doc-src/IsarImplementation/Thy/Logic.thy
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theory Logic
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imports Base
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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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  Derivations are relative to a logical theory, which declares type
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  constructors, constants, and axioms.  Theory declarations support
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  schematic polymorphism, which is strictly speaking outside the
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  logic.\footnote{This is the deeper logical reason, why the theory
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  context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"}
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  of the core calculus: type constructors, term constants, and facts
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  (proof constants) may involve arbitrary type schemes, but the type
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  of a locally fixed term parameter is also fixed!}
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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 "s = {c\<^isub>1,
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  \<dots>, c\<^isub>m}"}, it represents symbolic intersection.  Notationally, the
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  curly braces are omitted for singleton intersections, i.e.\ any
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  class @{text "c"} may be read as a sort @{text "{c}"}.  The ordering
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  on type classes is extended to sorts according to the meaning of
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  intersections: @{text "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff @{text
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  "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}.  The empty intersection @{text "{}"} refers to
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  the universal sort, which is the largest element wrt.\ the sort
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  order.  Thus @{text "{}"} represents the ``full sort'', not the
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  empty one!  The intersection of all (finitely many) classes declared
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  in the current theory is the least element wrt.\ the sort ordering.
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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, e.g.\
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  @{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}.
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  A \emph{schematic type variable} is a pair of an indexname and a
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  sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually
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  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: basic name, index, and sort constraint.  The core
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  logic handles type variables with the same name but different sorts
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  as different, although the type-inference layer (which is outside
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  the core) rejects anything like that.
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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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  written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.  For
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  @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"}
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  instead of @{text "()prop"}.  For @{text "k = 1"} the parentheses
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  are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}.
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  Further notation is provided for specific constructors, notably the
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  right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>,
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  \<beta>)fun"}.
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  The logical category \emph{type} is defined inductively over type
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  variables and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
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  (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>"}.
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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 appear as type
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  constructors in the syntax, but are expanded before entering the
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  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 any type constructor @{text "\<kappa>"}, and classes @{text
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  "c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> ::
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  (\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> ::
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  (\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq>
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  \<^vec>s\<^isub>2"} component-wise.
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  The key property of a coregular order-sorted algebra is that sort
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  constraints can be solved in a most general fashion: for each type
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  constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general
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  vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such
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  that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>,
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  \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}.
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  Consequently, type unification has most general solutions (modulo
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  equivalence of sorts), so type-inference produces primary types as
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  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: string} \\
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  @{index_ML_type sort: "class list"} \\
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  @{index_ML_type arity: "string * sort list * sort"} \\
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  @{index_ML_type typ} \\
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  @{index_ML Term.map_atyps: "(typ -> typ) -> typ -> typ"} \\
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  @{index_ML Term.fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
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  \end{mldecls}
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  \begin{mldecls}
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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: "Proof.context ->
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  (binding * int * mixfix) list -> theory -> theory"} \\
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  @{index_ML Sign.add_type_abbrev: "Proof.context ->
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  binding * string list * typ -> theory -> theory"} \\
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  @{index_ML Sign.primitive_class: "binding * 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 Type @{ML_type class} represents type classes.
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  \item Type @{ML_type sort} represents sorts, i.e.\ finite
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  intersections of classes.  The empty list @{ML "[]: sort"} refers to
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  the empty class intersection, i.e.\ the ``full sort''.
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  \item Type @{ML_type arity} represents type arities.  A triple
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  @{text "(\<kappa>, \<^vec>s, s) : arity"} represents @{text "\<kappa> ::
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  (\<^vec>s)s"} as described above.
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  \item Type @{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 Term.map_atyps}~@{text "f \<tau>"} applies the mapping @{text
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  "f"} to all atomic types (@{ML TFree}, @{ML TVar}) occurring in
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  @{text "\<tau>"}.
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  \item @{ML Term.fold_atyps}~@{text "f \<tau>"} iterates the operation
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  @{text "f"} over all occurrences of atomic types (@{ML TFree}, @{ML
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  TVar}) in @{text "\<tau>"}; the type structure is traversed from left to
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  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 type
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  @{text "\<tau>"} is of sort @{text "s"}.
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  \item @{ML Sign.add_types}~@{text "ctxt [(\<kappa>, k, mx), \<dots>]"} declares a
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  new type constructors @{text "\<kappa>"} with @{text "k"} arguments and
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  optional mixfix syntax.
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  \item @{ML Sign.add_type_abbrev}~@{text "ctxt (\<kappa>, \<^vec>\<alpha>, \<tau>)"}
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  defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"}.
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  \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
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  c\<^isub>n])"} declares a 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 the 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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  the arity @{text "\<kappa> :: (\<^vec>s)s"}.
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  \end{description}
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*}
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text %mlantiq {*
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  \begin{matharray}{rcl}
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  @{ML_antiquotation_def "class"} & : & @{text ML_antiquotation} \\
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  @{ML_antiquotation_def "sort"} & : & @{text ML_antiquotation} \\
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  @{ML_antiquotation_def "type_name"} & : & @{text ML_antiquotation} \\
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  @{ML_antiquotation_def "type_abbrev"} & : & @{text ML_antiquotation} \\
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  @{ML_antiquotation_def "nonterminal"} & : & @{text ML_antiquotation} \\
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  @{ML_antiquotation_def "typ"} & : & @{text ML_antiquotation} \\
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  \end{matharray}
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  \begin{rail}
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  'class' nameref
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  ;
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  'sort' sort
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  ;
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  ('type_name' | 'type_abbrev' | 'nonterminal') nameref
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  ;
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  'typ' type
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  ;
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  \end{rail}
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  \begin{description}
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  \item @{text "@{class c}"} inlines the internalized class @{text
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  "c"} --- as @{ML_type string} literal.
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  \item @{text "@{sort s}"} inlines the internalized sort @{text "s"}
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  --- as @{ML_type "string list"} literal.
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  \item @{text "@{type_name c}"} inlines the internalized type
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  constructor @{text "c"} --- as @{ML_type string} literal.
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  \item @{text "@{type_abbrev c}"} inlines the internalized type
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  abbreviation @{text "c"} --- as @{ML_type string} literal.
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  \item @{text "@{nonterminal c}"} inlines the internalized syntactic
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  type~/ grammar nonterminal @{text "c"} --- as @{ML_type string}
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  literal.
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  \item @{text "@{typ \<tau>}"} inlines the internalized type @{text "\<tau>"}
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  --- as constructor term for datatype @{ML_type typ}.
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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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  The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
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  with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
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  or \cite{paulson-ml2}), with the types being determined by the
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  corresponding binders.  In contrast, free variables and constants
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  have an explicit name and type in each occurrence.
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  \medskip A \emph{bound variable} is a natural number @{text "b"},
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  which accounts for the number of intermediate binders between the
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  variable occurrence in the body and its binding position.  For
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  example, the de-Bruijn term @{text "\<lambda>\<^bsub>bool\<^esub>. \<lambda>\<^bsub>bool\<^esub>. 1 \<and> 0"} would
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  correspond to @{text "\<lambda>x\<^bsub>bool\<^esub>. \<lambda>y\<^bsub>bool\<^esub>. x \<and> y"} in a named
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  representation.  Note that a bound variable may be represented by
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  different de-Bruijn indices at different occurrences, depending on
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  the nesting of abstractions.
