diff --git a/src/Doc/Implementation/Logic.thy b/src/Doc/Implementation/Logic.thy
--- a/src/Doc/Implementation/Logic.thy
+++ b/src/Doc/Implementation/Logic.thy
@@ -1,1258 +1,1364 @@
 (*:maxLineLen=78:*)
 
 theory Logic
 imports Base
 begin
 
 chapter \<open>Primitive logic \label{ch:logic}\<close>
 
 text \<open>
   The logical foundations of Isabelle/Isar are that of the Pure logic, which
   has been introduced as a Natural Deduction framework in @{cite paulson700}.
   This is essentially the same logic as ``\<open>\<lambda>HOL\<close>'' in the more abstract
   setting of Pure Type Systems (PTS) @{cite "Barendregt-Geuvers:2001"},
   although there are some key differences in the specific treatment of simple
   types in Isabelle/Pure.
 
   Following type-theoretic parlance, the Pure logic consists of three levels
   of \<open>\<lambda>\<close>-calculus with corresponding arrows, \<open>\<Rightarrow>\<close> for syntactic function space
   (terms depending on terms), \<open>\<And>\<close> for universal quantification (proofs
   depending on terms), and \<open>\<Longrightarrow>\<close> for implication (proofs depending on proofs).
 
   Derivations are relative to a logical theory, which declares type
   constructors, constants, and axioms. Theory declarations support schematic
   polymorphism, which is strictly speaking outside the logic.\<^footnote>\<open>This is the
   deeper logical reason, why the theory context \<open>\<Theta>\<close> is separate from the proof
   context \<open>\<Gamma>\<close> of the core calculus: type constructors, term constants, and
   facts (proof constants) may involve arbitrary type schemes, but the type of
   a locally fixed term parameter is also fixed!\<close>
 \<close>
 
 
 section \<open>Types \label{sec:types}\<close>
 
 text \<open>
   The language of types is an uninterpreted order-sorted first-order algebra;
   types are qualified by ordered type classes.
 
   \<^medskip>
   A \<^emph>\<open>type class\<close> is an abstract syntactic entity declared in the theory
   context. The \<^emph>\<open>subclass relation\<close> \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close> is specified by stating an
   acyclic generating relation; the transitive closure is maintained
   internally. The resulting relation is an ordering: reflexive, transitive,
   and antisymmetric.
 
   A \<^emph>\<open>sort\<close> is a list of type classes written as \<open>s = {c\<^sub>1, \<dots>, c\<^sub>m}\<close>, it
   represents symbolic intersection. Notationally, the curly braces are omitted
   for singleton intersections, i.e.\ any class \<open>c\<close> may be read as a sort
   \<open>{c}\<close>. The ordering on type classes is extended to sorts according to the
   meaning of intersections: \<open>{c\<^sub>1, \<dots> c\<^sub>m} \<subseteq> {d\<^sub>1, \<dots>, d\<^sub>n}\<close> iff \<open>\<forall>j. \<exists>i. c\<^sub>i \<subseteq>
   d\<^sub>j\<close>. The empty intersection \<open>{}\<close> refers to the universal sort, which is the
   largest element wrt.\ the sort order. Thus \<open>{}\<close> represents the ``full
   sort'', not the empty one! The intersection of all (finitely many) classes
   declared in the current theory is the least element wrt.\ the sort ordering.
 
   \<^medskip>
   A \<^emph>\<open>fixed type variable\<close> is a pair of a basic name (starting with a \<open>'\<close>
   character) and a sort constraint, e.g.\ \<open>('a, s)\<close> which is usually printed
   as \<open>\<alpha>\<^sub>s\<close>. A \<^emph>\<open>schematic type variable\<close> is a pair of an indexname and a sort
   constraint, e.g.\ \<open>(('a, 0), s)\<close> which is usually printed as \<open>?\<alpha>\<^sub>s\<close>.
 
   Note that \<^emph>\<open>all\<close> syntactic components contribute to the identity of type
   variables: basic name, index, and sort constraint. The core logic handles
   type variables with the same name but different sorts as different, although
   the type-inference layer (which is outside the core) rejects anything like
   that.
 
   A \<^emph>\<open>type constructor\<close> \<open>\<kappa>\<close> is a \<open>k\<close>-ary operator on types declared in the
   theory. Type constructor application is written postfix as \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>k)\<kappa>\<close>.
   For \<open>k = 0\<close> the argument tuple is omitted, e.g.\ \<open>prop\<close> instead of \<open>()prop\<close>.
   For \<open>k = 1\<close> the parentheses are omitted, e.g.\ \<open>\<alpha> list\<close> instead of
   \<open>(\<alpha>)list\<close>. Further notation is provided for specific constructors, notably
   the right-associative infix \<open>\<alpha> \<Rightarrow> \<beta>\<close> instead of \<open>(\<alpha>, \<beta>)fun\<close>.
 
   The logical category \<^emph>\<open>type\<close> is defined inductively over type variables and
   type constructors as follows: \<open>\<tau> = \<alpha>\<^sub>s | ?\<alpha>\<^sub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close>.
 
   A \<^emph>\<open>type abbreviation\<close> is a syntactic definition \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close> of an
   arbitrary type expression \<open>\<tau>\<close> over variables \<open>\<^vec>\<alpha>\<close>. Type abbreviations
   appear as type constructors in the syntax, but are expanded before entering
   the logical core.
 
   A \<^emph>\<open>type arity\<close> declares the image behavior of a type constructor wrt.\ the
   algebra of sorts: \<open>\<kappa> :: (s\<^sub>1, \<dots>, s\<^sub>k)s\<close> means that \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close> is of
   sort \<open>s\<close> if every argument type \<open>\<tau>\<^sub>i\<close> is of sort \<open>s\<^sub>i\<close>. Arity declarations
   are implicitly completed, i.e.\ \<open>\<kappa> :: (\<^vec>s)c\<close> entails \<open>\<kappa> ::
   (\<^vec>s)c'\<close> for any \<open>c' \<supseteq> c\<close>.
 
   \<^medskip>
   The sort algebra is always maintained as \<^emph>\<open>coregular\<close>, which means that type
   arities are consistent with the subclass relation: for any type constructor
   \<open>\<kappa>\<close>, and classes \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close>, and arities \<open>\<kappa> :: (\<^vec>s\<^sub>1)c\<^sub>1\<close> and \<open>\<kappa> ::
   (\<^vec>s\<^sub>2)c\<^sub>2\<close> holds \<open>\<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2\<close> component-wise.
 
   The key property of a coregular order-sorted algebra is that sort
   constraints can be solved in a most general fashion: for each type
   constructor \<open>\<kappa>\<close> and sort \<open>s\<close> there is a most general vector of argument
   sorts \<open>(s\<^sub>1, \<dots>, s\<^sub>k)\<close> such that a type scheme \<open>(\<alpha>\<^bsub>s\<^sub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^sub>k\<^esub>)\<kappa>\<close> is of
   sort \<open>s\<close>. Consequently, type unification has most general solutions (modulo
   equivalence of sorts), so type-inference produces primary types as expected
   @{cite "nipkow-prehofer"}.
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML_type class = string} \\
   @{define_ML_type sort = "class list"} \\
   @{define_ML_type arity = "string * sort list * sort"} \\
   @{define_ML_type typ} \\
   @{define_ML Term.map_atyps: "(typ -> typ) -> typ -> typ"} \\
   @{define_ML Term.fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
   \end{mldecls}
   \begin{mldecls}
   @{define_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
   @{define_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
   @{define_ML Sign.add_type: "Proof.context -> binding * int * mixfix -> theory -> theory"} \\
   @{define_ML Sign.add_type_abbrev: "Proof.context ->
   binding * string list * typ -> theory -> theory"} \\
   @{define_ML Sign.primitive_class: "binding * class list -> theory -> theory"} \\
   @{define_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
   @{define_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
   \end{mldecls}
 
   \<^descr> Type \<^ML_type>\<open>class\<close> represents type classes.
 
   \<^descr> Type \<^ML_type>\<open>sort\<close> represents sorts, i.e.\ finite intersections of
   classes. The empty list \<^ML>\<open>[]: sort\<close> refers to the empty class
   intersection, i.e.\ the ``full sort''.
 
   \<^descr> Type \<^ML_type>\<open>arity\<close> represents type arities. A triple \<open>(\<kappa>, \<^vec>s, s)
   : arity\<close> represents \<open>\<kappa> :: (\<^vec>s)s\<close> as described above.
 
   \<^descr> Type \<^ML_type>\<open>typ\<close> represents types; this is a datatype with constructors
   \<^ML>\<open>TFree\<close>, \<^ML>\<open>TVar\<close>, \<^ML>\<open>Type\<close>.
 
   \<^descr> \<^ML>\<open>Term.map_atyps\<close>~\<open>f \<tau>\<close> applies the mapping \<open>f\<close> to all atomic types
   (\<^ML>\<open>TFree\<close>, \<^ML>\<open>TVar\<close>) occurring in \<open>\<tau>\<close>.
 
   \<^descr> \<^ML>\<open>Term.fold_atyps\<close>~\<open>f \<tau>\<close> iterates the operation \<open>f\<close> over all
   occurrences of atomic types (\<^ML>\<open>TFree\<close>, \<^ML>\<open>TVar\<close>) in \<open>\<tau>\<close>; the type
   structure is traversed from left to right.
 
   \<^descr> \<^ML>\<open>Sign.subsort\<close>~\<open>thy (s\<^sub>1, s\<^sub>2)\<close> tests the subsort relation \<open>s\<^sub>1 \<subseteq>
   s\<^sub>2\<close>.
 
   \<^descr> \<^ML>\<open>Sign.of_sort\<close>~\<open>thy (\<tau>, s)\<close> tests whether type \<open>\<tau>\<close> is of sort \<open>s\<close>.
 
   \<^descr> \<^ML>\<open>Sign.add_type\<close>~\<open>ctxt (\<kappa>, k, mx)\<close> declares a new type constructors \<open>\<kappa>\<close>
   with \<open>k\<close> arguments and optional mixfix syntax.
 
   \<^descr> \<^ML>\<open>Sign.add_type_abbrev\<close>~\<open>ctxt (\<kappa>, \<^vec>\<alpha>, \<tau>)\<close> defines a new type
   abbreviation \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close>.
 
