doc-src/IsarImplementation/Thy/logic.thy
author wenzelm
Tue Sep 12 17:23:34 2006 +0200 (2006-09-12)
changeset 20519 d7ad1217c24a
parent 20514 5ede702cd2ca
child 20520 05fd007bdeb9
permissions -rw-r--r--
more on terms;
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(* $Id$ *)
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theory logic imports base begin
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chapter {* Primitive logic \label{ch:logic} *}
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text {*
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  The logical foundations of Isabelle/Isar are that of the Pure logic,
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  which has been introduced as a natural-deduction framework in
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  \cite{paulson700}.  This is essentially the same logic as ``@{text
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  "\<lambda>HOL"}'' in the more abstract setting of Pure Type Systems (PTS)
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  \cite{Barendregt-Geuvers:2001}, although there are some key
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  differences in the specific treatment of simple types in
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  Isabelle/Pure.
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  Following type-theoretic parlance, the Pure logic consists of three
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  levels of @{text "\<lambda>"}-calculus with corresponding arrows: @{text
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  "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
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  "\<And>"} for universal quantification (proofs depending on terms), and
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  @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
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  Pure derivations are relative to a logical theory, which declares
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  type constructors, term constants, and axioms.  Theory declarations
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  support schematic polymorphism, which is strictly speaking outside
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  the logic.\footnote{Incidently, this is the main logical reason, why
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  the theory context @{text "\<Theta>"} is separate from the context @{text
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  "\<Gamma>"} of the core calculus.}
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*}
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section {* Types \label{sec:types} *}
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text {*
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  The language of types is an uninterpreted order-sorted first-order
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  algebra; types are qualified by ordered type classes.
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  \medskip A \emph{type class} is an abstract syntactic entity
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  declared in the theory context.  The \emph{subclass relation} @{text
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  "c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic
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  generating relation; the transitive closure is maintained
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  internally.  The resulting relation is an ordering: reflexive,
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  transitive, and antisymmetric.
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  A \emph{sort} is a list of type classes written as @{text
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  "{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
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  intersection.  Notationally, the curly braces are omitted for
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  singleton intersections, i.e.\ any class @{text "c"} may be read as
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  a sort @{text "{c}"}.  The ordering on type classes is extended to
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  sorts according to the meaning of intersections: @{text
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  "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff
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  @{text "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}.  The empty intersection
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  @{text "{}"} refers to the universal sort, which is the largest
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  element wrt.\ the sort order.  The intersections of all (finitely
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  many) classes declared in the current theory are the minimal
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  elements wrt.\ the sort order.
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  \medskip A \emph{fixed type variable} is a pair of a basic name
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  (starting with a @{text "'"} character) and a sort constraint.  For
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  example, @{text "('a, s)"} which is usually printed as @{text
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  "\<alpha>\<^isub>s"}.  A \emph{schematic type variable} is a pair of an
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  indexname and a sort constraint.  For example, @{text "(('a, 0),
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  s)"} which is usually printed as @{text "?\<alpha>\<^isub>s"}.
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  Note that \emph{all} syntactic components contribute to the identity
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  of type variables, including the sort constraint.  The core logic
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  handles type variables with the same name but different sorts as
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  different, although some outer layers of the system make it hard to
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  produce anything like this.
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  A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
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  on types declared in the theory.  Type constructor application is
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  usually written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.
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  For @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text
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  "prop"} instead of @{text "()prop"}.  For @{text "k = 1"} the
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  parentheses are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text
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  "(\<alpha>)list"}.  Further notation is provided for specific constructors,
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  notably the right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of
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  @{text "(\<alpha>, \<beta>)fun"}.
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  A \emph{type} @{text "\<tau>"} is defined inductively over type variables
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  and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s |
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  ?\<alpha>\<^isub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
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  A \emph{type abbreviation} is a syntactic definition @{text
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  "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
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  variables @{text "\<^vec>\<alpha>"}.  Type abbreviations looks like type
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  constructors at the surface, but are fully expanded before entering
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  the logical core.
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  A \emph{type arity} declares the image behavior of a type
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  constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
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  s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)\<kappa>"} is
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  of sort @{text "s"} if every argument type @{text "\<tau>\<^isub>i"} is
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  of sort @{text "s\<^isub>i"}.  Arity declarations are implicitly
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  completed, i.e.\ @{text "\<kappa> :: (\<^vec>s)c"} entails @{text "\<kappa> ::
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  (\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}.