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  A \emph{loose variable} is a bound variable that is outside the
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  scope of local binders.  The types (and names) for loose variables
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  can be managed as a separate context, that is maintained as a stack
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  of hypothetical binders.  The core logic operates on closed terms,
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  without any loose variables.
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  A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
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  @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"} here.  A
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  \emph{schematic variable} is a pair of an indexname and a type,
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  e.g.\ @{text "((x, 0), \<tau>)"} which is likewise printed as @{text
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  "?x\<^isub>\<tau>"}.
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  \medskip A \emph{constant} is a pair of a basic name and a type,
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  e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text "c\<^isub>\<tau>"}
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  here.  Constants are declared in the context as polymorphic families
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  @{text "c :: \<sigma>"}, meaning that all substitution instances @{text
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  "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
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  The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"} wrt.\
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  the declaration @{text "c :: \<sigma>"} is defined as the codomain of the
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  matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>, ?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in
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  canonical order @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}, corresponding to the
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  left-to-right occurrences of the @{text "\<alpha>\<^isub>i"} in @{text "\<sigma>"}.
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  Within a given theory context, there is a one-to-one correspondence
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  between any constant @{text "c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1,
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  \<dots>, \<tau>\<^isub>n)"} of its type arguments.  For example, with @{text "plus :: \<alpha>
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  \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} corresponds to
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  @{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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  due to violation of type class restrictions.
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  \medskip An \emph{atomic term} is either a variable or constant.
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  The logical category \emph{term} is defined inductively over atomic
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  terms, with abstraction and application as follows: @{text "t = b |
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  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
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  converting between an external representation with named bound
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  variables.  Subsequently, we shall use the latter notation instead
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  of internal de-Bruijn representation.
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  The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a
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  term according to the structure of atomic terms, abstractions, and
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  applicatins:
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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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  component.  This means that different variables @{text
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  "x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after
9700a87f1cc2 misc tuning and clarification;
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   319
  type instantiation.  Type-inference rejects variables of the same
9700a87f1cc2 misc tuning and clarification;
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   320
  name, but different types.  In contrast, mixed instances of
9700a87f1cc2 misc tuning and clarification;
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diff changeset
   321
  polymorphic constants occur routinely.
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diff changeset
   322
5ede702cd2ca more on terms;
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diff changeset
   323
  \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
5ede702cd2ca more on terms;
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diff changeset
   324
  is the set of type variables occurring in @{text "t"}, but not in
34929
9700a87f1cc2 misc tuning and clarification;
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diff changeset
   325
  its type @{text "\<sigma>"}.  This means that the term implicitly depends
9700a87f1cc2 misc tuning and clarification;
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parents: 34921
diff changeset
   326
  on type arguments that are not accounted in the result type, i.e.\
9700a87f1cc2 misc tuning and clarification;
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parents: 34921
diff changeset
   327
  there are different type instances @{text "t\<vartheta> :: \<sigma>"} and
9700a87f1cc2 misc tuning and clarification;
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diff changeset
   328
  @{text "t\<vartheta>' :: \<sigma>"} with the same type.  This slightly
20543
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diff changeset
   329
  pathological situation notoriously demands additional care.
20514
5ede702cd2ca more on terms;
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diff changeset
   330
5ede702cd2ca more on terms;
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diff changeset
   331
  \medskip A \emph{term abbreviation} is a syntactic definition @{text
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diff changeset
   332
  "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"},
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diff changeset
   333
  without any hidden polymorphism.  A term abbreviation looks like a
20543
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diff changeset
   334
  constant in the syntax, but is expanded before entering the logical
wenzelm
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diff changeset
   335
  core.  Abbreviations are usually reverted when printing terms, using
wenzelm
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diff changeset
   336
  @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for higher-order rewriting.
20519
d7ad1217c24a more on terms;
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diff changeset
   337
d7ad1217c24a more on terms;
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diff changeset
   338
  \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
20537
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   339
  "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free
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diff changeset
   340
  renaming of bound variables; @{text "\<beta>"}-conversion contracts an
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diff changeset
   341
  abstraction applied to an argument term, substituting the argument
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d7ad1217c24a more on terms;
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   342
  in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
d7ad1217c24a more on terms;
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parents: 20514
diff changeset
   343
  "\<eta>"}-conversion contracts vacuous application-abstraction: @{text
d7ad1217c24a more on terms;
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   344
  "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
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   345
  does not occur in @{text "f"}.
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d7ad1217c24a more on terms;
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diff changeset
   346
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diff changeset
   347
  Terms are normally treated modulo @{text "\<alpha>"}-conversion, which is
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diff changeset
   348
  implicit in the de-Bruijn representation.  Names for bound variables
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diff changeset
   349
  in abstractions are maintained separately as (meaningless) comments,
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parents: 20521
diff changeset
   350
  mostly for parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-conversion is
28784
9495aec512e2 renamed "Rules" to "Object-level rules";
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parents: 28674
diff changeset
   351
  commonplace in various standard operations (\secref{sec:obj-rules})
9495aec512e2 renamed "Rules" to "Object-level rules";
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parents: 28674
diff changeset
   352
  that are based on higher-order unification and matching.
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diff changeset
   353
*}
2681f9e34390 "The Isabelle/Isar Implementation" manual;
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diff changeset
   354
20514
5ede702cd2ca more on terms;
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diff changeset
   355
text %mlref {*
5ede702cd2ca more on terms;
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diff changeset
   356
  \begin{mldecls}
5ede702cd2ca more on terms;
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diff changeset
   357
  @{index_ML_type term} \\
20519
d7ad1217c24a more on terms;
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diff changeset
   358
  @{index_ML "op aconv": "term * term -> bool"} \\
39846
cb6634eb8926 examples in Isabelle/HOL;
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parents: 39840
diff changeset
   359
  @{index_ML Term.map_types: "(typ -> typ) -> term -> term"} \\
cb6634eb8926 examples in Isabelle/HOL;
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parents: 39840
diff changeset
   360
  @{index_ML Term.fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   361
  @{index_ML Term.map_aterms: "(term -> term) -> term -> term"} \\
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   362
  @{index_ML Term.fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\
20547
wenzelm
parents: 20543
diff changeset
   363
  \end{mldecls}
wenzelm
parents: 20543
diff changeset
   364
  \begin{mldecls}
20514
5ede702cd2ca more on terms;
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parents: 20501
diff changeset
   365
  @{index_ML fastype_of: "term -> typ"} \\
20519
d7ad1217c24a more on terms;
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parents: 20514
diff changeset
   366
  @{index_ML lambda: "term -> term -> term"} \\
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   367
  @{index_ML betapply: "term * term -> term"} \\
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   368
  @{index_ML Sign.declare_const: "Proof.context ->
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   369
  (binding * typ) * mixfix -> theory -> term * theory"} \\
33174
1f2051f41335 adjusted to changes in corresponding ML code
haftmann
parents: 32833
diff changeset
   370
  @{index_ML Sign.add_abbrev: "string -> binding * term ->
24972
acafb18a47dc replaced Sign.add_consts_i by Sign.declare_const;
wenzelm
parents: 24828
diff changeset
   371
  theory -> (term * term) * theory"} \\
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   372
  @{index_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   373
  @{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
20514
5ede702cd2ca more on terms;
wenzelm
parents: 20501
diff changeset
   374
  \end{mldecls}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   375
20514
5ede702cd2ca more on terms;
wenzelm
parents: 20501
diff changeset
   376
  \begin{description}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   377
39864
wenzelm
parents: 39861
diff changeset
   378
  \item Type @{ML_type term} represents de-Bruijn terms, with comments
wenzelm
parents: 39861
diff changeset
   379
  in abstractions, and explicitly named free variables and constants;
20537
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wenzelm
parents: 20521
diff changeset
   380
  this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML
b6b49903db7e *** empty log message ***
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parents: 20521
diff changeset
   381
  Var}, @{ML Const}, @{ML Abs}, @{ML "op $"}.