   \<^descr> \<^ML>\<open>Sign.primitive_class\<close>~\<open>(c, [c\<^sub>1, \<dots>, c\<^sub>n])\<close> declares a new class \<open>c\<close>,
   together with class relations \<open>c \<subseteq> c\<^sub>i\<close>, for \<open>i = 1, \<dots>, n\<close>.
 
   \<^descr> \<^ML>\<open>Sign.primitive_classrel\<close>~\<open>(c\<^sub>1, c\<^sub>2)\<close> declares the class relation
   \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close>.
 
   \<^descr> \<^ML>\<open>Sign.primitive_arity\<close>~\<open>(\<kappa>, \<^vec>s, s)\<close> declares the arity \<open>\<kappa> ::
   (\<^vec>s)s\<close>.
 \<close>
 
 text %mlantiq \<open>
   \begin{matharray}{rcl}
   @{ML_antiquotation_def "class"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "sort"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "type_name"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "type_abbrev"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "nonterminal"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "typ"} & : & \<open>ML_antiquotation\<close> \\
+  @{ML_antiquotation_def "Type"} & : & \<open>ML_antiquotation\<close> \\
+  @{ML_antiquotation_def "Type_fn"} & : & \<open>ML_antiquotation\<close> \\
   \end{matharray}
 
   \<^rail>\<open>
   @@{ML_antiquotation class} embedded
   ;
   @@{ML_antiquotation sort} sort
   ;
   (@@{ML_antiquotation type_name} |
    @@{ML_antiquotation type_abbrev} |
    @@{ML_antiquotation nonterminal}) embedded
   ;
   @@{ML_antiquotation typ} type
+  ;
+  (@@{ML_antiquotation Type} | @@{ML_antiquotation Type_fn}) embedded
   \<close>
 
   \<^descr> \<open>@{class c}\<close> inlines the internalized class \<open>c\<close> --- as \<^ML_type>\<open>string\<close>
   literal.
 
   \<^descr> \<open>@{sort s}\<close> inlines the internalized sort \<open>s\<close> --- as \<^ML_type>\<open>string
   list\<close> literal.
 
   \<^descr> \<open>@{type_name c}\<close> inlines the internalized type constructor \<open>c\<close> --- as
   \<^ML_type>\<open>string\<close> literal.
 
   \<^descr> \<open>@{type_abbrev c}\<close> inlines the internalized type abbreviation \<open>c\<close> --- as
   \<^ML_type>\<open>string\<close> literal.
 
   \<^descr> \<open>@{nonterminal c}\<close> inlines the internalized syntactic type~/ grammar
   nonterminal \<open>c\<close> --- as \<^ML_type>\<open>string\<close> literal.
 
   \<^descr> \<open>@{typ \<tau>}\<close> inlines the internalized type \<open>\<tau>\<close> --- as constructor term for
   datatype \<^ML_type>\<open>typ\<close>.
+
+  \<^descr> \<open>@{Type source}\<close> refers to embedded source text to produce a type
+  constructor with name (formally checked) and arguments (inlined ML text).
+  The embedded \<open>source\<close> follows the syntax category @{syntax type_const}
+  defined below.
+
+  \<^descr> \<open>@{Type_fn source}\<close> is similar to \<open>@{Type source}\<close>, but produces a partial
+  ML function that matches against a type constructor pattern, following
+  syntax category @{syntax type_const_fn} below.
+
+
+  \<^rail>\<open>
+    @{syntax_def type_const}: @{syntax name} (@{syntax embedded_ml}*)
+    ;
+    @{syntax_def type_const_fn}: @{syntax type_const} @'=>' @{syntax embedded}
+    ;
+    @{syntax_def embedded_ml}: @{syntax embedded} |
+      @{syntax control_symbol} @{syntax embedded} | @'_'
+  \<close>
+
+  The text provided as @{syntax embedded_ml} is treated as regular Isabelle/ML
+  source, possibly with nested antiquotations. ML text that only consists of a
+  single antiquotation in compact control-cartouche notation, can be written
+  without an extra level of nesting embedded text (see the last example
+  below).
+\<close>
+
+text %mlex \<open>
+  Here are some minimal examples for type constructor antiquotations.
+\<close>
+
+ML \<open>
+  \<comment> \<open>type constructor without arguments:\<close>
+  val natT = \<^Type>\<open>nat\<close>;
+
+  \<comment> \<open>type constructor with arguments:\<close>
+  fun mk_funT (A, B) = \<^Type>\<open>fun A B\<close>;
+
+  \<comment> \<open>type constructor decomposition as partial function:\<close>
+  val dest_funT = \<^Type_fn>\<open>fun A B => \<open>(A, B)\<close>\<close>;
+
+  \<comment> \<open>nested type constructors:\<close>
+  fun mk_relT A = \<^Type>\<open>fun A \<^Type>\<open>fun A \<^Type>\<open>bool\<close>\<close>\<close>;
+
+  \<^assert> (mk_relT natT = \<^typ>\<open>nat \<Rightarrow> nat \<Rightarrow> bool\<close>);
 \<close>
 
 
 section \<open>Terms \label{sec:terms}\<close>
 
 text \<open>
   The language of terms is that of simply-typed \<open>\<lambda>\<close>-calculus with de-Bruijn
   indices for bound variables (cf.\ @{cite debruijn72} or @{cite
   "paulson-ml2"}), with the types being determined by the corresponding
   binders. In contrast, free variables and constants have an explicit name and
   type in each occurrence.
 
   \<^medskip>
   A \<^emph>\<open>bound variable\<close> is a natural number \<open>b\<close>, which accounts for the number
   of intermediate binders between the variable occurrence in the body and its
   binding position. For example, the de-Bruijn term \<open>\<lambda>\<^bsub>bool\<^esub>. \<lambda>\<^bsub>bool\<^esub>. 1 \<and> 0\<close>
   would correspond to \<open>\<lambda>x\<^bsub>bool\<^esub>. \<lambda>y\<^bsub>bool\<^esub>. x \<and> y\<close> in a named representation.
   Note that a bound variable may be represented by different de-Bruijn indices
   at different occurrences, depending on the nesting of abstractions.
 
   A \<^emph>\<open>loose variable\<close> is a bound variable that is outside the scope of local
   binders. The types (and names) for loose variables can be managed as a
   separate context, that is maintained as a stack of hypothetical binders. The
   core logic operates on closed terms, without any loose variables.
 
   A \<^emph>\<open>fixed variable\<close> is a pair of a basic name and a type, e.g.\ \<open>(x, \<tau>)\<close>
   which is usually printed \<open>x\<^sub>\<tau>\<close> here. A \<^emph>\<open>schematic variable\<close> is a pair of an
   indexname and a type, e.g.\ \<open>((x, 0), \<tau>)\<close> which is likewise printed as
   \<open>?x\<^sub>\<tau>\<close>.
 
   \<^medskip>
   A \<^emph>\<open>constant\<close> is a pair of a basic name and a type, e.g.\ \<open>(c, \<tau>)\<close> which is
   usually printed as \<open>c\<^sub>\<tau>\<close> here. Constants are declared in the context as
   polymorphic families \<open>c :: \<sigma>\<close>, meaning that all substitution instances \<open>c\<^sub>\<tau>\<close>
   for \<open>\<tau> = \<sigma>\<vartheta>\<close> are valid.
 
   The vector of \<^emph>\<open>type arguments\<close> of constant \<open>c\<^sub>\<tau>\<close> wrt.\ the declaration \<open>c
   :: \<sigma>\<close> is defined as the codomain of the matcher \<open>\<vartheta> = {?\<alpha>\<^sub>1 \<mapsto> \<tau>\<^sub>1,
   \<dots>, ?\<alpha>\<^sub>n \<mapsto> \<tau>\<^sub>n}\<close> presented in canonical order \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n)\<close>, corresponding
   to the left-to-right occurrences of the \<open>\<alpha>\<^sub>i\<close> in \<open>\<sigma>\<close>. Within a given theory
   context, there is a one-to-one correspondence between any constant \<open>c\<^sub>\<tau>\<close> and
   the application \<open>c(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n)\<close> of its type arguments. For example, with
   \<open>plus :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>\<close>, the instance \<open>plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>\<close> corresponds to
   \<open>plus(nat)\<close>.
 
   Constant declarations \<open>c :: \<sigma>\<close> may contain sort constraints for type
   variables in \<open>\<sigma>\<close>. These are observed by type-inference as expected, but
   \<^emph>\<open>ignored\<close> by the core logic. This means the primitive logic is able to
   reason with instances of polymorphic constants that the user-level
   type-checker would reject due to violation of type class restrictions.
 
   \<^medskip>
   An \<^emph>\<open>atomic term\<close> is either a variable or constant. The logical category
   \<^emph>\<open>term\<close> is defined inductively over atomic terms, with abstraction and
   application as follows: \<open>t = b | x\<^sub>\<tau> | ?x\<^sub>\<tau> | c\<^sub>\<tau> | \<lambda>\<^sub>\<tau>. t | t\<^sub>1 t\<^sub>2\<close>.
   Parsing and printing takes care of converting between an external
   representation with named bound variables. Subsequently, we shall use the
   latter notation instead of internal de-Bruijn representation.
 
   The inductive relation \<open>t :: \<tau>\<close> assigns a (unique) type to a term according
   to the structure of atomic terms, abstractions, and applications:
   \[
   \infer{\<open>a\<^sub>\<tau> :: \<tau>\<close>}{}
   \qquad
   \infer{\<open>(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>\<close>}{\<open>t :: \<sigma>\<close>}
   \qquad
   \infer{\<open>t u :: \<sigma>\<close>}{\<open>t :: \<tau> \<Rightarrow> \<sigma>\<close> & \<open>u :: \<tau>\<close>}
   \]
   A \<^emph>\<open>well-typed term\<close> is a term that can be typed according to these rules.
 
   Typing information can be omitted: type-inference is able to reconstruct the
   most general type of a raw term, while assigning most general types to all
   of its variables and constants. Type-inference depends on a context of type
   constraints for fixed variables, and declarations for polymorphic constants.
 
   The identity of atomic terms consists both of the name and the type
   component. This means that different variables \<open>x\<^bsub>\<tau>\<^sub>1\<^esub>\<close> and \<open>x\<^bsub>\<tau>\<^sub>2\<^esub>\<close> may
   become the same after type instantiation. Type-inference rejects variables
   of the same name, but different types. In contrast, mixed instances of
   polymorphic constants occur routinely.
 