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  \medskip The sort algebra is always maintained as \emph{coregular},
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  which means that type arities are consistent with the subclass
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  relation: for each type constructor @{text "\<kappa>"} and classes @{text
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  "c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "\<kappa> ::
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  (\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "\<kappa>
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  :: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq>
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  \<^vec>s\<^isub>2"} holds component-wise.
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  The key property of a coregular order-sorted algebra is that sort
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  constraints may be always solved in a most general fashion: for each
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  type constructor @{text "\<kappa>"} and sort @{text "s"} there is a most
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  general vector of argument sorts @{text "(s\<^isub>1, \<dots>,
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  s\<^isub>k)"} such that a type scheme @{text
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  "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is
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  of sort @{text "s"}.  Consequently, the unification problem on the
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  algebra of types has most general solutions (modulo renaming and
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  equivalence of sorts).  Moreover, the usual type-inference algorithm
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  will produce primary types as expected \cite{nipkow-prehofer}.
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*}
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text %mlref {*
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  \begin{mldecls}
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  @{index_ML_type class} \\
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  @{index_ML_type sort} \\
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  @{index_ML_type arity} \\
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  @{index_ML_type typ} \\
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  @{index_ML map_atyps: "(typ -> typ) -> typ -> typ"} \\
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  @{index_ML fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
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  @{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
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  @{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
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  @{index_ML Sign.add_types: "(bstring * int * mixfix) list -> theory -> theory"} \\
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  @{index_ML Sign.add_tyabbrs_i: "
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  (bstring * string list * typ * mixfix) list -> theory -> theory"} \\
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  @{index_ML Sign.primitive_class: "string * class list -> theory -> theory"} \\
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  @{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
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  @{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
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  \end{mldecls}
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  \begin{description}
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  \item @{ML_type class} represents type classes; this is an alias for
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  @{ML_type string}.
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  \item @{ML_type sort} represents sorts; this is an alias for
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  @{ML_type "class list"}.
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  \item @{ML_type arity} represents type arities; this is an alias for
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  triples of the form @{text "(\<kappa>, \<^vec>s, s)"} for @{text "\<kappa> ::
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  (\<^vec>s)s"} described above.
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  \item @{ML_type typ} represents types; this is a datatype with
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  constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
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  \item @{ML map_atyps}~@{text "f \<tau>"} applies mapping @{text "f"} to
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  all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text "\<tau>"}.
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  \item @{ML fold_atyps}~@{text "f \<tau>"} iterates operation @{text "f"}
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  over all occurrences of atoms (@{ML TFree}, @{ML TVar}) in @{text
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  "\<tau>"}; the type structure is traversed from left to right.
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  \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
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  tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
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  \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether a type
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  is of a given sort.
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  \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares new
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  type constructors @{text "\<kappa>"} with @{text "k"} arguments and
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  optional mixfix syntax.
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  \item @{ML Sign.add_tyabbrs_i}~@{text "[(\<kappa>, \<^vec>\<alpha>, \<tau>, mx), \<dots>]"}
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  defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"} with
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  optional mixfix syntax.
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  \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
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  c\<^isub>n])"} declares new class @{text "c"}, together with class
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  relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
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  \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
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  c\<^isub>2)"} declares class relation @{text "c\<^isub>1 \<subseteq>
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  c\<^isub>2"}.
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  \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
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  arity @{text "\<kappa> :: (\<^vec>s)s"}.
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  \end{description}
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*}
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section {* Terms \label{sec:terms} *}
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text {*
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  \glossary{Term}{FIXME}
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  The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
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  with de-Bruijn indices for bound variables
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  \cite{debruijn72,paulson-ml2}, and named free variables and
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  constants.  Terms with loose bound variables are usually considered
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  malformed.  The types of variables and constants is stored
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  explicitly at each occurrence in the term.
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  \medskip A \emph{bound variable} is a natural number @{text "b"},
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  which refers to the next binder that is @{text "b"} steps upwards
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  from the occurrence of @{text "b"} (counting from zero).  Bindings
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  may be introduced as abstractions within the term, or as a separate
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  context (an inside-out list).  This associates each bound variable
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  with a type.  A \emph{loose variables} is a bound variable that is
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  outside the current scope of local binders or the context.  For
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  example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"}
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  corresponds to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a named
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  representation.  Also note that the very same bound variable may get
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  different numbers at different occurrences.
