substituting the argument
in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
"\<eta>"}-conversion contracts vacuous application-abstraction: @{text
"\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
- @{text "0"} does not occur in @{text "f"}.
+ does not occur in @{text "f"}.
- Terms are almost always treated module @{text "\<alpha>"}-conversion, which
- is implicit in the de-Bruijn representation. The names in
- abstractions of bound variables are maintained only as a comment for
- parsing and printing. Full @{text "\<alpha>\<beta>\<eta>"}-equivalence is usually
- taken for granted higher rules (\secref{sec:rules}), anything
- depending on higher-order unification or rewriting.
+ Terms are normally treated modulo @{text "\<alpha>"}-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 @{text "\<alpha>\<beta>\<eta>"}-conversion is
+ commonplace in various higher operations (\secref{sec:rules}) that
+ are based on higher-order unification and matching.
*}
text %mlref {*
@@ -326,43 +332,43 @@
\begin{description}
- \item @{ML_type term} represents de-Bruijn terms with comments in
- abstractions for bound variable names. This is a datatype with
- constructors @{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}, @{ML
- Abs}, @{ML "op $"}.
+ \item @{ML_type term} represents de-Bruijn terms, with comments in
+ abstractions, and explicitly named free variables and constants;
+ this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML
+ Var}, @{ML Const}, @{ML Abs}, @{ML "op $"}.
\item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text
"\<alpha>"}-equivalence of two terms. This is the basic equality relation
on type @{ML_type term}; raw datatype equality should only be used
for operations related to parsing or printing!
- \item @{ML map_term_types}~@{text "f t"} applies mapping @{text "f"}
- to all types occurring in @{text "t"}.
+ \item @{ML map_term_types}~@{text "f t"} applies the mapping @{text
+ "f"} to all types occurring in @{text "t"}.
+
+ \item @{ML fold_types}~@{text "f t"} iterates the operation @{text
+ "f"} over all occurrences of types in @{text "t"}; the term
+ structure is traversed from left to right.
- \item @{ML fold_types}~@{text "f t"} iterates operation @{text "f"}
- over all occurrences of types in @{text "t"}; the term structure is
+ \item @{ML map_aterms}~@{text "f t"} applies the mapping @{text "f"}
+ to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
+ Const}) occurring in @{text "t"}.
+
+ \item @{ML fold_aterms}~@{text "f t"} iterates the operation @{text
+ "f"} over all occurrences of atomic terms (@{ML Bound}, @{ML Free},
+ @{ML Var}, @{ML Const}) in @{text "t"}; the term structure is
traversed from left to right.
- \item @{ML map_aterms}~@{text "f t"} applies mapping @{text "f"} to
- all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const})
- occurring in @{text "t"}.
-
- \item @{ML fold_aterms}~@{text "f t"} iterates operation @{text "f"}
- over all occurrences of atomic terms in (@{ML Bound}, @{ML Free},
- @{ML Var}, @{ML Const}) @{text "t"}; the term structure is traversed
- from left to right.
-
- \item @{ML fastype_of}~@{text "t"} recomputes the type of a
- well-formed term, while omitting any sanity checks. This operation
- is relatively slow.
+ \item @{ML fastype_of}~@{text "t"} determines the type of a
+ well-typed term. This operation is relatively slow, despite the
+ omission of any sanity checks.
\item @{ML lambda}~@{text "a b"} produces an abstraction @{text
- "\<lambda>a. b"}, where occurrences of the original (atomic) term @{text
- "a"} in the body @{text "b"} are replaced by bound variables.
+ "\<lambda>a. b"}, where occurrences of the atomic term @{text "a"} in the
+ body @{text "b"} are replaced by bound variables.
- \item @{ML betapply}~@{text "t u"} produces an application @{text "t
- u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} happens to
- be an abstraction.
+ \item @{ML betapply}~@{text "(t, u)"} produces an application @{text
+ "t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an
+ abstraction.
\item @{ML Sign.add_consts_i}~@{text "[(c, \<sigma>, mx), \<dots>]"} declares a
new constant @{text "c :: \<sigma>"} with optional mixfix syntax.
@@ -373,9 +379,9 @@
\item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
- convert between the two representations of constants, namely full
- type instance vs.\ compact type arguments form (depending on the
- most general declaration given in the context).
+ convert between the representations of polymorphic constants: the
+ full type instance vs.\ the compact type arguments form (depending
+ on the most general declaration given in the context).
\end{description}
*}
@@ -424,24 +430,23 @@
\emph{theorem} is a proven proposition (depending on a context of
hypotheses and the background theory). Primitive inferences include
plain natural deduction rules for the primary connectives @{text
- "\<And>"} and @{text "\<Longrightarrow>"} of the framework. There are separate (derived)
- rules for equality/equivalence @{text "\<equiv>"} and internal conjunction
- @{text "&"}.
+ "\<And>"} and @{text "\<Longrightarrow>"} of the framework. There is also a builtin
+ notion of equality/equivalence @{text "\<equiv>"}.