   \<^medskip>
   The \<^emph>\<open>hidden polymorphism\<close> of a term \<open>t :: \<sigma>\<close> is the set of type variables
   occurring in \<open>t\<close>, but not in its type \<open>\<sigma>\<close>. This means that the term
   implicitly depends on type arguments that are not accounted in the result
   type, i.e.\ there are different type instances \<open>t\<vartheta> :: \<sigma>\<close> and
   \<open>t\<vartheta>' :: \<sigma>\<close> with the same type. This slightly pathological
   situation notoriously demands additional care.
 
   \<^medskip>
   A \<^emph>\<open>term abbreviation\<close> is a syntactic definition \<open>c\<^sub>\<sigma> \<equiv> t\<close> of a closed term
   \<open>t\<close> of type \<open>\<sigma>\<close>, without any hidden polymorphism. A term abbreviation looks
   like a constant in the syntax, but is expanded before entering the logical
   core. Abbreviations are usually reverted when printing terms, using \<open>t \<rightarrow>
   c\<^sub>\<sigma>\<close> as rules for higher-order rewriting.
 
   \<^medskip>
   Canonical operations on \<open>\<lambda>\<close>-terms include \<open>\<alpha>\<beta>\<eta>\<close>-conversion: \<open>\<alpha>\<close>-conversion
   refers to capture-free renaming of bound variables; \<open>\<beta>\<close>-conversion contracts
   an abstraction applied to an argument term, substituting the argument in the
   body: \<open>(\<lambda>x. b)a\<close> becomes \<open>b[a/x]\<close>; \<open>\<eta>\<close>-conversion contracts vacuous
   application-abstraction: \<open>\<lambda>x. f x\<close> becomes \<open>f\<close>, provided that the bound
   variable does not occur in \<open>f\<close>.
 
   Terms are normally treated modulo \<open>\<alpha>\<close>-conversion, which is implicit in the
   de-Bruijn representation. Names for bound variables in abstractions are
   maintained separately as (meaningless) comments, mostly for parsing and
   printing. Full \<open>\<alpha>\<beta>\<eta>\<close>-conversion is commonplace in various standard
   operations (\secref{sec:obj-rules}) that are based on higher-order
   unification and matching.
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML_type term} \\
   @{define_ML_infix "aconv": "term * term -> bool"} \\
   @{define_ML Term.map_types: "(typ -> typ) -> term -> term"} \\
   @{define_ML Term.fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
   @{define_ML Term.map_aterms: "(term -> term) -> term -> term"} \\
   @{define_ML Term.fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\
   \end{mldecls}
   \begin{mldecls}
   @{define_ML fastype_of: "term -> typ"} \\
   @{define_ML lambda: "term -> term -> term"} \\
   @{define_ML betapply: "term * term -> term"} \\
   @{define_ML incr_boundvars: "int -> term -> term"} \\
   @{define_ML Sign.declare_const: "Proof.context ->
   (binding * typ) * mixfix -> theory -> term * theory"} \\
   @{define_ML Sign.add_abbrev: "string -> binding * term ->
   theory -> (term * term) * theory"} \\
   @{define_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\
   @{define_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
   \end{mldecls}
 
   \<^descr> Type \<^ML_type>\<open>term\<close> represents de-Bruijn terms, with comments in
   abstractions, and explicitly named free variables and constants; this is a
   datatype with constructors @{define_ML Bound}, @{define_ML Free}, @{define_ML
   Var}, @{define_ML Const}, @{define_ML Abs}, @{define_ML_infix "$"}.
 
   \<^descr> \<open>t\<close>~\<^ML_text>\<open>aconv\<close>~\<open>u\<close> checks \<open>\<alpha>\<close>-equivalence of two terms. This is the
   basic equality relation on type \<^ML_type>\<open>term\<close>; raw datatype equality
   should only be used for operations related to parsing or printing!
 
   \<^descr> \<^ML>\<open>Term.map_types\<close>~\<open>f t\<close> applies the mapping \<open>f\<close> to all types occurring
   in \<open>t\<close>.
 
   \<^descr> \<^ML>\<open>Term.fold_types\<close>~\<open>f t\<close> iterates the operation \<open>f\<close> over all
   occurrences of types in \<open>t\<close>; the term structure is traversed from left to
   right.
 
   \<^descr> \<^ML>\<open>Term.map_aterms\<close>~\<open>f t\<close> applies the mapping \<open>f\<close> to all atomic terms
   (\<^ML>\<open>Bound\<close>, \<^ML>\<open>Free\<close>, \<^ML>\<open>Var\<close>, \<^ML>\<open>Const\<close>) occurring in \<open>t\<close>.
 
   \<^descr> \<^ML>\<open>Term.fold_aterms\<close>~\<open>f t\<close> iterates the operation \<open>f\<close> over all
   occurrences of atomic terms (\<^ML>\<open>Bound\<close>, \<^ML>\<open>Free\<close>, \<^ML>\<open>Var\<close>, \<^ML>\<open>Const\<close>) in \<open>t\<close>; the term structure is traversed from left to right.
 
   \<^descr> \<^ML>\<open>fastype_of\<close>~\<open>t\<close> determines the type of a well-typed term. This
   operation is relatively slow, despite the omission of any sanity checks.
 
   \<^descr> \<^ML>\<open>lambda\<close>~\<open>a b\<close> produces an abstraction \<open>\<lambda>a. b\<close>, where occurrences of
   the atomic term \<open>a\<close> in the body \<open>b\<close> are replaced by bound variables.
 
   \<^descr> \<^ML>\<open>betapply\<close>~\<open>(t, u)\<close> produces an application \<open>t u\<close>, with topmost
   \<open>\<beta>\<close>-conversion if \<open>t\<close> is an abstraction.
 
   \<^descr> \<^ML>\<open>incr_boundvars\<close>~\<open>j\<close> increments a term's dangling bound variables by
   the offset \<open>j\<close>. This is required when moving a subterm into a context where
   it is enclosed by a different number of abstractions. Bound variables with a
   matching abstraction are unaffected.
 
   \<^descr> \<^ML>\<open>Sign.declare_const\<close>~\<open>ctxt ((c, \<sigma>), mx)\<close> declares a new constant \<open>c ::
   \<sigma>\<close> with optional mixfix syntax.
 
   \<^descr> \<^ML>\<open>Sign.add_abbrev\<close>~\<open>print_mode (c, t)\<close> introduces a new term
   abbreviation \<open>c \<equiv> t\<close>.
 
   \<^descr> \<^ML>\<open>Sign.const_typargs\<close>~\<open>thy (c, \<tau>)\<close> and \<^ML>\<open>Sign.const_instance\<close>~\<open>thy
   (c, [\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n])\<close> convert between two representations of polymorphic
   constants: full type instance vs.\ compact type arguments form.
 \<close>
 
 text %mlantiq \<open>
   \begin{matharray}{rcl}
   @{ML_antiquotation_def "const_name"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "const_abbrev"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "const"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "term"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "prop"} & : & \<open>ML_antiquotation\<close> \\
+  @{ML_antiquotation_def "Const"} & : & \<open>ML_antiquotation\<close> \\
+  @{ML_antiquotation_def "Const_"} & : & \<open>ML_antiquotation\<close> \\
+  @{ML_antiquotation_def "Const_fn"} & : & \<open>ML_antiquotation\<close> \\
   \end{matharray}
 
   \<^rail>\<open>
   (@@{ML_antiquotation const_name} |
    @@{ML_antiquotation const_abbrev}) embedded
   ;
   @@{ML_antiquotation const} ('(' (type + ',') ')')?
   ;
   @@{ML_antiquotation term} term
   ;
   @@{ML_antiquotation prop} prop
+  ;
+  (@@{ML_antiquotation Const} | @@{ML_antiquotation Const_} | @@{ML_antiquotation Const_fn})
+    embedded
   \<close>
 
   \<^descr> \<open>@{const_name c}\<close> inlines the internalized logical constant name \<open>c\<close> ---
   as \<^ML_type>\<open>string\<close> literal.
 
   \<^descr> \<open>@{const_abbrev c}\<close> inlines the internalized abbreviated constant name \<open>c\<close>
   --- as \<^ML_type>\<open>string\<close> literal.
 
   \<^descr> \<open>@{const c(\<^vec>\<tau>)}\<close> inlines the internalized constant \<open>c\<close> with precise
   type instantiation in the sense of \<^ML>\<open>Sign.const_instance\<close> --- as \<^ML>\<open>Const\<close> constructor term for datatype \<^ML_type>\<open>term\<close>.
 
   \<^descr> \<open>@{term t}\<close> inlines the internalized term \<open>t\<close> --- as constructor term for
   datatype \<^ML_type>\<open>term\<close>.
 