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  A \emph{fixed variable} is a pair of a basic name and a type.  For
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  example, @{text "(x, \<tau>)"} which is usually printed @{text
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  "x\<^isub>\<tau>"}.  A \emph{schematic variable} is a pair of an
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  indexname and a type.  For example, @{text "((x, 0), \<tau>)"} which is
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  usually printed as @{text "?x\<^isub>\<tau>"}.
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  \medskip A \emph{constant} is a atomic terms consisting of a basic
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  name and a type.  Constants are declared in the context as
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  polymorphic families @{text "c :: \<sigma>"}, meaning that any @{text
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  "c\<^isub>\<tau>"} is a valid constant for all substitution instances
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  @{text "\<tau> \<le> \<sigma>"}.
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  The list of \emph{type arguments} of @{text "c\<^isub>\<tau>"} wrt.\ the
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  declaration @{text "c :: \<sigma>"} is the codomain of the type matcher
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  presented in canonical order (according to the left-to-right
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  occurrences of type variables in in @{text "\<sigma>"}).  Thus @{text
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  "c\<^isub>\<tau>"} can be represented more compactly as @{text
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  "c(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}.  For example, the instance @{text
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  "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} of some @{text "plus :: \<alpha> \<Rightarrow> \<alpha>
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  \<Rightarrow> \<alpha>"} has the singleton list @{text "nat"} as type arguments, the
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  constant may be represented as @{text "plus(nat)"}.
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  Constant declarations @{text "c :: \<sigma>"} may contain sort constraints
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  for type variables in @{text "\<sigma>"}.  These are observed by
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  type-inference as expected, but \emph{ignored} by the core logic.
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  This means the primitive logic is able to reason with instances of
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  polymorphic constants that the user-level type-checker would reject.
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  \medskip A \emph{term} @{text "t"} is defined inductively over
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  variables and constants, with abstraction and application as
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  follows: @{text "t = b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> |
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  \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}.  Parsing and printing takes
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  care of converting between an external representation with named
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  bound variables.  Subsequently, we shall use the latter notation
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  instead of internal de-Bruijn representation.
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  The subsequent inductive relation @{text "t :: \<tau>"} assigns a
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  (unique) type to a term, using the special type constructor @{text
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  "(\<alpha>, \<beta>)fun"}, which is written @{text "\<alpha> \<Rightarrow> \<beta>"}.
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  \[
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  \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{}
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  \qquad
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  \infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}}
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  \qquad
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  \infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}}
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  \]
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  A \emph{well-typed term} is a term that can be typed according to these rules.
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  Typing information can be omitted: type-inference is able to
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  reconstruct the most general type of a raw term, while assigning
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  most general types to all of its variables and constants.
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  Type-inference depends on a context of type constraints for fixed
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  variables, and declarations for polymorphic constants.
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  The identity of atomic terms consists both of the name and the type.
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  Thus different entities @{text "c\<^bsub>\<tau>\<^isub>1\<^esub>"} and
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  @{text "c\<^bsub>\<tau>\<^isub>2\<^esub>"} may well identified by type
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  instantiation, by mapping @{text "\<tau>\<^isub>1"} and @{text
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  "\<tau>\<^isub>2"} to the same @{text "\<tau>"}.  Although,
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  different type instances of constants of the same basic name are
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  commonplace, this rarely happens for variables: type-inference
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  always demands ``consistent'' type constraints.
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  \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
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  is the set of type variables occurring in @{text "t"}, but not in
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  @{text "\<sigma>"}.  This means that the term implicitly depends on the
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  values of various type variables that are not visible in the overall
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  type, i.e.\ there are different type instances @{text "t\<vartheta>
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  :: \<sigma>"} and @{text "t\<vartheta>' :: \<sigma>"} with the same type.  This
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  slightly pathological situation is apt to cause strange effects.
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  \medskip A \emph{term abbreviation} is a syntactic definition @{text
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  "c\<^isub>\<sigma> \<equiv> t"} of an arbitrary closed term @{text "t"} of type
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  @{text "\<sigma>"} without any hidden polymorphism.  A term abbreviation
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  looks like a constant at the surface, but is fully expanded before
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  entering the logical core.  Abbreviations are usually reverted when
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  printing terms, using rules @{text "t \<rightarrow> c\<^isub>\<sigma>"} has a
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  higher-order term rewrite system.
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  \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
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  "\<alpha>\<beta>\<eta>"}-conversion. @{text "\<alpha>"}-conversion refers to capture-free
wenzelm@20519
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  renaming of bound variables; @{text "\<beta>"}-conversion contracts an
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  abstraction applied to some argument term, substituting the argument
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  in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
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  "\<eta>"}-conversion contracts vacuous application-abstraction: @{text
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  "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
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  @{text "0"} does not occur in @{text "f"}.