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   382
36166
da7b40aa2215 made SML/NJ happy;
wenzelm
parents: 36134
diff changeset
   383
  \item @{text "t"}~@{ML_text aconv}~@{text "u"} checks @{text
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   384
  "\<alpha>"}-equivalence of two terms.  This is the basic equality relation
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   385
  on type @{ML_type term}; raw datatype equality should only be used
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   386
  for operations related to parsing or printing!
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   387
39846
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   388
  \item @{ML Term.map_types}~@{text "f t"} applies the mapping @{text
20537
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parents: 20521
diff changeset
   389
  "f"} to all types occurring in @{text "t"}.
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   390
39846
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   391
  \item @{ML Term.fold_types}~@{text "f t"} iterates the operation
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   392
  @{text "f"} over all occurrences of types in @{text "t"}; the term
20537
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wenzelm
parents: 20521
diff changeset
   393
  structure is traversed from left to right.
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   394
39846
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   395
  \item @{ML Term.map_aterms}~@{text "f t"} applies the mapping @{text
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   396
  "f"} to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
20537
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wenzelm
parents: 20521
diff changeset
   397
  Const}) occurring in @{text "t"}.
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   398
39846
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   399
  \item @{ML Term.fold_aterms}~@{text "f t"} iterates the operation
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   400
  @{text "f"} over all occurrences of atomic terms (@{ML Bound}, @{ML
cb6634eb8926 examples in Isabelle/HOL;
wenzelm
parents: 39840
diff changeset
   401
  Free}, @{ML Var}, @{ML Const}) in @{text "t"}; the term structure is
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   402
  traversed from left to right.
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   403
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   404
  \item @{ML fastype_of}~@{text "t"} determines the type of a
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   405
  well-typed term.  This operation is relatively slow, despite the
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   406
  omission of any sanity checks.
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   407
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   408
  \item @{ML lambda}~@{text "a b"} produces an abstraction @{text
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   409
  "\<lambda>a. b"}, where occurrences of the atomic term @{text "a"} in the
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   410
  body @{text "b"} are replaced by bound variables.
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   411
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   412
  \item @{ML betapply}~@{text "(t, u)"} produces an application @{text
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   413
  "t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   414
  abstraction.
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   415
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   416
  \item @{ML Sign.declare_const}~@{text "ctxt ((c, \<sigma>), mx)"} declares
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   417
  a new constant @{text "c :: \<sigma>"} with optional mixfix syntax.
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   418
33174
1f2051f41335 adjusted to changes in corresponding ML code
haftmann
parents: 32833
diff changeset
   419
  \item @{ML Sign.add_abbrev}~@{text "print_mode (c, t)"}
21827
0b1d07f79c1e updated;
wenzelm
parents: 21324
diff changeset
   420
  introduces a new term abbreviation @{text "c \<equiv> t"}.
20519
d7ad1217c24a more on terms;
wenzelm
parents: 20514
diff changeset
   421
20520
wenzelm
parents: 20519
diff changeset
   422
  \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
wenzelm
parents: 20519
diff changeset
   423
  Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
20543
wenzelm
parents: 20542
diff changeset
   424
  convert between two representations of polymorphic constants: full
wenzelm
parents: 20542
diff changeset
   425
  type instance vs.\ compact type arguments form.
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   426
20514
5ede702cd2ca more on terms;
wenzelm
parents: 20501
diff changeset
   427
  \end{description}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   428
*}
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   429
39832
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   430
text %mlantiq {*
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   431
  \begin{matharray}{rcl}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   432
  @{ML_antiquotation_def "const_name"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   433
  @{ML_antiquotation_def "const_abbrev"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   434
  @{ML_antiquotation_def "const"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   435
  @{ML_antiquotation_def "term"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   436
  @{ML_antiquotation_def "prop"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   437
  \end{matharray}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   438
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   439
  \begin{rail}
40255
9ffbc25e1606 eliminated obsolete \_ escapes in rail environments;
wenzelm
parents: 40126
diff changeset
   440
  ('const_name' | 'const_abbrev') nameref
39832
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   441
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   442
  'const' ('(' (type + ',') ')')?
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   443
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   444
  'term' term
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   445
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   446
  'prop' prop
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   447
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   448
  \end{rail}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   449
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   450
  \begin{description}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   451
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   452
  \item @{text "@{const_name c}"} inlines the internalized logical
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   453
  constant name @{text "c"} --- as @{ML_type string} literal.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   454
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   455
  \item @{text "@{const_abbrev c}"} inlines the internalized
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   456
  abbreviated constant name @{text "c"} --- as @{ML_type string}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   457
  literal.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   458
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   459
  \item @{text "@{const c(\<^vec>\<tau>)}"} inlines the internalized
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   460
  constant @{text "c"} with precise type instantiation in the sense of
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   461
  @{ML Sign.const_instance} --- as @{ML Const} constructor term for
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   462
  datatype @{ML_type term}.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   463
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   464
  \item @{text "@{term t}"} inlines the internalized term @{text "t"}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   465
  --- as constructor term for datatype @{ML_type term}.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   466
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   467
  \item @{text "@{prop \<phi>}"} inlines the internalized proposition
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   468
  @{text "\<phi>"} --- as constructor term for datatype @{ML_type term}.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   469
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   470
  \end{description}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   471
*}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   472
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   473
20451
27ea2ba48fa3 misc cleanup;
wenzelm
parents: 20450
diff changeset
   474
section {* Theorems \label{sec:thms} *}
18537
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wenzelm
parents:
diff changeset
   475
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   476
text {*
20543
wenzelm
parents: 20542
diff changeset
   477
  A \emph{proposition} is a well-typed term of type @{text "prop"}, a
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   478
  \emph{theorem} is a proven proposition (depending on a context of
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   479
  hypotheses and the background theory).  Primitive inferences include
29774
wenzelm
parents: 29771
diff changeset
   480
  plain Natural Deduction rules for the primary connectives @{text
20537
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wenzelm
parents: 20521
diff changeset
   481
  "\<And>"} and @{text "\<Longrightarrow>"} of the framework.  There is also a builtin
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wenzelm
parents: 20521
diff changeset
   482
  notion of equality/equivalence @{text "\<equiv>"}.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   483
*}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   484
29758
7a3b5bbed313 removed rudiments of glossary;
wenzelm
parents: 29755
diff changeset
   485
26872
336dfd860744 fixed some labels;
wenzelm
parents: 24972
diff changeset
   486
subsection {* Primitive connectives and rules \label{sec:prim-rules} *}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   487
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   488
text {*
20543
wenzelm
parents: 20542
diff changeset
   489
  The theory @{text "Pure"} contains constant declarations for the
wenzelm
parents: 20542
diff changeset
   490
  primitive connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of
wenzelm
parents: 20542
diff changeset
   491
  the logical framework, see \figref{fig:pure-connectives}.  The
wenzelm
parents: 20542
diff changeset
   492
  derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is
wenzelm
parents: 20542
diff changeset
   493
  defined inductively by the primitive inferences given in
wenzelm
parents: 20542
diff changeset
   494
  \figref{fig:prim-rules}, with the global restriction that the
wenzelm
parents: 20542
diff changeset
   495
  hypotheses must \emph{not} contain any schematic variables.  The
wenzelm
parents: 20542
diff changeset
   496
  builtin equality is conceptually axiomatized as shown in
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   497
  \figref{fig:pure-equality}, although the implementation works
20543
wenzelm
parents: 20542
diff changeset
   498
  directly with derived inferences.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   499
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   500
  \begin{figure}[htb]
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   501
  \begin{center}
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   502
  \begin{tabular}{ll}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   503
  @{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   504
  @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   505
  @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   506
  \end{tabular}
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   507
  \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   508
  \end{center}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   509
  \end{figure}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   510
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   511
  \begin{figure}[htb]
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   512
  \begin{center}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   513
  \[
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   514
  \infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}}
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   515
  \qquad
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   516
  \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{}
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   517
  \]
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   518
  \[
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   519
  \infer[@{text "(\<And>\<dash>intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   520
  \qquad
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   521
  \infer[@{text "(\<And>\<dash>elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   522
  \]
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   523
  \[
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   524
  \infer[@{text "(\<Longrightarrow>\<dash>intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   525
  \qquad
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   526
  \infer[@{text "(\<Longrightarrow>\<dash>elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   527
  \]
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   528
  \caption{Primitive inferences of Pure}\label{fig:prim-rules}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   529
  \end{center}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   530
  \end{figure}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   531
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   532
  \begin{figure}[htb]
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   533
  \begin{center}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   534
  \begin{tabular}{ll}
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   535
  @{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   536
  @{text "\<turnstile> x \<equiv> x"} & reflexivity \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   537
  @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   538
  @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   539
  @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   540
  \end{tabular}
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   541
  \caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality}
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   542
  \end{center}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   543
  \end{figure}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   544
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   545
  The introduction and elimination rules for @{text "\<And>"} and @{text
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   546
  "\<Longrightarrow>"} are analogous to formation of dependently typed @{text
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   547
  "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
20543
wenzelm
parents: 20542
diff changeset
   548
  are irrelevant in the Pure logic, though; they cannot occur within
wenzelm
parents: 20542
diff changeset
   549
  propositions.  The system provides a runtime option to record
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   550
  explicit proof terms for primitive inferences.  Thus all three
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   551
  levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   552
  terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   553
  \cite{Berghofer-Nipkow:2000:TPHOL}).
20491
wenzelm
parents: 20480
diff changeset
   554
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   555
  Observe that locally fixed parameters (as in @{text
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   556
  "\<And>\<dash>intro"}) need not be recorded in the hypotheses, because
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   557
  the simple syntactic types of Pure are always inhabitable.
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   558
  ``Assumptions'' @{text "x :: \<tau>"} for type-membership are only
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   559
  present as long as some @{text "x\<^isub>\<tau>"} occurs in the statement
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   560
  body.\footnote{This is the key difference to ``@{text "\<lambda>HOL"}'' in
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   561
  the PTS framework \cite{Barendregt-Geuvers:2001}, where hypotheses
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   562
  @{text "x : A"} are treated uniformly for propositions and types.}
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   563
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   564
  \medskip The axiomatization of a theory is implicitly closed by
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   565
  forming all instances of type and term variables: @{text "\<turnstile>
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   566
  A\<vartheta>"} holds for any substitution instance of an axiom
20543
wenzelm
parents: 20542
diff changeset
   567
  @{text "\<turnstile> A"}.  By pushing substitutions through derivations
wenzelm
parents: 20542
diff changeset
   568
  inductively, we also get admissible @{text "generalize"} and @{text
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   569
  "instantiate"} rules as shown in \figref{fig:subst-rules}.
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   570
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   571
  \begin{figure}[htb]
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   572
  \begin{center}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   573
  \[
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   574
  \infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   575
  \quad
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   576
  \infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   577
  \]
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   578
  \[
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   579
  \infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   580
  \quad
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   581
  \infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   582
  \]
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   583
  \caption{Admissible substitution rules}\label{fig:subst-rules}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   584
  \end{center}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   585
  \end{figure}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   586
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   587
  Note that @{text "instantiate"} does not require an explicit
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   588
  side-condition, because @{text "\<Gamma>"} may never contain schematic
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   589
  variables.
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   590
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   591
  In principle, variables could be substituted in hypotheses as well,
20543
wenzelm
parents: 20542
diff changeset
   592
  but this would disrupt the monotonicity of reasoning: deriving
wenzelm
parents: 20542
diff changeset
   593
  @{text "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is
wenzelm
parents: 20542
diff changeset
   594
  correct, but @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold:
wenzelm
parents: 20542
diff changeset
   595
  the result belongs to a different proof context.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   596
20543
wenzelm
parents: 20542
diff changeset
   597
  \medskip An \emph{oracle} is a function that produces axioms on the
wenzelm
parents: 20542
diff changeset
   598
  fly.  Logically, this is an instance of the @{text "axiom"} rule
wenzelm
parents: 20542
diff changeset
   599
  (\figref{fig:prim-rules}), but there is an operational difference.
wenzelm
parents: 20542
diff changeset
   600
  The system always records oracle invocations within derivations of
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   601
  theorems by a unique tag.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   602
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   603
  Axiomatizations should be limited to the bare minimum, typically as
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   604
  part of the initial logical basis of an object-logic formalization.
20543
wenzelm
parents: 20542
diff changeset
   605
  Later on, theories are usually developed in a strictly definitional
wenzelm
parents: 20542
diff changeset
   606
  fashion, by stating only certain equalities over new constants.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   607
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   608
  A \emph{simple definition} consists of a constant declaration @{text
20543
wenzelm
parents: 20542
diff changeset
   609
  "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text "t
wenzelm
parents: 20542
diff changeset
   610
  :: \<sigma>"} is a closed term without any hidden polymorphism.  The RHS
wenzelm
parents: 20542
diff changeset
   611
  may depend on further defined constants, but not @{text "c"} itself.