*}
-subsection {* Standard connectives and rules *}
+subsection {* Primitive connectives and rules *}
text {*
- The basic theory is called @{text "Pure"}, it contains declarations
- for the standard logical connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and
- @{text "\<equiv>"} of the framework, see \figref{fig:pure-connectives}.
- The derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is
- defined inductively by the primitive inferences given in
- \figref{fig:prim-rules}, with the global syntactic restriction that
- hypotheses may never contain schematic variables. The builtin
- equality is conceptually axiomatized shown in
+ The theory @{text "Pure"} contains declarations for the standard
+ connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of the logical
+ framework, see \figref{fig:pure-connectives}. The derivability
+ judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined
+ inductively by the primitive inferences given in
+ \figref{fig:prim-rules}, with the global restriction that hypotheses
+ @{text "\<Gamma>"} may \emph{not} contain schematic variables. The builtin
+ equality is conceptually axiomatized as shown in
\figref{fig:pure-equality}, although the implementation works
- directly with (derived) inference rules.
+ directly with derived inference rules.
\begin{figure}[htb]
\begin{center}
@@ -450,7 +455,7 @@
@{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
@{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
\end{tabular}
- \caption{Standard connectives of Pure}\label{fig:pure-connectives}
+ \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
\end{center}
\end{figure}
@@ -462,9 +467,9 @@
\infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{}
\]
\[
- \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}{@{text "\<Gamma> \<turnstile> b x"} & @{text "x \<notin> \<Gamma>"}}
+ \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}}
\qquad
- \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b a"}}{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}
+ \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}
\]
\[
\infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
@@ -478,34 +483,34 @@
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
- @{text "\<turnstile> (\<lambda>x. b x) a \<equiv> b a"} & @{text "\<beta>"}-conversion \\
+ @{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\
@{text "\<turnstile> x \<equiv> x"} & reflexivity \\
@{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\
@{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
- @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & coincidence with equivalence \\
+ @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\
\end{tabular}
- \caption{Conceptual axiomatization of builtin equality}\label{fig:pure-equality}
+ \caption{Conceptual axiomatization of @{text "\<equiv>"}}\label{fig:pure-equality}
\end{center}
\end{figure}
The introduction and elimination rules for @{text "\<And>"} and @{text
- "\<Longrightarrow>"} are analogous to formation of (dependently typed) @{text
+ "\<Longrightarrow>"} are analogous to formation of dependently typed @{text
"\<lambda>"}-terms representing the underlying proof objects. Proof terms
- are \emph{irrelevant} in the Pure logic, they may never occur within
- propositions, i.e.\ the @{text "\<Longrightarrow>"} arrow is non-dependent. The
- system provides a runtime option to record explicit proof terms for
- primitive inferences, cf.\ \cite{Berghofer-Nipkow:2000:TPHOL}. Thus
- the three-fold @{text "\<lambda>"}-structure can be made explicit.
+ are irrelevant in the Pure logic, though, they may never occur
+ within propositions. The system provides a runtime option to record
+ explicit proof terms for primitive inferences. Thus all three
+ levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for
+ terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\
+ \cite{Berghofer-Nipkow:2000:TPHOL}).
- Observe that locally fixed parameters (as used in rule @{text
- "\<And>_intro"}) need not be recorded in the hypotheses, because the
- simple syntactic types of Pure are always inhabitable. The typing
- ``assumption'' @{text "x :: \<tau>"} is logically vacuous, it disappears
- automatically whenever the statement body ceases to mention variable
- @{text "x\<^isub>\<tau>"}.\footnote{This greatly simplifies many basic
- reasoning steps, and is the key difference to the formulation of
- this logic as ``@{text "\<lambda>HOL"}'' in the PTS framework
- \cite{Barendregt-Geuvers:2001}.}
+ Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need
+ not be recorded in the hypotheses, because the simple syntactic
+ types of Pure are always inhabitable. Typing ``assumptions'' @{text
+ "x :: \<tau>"} are (implicitly) present only with occurrences of @{text
+ "x\<^isub>\<tau>"} in the statement body.\footnote{This is the key
+ difference ``@{text "\<lambda>HOL"}'' in the PTS framework
+ \cite{Barendregt-Geuvers:2001}, where @{text "x : A"} hypotheses are
+ treated explicitly for types, in the same way as propositions.}
\medskip FIXME @{text "\<alpha>\<beta>\<eta>"}-equivalence and primitive definitions
@@ -514,13 +519,11 @@
"\<alpha>\<beta>\<eta>"}-equivalence on terms, while coinciding with bi-implication.
\medskip The axiomatization of a theory is implicitly closed by
- forming all instances of type and term variables: @{text "\<turnstile> A\<vartheta>"} for
- any substitution instance of axiom @{text "\<turnstile> A"}. By pushing
- substitution through derivations inductively, we get admissible
- substitution rules for theorems shown in \figref{fig:subst-rules}.
- Alternatively, the term substitution rules could be derived from
- @{text "\<And>_intro/elim"}. The versions for types are genuine
- admissible rules, due to the lack of true polymorphism in the logic.