   \<^descr> \<open>@{prop \<phi>}\<close> inlines the internalized proposition \<open>\<phi>\<close> --- as constructor
   term for datatype \<^ML_type>\<open>term\<close>.
+
+  \<^descr> \<open>@{Const source}\<close> refers to embedded source text to produce a term
+  constructor with name (formally checked), type arguments and term arguments
+  (inlined ML text). The embedded \<open>source\<close> follows the syntax category
+  @{syntax term_const} defined below, using @{syntax embedded_ml} from
+  antiquotation @{ML_antiquotation Type} (\secref{sec:types}).
+
+  \<^descr> \<open>@{Const_ source}\<close> is similar to \<open>@{Const source}\<close> for patterns instead of
+  closed values. This distinction is required due to redundant type
+  information within term constants: subtrees with duplicate ML pattern
+  variable need to be suppressed (replaced by dummy patterns). The embedded \<open>source\<close> follows
+  the syntax category @{syntax term_const} defined below.
+
+  \<^descr> \<open>@{Const_fn source}\<close> is similar to \<open>@{Const_ source}\<close>, but produces a
+  proper \<^verbatim>\<open>fn\<close> expression with function body. The embedded \<open>source\<close> follows
+  the syntax category @{syntax term_const_fn} defined below.
+
+
+  \<^rail>\<open>
+    @{syntax_def term_const}:
+      @{syntax name} (@{syntax embedded_ml}*) (@{syntax for_args})?
+    ;
+    @{syntax_def type_const_fn}: @{ syntax term_const} @'=>' @{syntax embedded}
+    ;
+    @{syntax_def for_args}: @'for' (@{syntax embedded_ml}*)
+  \<close>
+\<close>
+
+text %mlex \<open>
+  Here are some minimal examples for term constant antiquotations. Type
+  arguments for constants are analogous to type constructors
+  (\secref{sec:types}). Term arguments are different: a flexible number of
+  arguments is inserted via curried applications (\<^ML>\<open>op $\<close>).
+\<close>
+
+ML \<open>
+  \<comment> \<open>constant without type argument:\<close>
+  fun mk_conj (A, B) = \<^Const>\<open>conj for A B\<close>;
+  val dest_conj = \<^Const_fn>\<open>conj for A B => \<open>(A, B)\<close>\<close>;
+
+  \<comment> \<open>constant with type argument:\<close>
+  fun mk_eq T (t, u) = \<^Const>\<open>HOL.eq T for t u\<close>;
+  val dest_eq = \<^Const_fn>\<open>HOL.eq T for t u => \<open>(T, (t, u))\<close>\<close>;
+
+  \<comment> \<open>constant with variable number of term arguments:\<close>
+  val c = Const (\<^const_name>\<open>conj\<close>, \<^typ>\<open>bool \<Rightarrow> bool \<Rightarrow> bool\<close>);
+  val P = \<^term>\<open>P::bool\<close>;
+  val Q = \<^term>\<open>Q::bool\<close>;
+  \<^assert> (\<^Const>\<open>conj\<close> = c);
+  \<^assert> (\<^Const>\<open>conj for P\<close> = c $ P);
+  \<^assert> (\<^Const>\<open>conj for P Q\<close> = c $ P $ Q);
 \<close>
 
 
 section \<open>Theorems \label{sec:thms}\<close>
 
 text \<open>
   A \<^emph>\<open>proposition\<close> is a well-typed term of type \<open>prop\<close>, a \<^emph>\<open>theorem\<close> is a
   proven proposition (depending on a context of hypotheses and the background
   theory). Primitive inferences include plain Natural Deduction rules for the
   primary connectives \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> of the framework. There is also a builtin
   notion of equality/equivalence \<open>\<equiv>\<close>.
 \<close>
 
 
 subsection \<open>Primitive connectives and rules \label{sec:prim-rules}\<close>
 
 text \<open>
   The theory \<open>Pure\<close> contains constant declarations for the primitive
   connectives \<open>\<And>\<close>, \<open>\<Longrightarrow>\<close>, and \<open>\<equiv>\<close> of the logical framework, see
   \figref{fig:pure-connectives}. The derivability judgment \<open>A\<^sub>1, \<dots>, A\<^sub>n \<turnstile> B\<close>
   is defined inductively by the primitive inferences given in
   \figref{fig:prim-rules}, with the global restriction that the hypotheses
   must \<^emph>\<open>not\<close> contain any schematic variables. The builtin equality is
   conceptually axiomatized as shown in \figref{fig:pure-equality}, although
   the implementation works directly with derived inferences.
 
   \begin{figure}[htb]
   \begin{center}
   \begin{tabular}{ll}
   \<open>all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop\<close> & universal quantification (binder \<open>\<And>\<close>) \\
   \<open>\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop\<close> & implication (right associative infix) \\
   \<open>\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop\<close> & equality relation (infix) \\
   \end{tabular}
   \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
   \end{center}
   \end{figure}
 
   \begin{figure}[htb]
   \begin{center}
   \[
   \infer[\<open>(axiom)\<close>]{\<open>\<turnstile> A\<close>}{\<open>A \<in> \<Theta>\<close>}
   \qquad
   \infer[\<open>(assume)\<close>]{\<open>A \<turnstile> A\<close>}{}
   \]
   \[
   \infer[\<open>(\<And>\<hyphen>intro)\<close>]{\<open>\<Gamma> \<turnstile> \<And>x. B[x]\<close>}{\<open>\<Gamma> \<turnstile> B[x]\<close> & \<open>x \<notin> \<Gamma>\<close>}
   \qquad
   \infer[\<open>(\<And>\<hyphen>elim)\<close>]{\<open>\<Gamma> \<turnstile> B[a]\<close>}{\<open>\<Gamma> \<turnstile> \<And>x. B[x]\<close>}
   \]
   \[
   \infer[\<open>(\<Longrightarrow>\<hyphen>intro)\<close>]{\<open>\<Gamma> - A \<turnstile> A \<Longrightarrow> B\<close>}{\<open>\<Gamma> \<turnstile> B\<close>}
   \qquad
   \infer[\<open>(\<Longrightarrow>\<hyphen>elim)\<close>]{\<open>\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B\<close>}{\<open>\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B\<close> & \<open>\<Gamma>\<^sub>2 \<turnstile> A\<close>}
   \]
   \caption{Primitive inferences of Pure}\label{fig:prim-rules}
   \end{center}
   \end{figure}
 
   \begin{figure}[htb]
   \begin{center}
   \begin{tabular}{ll}
   \<open>\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]\<close> & \<open>\<beta>\<close>-conversion \\
   \<open>\<turnstile> x \<equiv> x\<close> & reflexivity \\
   \<open>\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y\<close> & substitution \\
   \<open>\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g\<close> & extensionality \\
   \<open>\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B\<close> & logical equivalence \\
   \end{tabular}
   \caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality}
   \end{center}
   \end{figure}
 
   The introduction and elimination rules for \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> are analogous to
   formation of dependently typed \<open>\<lambda>\<close>-terms representing the underlying proof
   objects. Proof terms are irrelevant in the Pure logic, though; they cannot
   occur within propositions. The system provides a runtime option to record
   explicit proof terms for primitive inferences, see also
   \secref{sec:proof-terms}. Thus all three levels of \<open>\<lambda>\<close>-calculus become
   explicit: \<open>\<Rightarrow>\<close> for terms, and \<open>\<And>/\<Longrightarrow>\<close> for proofs (cf.\ @{cite
   "Berghofer-Nipkow:2000:TPHOL"}).
 
   Observe that locally fixed parameters (as in \<open>\<And>\<hyphen>intro\<close>) need not be recorded
   in the hypotheses, because the simple syntactic types of Pure are always
   inhabitable. ``Assumptions'' \<open>x :: \<tau>\<close> for type-membership are only present
   as long as some \<open>x\<^sub>\<tau>\<close> occurs in the statement body.\<^footnote>\<open>This is the key
   difference to ``\<open>\<lambda>HOL\<close>'' in the PTS framework @{cite
   "Barendregt-Geuvers:2001"}, where hypotheses \<open>x : A\<close> are treated uniformly
   for propositions and types.\<close>
 
   \<^medskip>
   The axiomatization of a theory is implicitly closed by forming all instances
   of type and term variables: \<open>\<turnstile> A\<vartheta>\<close> holds for any substitution
   instance of an axiom \<open>\<turnstile> A\<close>. By pushing substitutions through derivations
   inductively, we also get admissible \<open>generalize\<close> and \<open>instantiate\<close> rules as
   shown in \figref{fig:subst-rules}.
 
   \begin{figure}[htb]
   \begin{center}
   \[
   \infer{\<open>\<Gamma> \<turnstile> B[?\<alpha>]\<close>}{\<open>\<Gamma> \<turnstile> B[\<alpha>]\<close> & \<open>\<alpha> \<notin> \<Gamma>\<close>}
   \quad
   \infer[\quad\<open>(generalize)\<close>]{\<open>\<Gamma> \<turnstile> B[?x]\<close>}{\<open>\<Gamma> \<turnstile> B[x]\<close> & \<open>x \<notin> \<Gamma>\<close>}
   \]
   \[
   \infer{\<open>\<Gamma> \<turnstile> B[\<tau>]\<close>}{\<open>\<Gamma> \<turnstile> B[?\<alpha>]\<close>}
   \quad
   \infer[\quad\<open>(instantiate)\<close>]{\<open>\<Gamma> \<turnstile> B[t]\<close>}{\<open>\<Gamma> \<turnstile> B[?x]\<close>}
   \]
   \caption{Admissible substitution rules}\label{fig:subst-rules}
   \end{center}
   \end{figure}
 
   Note that \<open>instantiate\<close> does not require an explicit side-condition, because
   \<open>\<Gamma>\<close> may never contain schematic variables.
 
   In principle, variables could be substituted in hypotheses as well, but this
   would disrupt the monotonicity of reasoning: deriving \<open>\<Gamma>\<vartheta> \<turnstile>
   B\<vartheta>\<close> from \<open>\<Gamma> \<turnstile> B\<close> is correct, but \<open>\<Gamma>\<vartheta> \<supseteq> \<Gamma>\<close> does not
   necessarily hold: the result belongs to a different proof context.
 
   \<^medskip> An \<^emph>\<open>oracle\<close> is a function that produces axioms on the fly. Logically,
   this is an instance of the \<open>axiom\<close> rule (\figref{fig:prim-rules}), but there
   is an operational difference. The inference kernel records oracle
   invocations within derivations of theorems by a unique tag. This also
   includes implicit type-class reasoning via the order-sorted algebra of class
   relations and type arities (see also @{command_ref instantiation} and
   @{command_ref instance}).
 
   Axiomatizations should be limited to the bare minimum, typically as part of
   the initial logical basis of an object-logic formalization. Later on,
   theories are usually developed in a strictly definitional fashion, by
   stating only certain equalities over new constants.
 
   A \<^emph>\<open>simple definition\<close> consists of a constant declaration \<open>c :: \<sigma>\<close> together
   with an axiom \<open>\<turnstile> c \<equiv> t\<close>, where \<open>t :: \<sigma>\<close> is a closed term without any hidden
   polymorphism. The RHS may depend on further defined constants, but not \<open>c\<close>
   itself. Definitions of functions may be presented as \<open>c \<^vec>x \<equiv> t\<close>
   instead of the puristic \<open>c \<equiv> \<lambda>\<^vec>x. t\<close>.
 