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wenzelm@20519
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  Terms are almost always treated module @{text "\<alpha>"}-conversion, which
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  is implicit in the de-Bruijn representation.  The names in
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  abstractions of bound variables are maintained only as a comment for
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  parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-equivalence is usually
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  taken for granted higher rules (\secref{sec:rules}), anything
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  depending on higher-order unification or rewriting.
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*}
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text %mlref {*
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  \begin{mldecls}
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  @{index_ML_type term} \\
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  @{index_ML "op aconv": "term * term -> bool"} \\
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  @{index_ML map_term_types: "(typ -> typ) -> term -> term"} \\  %FIXME rename map_types
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  @{index_ML fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
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  @{index_ML map_aterms: "(term -> term) -> term -> term"} \\
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  @{index_ML fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\
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  @{index_ML fastype_of: "term -> typ"} \\
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  @{index_ML lambda: "term -> term -> term"} \\
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  @{index_ML betapply: "term * term -> term"} \\
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  @{index_ML Sign.add_consts_i: "(bstring * typ * mixfix) list -> theory -> theory"} \\
wenzelm@20519
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  @{index_ML Sign.add_abbrevs: "string * bool ->
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  ((bstring * mixfix) * term) list -> theory -> theory"} \\
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  @{index_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\
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  @{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
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  \end{mldecls}
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  \begin{description}
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  \item @{ML_type term} represents de-Bruijn terms with comments in
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  abstractions for bound variable names.  This is a datatype with
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  constructors @{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}, @{ML
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  Abs}, @{ML "op $"}.
wenzelm@20519
   333
wenzelm@20519
   334
  \item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text
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  "\<alpha>"}-equivalence of two terms.  This is the basic equality relation
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   336
  on type @{ML_type term}; raw datatype equality should only be used
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  for operations related to parsing or printing!
wenzelm@20519
   338
wenzelm@20519
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  \item @{ML map_term_types}~@{text "f t"} applies mapping @{text "f"}
wenzelm@20519
   340
  to all types occurring in @{text "t"}.
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   341
wenzelm@20519
   342
  \item @{ML fold_types}~@{text "f t"} iterates operation @{text "f"}
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   343
  over all occurrences of types in @{text "t"}; the term structure is
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  traversed from left to right.
wenzelm@20519
   345
wenzelm@20519
   346
  \item @{ML map_aterms}~@{text "f t"} applies mapping @{text "f"} to
wenzelm@20519
   347
  all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const})
wenzelm@20519
   348
  occurring in @{text "t"}.
wenzelm@20519
   349
wenzelm@20519
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  \item @{ML fold_aterms}~@{text "f t"} iterates operation @{text "f"}
wenzelm@20519
   351
  over all occurrences of atomic terms in (@{ML Bound}, @{ML Free},
wenzelm@20519
   352
  @{ML Var}, @{ML Const}) @{text "t"}; the term structure is traversed
wenzelm@20519
   353
  from left to right.
wenzelm@20519
   354
wenzelm@20519
   355
  \item @{ML fastype_of}~@{text "t"} recomputes the type of a
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   356
  well-formed term, while omitting any sanity checks.  This operation
wenzelm@20519
   357
  is relatively slow.
wenzelm@20519
   358
wenzelm@20519
   359
  \item @{ML lambda}~@{text "a b"} produces an abstraction @{text
wenzelm@20519
   360
  "\<lambda>a. b"}, where occurrences of the original (atomic) term @{text
wenzelm@20519
   361
  "a"} are replaced by bound variables.
wenzelm@20519
   362
wenzelm@20519
   363
  \item @{ML betapply}~@{text "t u"} produces an application @{text "t
wenzelm@20519
   364
  u"}, with topmost @{text "\<beta>"}-conversion @{text "t"} is an
wenzelm@20519
   365
  abstraction.
wenzelm@20519
   366
wenzelm@20519
   367
  \item @{ML Sign.add_consts_i}~@{text "[(c, \<sigma>, mx), \<dots>]"} declares a
wenzelm@20519
   368
  new constant @{text "c :: \<sigma>"} with optional mixfix syntax.
wenzelm@20519
   369
wenzelm@20519
   370
  \item @{ML Sign.add_abbrevs}~@{text "print_mode [((c, t), mx), \<dots>]"}
wenzelm@20519
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  declares a new term abbreviation @{text "c \<equiv> t"} with optional
wenzelm@20519
   372
  mixfix syntax.