+ forming all instances of type and term variables: @{text "\<turnstile>
+ A\<vartheta>"} holds for any substitution instance of an axiom
+ @{text "\<turnstile> A"}. By pushing substitution through derivations
+ inductively, we get admissible @{text "generalize"} and @{text
+ "instance"} rules shown in \figref{fig:subst-rules}.
\begin{figure}[htb]
\begin{center}
@@ -538,11 +541,15 @@
\end{center}
\end{figure}
- Since @{text "\<Gamma>"} may never contain any schematic variables, the
- @{text "instantiate"} do not require an explicit side-condition. In
- principle, variables could be substituted in hypotheses as well, but
- this could disrupt monotonicity of the basic calculus: derivations
- could leave the current proof context.
+ Note that @{text "instantiate"} does not require an explicit
+ side-condition, because @{text "\<Gamma>"} may never contain schematic
+ variables.
+
+ In principle, variables could be substituted in hypotheses as well,
+ but this would disrupt monotonicity reasoning: deriving @{text
+ "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is correct, but
+ @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold --- the result
+ belongs to a different proof context.
*}
text %mlref {*
@@ -567,16 +574,16 @@
subsection {* Auxiliary connectives *}
text {*
- Pure also provides various auxiliary connectives based on primitive
- definitions, see \figref{fig:pure-aux}. These are normally not
- exposed to the user, but appear in internal encodings only.
+ Theory @{text "Pure"} also defines a few auxiliary connectives, see
+ \figref{fig:pure-aux}. These are normally not exposed to the user,
+ but appear in internal encodings only.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
@{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\
@{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex]
- @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}) \\
+ @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, hidden) \\
@{text "#A \<equiv> A"} \\[1ex]
@{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\
@{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex]
@@ -587,39 +594,38 @@
\end{center}
\end{figure}
- Conjunction as an explicit connective allows to treat both
- simultaneous assumptions and conclusions uniformly. The definition
- allows to derive the usual introduction @{text "\<turnstile> A \<Longrightarrow> B \<Longrightarrow> A & B"},
- and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}. For
- example, several claims may be stated at the same time, which is
- intermediately represented as an assumption, but the user only
- encounters several sub-goals, and several resulting facts in the
- very end (cf.\ \secref{sec:tactical-goals}).
+ Derived conjunction rules include introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A &
+ B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}.
+ Conjunction allows to treat simultaneous assumptions and conclusions
+ uniformly. For example, multiple claims are intermediately
+ represented as explicit conjunction, but this is usually refined
+ into separate sub-goals before the user continues the proof; the
+ final result is projected into a list of theorems (cf.\
+ \secref{sec:tactical-goals}).
- The @{text "#"} marker allows complex propositions (nested @{text
- "\<And>"} and @{text "\<Longrightarrow>"}) to appear formally as atomic, without changing
- the meaning: @{text "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are
- interchangeable. See \secref{sec:tactical-goals} for specific
- operations.
+ The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex
+ propositions appear as atomic, without changing the meaning: @{text
+ "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable. See
+ \secref{sec:tactical-goals} for specific operations.
- The @{text "TERM"} marker turns any well-formed term into a
- derivable proposition: @{text "\<turnstile> TERM t"} holds
- unconditionally. Despite its logically vacous meaning, this is
- occasionally useful to treat syntactic terms and proven propositions
- uniformly, as in a type-theoretic framework.
+ The @{text "term"} marker turns any well-formed term into a
+ derivable proposition: @{text "\<turnstile> TERM t"} holds unconditionally.
+ Although this is logically vacuous, it allows to treat terms and
+ proofs uniformly, similar to a type-theoretic framework.
- The @{text "TYPE"} constructor (which is the canonical
- representative of the unspecified type @{text "\<alpha> itself"}) injects
- the language of types into that of terms. There is specific
- notation @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau>
+ The @{text "TYPE"} constructor is the canonical representative of
+ the unspecified type @{text "\<alpha> itself"}; it essentially injects the
+ language of types into that of terms. There is specific notation
+ @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau>
itself\<^esub>"}.
- Although being devoid of any particular meaning, the term @{text
- "TYPE(\<tau>)"} is able to carry the type @{text "\<tau>"} formally. @{text
- "TYPE(\<alpha>)"} may be used as an additional formal argument in primitive
- definitions, in order to avoid hidden polymorphism (cf.\
- \secref{sec:terms}). For example, @{text "c TYPE(\<alpha>) \<equiv> A[\<alpha>]"} turns
- out as a formally correct definition of some proposition @{text "A"}
- that depends on an additional type argument.
+ Although being devoid of any particular meaning, the @{text
+ "TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term
+ language. In particular, @{text "TYPE(\<alpha>)"} may be used as formal
+ argument in primitive definitions, in order to circumvent hidden
+ polymorphism (cf.\ \secref{sec:terms}). For example, @{text "c
+ TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of
+ a proposition @{text "A"} that depends on an additional type
+ argument, which is essentially a predicate on types.
*}
text %mlref {*