doc-src/IsarImplementation/Thy/logic.thy
changeset 20537 b6b49903db7e
parent 20521 189811b39869
child 20542 a54ca4e90874
     1.1 --- a/doc-src/IsarImplementation/Thy/logic.thy	Thu Sep 14 15:27:08 2006 +0200
     1.2 +++ b/doc-src/IsarImplementation/Thy/logic.thy	Thu Sep 14 15:51:20 2006 +0200
     1.3 @@ -20,12 +20,12 @@
     1.4    "\<And>"} for universal quantification (proofs depending on terms), and
     1.5    @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
     1.6  
     1.7 -  Pure derivations are relative to a logical theory, which declares
     1.8 -  type constructors, term constants, and axioms.  Theory declarations
     1.9 -  support schematic polymorphism, which is strictly speaking outside
    1.10 -  the logic.\footnote{Incidently, this is the main logical reason, why
    1.11 -  the theory context @{text "\<Theta>"} is separate from the context @{text
    1.12 -  "\<Gamma>"} of the core calculus.}
    1.13 +  Derivations are relative to a logical theory, which declares type
    1.14 +  constructors, constants, and axioms.  Theory declarations support
    1.15 +  schematic polymorphism, which is strictly speaking outside the
    1.16 +  logic.\footnote{This is the deeper logical reason, why the theory
    1.17 +  context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"}
    1.18 +  of the core calculus.}
    1.19  *}
    1.20  
    1.21  
    1.22 @@ -42,8 +42,8 @@
    1.23    internally.  The resulting relation is an ordering: reflexive,
    1.24    transitive, and antisymmetric.
    1.25  
    1.26 -  A \emph{sort} is a list of type classes written as @{text
    1.27 -  "{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
    1.28 +  A \emph{sort} is a list of type classes written as @{text "s =
    1.29 +  {c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
    1.30    intersection.  Notationally, the curly braces are omitted for
    1.31    singleton intersections, i.e.\ any class @{text "c"} may be read as
    1.32    a sort @{text "{c}"}.  The ordering on type classes is extended to
    1.33 @@ -56,11 +56,11 @@
    1.34    elements wrt.\ the sort order.
    1.35  
    1.36    \medskip A \emph{fixed type variable} is a pair of a basic name
    1.37 -  (starting with a @{text "'"} character) and a sort constraint.  For
    1.38 -  example, @{text "('a, s)"} which is usually printed as @{text
    1.39 -  "\<alpha>\<^isub>s"}.  A \emph{schematic type variable} is a pair of an
    1.40 -  indexname and a sort constraint.  For example, @{text "(('a, 0),
    1.41 -  s)"} which is usually printed as @{text "?\<alpha>\<^isub>s"}.
    1.42 +  (starting with a @{text "'"} character) and a sort constraint, e.g.\
    1.43 +  @{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}.
    1.44 +  A \emph{schematic type variable} is a pair of an indexname and a
    1.45 +  sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually
    1.46 +  printed as @{text "?\<alpha>\<^isub>s"}.
    1.47  
    1.48    Note that \emph{all} syntactic components contribute to the identity
    1.49    of type variables, including the sort constraint.  The core logic
    1.50 @@ -70,23 +70,23 @@
    1.51  
    1.52    A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
    1.53    on types declared in the theory.  Type constructor application is
    1.54 -  usually written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.
    1.55 -  For @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text
    1.56 -  "prop"} instead of @{text "()prop"}.  For @{text "k = 1"} the
    1.57 -  parentheses are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text
    1.58 -  "(\<alpha>)list"}.  Further notation is provided for specific constructors,
    1.59 -  notably the right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of
    1.60 -  @{text "(\<alpha>, \<beta>)fun"}.
    1.61 +  written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.  For
    1.62 +  @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"}
    1.63 +  instead of @{text "()prop"}.  For @{text "k = 1"} the parentheses
    1.64 +  are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}.
    1.65 +  Further notation is provided for specific constructors, notably the
    1.66 +  right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>,
    1.67 +  \<beta>)fun"}.