wenzelm@20519
   373
wenzelm@20519
   374
  \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} produces the
wenzelm@20519
   375
  type arguments of the instance @{text "c\<^isub>\<tau>"} wrt.\ its
wenzelm@20519
   376
  declaration in the theory.
wenzelm@20519
   377
wenzelm@20519
   378
  \item @{ML Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>,
wenzelm@20519
   379
  \<tau>\<^isub>n])"} produces the full instance @{text "c(\<tau>\<^isub>1, \<dots>,
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   380
  \<tau>\<^isub>n)"} wrt.\ its declaration in the theory.
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   382
  \end{description}
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   383
*}
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   385
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   386
section {* Theorems \label{sec:thms} *}
wenzelm@18537
   387
wenzelm@18537
   388
text {*
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   389
  \glossary{Proposition}{A \seeglossary{term} of \seeglossary{type}
wenzelm@20501
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  @{text "prop"}.  Internally, there is nothing special about
wenzelm@20501
   391
  propositions apart from their type, but the concrete syntax enforces
wenzelm@20501
   392
  a clear distinction.  Propositions are structured via implication
wenzelm@20501
   393
  @{text "A \<Longrightarrow> B"} or universal quantification @{text "\<And>x. B x"} ---
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  anything else is considered atomic.  The canonical form for
wenzelm@20501
   395
  propositions is that of a \seeglossary{Hereditary Harrop Formula}. FIXME}
wenzelm@20480
   396
wenzelm@20501
   397
  \glossary{Theorem}{A proven proposition within a certain theory and
wenzelm@20501
   398
  proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are
wenzelm@20501
   399
  rarely spelled out explicitly.  Theorems are usually normalized
wenzelm@20501
   400
  according to the \seeglossary{HHF} format. FIXME}
wenzelm@20480
   401
wenzelm@20519
   402
  \glossary{Fact}{Sometimes used interchangeably for
wenzelm@20501
   403
  \seeglossary{theorem}.  Strictly speaking, a list of theorems,
wenzelm@20501
   404
  essentially an extra-logical conjunction.  Facts emerge either as
wenzelm@20501
   405
  local assumptions, or as results of local goal statements --- both
wenzelm@20501
   406
  may be simultaneous, hence the list representation. FIXME}
wenzelm@18537
   407
wenzelm@20501
   408
  \glossary{Schematic variable}{FIXME}
wenzelm@20501
   409
wenzelm@20501
   410
  \glossary{Fixed variable}{A variable that is bound within a certain
wenzelm@20501
   411
  proof context; an arbitrary-but-fixed entity within a portion of
wenzelm@20501
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  proof text. FIXME}
wenzelm@18537
   413
wenzelm@20501
   414
  \glossary{Free variable}{Synonymous for \seeglossary{fixed
wenzelm@20501
   415
  variable}. FIXME}
wenzelm@20501
   416
wenzelm@20501
   417
  \glossary{Bound variable}{FIXME}
wenzelm@18537
   418
wenzelm@20501
   419
  \glossary{Variable}{See \seeglossary{schematic variable},
wenzelm@20501
   420
  \seeglossary{fixed variable}, \seeglossary{bound variable}, or
wenzelm@20501
   421
  \seeglossary{type variable}.  The distinguishing feature of
wenzelm@20501
   422
  different variables is their binding scope. FIXME}
wenzelm@18537
   423
wenzelm@20501
   424
  A \emph{proposition} is a well-formed term of type @{text "prop"}.
wenzelm@20501
   425
  The connectives of minimal logic are declared as constants of the
wenzelm@20501
   426
  basic theory:
wenzelm@18537
   427
wenzelm@20501
   428
  \smallskip
wenzelm@20501
   429
  \begin{tabular}{ll}
wenzelm@20501
   430
  @{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\
wenzelm@20501
   431
  @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
wenzelm@20501
   432
  \end{tabular}
wenzelm@18537
   433
wenzelm@20501
   434
  \medskip A \emph{theorem} is a proven proposition, depending on a
wenzelm@20501
   435
  collection of assumptions, and axioms from the theory context.  The
wenzelm@20501
   436
  judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined
wenzelm@20501
   437
  inductively by the primitive inferences given in
wenzelm@20501
   438
  \figref{fig:prim-rules}; there is a global syntactic restriction
wenzelm@20501
   439
  that the hypotheses may not contain schematic variables.