   An \<^emph>\<open>overloaded definition\<close> consists of a collection of axioms for the same
   constant, with zero or one equations \<open>c((\<^vec>\<alpha>)\<kappa>) \<equiv> t\<close> for each type
   constructor \<open>\<kappa>\<close> (for distinct variables \<open>\<^vec>\<alpha>\<close>). The RHS may mention
   previously defined constants as above, or arbitrary constants \<open>d(\<alpha>\<^sub>i)\<close> for
   some \<open>\<alpha>\<^sub>i\<close> projected from \<open>\<^vec>\<alpha>\<close>. Thus overloaded definitions
   essentially work by primitive recursion over the syntactic structure of a
   single type argument. See also @{cite \<open>\S4.3\<close>
   "Haftmann-Wenzel:2006:classes"}.
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML Logic.all: "term -> term -> term"} \\
   @{define_ML Logic.mk_implies: "term * term -> term"} \\
   \end{mldecls}
   \begin{mldecls}
   @{define_ML_type ctyp} \\
   @{define_ML_type cterm} \\
   @{define_ML Thm.ctyp_of: "Proof.context -> typ -> ctyp"} \\
   @{define_ML Thm.cterm_of: "Proof.context -> term -> cterm"} \\
   @{define_ML Thm.apply: "cterm -> cterm -> cterm"} \\
   @{define_ML Thm.lambda: "cterm -> cterm -> cterm"} \\
   @{define_ML Thm.all: "Proof.context -> cterm -> cterm -> cterm"} \\
   @{define_ML Drule.mk_implies: "cterm * cterm -> cterm"} \\
   \end{mldecls}
   \begin{mldecls}
   @{define_ML_type thm} \\
   @{define_ML Thm.transfer: "theory -> thm -> thm"} \\
   @{define_ML Thm.assume: "cterm -> thm"} \\
   @{define_ML Thm.forall_intr: "cterm -> thm -> thm"} \\
   @{define_ML Thm.forall_elim: "cterm -> thm -> thm"} \\
   @{define_ML Thm.implies_intr: "cterm -> thm -> thm"} \\
   @{define_ML Thm.implies_elim: "thm -> thm -> thm"} \\
   @{define_ML Thm.generalize: "Names.set * Names.set -> int -> thm -> thm"} \\
   @{define_ML Thm.instantiate: "ctyp TVars.table * cterm Vars.table -> thm -> thm"} \\
   @{define_ML Thm.add_axiom: "Proof.context ->
   binding * term -> theory -> (string * thm) * theory"} \\
   @{define_ML Thm.add_oracle: "binding * ('a -> cterm) -> theory ->
   (string * ('a -> thm)) * theory"} \\
   @{define_ML Thm.add_def: "Defs.context -> bool -> bool ->
   binding * term -> theory -> (string * thm) * theory"} \\
   \end{mldecls}
   \begin{mldecls}
   @{define_ML Theory.add_deps: "Defs.context -> string ->
   Defs.entry -> Defs.entry list -> theory -> theory"} \\
   @{define_ML Thm_Deps.all_oracles: "thm list -> Proofterm.oracle list"} \\
   \end{mldecls}
 
   \<^descr> \<^ML>\<open>Logic.all\<close>~\<open>a B\<close> produces a Pure quantification \<open>\<And>a. B\<close>, where
   occurrences of the atomic term \<open>a\<close> in the body proposition \<open>B\<close> are replaced
   by bound variables. (See also \<^ML>\<open>lambda\<close> on terms.)
 
   \<^descr> \<^ML>\<open>Logic.mk_implies\<close>~\<open>(A, B)\<close> produces a Pure implication \<open>A \<Longrightarrow> B\<close>.
 
   \<^descr> Types \<^ML_type>\<open>ctyp\<close> and \<^ML_type>\<open>cterm\<close> represent certified types and
   terms, respectively. These are abstract datatypes that guarantee that its
   values have passed the full well-formedness (and well-typedness) checks,
   relative to the declarations of type constructors, constants etc.\ in the
   background theory. The abstract types \<^ML_type>\<open>ctyp\<close> and \<^ML_type>\<open>cterm\<close>
   are part of the same inference kernel that is mainly responsible for
   \<^ML_type>\<open>thm\<close>. Thus syntactic operations on \<^ML_type>\<open>ctyp\<close> and \<^ML_type>\<open>cterm\<close> are located in the \<^ML_structure>\<open>Thm\<close> module, even though theorems
   are not yet involved at that stage.
 
   \<^descr> \<^ML>\<open>Thm.ctyp_of\<close>~\<open>ctxt \<tau>\<close> and \<^ML>\<open>Thm.cterm_of\<close>~\<open>ctxt t\<close> explicitly
   check types and terms, respectively. This also involves some basic
   normalizations, such expansion of type and term abbreviations from the
   underlying theory context. Full re-certification is relatively slow and
   should be avoided in tight reasoning loops.
 
   \<^descr> \<^ML>\<open>Thm.apply\<close>, \<^ML>\<open>Thm.lambda\<close>, \<^ML>\<open>Thm.all\<close>, \<^ML>\<open>Drule.mk_implies\<close>
   etc.\ compose certified terms (or propositions) incrementally. This is
   equivalent to \<^ML>\<open>Thm.cterm_of\<close> after unchecked \<^ML_infix>\<open>$\<close>, \<^ML>\<open>lambda\<close>,
   \<^ML>\<open>Logic.all\<close>, \<^ML>\<open>Logic.mk_implies\<close> etc., but there can be a big
   difference in performance when large existing entities are composed by a few
   extra constructions on top. There are separate operations to decompose
   certified terms and theorems to produce certified terms again.
 
   \<^descr> Type \<^ML_type>\<open>thm\<close> represents proven propositions. This is an abstract
   datatype that guarantees that its values have been constructed by basic
   principles of the \<^ML_structure>\<open>Thm\<close> module. Every \<^ML_type>\<open>thm\<close> value
   refers its background theory, cf.\ \secref{sec:context-theory}.
 
   \<^descr> \<^ML>\<open>Thm.transfer\<close>~\<open>thy thm\<close> transfers the given theorem to a \<^emph>\<open>larger\<close>
   theory, see also \secref{sec:context}. This formal adjustment of the
   background context has no logical significance, but is occasionally required
   for formal reasons, e.g.\ when theorems that are imported from more basic
   theories are used in the current situation.
 
   \<^descr> \<^ML>\<open>Thm.assume\<close>, \<^ML>\<open>Thm.forall_intr\<close>, \<^ML>\<open>Thm.forall_elim\<close>, \<^ML>\<open>Thm.implies_intr\<close>, and \<^ML>\<open>Thm.implies_elim\<close> correspond to the primitive
   inferences of \figref{fig:prim-rules}.
 
   \<^descr> \<^ML>\<open>Thm.generalize\<close>~\<open>(\<^vec>\<alpha>, \<^vec>x)\<close> corresponds to the
   \<open>generalize\<close> rules of \figref{fig:subst-rules}. Here collections of type and
   term variables are generalized simultaneously, specified by the given sets of
   basic names.
 
   \<^descr> \<^ML>\<open>Thm.instantiate\<close>~\<open>(\<^vec>\<alpha>\<^sub>s, \<^vec>x\<^sub>\<tau>)\<close> corresponds to the
   \<open>instantiate\<close> rules of \figref{fig:subst-rules}. Type variables are
   substituted before term variables. Note that the types in \<open>\<^vec>x\<^sub>\<tau>\<close> refer
   to the instantiated versions.
 
   \<^descr> \<^ML>\<open>Thm.add_axiom\<close>~\<open>ctxt (name, A)\<close> declares an arbitrary proposition as
   axiom, and retrieves it as a theorem from the resulting theory, cf.\ \<open>axiom\<close>
   in \figref{fig:prim-rules}. Note that the low-level representation in the
   axiom table may differ slightly from the returned theorem.
 
   \<^descr> \<^ML>\<open>Thm.add_oracle\<close>~\<open>(binding, oracle)\<close> produces a named oracle rule,
   essentially generating arbitrary axioms on the fly, cf.\ \<open>axiom\<close> in
   \figref{fig:prim-rules}.
 
   \<^descr> \<^ML>\<open>Thm.add_def\<close>~\<open>ctxt unchecked overloaded (name, c \<^vec>x \<equiv> t)\<close>
   states a definitional axiom for an existing constant \<open>c\<close>. Dependencies are
   recorded via \<^ML>\<open>Theory.add_deps\<close>, unless the \<open>unchecked\<close> option is set.
   Note that the low-level representation in the axiom table may differ
   slightly from the returned theorem.
 
   \<^descr> \<^ML>\<open>Theory.add_deps\<close>~\<open>ctxt name c\<^sub>\<tau> \<^vec>d\<^sub>\<sigma>\<close> declares dependencies of
   a named specification for constant \<open>c\<^sub>\<tau>\<close>, relative to existing
   specifications for constants \<open>\<^vec>d\<^sub>\<sigma>\<close>. This also works for type
   constructors.
 
   \<^descr> \<^ML>\<open>Thm_Deps.all_oracles\<close>~\<open>thms\<close> returns all oracles used in the
   internal derivation of the given theorems; this covers the full graph of
   transitive dependencies. The result contains an authentic oracle name; if
   @{ML Proofterm.proofs} was at least @{ML 1} during the oracle inference, it
   also contains the position of the oracle invocation and its proposition. See
   also the command @{command_ref "thm_oracles"}.
 \<close>
 
 text %mlantiq \<open>
   \begin{matharray}{rcl}
   @{ML_antiquotation_def "ctyp"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "cterm"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "cprop"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "thm"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "thms"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "lemma"} & : & \<open>ML_antiquotation\<close> \\
   @{ML_antiquotation_def "oracle_name"} & : & \<open>ML_antiquotation\<close> \\
   \end{matharray}
 
   \<^rail>\<open>
   @@{ML_antiquotation ctyp} typ
   ;
   @@{ML_antiquotation cterm} term
   ;
   @@{ML_antiquotation cprop} prop
   ;
   @@{ML_antiquotation thm} thm
   ;
   @@{ML_antiquotation thms} thms
   ;
   @@{ML_antiquotation lemma} ('(' @'open' ')')? ((prop +) + @'and') \<newline>
     @{syntax for_fixes} @'by' method method?
   ;
   @@{ML_antiquotation oracle_name} embedded
   \<close>
 
   \<^descr> \<open>@{ctyp \<tau>}\<close> produces a certified type wrt.\ the current background theory
   --- as abstract value of type \<^ML_type>\<open>ctyp\<close>.
 
   \<^descr> \<open>@{cterm t}\<close> and \<open>@{cprop \<phi>}\<close> produce a certified term wrt.\ the current
   background theory --- as abstract value of type \<^ML_type>\<open>cterm\<close>.
 