    1.68    
    1.69 -  A \emph{type} @{text "\<tau>"} is defined inductively over type variables
    1.70 -  and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s |
    1.71 -  ?\<alpha>\<^isub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
    1.72 +  A \emph{type} is defined inductively over type variables and type
    1.73 +  constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
    1.74 +  (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
    1.75  
    1.76    A \emph{type abbreviation} is a syntactic definition @{text
    1.77    "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
    1.78 -  variables @{text "\<^vec>\<alpha>"}.  Type abbreviations looks like type
    1.79 -  constructors at the surface, but are fully expanded before entering
    1.80 -  the logical core.
    1.81 +  variables @{text "\<^vec>\<alpha>"}.  Type abbreviations appear as type
    1.82 +  constructors in the syntax, but are expanded before entering the
    1.83 +  logical core.
    1.84  
    1.85    A \emph{type arity} declares the image behavior of a type
    1.86    constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
    1.87 @@ -98,22 +98,22 @@
    1.88  
    1.89    \medskip The sort algebra is always maintained as \emph{coregular},
    1.90    which means that type arities are consistent with the subclass
    1.91 -  relation: for each type constructor @{text "\<kappa>"} and classes @{text
    1.92 -  "c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "\<kappa> ::
    1.93 -  (\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "\<kappa>
    1.94 -  :: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq>
    1.95 -  \<^vec>s\<^isub>2"} holds component-wise.
    1.96 +  relation: for any type constructor @{text "\<kappa>"}, and classes @{text
    1.97 +  "c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> ::
    1.98 +  (\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> ::
    1.99 +  (\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq>
   1.100 +  \<^vec>s\<^isub>2"} component-wise.
   1.101  
   1.102    The key property of a coregular order-sorted algebra is that sort
   1.103 -  constraints may be always solved in a most general fashion: for each
   1.104 -  type constructor @{text "\<kappa>"} and sort @{text "s"} there is a most
   1.105 -  general vector of argument sorts @{text "(s\<^isub>1, \<dots>,
   1.106 -  s\<^isub>k)"} such that a type scheme @{text
   1.107 -  "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is
   1.108 -  of sort @{text "s"}.  Consequently, the unification problem on the
   1.109 -  algebra of types has most general solutions (modulo renaming and
   1.110 -  equivalence of sorts).  Moreover, the usual type-inference algorithm
   1.111 -  will produce primary types as expected \cite{nipkow-prehofer}.
   1.112 +  constraints can be solved in a most general fashion: for each type
   1.113 +  constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general
   1.114 +  vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such
   1.115 +  that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>,
   1.116 +  \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}.
   1.117 +  Consequently, unification on the algebra of types has most general
   1.118 +  solutions (modulo equivalence of sorts).  This means that
   1.119 +  type-inference will produce primary types as expected
   1.120 +  \cite{nipkow-prehofer}.
   1.121  *}
   1.122  
   1.123  text %mlref {*
   1.124 @@ -149,20 +149,21 @@
   1.125    \item @{ML_type typ} represents types; this is a datatype with
   1.126    constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
   1.127  
   1.128 -  \item @{ML map_atyps}~@{text "f \<tau>"} applies mapping @{text "f"} to
   1.129 -  all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text "\<tau>"}.
   1.130 +  \item @{ML map_atyps}~@{text "f \<tau>"} applies the mapping @{text "f"}
   1.131 +  to all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text
   1.132 +  "\<tau>"}.
   1.133  
   1.134 -  \item @{ML fold_atyps}~@{text "f \<tau>"} iterates operation @{text "f"}
   1.135 -  over all occurrences of atoms (@{ML TFree}, @{ML TVar}) in @{text
   1.136 -  "\<tau>"}; the type structure is traversed from left to right.
   1.137 +  \item @{ML fold_atyps}~@{text "f \<tau>"} iterates the operation @{text
   1.138 +  "f"} over all occurrences of atomic types (@{ML TFree}, @{ML TVar})
   1.139 +  in @{text "\<tau>"}; the type structure is traversed from left to right.
   1.140  
   1.141    \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
   1.142    tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
   1.143  
   1.144 -  \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether a type
   1.145 -  is of a given sort.
   1.146 +  \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether type
   1.147 +  @{text "\<tau>"} is of sort @{text "s"}.