wenzelm
parents: 20542
diff changeset
   612
  Definitions of functions may be presented as @{text "c \<^vec>x \<equiv>
wenzelm
parents: 20542
diff changeset
   613
  t"} instead of the puristic @{text "c \<equiv> \<lambda>\<^vec>x. t"}.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   614
20543
wenzelm
parents: 20542
diff changeset
   615
  An \emph{overloaded definition} consists of a collection of axioms
wenzelm
parents: 20542
diff changeset
   616
  for the same constant, with zero or one equations @{text
wenzelm
parents: 20542
diff changeset
   617
  "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type constructor @{text "\<kappa>"} (for
wenzelm
parents: 20542
diff changeset
   618
  distinct variables @{text "\<^vec>\<alpha>"}).  The RHS may mention
wenzelm
parents: 20542
diff changeset
   619
  previously defined constants as above, or arbitrary constants @{text
wenzelm
parents: 20542
diff changeset
   620
  "d(\<alpha>\<^isub>i)"} for some @{text "\<alpha>\<^isub>i"} projected from @{text
wenzelm
parents: 20542
diff changeset
   621
  "\<^vec>\<alpha>"}.  Thus overloaded definitions essentially work by
wenzelm
parents: 20542
diff changeset
   622
  primitive recursion over the syntactic structure of a single type
39840
3eb0694e6fcb more refs;
wenzelm
parents: 39832
diff changeset
   623
  argument.  See also \cite[\S4.3]{Haftmann-Wenzel:2006:classes}.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   624
*}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   625
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   626
text %mlref {*
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   627
  \begin{mldecls}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   628
  @{index_ML_type ctyp} \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   629
  @{index_ML_type cterm} \\
20547
wenzelm
parents: 20543
diff changeset
   630
  @{index_ML Thm.ctyp_of: "theory -> typ -> ctyp"} \\
wenzelm
parents: 20543
diff changeset
   631
  @{index_ML Thm.cterm_of: "theory -> term -> cterm"} \\
wenzelm
parents: 20543
diff changeset
   632
  \end{mldecls}
wenzelm
parents: 20543
diff changeset
   633
  \begin{mldecls}
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   634
  @{index_ML_type thm} \\
32833
f3716d1a2e48 explicitly Unsynchronized;
wenzelm
parents: 30552
diff changeset
   635
  @{index_ML proofs: "int Unsynchronized.ref"} \\
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   636
  @{index_ML Thm.assume: "cterm -> thm"} \\
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   637
  @{index_ML Thm.forall_intr: "cterm -> thm -> thm"} \\
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   638
  @{index_ML Thm.forall_elim: "cterm -> thm -> thm"} \\
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   639
  @{index_ML Thm.implies_intr: "cterm -> thm -> thm"} \\
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   640
  @{index_ML Thm.implies_elim: "thm -> thm -> thm"} \\
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   641
  @{index_ML Thm.generalize: "string list * string list -> int -> thm -> thm"} \\
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   642
  @{index_ML Thm.instantiate: "(ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm"} \\
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   643
  @{index_ML Thm.add_axiom: "Proof.context ->
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   644
  binding * term -> theory -> (string * thm) * theory"} \\
39821
bf164c153d10 minor tuning and updating;
wenzelm
parents: 39281
diff changeset
   645
  @{index_ML Thm.add_oracle: "binding * ('a -> cterm) -> theory ->
bf164c153d10 minor tuning and updating;
wenzelm
parents: 39281
diff changeset
   646
  (string * ('a -> thm)) * theory"} \\
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   647
  @{index_ML Thm.add_def: "Proof.context -> bool -> bool ->
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   648
  binding * term -> theory -> (string * thm) * theory"} \\
20547
wenzelm
parents: 20543
diff changeset
   649
  \end{mldecls}
wenzelm
parents: 20543
diff changeset
   650
  \begin{mldecls}
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   651
  @{index_ML Theory.add_deps: "Proof.context -> string ->
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   652
  string * typ -> (string * typ) list -> theory -> theory"} \\
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   653
  \end{mldecls}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   654
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   655
  \begin{description}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   656
39864
wenzelm
parents: 39861
diff changeset
   657
  \item Types @{ML_type ctyp} and @{ML_type cterm} represent certified
wenzelm
parents: 39861
diff changeset
   658
  types and terms, respectively.  These are abstract datatypes that
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   659
  guarantee that its values have passed the full well-formedness (and
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   660
  well-typedness) checks, relative to the declarations of type
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   661
  constructors, constants etc. in the theory.
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   662
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   663
  \item @{ML Thm.ctyp_of}~@{text "thy \<tau>"} and @{ML
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   664
  Thm.cterm_of}~@{text "thy t"} explicitly checks types and terms,
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   665
  respectively.  This also involves some basic normalizations, such
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   666
  expansion of type and term abbreviations from the theory context.
20547
wenzelm
parents: 20543
diff changeset
   667
wenzelm
parents: 20543
diff changeset
   668
  Re-certification is relatively slow and should be avoided in tight
wenzelm
parents: 20543
diff changeset
   669
  reasoning loops.  There are separate operations to decompose
wenzelm
parents: 20543
diff changeset
   670
  certified entities (including actual theorems).
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   671
39864
wenzelm
parents: 39861
diff changeset
   672
  \item Type @{ML_type thm} represents proven propositions.  This is
wenzelm
parents: 39861
diff changeset
   673
  an abstract datatype that guarantees that its values have been
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   674
  constructed by basic principles of the @{ML_struct Thm} module.
39281
148b78fb70d8 fixed antiquotation;
wenzelm
parents: 36345
diff changeset
   675
  Every @{ML_type thm} value contains a sliding back-reference to the
20543
wenzelm
parents: 20542
diff changeset
   676
  enclosing theory, cf.\ \secref{sec:context-theory}.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   677
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   678
  \item @{ML proofs} specifies the detail of proof recording within
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   679
  @{ML_type thm} values: @{ML 0} records only the names of oracles,
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   680
  @{ML 1} records oracle names and propositions, @{ML 2} additionally
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   681
  records full proof terms.  Officially named theorems that contribute
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   682
  to a result are recorded in any case.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   683
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   684
  \item @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   685
  Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim}
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   686
  correspond to the primitive inferences of \figref{fig:prim-rules}.
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   687
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   688
  \item @{ML Thm.generalize}~@{text "(\<^vec>\<alpha>, \<^vec>x)"}
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   689
  corresponds to the @{text "generalize"} rules of
20543
wenzelm
parents: 20542
diff changeset
   690
  \figref{fig:subst-rules}.  Here collections of type and term
wenzelm
parents: 20542
diff changeset
   691
  variables are generalized simultaneously, specified by the given
wenzelm
parents: 20542
diff changeset
   692
  basic names.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   693
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   694
  \item @{ML Thm.instantiate}~@{text "(\<^vec>\<alpha>\<^isub>s,
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   695
  \<^vec>x\<^isub>\<tau>)"} corresponds to the @{text "instantiate"} rules
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   696
  of \figref{fig:subst-rules}.  Type variables are substituted before
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   697
  term variables.  Note that the types in @{text "\<^vec>x\<^isub>\<tau>"}
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   698
  refer to the instantiated versions.