   \<^descr> \<open>@{thm a}\<close> produces a singleton fact --- as abstract value of type
   \<^ML_type>\<open>thm\<close>.
 
   \<^descr> \<open>@{thms a}\<close> produces a general fact --- as abstract value of type
   \<^ML_type>\<open>thm list\<close>.
 
   \<^descr> \<open>@{lemma \<phi> by meth}\<close> produces a fact that is proven on the spot according
   to the minimal proof, which imitates a terminal Isar proof. The result is an
   abstract value of type \<^ML_type>\<open>thm\<close> or \<^ML_type>\<open>thm list\<close>, depending on
   the number of propositions given here.
 
   The internal derivation object lacks a proper theorem name, but it is
   formally closed, unless the \<open>(open)\<close> option is specified (this may impact
   performance of applications with proof terms).
 
   Since ML antiquotations are always evaluated at compile-time, there is no
   run-time overhead even for non-trivial proofs. Nonetheless, the
   justification is syntactically limited to a single @{command "by"} step.
   More complex Isar proofs should be done in regular theory source, before
   compiling the corresponding ML text that uses the result.
 
   \<^descr> \<open>@{oracle_name a}\<close> inlines the internalized oracle name \<open>a\<close> --- as
   \<^ML_type>\<open>string\<close> literal.
 \<close>
 
 
 subsection \<open>Auxiliary connectives \label{sec:logic-aux}\<close>
 
 text \<open>
   Theory \<open>Pure\<close> provides a few auxiliary connectives that are defined on top
   of the primitive ones, see \figref{fig:pure-aux}. These special constants
   are useful in certain internal encodings, and are normally not directly
   exposed to the user.
 
   \begin{figure}[htb]
   \begin{center}
   \begin{tabular}{ll}
   \<open>conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop\<close> & (infix \<open>&&&\<close>) \\
   \<open>\<turnstile> A &&& B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)\<close> \\[1ex]
   \<open>prop :: prop \<Rightarrow> prop\<close> & (prefix \<open>#\<close>, suppressed) \\
   \<open>#A \<equiv> A\<close> \\[1ex]
   \<open>term :: \<alpha> \<Rightarrow> prop\<close> & (prefix \<open>TERM\<close>) \\
   \<open>term x \<equiv> (\<And>A. A \<Longrightarrow> A)\<close> \\[1ex]
   \<open>type :: \<alpha> itself\<close> & (prefix \<open>TYPE\<close>) \\
   \<open>(unspecified)\<close> \\
   \end{tabular}
   \caption{Definitions of auxiliary connectives}\label{fig:pure-aux}
   \end{center}
   \end{figure}
 
   The introduction \<open>A \<Longrightarrow> B \<Longrightarrow> A &&& B\<close>, and eliminations (projections) \<open>A &&& B
   \<Longrightarrow> A\<close> and \<open>A &&& B \<Longrightarrow> B\<close> are available as derived rules. Conjunction allows to
   treat simultaneous assumptions and conclusions uniformly, e.g.\ consider \<open>A
   \<Longrightarrow> B \<Longrightarrow> C &&& D\<close>. In particular, the goal mechanism represents multiple claims
   as explicit conjunction internally, but this is refined (via backwards
   introduction) into separate sub-goals before the user commences the proof;
   the final result is projected into a list of theorems using eliminations
   (cf.\ \secref{sec:tactical-goals}).
 
   The \<open>prop\<close> marker (\<open>#\<close>) makes arbitrarily complex propositions appear as
   atomic, without changing the meaning: \<open>\<Gamma> \<turnstile> A\<close> and \<open>\<Gamma> \<turnstile> #A\<close> are
   interchangeable. See \secref{sec:tactical-goals} for specific operations.
 
   The \<open>term\<close> marker turns any well-typed term into a derivable proposition: \<open>\<turnstile>
   TERM t\<close> holds unconditionally. Although this is logically vacuous, it allows
   to treat terms and proofs uniformly, similar to a type-theoretic framework.
 
   The \<open>TYPE\<close> constructor is the canonical representative of the unspecified
   type \<open>\<alpha> itself\<close>; it essentially injects the language of types into that of
   terms. There is specific notation \<open>TYPE(\<tau>)\<close> for \<open>TYPE\<^bsub>\<tau> itself\<^esub>\<close>. Although
   being devoid of any particular meaning, the term \<open>TYPE(\<tau>)\<close> accounts for the
   type \<open>\<tau>\<close> within the term language. In particular, \<open>TYPE(\<alpha>)\<close> may be used as
   formal argument in primitive definitions, in order to circumvent hidden
   polymorphism (cf.\ \secref{sec:terms}). For example, \<open>c TYPE(\<alpha>) \<equiv> A[\<alpha>]\<close>
   defines \<open>c :: \<alpha> itself \<Rightarrow> prop\<close> in terms of a proposition \<open>A\<close> that depends on
   an additional type argument, which is essentially a predicate on types.
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML Conjunction.intr: "thm -> thm -> thm"} \\
   @{define_ML Conjunction.elim: "thm -> thm * thm"} \\
   @{define_ML Drule.mk_term: "cterm -> thm"} \\
   @{define_ML Drule.dest_term: "thm -> cterm"} \\
   @{define_ML Logic.mk_type: "typ -> term"} \\
   @{define_ML Logic.dest_type: "term -> typ"} \\
   \end{mldecls}
 
   \<^descr> \<^ML>\<open>Conjunction.intr\<close> derives \<open>A &&& B\<close> from \<open>A\<close> and \<open>B\<close>.
 
   \<^descr> \<^ML>\<open>Conjunction.elim\<close> derives \<open>A\<close> and \<open>B\<close> from \<open>A &&& B\<close>.
 
   \<^descr> \<^ML>\<open>Drule.mk_term\<close> derives \<open>TERM t\<close>.
 
   \<^descr> \<^ML>\<open>Drule.dest_term\<close> recovers term \<open>t\<close> from \<open>TERM t\<close>.
 
   \<^descr> \<^ML>\<open>Logic.mk_type\<close>~\<open>\<tau>\<close> produces the term \<open>TYPE(\<tau>)\<close>.
 
   \<^descr> \<^ML>\<open>Logic.dest_type\<close>~\<open>TYPE(\<tau>)\<close> recovers the type \<open>\<tau>\<close>.
 \<close>
 
 
 subsection \<open>Sort hypotheses\<close>
 
 text \<open>
   Type variables are decorated with sorts, as explained in \secref{sec:types}.
   This constrains type instantiation to certain ranges of types: variable
   \<open>\<alpha>\<^sub>s\<close> may only be assigned to types \<open>\<tau>\<close> that belong to sort \<open>s\<close>. Within the
   logic, sort constraints act like implicit preconditions on the result \<open>\<lparr>\<alpha>\<^sub>1
   : s\<^sub>1\<rparr>, \<dots>, \<lparr>\<alpha>\<^sub>n : s\<^sub>n\<rparr>, \<Gamma> \<turnstile> \<phi>\<close> where the type variables \<open>\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n\<close> cover
   the propositions \<open>\<Gamma>\<close>, \<open>\<phi>\<close>, as well as the proof of \<open>\<Gamma> \<turnstile> \<phi>\<close>.
 
   These \<^emph>\<open>sort hypothesis\<close> of a theorem are passed monotonically through
   further derivations. They are redundant, as long as the statement of a
   theorem still contains the type variables that are accounted here. The
   logical significance of sort hypotheses is limited to the boundary case
   where type variables disappear from the proposition, e.g.\ \<open>\<lparr>\<alpha>\<^sub>s : s\<rparr> \<turnstile> \<phi>\<close>.
   Since such dangling type variables can be renamed arbitrarily without
   changing the proposition \<open>\<phi>\<close>, the inference kernel maintains sort hypotheses
   in anonymous form \<open>s \<turnstile> \<phi>\<close>.
 
   In most practical situations, such extra sort hypotheses may be stripped in
   a final bookkeeping step, e.g.\ at the end of a proof: they are typically
   left over from intermediate reasoning with type classes that can be
   satisfied by some concrete type \<open>\<tau>\<close> of sort \<open>s\<close> to replace the hypothetical
   type variable \<open>\<alpha>\<^sub>s\<close>.
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML Thm.extra_shyps: "thm -> sort list"} \\
   @{define_ML Thm.strip_shyps: "thm -> thm"} \\
   \end{mldecls}
 
   \<^descr> \<^ML>\<open>Thm.extra_shyps\<close>~\<open>thm\<close> determines the extraneous sort hypotheses of
   the given theorem, i.e.\ the sorts that are not present within type
   variables of the statement.
 
   \<^descr> \<^ML>\<open>Thm.strip_shyps\<close>~\<open>thm\<close> removes any extraneous sort hypotheses that
   can be witnessed from the type signature.
 \<close>
 
 text %mlex \<open>
   The following artificial example demonstrates the derivation of \<^prop>\<open>False\<close> with a pending sort hypothesis involving a logically empty sort.
 \<close>
 
 class empty =
   assumes bad: "\<And>(x::'a) y. x \<noteq> y"
 
 theorem (in empty) false: False
   using bad by blast
 
 ML_val \<open>\<^assert> (Thm.extra_shyps @{thm false} = [\<^sort>\<open>empty\<close>])\<close>
 
 text \<open>
   Thanks to the inference kernel managing sort hypothesis according to their
   logical significance, this example is merely an instance of \<^emph>\<open>ex falso
   quodlibet consequitur\<close> --- not a collapse of the logical framework!
 \<close>
 
 
 section \<open>Object-level rules \label{sec:obj-rules}\<close>
 
 text \<open>
   The primitive inferences covered so far mostly serve foundational purposes.
   User-level reasoning usually works via object-level rules that are
   represented as theorems of Pure. Composition of rules involves
   \<^emph>\<open>backchaining\<close>, \<^emph>\<open>higher-order unification\<close> modulo \<open>\<alpha>\<beta>\<eta>\<close>-conversion of
   \<open>\<lambda>\<close>-terms, and so-called \<^emph>\<open>lifting\<close> of rules into a context of \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close>
   connectives. Thus the full power of higher-order Natural Deduction in
   Isabelle/Pure becomes readily available.
 \<close>
 
 
 subsection \<open>Hereditary Harrop Formulae\<close>
 
 text \<open>
   The idea of object-level rules is to model Natural Deduction inferences in
   the style of Gentzen @{cite "Gentzen:1935"}, but we allow arbitrary nesting
   similar to @{cite "Schroeder-Heister:1984"}. The most basic rule format is
   that of a \<^emph>\<open>Horn Clause\<close>:
   \[
   \infer{\<open>A\<close>}{\<open>A\<^sub>1\<close> & \<open>\<dots>\<close> & \<open>A\<^sub>n\<close>}
   \]
   where \<open>A, A\<^sub>1, \<dots>, A\<^sub>n\<close> are atomic propositions of the framework, usually of
   the form \<open>Trueprop B\<close>, where \<open>B\<close> is a (compound) object-level statement.
   This object-level inference corresponds to an iterated implication in Pure
   like this:
   \[
   \<open>A\<^sub>1 \<Longrightarrow> \<dots> A\<^sub>n \<Longrightarrow> A\<close>
   \]
   As an example consider conjunction introduction: \<open>A \<Longrightarrow> B \<Longrightarrow> A \<and> B\<close>. Any
   parameters occurring in such rule statements are conceptionally treated as
   arbitrary:
   \[
   \<open>\<And>x\<^sub>1 \<dots> x\<^sub>m. A\<^sub>1 x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> \<dots> A\<^sub>n x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> A x\<^sub>1 \<dots> x\<^sub>m\<close>
   \]
 