   1.148  
   1.149 -  \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares new
   1.150 +  \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares a new
   1.151    type constructors @{text "\<kappa>"} with @{text "k"} arguments and
   1.152    optional mixfix syntax.
   1.153  
   1.154 @@ -171,7 +172,7 @@
   1.155    optional mixfix syntax.
   1.156  
   1.157    \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
   1.158 -  c\<^isub>n])"} declares new class @{text "c"}, together with class
   1.159 +  c\<^isub>n])"} declares a new class @{text "c"}, together with class
   1.160    relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
   1.161  
   1.162    \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
   1.163 @@ -179,7 +180,7 @@
   1.164    c\<^isub>2"}.
   1.165  
   1.166    \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
   1.167 -  arity @{text "\<kappa> :: (\<^vec>s)s"}.
   1.168 +  the arity @{text "\<kappa> :: (\<^vec>s)s"}.
   1.169  
   1.170    \end{description}
   1.171  *}
   1.172 @@ -193,62 +194,66 @@
   1.173  
   1.174    The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
   1.175    with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
   1.176 -  or \cite{paulson-ml2}), and named free variables and constants.
   1.177 -  Terms with loose bound variables are usually considered malformed.
   1.178 -  The types of variables and constants is stored explicitly at each
   1.179 -  occurrence in the term.
   1.180 +  or \cite{paulson-ml2}), with the types being determined determined
   1.181 +  by the corresponding binders.  In contrast, free variables and
   1.182 +  constants are have an explicit name and type in each occurrence.
   1.183  
   1.184    \medskip A \emph{bound variable} is a natural number @{text "b"},
   1.185 -  which refers to the next binder that is @{text "b"} steps upwards
   1.186 -  from the occurrence of @{text "b"} (counting from zero).  Bindings
   1.187 -  may be introduced as abstractions within the term, or as a separate
   1.188 -  context (an inside-out list).  This associates each bound variable
   1.189 -  with a type.  A \emph{loose variables} is a bound variable that is
   1.190 -  outside the current scope of local binders or the context.  For
   1.191 +  which accounts for the number of intermediate binders between the
   1.192 +  variable occurrence in the body and its binding position.  For
   1.193    example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"}
   1.194 -  corresponds to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a named
   1.195 -  representation.  Also note that the very same bound variable may get
   1.196 -  different numbers at different occurrences.
   1.197 +  would correspond to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a
   1.198 +  named representation.  Note that a bound variable may be represented
   1.199 +  by different de-Bruijn indices at different occurrences, depending
   1.200 +  on the nesting of abstractions.
   1.201 +
   1.202 +  A \emph{loose variables} is a bound variable that is outside the
   1.203 +  scope of local binders.  The types (and names) for loose variables
   1.204 +  can be managed as a separate context, that is maintained inside-out
   1.205 +  like a stack of hypothetical binders.  The core logic only operates
   1.206 +  on closed terms, without any loose variables.
   1.207  
   1.208 -  A \emph{fixed variable} is a pair of a basic name and a type.  For
   1.209 -  example, @{text "(x, \<tau>)"} which is usually printed @{text
   1.210 -  "x\<^isub>\<tau>"}.  A \emph{schematic variable} is a pair of an
   1.211 -  indexname and a type.  For example, @{text "((x, 0), \<tau>)"} which is
   1.212 -  usually printed as @{text "?x\<^isub>\<tau>"}.
   1.213 +  A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
   1.214 +  @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}.  A
   1.215 +  \emph{schematic variable} is a pair of an indexname and a type,
   1.216 +  e.g.\ @{text "((x, 0), \<tau>)"} which is usually printed as @{text
   1.217 +  "?x\<^isub>\<tau>"}.
   1.218  
   1.219 -  \medskip A \emph{constant} is a atomic terms consisting of a basic
   1.220 -  name and a type.  Constants are declared in the context as
   1.221 -  polymorphic families @{text "c :: \<sigma>"}, meaning that any @{text
   1.222 -  "c\<^isub>\<tau>"} is a valid constant for all substitution instances
   1.223 -  @{text "\<tau> \<le> \<sigma>"}.