wenzelm@18537
   440
wenzelm@20501
   441
  \begin{figure}[htb]
wenzelm@20501
   442
  \begin{center}
wenzelm@20498
   443
  \[
wenzelm@20498
   444
  \infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}}
wenzelm@20498
   445
  \qquad
wenzelm@20498
   446
  \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{}
wenzelm@20498
   447
  \]
wenzelm@20498
   448
  \[
wenzelm@20498
   449
  \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}{@{text "\<Gamma> \<turnstile> b x"} & @{text "x \<notin> \<Gamma>"}}
wenzelm@20498
   450
  \qquad
wenzelm@20498
   451
  \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b a"}}{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}
wenzelm@20498
   452
  \]
wenzelm@20498
   453
  \[
wenzelm@20498
   454
  \infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
wenzelm@20498
   455
  \qquad
wenzelm@20498
   456
  \infer[@{text "(\<Longrightarrow>_elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}}
wenzelm@20498
   457
  \]
wenzelm@20501
   458
  \caption{Primitive inferences of the Pure logic}\label{fig:prim-rules}
wenzelm@20501
   459
  \end{center}
wenzelm@20501
   460
  \end{figure}
wenzelm@18537
   461
wenzelm@20501
   462
  The introduction and elimination rules for @{text "\<And>"} and @{text
wenzelm@20501
   463
  "\<Longrightarrow>"} are analogous to formation of (dependently typed) @{text
wenzelm@20501
   464
  "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
wenzelm@20501
   465
  are \emph{irrelevant} in the Pure logic, they may never occur within
wenzelm@20501
   466
  propositions, i.e.\ the @{text "\<Longrightarrow>"} arrow of the framework is a
wenzelm@20501
   467
  non-dependent one.
wenzelm@20491
   468
wenzelm@20501
   469
  Also note that fixed parameters as in @{text "\<And>_intro"} need not be
wenzelm@20501
   470
  recorded in the context @{text "\<Gamma>"}, since syntactic types are
wenzelm@20501
   471
  always inhabitable.  An ``assumption'' @{text "x :: \<tau>"} is logically
wenzelm@20501
   472
  vacuous, because @{text "\<tau>"} is always non-empty.  This is the deeper
wenzelm@20501
   473
  reason why @{text "\<Gamma>"} only consists of hypothetical proofs, but no
wenzelm@20501
   474
  hypothetical terms.
wenzelm@20501
   475
wenzelm@20501
   476
  The corresponding proof terms are left implicit in the classic
wenzelm@20501
   477
  ``LCF-approach'', although they could be exploited separately
wenzelm@20501
   478
  \cite{Berghofer-Nipkow:2000}.  The implementation provides a runtime
wenzelm@20501
   479
  option to control the generation of full proof terms.
wenzelm@20501
   480
wenzelm@20501
   481
  \medskip The axiomatization of a theory is implicitly closed by
wenzelm@20501
   482
  forming all instances of type and term variables: @{text "\<turnstile> A\<theta>"} for
wenzelm@20514
   483
  any substitution instance of axiom @{text "\<turnstile> A"}.  By pushing
wenzelm@20501
   484
  substitution through derivations inductively, we get admissible
wenzelm@20501
   485
  substitution rules for theorems shown in \figref{fig:subst-rules}.
wenzelm@20501
   486
wenzelm@20501
   487
  \begin{figure}[htb]
wenzelm@20501
   488
  \begin{center}
wenzelm@20498
   489
  \[
wenzelm@20501
   490
  \infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}}
wenzelm@20501
   491
  \quad
wenzelm@20501
   492
  \infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}}
wenzelm@20498
   493
  \]
wenzelm@20498
   494
  \[
wenzelm@20501
   495
  \infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}
wenzelm@20501
   496
  \quad
wenzelm@20501
   497
  \infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}}
wenzelm@20498
   498
  \]
wenzelm@20501
   499
  \caption{Admissible substitution rules}\label{fig:subst-rules}
wenzelm@20501
   500
  \end{center}
wenzelm@20501
   501
  \end{figure}
wenzelm@18537
   502
wenzelm@20498
   503
  Note that @{text "instantiate_term"} could be derived using @{text
wenzelm@20498
   504
  "\<And>_intro/elim"}, but this is not how it is implemented.  The type
wenzelm@20501
   505
  instantiation rule is a genuine admissible one, due to the lack of
wenzelm@20501
   506
  true polymorphism in the logic.