   Nesting of rules means that the positions of \<open>A\<^sub>i\<close> may again hold compound
   rules, not just atomic propositions. Propositions of this format are called
   \<^emph>\<open>Hereditary Harrop Formulae\<close> in the literature @{cite "Miller:1991"}. Here
   we give an inductive characterization as follows:
 
   \<^medskip>
   \begin{tabular}{ll}
   \<open>\<^bold>x\<close> & set of variables \\
   \<open>\<^bold>A\<close> & set of atomic propositions \\
   \<open>\<^bold>H  =  \<And>\<^bold>x\<^sup>*. \<^bold>H\<^sup>* \<Longrightarrow> \<^bold>A\<close> & set of Hereditary Harrop Formulas \\
   \end{tabular}
   \<^medskip>
 
   Thus we essentially impose nesting levels on propositions formed from \<open>\<And>\<close>
   and \<open>\<Longrightarrow>\<close>. At each level there is a prefix of parameters and compound
   premises, concluding an atomic proposition. Typical examples are
   \<open>\<longrightarrow>\<close>-introduction \<open>(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B\<close> or mathematical induction \<open>P 0 \<Longrightarrow> (\<And>n. P n
   \<Longrightarrow> P (Suc n)) \<Longrightarrow> P n\<close>. Even deeper nesting occurs in well-founded induction
   \<open>(\<And>x. (\<And>y. y \<prec> x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x\<close>, but this already marks the limit of
   rule complexity that is usually seen in practice.
 
   \<^medskip>
   Regular user-level inferences in Isabelle/Pure always maintain the following
   canonical form of results:
 
   \<^item> Normalization by \<open>(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)\<close>, which is a theorem of
   Pure, means that quantifiers are pushed in front of implication at each
   level of nesting. The normal form is a Hereditary Harrop Formula.
 
   \<^item> The outermost prefix of parameters is represented via schematic variables:
   instead of \<open>\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x\<close> we have \<open>\<^vec>H
   ?\<^vec>x \<Longrightarrow> A ?\<^vec>x\<close>. Note that this representation looses information
   about the order of parameters, and vacuous quantifiers vanish automatically.
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML Simplifier.norm_hhf: "Proof.context -> thm -> thm"} \\
   \end{mldecls}
 
   \<^descr> \<^ML>\<open>Simplifier.norm_hhf\<close>~\<open>ctxt thm\<close> normalizes the given theorem
   according to the canonical form specified above. This is occasionally
   helpful to repair some low-level tools that do not handle Hereditary Harrop
   Formulae properly.
 \<close>
 
 
 subsection \<open>Rule composition\<close>
 
 text \<open>
   The rule calculus of Isabelle/Pure provides two main inferences: @{inference
   resolution} (i.e.\ back-chaining of rules) and @{inference assumption}
   (i.e.\ closing a branch), both modulo higher-order unification. There are
   also combined variants, notably @{inference elim_resolution} and @{inference
   dest_resolution}.
 
   To understand the all-important @{inference resolution} principle, we first
   consider raw @{inference_def composition} (modulo higher-order unification
   with substitution \<open>\<vartheta>\<close>):
   \[
   \infer[(@{inference_def composition})]{\<open>\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>\<close>}
   {\<open>\<^vec>A \<Longrightarrow> B\<close> & \<open>B' \<Longrightarrow> C\<close> & \<open>B\<vartheta> = B'\<vartheta>\<close>}
   \]
   Here the conclusion of the first rule is unified with the premise of the
   second; the resulting rule instance inherits the premises of the first and
   conclusion of the second. Note that \<open>C\<close> can again consist of iterated
   implications. We can also permute the premises of the second rule
   back-and-forth in order to compose with \<open>B'\<close> in any position (subsequently
   we shall always refer to position 1 w.l.o.g.).
 
   In @{inference composition} the internal structure of the common part \<open>B\<close>
   and \<open>B'\<close> is not taken into account. For proper @{inference resolution} we
   require \<open>B\<close> to be atomic, and explicitly observe the structure \<open>\<And>\<^vec>x.
   \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x\<close> of the premise of the second rule. The idea
   is to adapt the first rule by ``lifting'' it into this context, by means of
   iterated application of the following inferences:
   \[
   \infer[(@{inference_def imp_lift})]{\<open>(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)\<close>}{\<open>\<^vec>A \<Longrightarrow> B\<close>}
   \]
   \[
   \infer[(@{inference_def all_lift})]{\<open>(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))\<close>}{\<open>\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a\<close>}
   \]
   By combining raw composition with lifting, we get full @{inference
   resolution} as follows:
   \[
   \infer[(@{inference_def resolution})]
   {\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>\<close>}
   {\begin{tabular}{l}
     \<open>\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a\<close> \\
     \<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C\<close> \\
     \<open>(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>\<close> \\
    \end{tabular}}
   \]
 
   Continued resolution of rules allows to back-chain a problem towards more
   and sub-problems. Branches are closed either by resolving with a rule of 0
   premises, or by producing a ``short-circuit'' within a solved situation
   (again modulo unification):
   \[
   \infer[(@{inference_def assumption})]{\<open>C\<vartheta>\<close>}
   {\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C\<close> & \<open>A\<vartheta> = H\<^sub>i\<vartheta>\<close>~~\mbox{(for some~\<open>i\<close>)}}
   \]
 
   %FIXME @{inference_def elim_resolution}, @{inference_def dest_resolution}
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML_infix "RSN": "thm * (int * thm) -> thm"} \\
   @{define_ML_infix "RS": "thm * thm -> thm"} \\
 
   @{define_ML_infix "RLN": "thm list * (int * thm list) -> thm list"} \\
   @{define_ML_infix "RL": "thm list * thm list -> thm list"} \\
 
   @{define_ML_infix "MRS": "thm list * thm -> thm"} \\
   @{define_ML_infix "OF": "thm * thm list -> thm"} \\
   \end{mldecls}
 
   \<^descr> \<open>rule\<^sub>1 RSN (i, rule\<^sub>2)\<close> resolves the conclusion of \<open>rule\<^sub>1\<close> with the
   \<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>, according to the @{inference resolution}
   principle explained above. Unless there is precisely one resolvent it raises
   exception \<^ML>\<open>THM\<close>.
 
   This corresponds to the rule attribute @{attribute THEN} in Isar source
   language.
 
   \<^descr> \<open>rule\<^sub>1 RS rule\<^sub>2\<close> abbreviates \<open>rule\<^sub>1 RSN (1, rule\<^sub>2)\<close>.
 
   \<^descr> \<open>rules\<^sub>1 RLN (i, rules\<^sub>2)\<close> joins lists of rules. For every \<open>rule\<^sub>1\<close> in
   \<open>rules\<^sub>1\<close> and \<open>rule\<^sub>2\<close> in \<open>rules\<^sub>2\<close>, it resolves the conclusion of \<open>rule\<^sub>1\<close>
   with the \<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>, accumulating multiple results in one
   big list. Note that such strict enumerations of higher-order unifications
   can be inefficient compared to the lazy variant seen in elementary tactics
   like \<^ML>\<open>resolve_tac\<close>.
 
   \<^descr> \<open>rules\<^sub>1 RL rules\<^sub>2\<close> abbreviates \<open>rules\<^sub>1 RLN (1, rules\<^sub>2)\<close>.
 
   \<^descr> \<open>[rule\<^sub>1, \<dots>, rule\<^sub>n] MRS rule\<close> resolves \<open>rule\<^sub>i\<close> against premise \<open>i\<close> of
   \<open>rule\<close>, for \<open>i = n, \<dots>, 1\<close>. By working from right to left, newly emerging
   premises are concatenated in the result, without interfering.
 
   \<^descr> \<open>rule OF rules\<close> is an alternative notation for \<open>rules MRS rule\<close>, which
   makes rule composition look more like function application. Note that the
   argument \<open>rules\<close> need not be atomic.
 
   This corresponds to the rule attribute @{attribute OF} in Isar source
   language.
 \<close>
 
 
 section \<open>Proof terms \label{sec:proof-terms}\<close>
 
 text \<open>
   The Isabelle/Pure inference kernel can record the proof of each theorem as a
   proof term that contains all logical inferences in detail. Rule composition
   by resolution (\secref{sec:obj-rules}) and type-class reasoning is broken
   down to primitive rules of the logical framework. The proof term can be
   inspected by a separate proof-checker, for example.
 
   According to the well-known \<^emph>\<open>Curry-Howard isomorphism\<close>, a proof can be
   viewed as a \<open>\<lambda>\<close>-term. Following this idea, proofs in Isabelle are internally
   represented by a datatype similar to the one for terms described in
   \secref{sec:terms}. On top of these syntactic terms, two more layers of
   \<open>\<lambda>\<close>-calculus are added, which correspond to \<open>\<And>x :: \<alpha>. B x\<close> and \<open>A \<Longrightarrow> B\<close>
   according to the propositions-as-types principle. The resulting 3-level
   \<open>\<lambda>\<close>-calculus resembles ``\<open>\<lambda>HOL\<close>'' in the more abstract setting of Pure Type
   Systems (PTS) @{cite "Barendregt-Geuvers:2001"}, if some fine points like
   schematic polymorphism and type classes are ignored.
 