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   699
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   700
  \item @{ML Thm.add_axiom}~@{text "ctxt (name, A)"} declares an
35927
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   701
  arbitrary proposition as axiom, and retrieves it as a theorem from
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   702
  the resulting theory, cf.\ @{text "axiom"} in
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   703
  \figref{fig:prim-rules}.  Note that the low-level representation in
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   704
  the axiom table may differ slightly from the returned theorem.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   705
30288
a32700e45ab3 Thm.add_oracle interface: replaced old bstring by binding;
wenzelm
parents: 30272
diff changeset
   706
  \item @{ML Thm.add_oracle}~@{text "(binding, oracle)"} produces a named
28290
4cc2b6046258 simplified oracle interface;
wenzelm
parents: 28110
diff changeset
   707
  oracle rule, essentially generating arbitrary axioms on the fly,
4cc2b6046258 simplified oracle interface;
wenzelm
parents: 28110
diff changeset
   708
  cf.\ @{text "axiom"} in \figref{fig:prim-rules}.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   709
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   710
  \item @{ML Thm.add_def}~@{text "ctxt unchecked overloaded (name, c
35927
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   711
  \<^vec>x \<equiv> t)"} states a definitional axiom for an existing constant
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   712
  @{text "c"}.  Dependencies are recorded via @{ML Theory.add_deps},
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   713
  unless the @{text "unchecked"} option is set.  Note that the
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   714
  low-level representation in the axiom table may differ slightly from
343d5b0df29a updated Thm.add_axiom/add_def;
wenzelm
parents: 34929
diff changeset
   715
  the returned theorem.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   716
42401
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   717
  \item @{ML Theory.add_deps}~@{text "ctxt name c\<^isub>\<tau> \<^vec>d\<^isub>\<sigma>"}
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   718
  declares dependencies of a named specification for constant @{text
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   719
  "c\<^isub>\<tau>"}, relative to existing specifications for constants @{text
9bfaf6819291 updated some theory primitives, which now depend on auxiliary context;
wenzelm
parents: 40255
diff changeset
   720
  "\<^vec>d\<^isub>\<sigma>"}.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   721
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   722
  \end{description}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   723
*}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   724
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   725
39832
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   726
text %mlantiq {*
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   727
  \begin{matharray}{rcl}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   728
  @{ML_antiquotation_def "ctyp"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   729
  @{ML_antiquotation_def "cterm"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   730
  @{ML_antiquotation_def "cprop"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   731
  @{ML_antiquotation_def "thm"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   732
  @{ML_antiquotation_def "thms"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   733
  @{ML_antiquotation_def "lemma"} & : & @{text ML_antiquotation} \\
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   734
  \end{matharray}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   735
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   736
  \begin{rail}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   737
  'ctyp' typ
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   738
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   739
  'cterm' term
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   740
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   741
  'cprop' prop
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   742
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   743
  'thm' thmref
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   744
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   745
  'thms' thmrefs
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   746
  ;
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   747
  'lemma' ('(open)')? ((prop +) + 'and') \\ 'by' method method?
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   748
  \end{rail}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   749
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   750
  \begin{description}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   751
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   752
  \item @{text "@{ctyp \<tau>}"} produces a certified type wrt.\ the
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   753
  current background theory --- as abstract value of type @{ML_type
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   754
  ctyp}.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   755
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   756
  \item @{text "@{cterm t}"} and @{text "@{cprop \<phi>}"} produce a
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   757
  certified term wrt.\ the current background theory --- as abstract
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   758
  value of type @{ML_type cterm}.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   759
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   760
  \item @{text "@{thm a}"} produces a singleton fact --- as abstract
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   761
  value of type @{ML_type thm}.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   762
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   763
  \item @{text "@{thms a}"} produces a general fact --- as abstract
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   764
  value of type @{ML_type "thm list"}.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   765
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   766
  \item @{text "@{lemma \<phi> by meth}"} produces a fact that is proven on
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   767
  the spot according to the minimal proof, which imitates a terminal
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   768
  Isar proof.  The result is an abstract value of type @{ML_type thm}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   769
  or @{ML_type "thm list"}, depending on the number of propositions
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   770
  given here.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   771
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   772
  The internal derivation object lacks a proper theorem name, but it
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   773
  is formally closed, unless the @{text "(open)"} option is specified
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   774
  (this may impact performance of applications with proof terms).
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   775
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   776
  Since ML antiquotations are always evaluated at compile-time, there
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   777
  is no run-time overhead even for non-trivial proofs.  Nonetheless,
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   778
  the justification is syntactically limited to a single @{command
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   779
  "by"} step.  More complex Isar proofs should be done in regular
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   780
  theory source, before compiling the corresponding ML text that uses
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   781
  the result.
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   782
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   783
  \end{description}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   784
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   785
*}
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   786
1080dee73a53 various concrete ML antiquotations;
wenzelm
parents: 39821
diff changeset
   787
40126
916cb4a28ffd misc tuning;
wenzelm
parents: 39864
diff changeset
   788
subsection {* Auxiliary definitions \label{sec:logic-aux} *}
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   789
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   790
text {*
20543
wenzelm
parents: 20542
diff changeset
   791
  Theory @{text "Pure"} provides a few auxiliary definitions, see
wenzelm
parents: 20542
diff changeset
   792
  \figref{fig:pure-aux}.  These special constants are normally not
wenzelm
parents: 20542
diff changeset
   793
  exposed to the user, but appear in internal encodings.
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   794
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   795
  \begin{figure}[htb]
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   796
  \begin{center}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   797
  \begin{tabular}{ll}
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   798
  @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&&&"}) \\
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   799
  @{text "\<turnstile> A &&& B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex]
20543
wenzelm
parents: 20542
diff changeset
   800
  @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, suppressed) \\
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   801
  @{text "#A \<equiv> A"} \\[1ex]
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   802
  @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   803
  @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex]
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   804
  @{text "TYPE :: \<alpha> itself"} & (prefix @{text "TYPE"}) \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   805
  @{text "(unspecified)"} \\
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   806
  \end{tabular}
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   807
  \caption{Definitions of auxiliary connectives}\label{fig:pure-aux}
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   808
  \end{center}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   809
  \end{figure}
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   810
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   811
  The introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A &&& B"}, and eliminations
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   812
  (projections) @{text "A &&& B \<Longrightarrow> A"} and @{text "A &&& B \<Longrightarrow> B"} are
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   813
  available as derived rules.  Conjunction allows to treat
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   814
  simultaneous assumptions and conclusions uniformly, e.g.\ consider
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   815
  @{text "A \<Longrightarrow> B \<Longrightarrow> C &&& D"}.  In particular, the goal mechanism
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   816
  represents multiple claims as explicit conjunction internally, but
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   817
  this is refined (via backwards introduction) into separate sub-goals
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   818
  before the user commences the proof; the final result is projected
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   819
  into a list of theorems using eliminations (cf.\
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   820
  \secref{sec:tactical-goals}).
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   821
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   822
  The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   823
  propositions appear as atomic, without changing the meaning: @{text
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   824
  "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable.  See
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   825
  \secref{sec:tactical-goals} for specific operations.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   826
20543
wenzelm
parents: 20542
diff changeset
   827
  The @{text "term"} marker turns any well-typed term into a derivable
wenzelm
parents: 20542
diff changeset
   828
  proposition: @{text "\<turnstile> TERM t"} holds unconditionally.  Although
wenzelm
parents: 20542
diff changeset
   829
  this is logically vacuous, it allows to treat terms and proofs
wenzelm
parents: 20542
diff changeset
   830
  uniformly, similar to a type-theoretic framework.
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   831
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   832
  The @{text "TYPE"} constructor is the canonical representative of
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   833
  the unspecified type @{text "\<alpha> itself"}; it essentially injects the
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   834
  language of types into that of terms.  There is specific notation
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   835
  @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau>
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   836
 itself\<^esub>"}.
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   837
  Although being devoid of any particular meaning, the term @{text
20537
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   838
  "TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   839
  language.  In particular, @{text "TYPE(\<alpha>)"} may be used as formal
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   840
  argument in primitive definitions, in order to circumvent hidden
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   841
  polymorphism (cf.\ \secref{sec:terms}).  For example, @{text "c
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   842
  TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   843
  a proposition @{text "A"} that depends on an additional type
b6b49903db7e *** empty log message ***
wenzelm
parents: 20521
diff changeset
   844
  argument, which is essentially a predicate on types.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   845
*}
20501
de0b523b0d62 more rules;
wenzelm
parents: 20498
diff changeset
   846
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   847
text %mlref {*
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   848
  \begin{mldecls}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   849
  @{index_ML Conjunction.intr: "thm -> thm -> thm"} \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   850
  @{index_ML Conjunction.elim: "thm -> thm * thm"} \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   851
  @{index_ML Drule.mk_term: "cterm -> thm"} \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   852
  @{index_ML Drule.dest_term: "thm -> cterm"} \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   853
  @{index_ML Logic.mk_type: "typ -> term"} \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   854
  @{index_ML Logic.dest_type: "term -> typ"} \\
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   855
  \end{mldecls}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   856
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   857
  \begin{description}
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   858
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   859
  \item @{ML Conjunction.intr} derives @{text "A &&& B"} from @{text
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   860
  "A"} and @{text "B"}.