wenzelm@20498
   507
wenzelm@20501
   508
  Since @{text "\<Gamma>"} may never contain any schematic variables, the
wenzelm@20501
   509
  @{text "instantiate"} do not require an explicit side-condition.  In
wenzelm@20501
   510
  principle, variables could be substituted in hypotheses as well, but
wenzelm@20501
   511
  this could disrupt monotonicity of the basic calculus: derivations
wenzelm@20501
   512
  could leave the current proof context.
wenzelm@20498
   513
wenzelm@20501
   514
  \medskip The framework also provides builtin equality @{text "\<equiv>"},
wenzelm@20501
   515
  which is conceptually axiomatized shown in \figref{fig:equality},
wenzelm@20501
   516
  although the implementation provides derived rules directly:
wenzelm@20501
   517
wenzelm@20501
   518
  \begin{figure}[htb]
wenzelm@20501
   519
  \begin{center}
wenzelm@20498
   520
  \begin{tabular}{ll}
wenzelm@20498
   521
  @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
wenzelm@20501
   522
  @{text "\<turnstile> (\<lambda>x. b x) a \<equiv> b a"} & @{text "\<beta>"}-conversion \\
wenzelm@20498
   523
  @{text "\<turnstile> x \<equiv> x"} & reflexivity law \\
wenzelm@20498
   524
  @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution law \\
wenzelm@20498
   525
  @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
wenzelm@20498
   526
  @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & coincidence with equivalence \\
wenzelm@20498
   527
  \end{tabular}
wenzelm@20501
   528
  \caption{Conceptual axiomatization of equality.}\label{fig:equality}
wenzelm@20501
   529
  \end{center}
wenzelm@20501
   530
  \end{figure}
wenzelm@20501
   531
wenzelm@20501
   532
  Since the basic representation of terms already accounts for @{text
wenzelm@20501
   533
  "\<alpha>"}-conversion, Pure equality essentially acts like @{text
wenzelm@20501
   534
  "\<alpha>\<beta>\<eta>"}-equivalence on terms, while coinciding with bi-implication.
wenzelm@20498
   535
wenzelm@20498
   536
wenzelm@20501
   537
  \medskip Conjunction is defined in Pure as a derived connective, see
wenzelm@20501
   538
  \figref{fig:conjunction}.  This is occasionally useful to represent
wenzelm@20501
   539
  simultaneous statements behind the scenes --- framework conjunction
wenzelm@20501
   540
  is usually not exposed to the user.
wenzelm@20498
   541
wenzelm@20501
   542
  \begin{figure}[htb]
wenzelm@20501
   543
  \begin{center}
wenzelm@20501
   544
  \begin{tabular}{ll}
wenzelm@20501
   545
  @{text "& :: prop \<Rightarrow> prop \<Rightarrow> prop"} & conjunction (hidden) \\
wenzelm@20501
   546
  @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\
wenzelm@20501
   547
  \end{tabular}
wenzelm@20501
   548
  \caption{Definition of conjunction.}\label{fig:equality}
wenzelm@20501
   549
  \end{center}
wenzelm@20501
   550
  \end{figure}
wenzelm@20501
   551
wenzelm@20501
   552
  The definition allows to derive the usual introduction @{text "\<turnstile> A \<Longrightarrow>
wenzelm@20501
   553
  B \<Longrightarrow> A & B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B
wenzelm@20501
   554
  \<Longrightarrow> B"}.
wenzelm@20491
   555
*}
wenzelm@18537
   556
wenzelm@20480
   557
wenzelm@20491
   558
section {* Rules \label{sec:rules} *}
wenzelm@18537
   559
wenzelm@18537
   560
text {*
wenzelm@18537
   561
wenzelm@18537
   562
FIXME
wenzelm@18537
   563
wenzelm@20491
   564
  A \emph{rule} is any Pure theorem in HHF normal form; there is a
wenzelm@20491
   565
  separate calculus for rule composition, which is modeled after
wenzelm@20491
   566
  Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
wenzelm@20491
   567
  rules to be nested arbitrarily, similar to \cite{extensions91}.
wenzelm@20491
   568
wenzelm@20491
   569
  Normally, all theorems accessible to the user are proper rules.
wenzelm@20491
   570
  Low-level inferences are occasional required internally, but the
wenzelm@20491
   571
  result should be always presented in canonical form.  The higher
wenzelm@20491
   572
  interfaces of Isabelle/Isar will always produce proper rules.  It is
wenzelm@20491
   573
  important to maintain this invariant in add-on applications!