   \<^medskip>
   \<^emph>\<open>Proof abstractions\<close> of the form \<open>\<^bold>\<lambda>x :: \<alpha>. prf\<close> or \<open>\<^bold>\<lambda>p : A. prf\<close>
   correspond to introduction of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>, and \<^emph>\<open>proof applications\<close> of the form
   \<open>p \<cdot> t\<close> or \<open>p \<bullet> q\<close> correspond to elimination of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>. Actual types \<open>\<alpha>\<close>,
   propositions \<open>A\<close>, and terms \<open>t\<close> might be suppressed and reconstructed from
   the overall proof term.
 
   \<^medskip>
   Various atomic proofs indicate special situations within the proof
   construction as follows.
 
   A \<^emph>\<open>bound proof variable\<close> is a natural number \<open>b\<close> that acts as de-Bruijn
   index for proof term abstractions.
 
   A \<^emph>\<open>minimal proof\<close> ``\<open>?\<close>'' is a dummy proof term. This indicates some
   unrecorded part of the proof.
 
   \<open>Hyp A\<close> refers to some pending hypothesis by giving its proposition. This
   indicates an open context of implicit hypotheses, similar to loose bound
   variables or free variables within a term (\secref{sec:terms}).
 
   An \<^emph>\<open>axiom\<close> or \<^emph>\<open>oracle\<close> \<open>a : A[\<^vec>\<tau>]\<close> refers some postulated \<open>proof
   constant\<close>, which is subject to schematic polymorphism of theory content, and
   the particular type instantiation may be given explicitly. The vector of
   types \<open>\<^vec>\<tau>\<close> refers to the schematic type variables in the generic
   proposition \<open>A\<close> in canonical order.
 
   A \<^emph>\<open>proof promise\<close> \<open>a : A[\<^vec>\<tau>]\<close> is a placeholder for some proof of
   polymorphic proposition \<open>A\<close>, with explicit type instantiation as given by
   the vector \<open>\<^vec>\<tau>\<close>, as above. Unlike axioms or oracles, proof promises
   may be \<^emph>\<open>fulfilled\<close> eventually, by substituting \<open>a\<close> by some particular proof
   \<open>q\<close> at the corresponding type instance. This acts like Hindley-Milner
   \<open>let\<close>-polymorphism: a generic local proof definition may get used at
   different type instances, and is replaced by the concrete instance
   eventually.
 
   A \<^emph>\<open>named theorem\<close> wraps up some concrete proof as a closed formal entity,
   in the manner of constant definitions for proof terms. The \<^emph>\<open>proof body\<close> of
   such boxed theorems involves some digest about oracles and promises
   occurring in the original proof. This allows the inference kernel to manage
   this critical information without the full overhead of explicit proof terms.
 \<close>
 
 
 subsection \<open>Reconstructing and checking proof terms\<close>
 
 text \<open>
   Fully explicit proof terms can be large, but most of this information is
   redundant and can be reconstructed from the context. Therefore, the
   Isabelle/Pure inference kernel records only \<^emph>\<open>implicit\<close> proof terms, by
   omitting all typing information in terms, all term and type labels of proof
   abstractions, and some argument terms of applications \<open>p \<cdot> t\<close> (if possible).
 
   There are separate operations to reconstruct the full proof term later on,
   using \<^emph>\<open>higher-order pattern unification\<close> @{cite "nipkow-patterns" and
   "Berghofer-Nipkow:2000:TPHOL"}.
 
   The \<^emph>\<open>proof checker\<close> expects a fully reconstructed proof term, and can turn
   it into a theorem by replaying its primitive inferences within the kernel.
 \<close>
 
 
 subsection \<open>Concrete syntax of proof terms\<close>
 
 text \<open>
   The concrete syntax of proof terms is a slight extension of the regular
   inner syntax of Isabelle/Pure @{cite "isabelle-isar-ref"}. Its main
   syntactic category @{syntax (inner) proof} is defined as follows:
 
   \begin{center}
   \begin{supertabular}{rclr}
 
   @{syntax_def (inner) proof} & = & \<open>\<^bold>\<lambda>\<close> \<open>params\<close> \<^verbatim>\<open>.\<close> \<open>proof\<close> \\
     & \<open>|\<close> & \<open>proof\<close> \<open>\<cdot>\<close> \<open>any\<close> \\
     & \<open>|\<close> & \<open>proof\<close> \<open>\<bullet>\<close> \<open>proof\<close> \\
     & \<open>|\<close> & \<open>id  |  longid\<close> \\
   \\
 
   \<open>param\<close> & = & \<open>idt\<close> \\
     & \<open>|\<close> & \<open>idt\<close> \<^verbatim>\<open>:\<close> \<open>prop\<close> \\
     & \<open>|\<close> & \<^verbatim>\<open>(\<close> \<open>param\<close> \<^verbatim>\<open>)\<close> \\
   \\
 
   \<open>params\<close> & = & \<open>param\<close> \\
     & \<open>|\<close> & \<open>param\<close> \<open>params\<close> \\
 
   \end{supertabular}
   \end{center}
 
   Implicit term arguments in partial proofs are indicated by ``\<open>_\<close>''. Type
   arguments for theorems and axioms may be specified using \<open>p \<cdot> TYPE(type)\<close>
   (they must appear before any other term argument of a theorem or axiom, but
   may be omitted altogether).
 
   \<^medskip>
   There are separate read and print operations for proof terms, in order to
   avoid conflicts with the regular term language.
 \<close>
 
 text %mlref \<open>
   \begin{mldecls}
   @{define_ML_type proof} \\
   @{define_ML_type proof_body} \\
   @{define_ML Proofterm.proofs: "int Unsynchronized.ref"} \\
   @{define_ML Proofterm.reconstruct_proof:
   "theory -> term -> proof -> proof"} \\
   @{define_ML Proofterm.expand_proof: "theory ->
   (Proofterm.thm_header -> string option) -> proof -> proof"} \\
   @{define_ML Proof_Checker.thm_of_proof: "theory -> proof -> thm"} \\
   @{define_ML Proof_Syntax.read_proof: "theory -> bool -> bool -> string -> proof"} \\
   @{define_ML Proof_Syntax.pretty_proof: "Proof.context -> proof -> Pretty.T"} \\
   \end{mldecls}
 
   \<^descr> Type \<^ML_type>\<open>proof\<close> represents proof terms; this is a datatype with
   constructors @{define_ML Abst}, @{define_ML AbsP}, @{define_ML_infix "%"},
   @{define_ML_infix "%%"}, @{define_ML PBound}, @{define_ML MinProof}, @{define_ML
   Hyp}, @{define_ML PAxm}, @{define_ML Oracle}, @{define_ML PThm} as explained
   above. %FIXME PClass (!?)
 
   \<^descr> Type \<^ML_type>\<open>proof_body\<close> represents the nested proof information of a
   named theorem, consisting of a digest of oracles and named theorem over some
   proof term. The digest only covers the directly visible part of the proof:
   in order to get the full information, the implicit graph of nested theorems
   needs to be traversed (e.g.\ using \<^ML>\<open>Proofterm.fold_body_thms\<close>).
 
   \<^descr> \<^ML>\<open>Thm.proof_of\<close>~\<open>thm\<close> and \<^ML>\<open>Thm.proof_body_of\<close>~\<open>thm\<close> produce the
   proof term or proof body (with digest of oracles and theorems) from a given
   theorem. Note that this involves a full join of internal futures that
   fulfill pending proof promises, and thus disrupts the natural bottom-up
   construction of proofs by introducing dynamic ad-hoc dependencies. Parallel
   performance may suffer by inspecting proof terms at run-time.
 
   \<^descr> \<^ML>\<open>Proofterm.proofs\<close> specifies the detail of proof recording within
   \<^ML_type>\<open>thm\<close> values produced by the inference kernel: \<^ML>\<open>0\<close> records only
   the names of oracles, \<^ML>\<open>1\<close> records oracle names and propositions, \<^ML>\<open>2\<close>
   additionally records full proof terms. Officially named theorems that
   contribute to a result are recorded in any case.
 
   \<^descr> \<^ML>\<open>Proofterm.reconstruct_proof\<close>~\<open>thy prop prf\<close> turns the implicit
   proof term \<open>prf\<close> into a full proof of the given proposition.
 
   Reconstruction may fail if \<open>prf\<close> is not a proof of \<open>prop\<close>, or if it does not
   contain sufficient information for reconstruction. Failure may only happen
   for proofs that are constructed manually, but not for those produced
   automatically by the inference kernel.
 
   \<^descr> \<^ML>\<open>Proofterm.expand_proof\<close>~\<open>thy expand prf\<close> reconstructs and expands
   the proofs of nested theorems according to the given \<open>expand\<close> function: a
   result of @{ML \<open>SOME ""\<close>} means full expansion, @{ML \<open>SOME\<close>}~\<open>name\<close> means to
   keep the theorem node but replace its internal name, @{ML NONE} means no
   change.
 
   \<^descr> \<^ML>\<open>Proof_Checker.thm_of_proof\<close>~\<open>thy prf\<close> turns the given (full) proof
   into a theorem, by replaying it using only primitive rules of the inference
   kernel.
 
   \<^descr> \<^ML>\<open>Proof_Syntax.read_proof\<close>~\<open>thy b\<^sub>1 b\<^sub>2 s\<close> reads in a proof term. The
   Boolean flags indicate the use of sort and type information. Usually, typing
   information is left implicit and is inferred during proof reconstruction.
   %FIXME eliminate flags!?
 
   \<^descr> \<^ML>\<open>Proof_Syntax.pretty_proof\<close>~\<open>ctxt prf\<close> pretty-prints the given proof
   term.
 \<close>
 
 text %mlex \<open>
   \<^item> \<^file>\<open>~~/src/HOL/Proofs/ex/Proof_Terms.thy\<close> provides basic examples involving
   proof terms.
 
   \<^item> \<^file>\<open>~~/src/HOL/Proofs/ex/XML_Data.thy\<close> demonstrates export and import of
   proof terms via XML/ML data representation.
 \<close>
 
 end