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   861
20543
wenzelm
parents: 20542
diff changeset
   862
  \item @{ML Conjunction.elim} derives @{text "A"} and @{text "B"}
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   863
  from @{text "A &&& B"}.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   864
20543
wenzelm
parents: 20542
diff changeset
   865
  \item @{ML Drule.mk_term} derives @{text "TERM t"}.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   866
20543
wenzelm
parents: 20542
diff changeset
   867
  \item @{ML Drule.dest_term} recovers term @{text "t"} from @{text
wenzelm
parents: 20542
diff changeset
   868
  "TERM t"}.
20542
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   869
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   870
  \item @{ML Logic.mk_type}~@{text "\<tau>"} produces the term @{text
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   871
  "TYPE(\<tau>)"}.
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   872
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   873
  \item @{ML Logic.dest_type}~@{text "TYPE(\<tau>)"} recovers the type
a54ca4e90874 more on theorems;
wenzelm
parents: 20537
diff changeset
   874
  @{text "\<tau>"}.
20521
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   875
189811b39869 more on theorems;
wenzelm
parents: 20520
diff changeset
   876
  \end{description}
20491
wenzelm
parents: 20480
diff changeset
   877
*}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   878
20480
4e0522d38968 more on types and type classes;
wenzelm
parents: 20477
diff changeset
   879
28784
9495aec512e2 renamed "Rules" to "Object-level rules";
wenzelm
parents: 28674
diff changeset
   880
section {* Object-level rules \label{sec:obj-rules} *}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
   881
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   882
text {*
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   883
  The primitive inferences covered so far mostly serve foundational
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   884
  purposes.  User-level reasoning usually works via object-level rules
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   885
  that are represented as theorems of Pure.  Composition of rules
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   886
  involves \emph{backchaining}, \emph{higher-order unification} modulo
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   887
  @{text "\<alpha>\<beta>\<eta>"}-conversion of @{text "\<lambda>"}-terms, and so-called
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   888
  \emph{lifting} of rules into a context of @{text "\<And>"} and @{text
29774
wenzelm
parents: 29771
diff changeset
   889
  "\<Longrightarrow>"} connectives.  Thus the full power of higher-order Natural
wenzelm
parents: 29771
diff changeset
   890
  Deduction in Isabelle/Pure becomes readily available.
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   891
*}
20491
wenzelm
parents: 20480
diff changeset
   892
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   893
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   894
subsection {* Hereditary Harrop Formulae *}
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   895
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   896
text {*
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   897
  The idea of object-level rules is to model Natural Deduction
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   898
  inferences in the style of Gentzen \cite{Gentzen:1935}, but we allow
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   899
  arbitrary nesting similar to \cite{extensions91}.  The most basic
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   900
  rule format is that of a \emph{Horn Clause}:
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   901
  \[
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   902
  \infer{@{text "A"}}{@{text "A\<^sub>1"} & @{text "\<dots>"} & @{text "A\<^sub>n"}}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   903
  \]
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   904
  where @{text "A, A\<^sub>1, \<dots>, A\<^sub>n"} are atomic propositions
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   905
  of the framework, usually of the form @{text "Trueprop B"}, where
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   906
  @{text "B"} is a (compound) object-level statement.  This
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   907
  object-level inference corresponds to an iterated implication in
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   908
  Pure like this:
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   909
  \[
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   910
  @{text "A\<^sub>1 \<Longrightarrow> \<dots> A\<^sub>n \<Longrightarrow> A"}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   911
  \]
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   912
  As an example consider conjunction introduction: @{text "A \<Longrightarrow> B \<Longrightarrow> A \<and>
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   913
  B"}.  Any parameters occurring in such rule statements are
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   914
  conceptionally treated as arbitrary:
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   915
  \[
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   916
  @{text "\<And>x\<^sub>1 \<dots> x\<^sub>m. A\<^sub>1 x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> \<dots> A\<^sub>n x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> A x\<^sub>1 \<dots> x\<^sub>m"}
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   917
  \]
20491
wenzelm
parents: 20480
diff changeset
   918
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   919
  Nesting of rules means that the positions of @{text "A\<^sub>i"} may
29770
wenzelm
parents: 29769
diff changeset
   920
  again hold compound rules, not just atomic propositions.
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   921
  Propositions of this format are called \emph{Hereditary Harrop
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   922
  Formulae} in the literature \cite{Miller:1991}.  Here we give an
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   923
  inductive characterization as follows:
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   924
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   925
  \medskip
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   926
  \begin{tabular}{ll}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   927
  @{text "\<^bold>x"} & set of variables \\
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   928
  @{text "\<^bold>A"} & set of atomic propositions \\
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   929
  @{text "\<^bold>H  =  \<And>\<^bold>x\<^sup>*. \<^bold>H\<^sup>* \<Longrightarrow> \<^bold>A"} & set of Hereditary Harrop Formulas \\
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   930
  \end{tabular}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   931
  \medskip
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   932
39861
b8d89db3e238 use continental paragraph style, which works better with mixture of (in)formal text;
wenzelm
parents: 39846
diff changeset
   933
  Thus we essentially impose nesting levels on propositions formed
b8d89db3e238 use continental paragraph style, which works better with mixture of (in)formal text;
wenzelm
parents: 39846
diff changeset
   934
  from @{text "\<And>"} and @{text "\<Longrightarrow>"}.  At each level there is a prefix
b8d89db3e238 use continental paragraph style, which works better with mixture of (in)formal text;
wenzelm
parents: 39846
diff changeset
   935
  of parameters and compound premises, concluding an atomic
29770
wenzelm
parents: 29769
diff changeset
   936
  proposition.  Typical examples are @{text "\<longrightarrow>"}-introduction @{text
wenzelm
parents: 29769
diff changeset
   937
  "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B"} or mathematical induction @{text "P 0 \<Longrightarrow> (\<And>n. P n
wenzelm
parents: 29769
diff changeset
   938
  \<Longrightarrow> P (Suc n)) \<Longrightarrow> P n"}.  Even deeper nesting occurs in well-founded
wenzelm
parents: 29769
diff changeset
   939
  induction @{text "(\<And>x. (\<And>y. y \<prec> x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x"}, but this
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   940
  already marks the limit of rule complexity that is usually seen in
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
   941
  practice.
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   942
29770
wenzelm
parents: 29769
diff changeset
   943
  \medskip Regular user-level inferences in Isabelle/Pure always
wenzelm
parents: 29769
diff changeset
   944
  maintain the following canonical form of results:
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   945
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   946
  \begin{itemize}
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
   947
29774
wenzelm
parents: 29771
diff changeset
   948
  \item Normalization by @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"},
wenzelm
parents: 29771
diff changeset
   949
  which is a theorem of Pure, means that quantifiers are pushed in
wenzelm
parents: 29771
diff changeset
   950
  front of implication at each level of nesting.  The normal form is a
wenzelm
parents: 29771
diff changeset
   951
  Hereditary Harrop Formula.