wenzelm@20491
   574
wenzelm@20491
   575
  There are two main principles of rule composition: @{text
wenzelm@20491
   576
  "resolution"} (i.e.\ backchaining of rules) and @{text
wenzelm@20491
   577
  "by-assumption"} (i.e.\ closing a branch); both principles are
wenzelm@20519
   578
  combined in the variants of @{text "elim-resolution"} and @{text
wenzelm@20491
   579
  "dest-resolution"}.  Raw @{text "composition"} is occasionally
wenzelm@20491
   580
  useful as well, also it is strictly speaking outside of the proper
wenzelm@20491
   581
  rule calculus.
wenzelm@20491
   582
wenzelm@20491
   583
  Rules are treated modulo general higher-order unification, which is
wenzelm@20491
   584
  unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion
wenzelm@20491
   585
  on @{text "\<lambda>"}-terms.  Moreover, propositions are understood modulo
wenzelm@20491
   586
  the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.
wenzelm@20491
   587
wenzelm@20491
   588
  This means that any operations within the rule calculus may be
wenzelm@20491
   589
  subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions.  It is common
wenzelm@20491
   590
  practice not to contract or expand unnecessarily.  Some mechanisms
wenzelm@20491
   591
  prefer an one form, others the opposite, so there is a potential
wenzelm@20491
   592
  danger to produce some oscillation!
wenzelm@20491
   593
wenzelm@20491
   594
  Only few operations really work \emph{modulo} HHF conversion, but
wenzelm@20491
   595
  expect a normal form: quantifiers @{text "\<And>"} before implications
wenzelm@20491
   596
  @{text "\<Longrightarrow>"} at each level of nesting.
wenzelm@20491
   597
wenzelm@18537
   598
\glossary{Hereditary Harrop Formula}{The set of propositions in HHF
wenzelm@18537
   599
format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow>
wenzelm@18537
   600
A)"}, for variables @{text "x"} and atomic propositions @{text "A"}.
wenzelm@18537
   601
Any proposition may be put into HHF form by normalizing with the rule
wenzelm@18537
   602
@{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.  In Isabelle, the outermost
wenzelm@18537
   603
quantifier prefix is represented via \seeglossary{schematic
wenzelm@18537
   604
variables}, such that the top-level structure is merely that of a
wenzelm@18537
   605
\seeglossary{Horn Clause}}.
wenzelm@18537
   606
wenzelm@18537
   607
\glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
wenzelm@18537
   608
wenzelm@20498
   609
wenzelm@20498
   610
  \[
wenzelm@20498
   611
  \infer[@{text "(assumption)"}]{@{text "C\<vartheta>"}}
wenzelm@20498
   612
  {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text i})}}
wenzelm@20498
   613
  \]
wenzelm@20498
   614
wenzelm@20498
   615
wenzelm@20498
   616
  \[
wenzelm@20498
   617
  \infer[@{text "(compose)"}]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}}
wenzelm@20498
   618
  {@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}}
wenzelm@20498
   619
  \]
wenzelm@20498
   620
wenzelm@20498
   621
wenzelm@20498
   622
  \[
wenzelm@20498
   623
  \infer[@{text "(\<And>_lift)"}]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}}
wenzelm@20498
   624
  \]
wenzelm@20498
   625
  \[
wenzelm@20498
   626
  \infer[@{text "(\<Longrightarrow>_lift)"}]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}}
wenzelm@20498
   627
  \]
wenzelm@20498
   628
wenzelm@20498
   629
  The @{text resolve} scheme is now acquired from @{text "\<And>_lift"},
wenzelm@20498
   630
  @{text "\<Longrightarrow>_lift"}, and @{text compose}.
wenzelm@20498
   631
wenzelm@20498
   632
  \[
wenzelm@20498
   633
  \infer[@{text "(resolution)"}]
wenzelm@20498
   634
  {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}}
wenzelm@20498
   635
  {\begin{tabular}{l}
wenzelm@20498
   636
    @{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\
wenzelm@20498
   637
    @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\
wenzelm@20498
   638
    @{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\
wenzelm@20498
   639
   \end{tabular}}
wenzelm@20498
   640
  \]
wenzelm@20498
   641
wenzelm@20498
   642
wenzelm@20498
   643
  FIXME @{text "elim_resolution"}, @{text "dest_resolution"}
wenzelm@18537
   644
*}
wenzelm@18537
   645
wenzelm@20498
   646
wenzelm@18537
   647
end