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   952
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   953
  \item The outermost prefix of parameters is represented via
29770
wenzelm
parents: 29769
diff changeset
   954
  schematic variables: instead of @{text "\<And>\<^vec>x. \<^vec>H \<^vec>x
29774
wenzelm
parents: 29771
diff changeset
   955
  \<Longrightarrow> A \<^vec>x"} we have @{text "\<^vec>H ?\<^vec>x \<Longrightarrow> A ?\<^vec>x"}.
wenzelm
parents: 29771
diff changeset
   956
  Note that this representation looses information about the order of
wenzelm
parents: 29771
diff changeset
   957
  parameters, and vacuous quantifiers vanish automatically.
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   958
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   959
  \end{itemize}
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   960
*}
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   961
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   962
text %mlref {*
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   963
  \begin{mldecls}
30552
58db56278478 provide Simplifier.norm_hhf(_protect) as regular simplifier operation;
wenzelm
parents: 30355
diff changeset
   964
  @{index_ML Simplifier.norm_hhf: "thm -> thm"} \\
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   965
  \end{mldecls}
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   966
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   967
  \begin{description}
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   968
30552
58db56278478 provide Simplifier.norm_hhf(_protect) as regular simplifier operation;
wenzelm
parents: 30355
diff changeset
   969
  \item @{ML Simplifier.norm_hhf}~@{text thm} normalizes the given
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   970
  theorem according to the canonical form specified above.  This is
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   971
  occasionally helpful to repair some low-level tools that do not
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   972
  handle Hereditary Harrop Formulae properly.
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   973
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   974
  \end{description}
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   975
*}
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   976
29769
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   977
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   978
subsection {* Rule composition *}
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   979
03634a9e91ae improved section on "Hereditary Harrop Formulae";
wenzelm
parents: 29768
diff changeset
   980
text {*
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   981
  The rule calculus of Isabelle/Pure provides two main inferences:
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   982
  @{inference resolution} (i.e.\ back-chaining of rules) and
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   983
  @{inference assumption} (i.e.\ closing a branch), both modulo
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   984
  higher-order unification.  There are also combined variants, notably
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   985
  @{inference elim_resolution} and @{inference dest_resolution}.
20491
wenzelm
parents: 20480
diff changeset
   986
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   987
  To understand the all-important @{inference resolution} principle,
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   988
  we first consider raw @{inference_def composition} (modulo
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   989
  higher-order unification with substitution @{text "\<vartheta>"}):
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   990
  \[
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   991
  \infer[(@{inference_def composition})]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   992
  {@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}}
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
   993
  \]
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   994
  Here the conclusion of the first rule is unified with the premise of
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   995
  the second; the resulting rule instance inherits the premises of the
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   996
  first and conclusion of the second.  Note that @{text "C"} can again
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   997
  consist of iterated implications.  We can also permute the premises
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   998
  of the second rule back-and-forth in order to compose with @{text
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
   999
  "B'"} in any position (subsequently we shall always refer to
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1000
  position 1 w.l.o.g.).
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1001
29774
wenzelm
parents: 29771
diff changeset
  1002
  In @{inference composition} the internal structure of the common
wenzelm
parents: 29771
diff changeset
  1003
  part @{text "B"} and @{text "B'"} is not taken into account.  For
wenzelm
parents: 29771
diff changeset
  1004
  proper @{inference resolution} we require @{text "B"} to be atomic,
wenzelm
parents: 29771
diff changeset
  1005
  and explicitly observe the structure @{text "\<And>\<^vec>x. \<^vec>H
wenzelm
parents: 29771
diff changeset
  1006
  \<^vec>x \<Longrightarrow> B' \<^vec>x"} of the premise of the second rule.  The
wenzelm
parents: 29771
diff changeset
  1007
  idea is to adapt the first rule by ``lifting'' it into this context,
wenzelm
parents: 29771
diff changeset
  1008
  by means of iterated application of the following inferences:
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1009
  \[
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1010
  \infer[(@{inference_def imp_lift})]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1011
  \]
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1012
  \[
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1013
  \infer[(@{inference_def all_lift})]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}}
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1014
  \]
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1015
  By combining raw composition with lifting, we get full @{inference
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1016
  resolution} as follows:
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1017
  \[
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1018
  \infer[(@{inference_def resolution})]
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1019
  {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}}
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1020
  {\begin{tabular}{l}
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1021
    @{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1022
    @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1023
    @{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1024
   \end{tabular}}
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1025
  \]
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1026
29774
wenzelm
parents: 29771
diff changeset
  1027
  Continued resolution of rules allows to back-chain a problem towards
wenzelm
parents: 29771
diff changeset
  1028
  more and sub-problems.  Branches are closed either by resolving with
wenzelm
parents: 29771
diff changeset
  1029
  a rule of 0 premises, or by producing a ``short-circuit'' within a
wenzelm
parents: 29771
diff changeset
  1030
  solved situation (again modulo unification):
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1031
  \[
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1032
  \infer[(@{inference_def assumption})]{@{text "C\<vartheta>"}}
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1033
  {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text i})}}
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1034
  \]
20498
825a8d2335ce more rules;
wenzelm
parents: 20494
diff changeset
  1035
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1036
  FIXME @{inference_def elim_resolution}, @{inference_def dest_resolution}
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
  1037
*}
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
  1038
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1039
text %mlref {*
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1040
  \begin{mldecls}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1041
  @{index_ML "op RS": "thm * thm -> thm"} \\
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1042
  @{index_ML "op OF": "thm * thm list -> thm"} \\
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1043
  \end{mldecls}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1044
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1045
  \begin{description}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1046
34929
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
  1047
  \item @{text "rule\<^sub>1 RS rule\<^sub>2"} resolves @{text "rule\<^sub>1"} with @{text
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
  1048
  "rule\<^sub>2"} according to the @{inference resolution} principle
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
  1049
  explained above.  Note that the corresponding rule attribute in the
9700a87f1cc2 misc tuning and clarification;
wenzelm
parents: 34921
diff changeset
  1050
  Isar language is called @{attribute THEN}.
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1051
29771
aa1d3b5d1b5e improved section "Rule composition";
wenzelm
parents: 29770
diff changeset
  1052
  \item @{text "rule OF rules"} resolves a list of rules with the
29774
wenzelm
parents: 29771
diff changeset
  1053
  first rule, addressing its premises @{text "1, \<dots>, length rules"}
wenzelm
parents: 29771
diff changeset
  1054
  (operating from last to first).  This means the newly emerging
wenzelm
parents: 29771
diff changeset
  1055
  premises are all concatenated, without interfering.  Also note that
wenzelm
parents: 29771
diff changeset
  1056
  compared to @{text "RS"}, the rule argument order is swapped: @{text
wenzelm
parents: 29771
diff changeset
  1057
  "rule\<^sub>1 RS rule\<^sub>2 = rule\<^sub>2 OF [rule\<^sub>1]"}.
29768
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1058
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1059
  \end{description}
64a50ff3f308 more on object-level rules;
wenzelm
parents: 29761
diff changeset
  1060
*}
30272
2d612824e642 regenerated document;
wenzelm
parents: 30270
diff changeset
  1061
18537
2681f9e34390 "The Isabelle/Isar Implementation" manual;
wenzelm
parents:
diff changeset
  1062
end