   1.224 +  \medskip A \emph{constant} is a pair of a basic name and a type,
   1.225 +  e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text
   1.226 +  "c\<^isub>\<tau>"}.  Constants are declared in the context as polymorphic
   1.227 +  families @{text "c :: \<sigma>"}, meaning that valid all substitution
   1.228 +  instances @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
   1.229  
   1.230 -  The list of \emph{type arguments} of @{text "c\<^isub>\<tau>"} wrt.\ the
   1.231 -  declaration @{text "c :: \<sigma>"} is the codomain of the type matcher
   1.232 -  presented in canonical order (according to the left-to-right
   1.233 -  occurrences of type variables in in @{text "\<sigma>"}).  Thus @{text
   1.234 -  "c\<^isub>\<tau>"} can be represented more compactly as @{text
   1.235 -  "c(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}.  For example, the instance @{text
   1.236 -  "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} of some @{text "plus :: \<alpha> \<Rightarrow> \<alpha>
   1.237 -  \<Rightarrow> \<alpha>"} has the singleton list @{text "nat"} as type arguments, the
   1.238 -  constant may be represented as @{text "plus(nat)"}.
   1.239 +  The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"}
   1.240 +  wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of
   1.241 +  the matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>,
   1.242 +  ?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in canonical order @{text
   1.243 +  "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}.  Within a given theory context,
   1.244 +  there is a one-to-one correspondence between any constant @{text
   1.245 +  "c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1, \<dots>,
   1.246 +  \<tau>\<^isub>n)"} of its type arguments.  For example, with @{text "plus
   1.247 +  :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow>
   1.248 +  nat\<^esub>"} corresponds to @{text "plus(nat)"}.
   1.249  
   1.250    Constant declarations @{text "c :: \<sigma>"} may contain sort constraints
   1.251    for type variables in @{text "\<sigma>"}.  These are observed by
   1.252    type-inference as expected, but \emph{ignored} by the core logic.
   1.253    This means the primitive logic is able to reason with instances of
   1.254 -  polymorphic constants that the user-level type-checker would reject.
   1.255 +  polymorphic constants that the user-level type-checker would reject
   1.256 +  due to violation of type class restrictions.
   1.257  
   1.258 -  \medskip A \emph{term} @{text "t"} is defined inductively over
   1.259 -  variables and constants, with abstraction and application as
   1.260 -  follows: @{text "t = b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> |
   1.261 -  \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}.  Parsing and printing takes
   1.262 -  care of converting between an external representation with named
   1.263 -  bound variables.  Subsequently, we shall use the latter notation
   1.264 -  instead of internal de-Bruijn representation.
   1.265 +  \medskip A \emph{term} is defined inductively over variables and
   1.266 +  constants, with abstraction and application as follows: @{text "t =
   1.267 +  b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t |
   1.268 +  t\<^isub>1 t\<^isub>2"}.  Parsing and printing takes care of
   1.269 +  converting between an external representation with named bound
   1.270 +  variables.  Subsequently, we shall use the latter notation instead
   1.271 +  of internal de-Bruijn representation.
   1.272  
   1.273 -  The subsequent inductive relation @{text "t :: \<tau>"} assigns a
   1.274 -  (unique) type to a term, using the special type constructor @{text
   1.275 -  "(\<alpha>, \<beta>)fun"}, which is written @{text "\<alpha> \<Rightarrow> \<beta>"}.
   1.276 +  The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a
   1.277 +  term according to the structure of atomic terms, abstractions, and
   1.278 +  applicatins:
   1.279    \[
   1.280    \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{}
   1.281    \qquad
   1.282 @@ -264,46 +269,47 @@
   1.283    Type-inference depends on a context of type constraints for fixed
   1.284    variables, and declarations for polymorphic constants.
   1.285  
   1.286 -  The identity of atomic terms consists both of the name and the type.
   1.287 -  Thus different entities @{text "c\<^bsub>\<tau>\<^isub>1\<^esub>"} and
   1.288 -  @{text "c\<^bsub>\<tau>\<^isub>2\<^esub>"} may well identified by type
   1.289 -  instantiation, by mapping @{text "\<tau>\<^isub>1"} and @{text
   1.290 -  "\<tau>\<^isub>2"} to the same @{text "\<tau>"}.  Although,
   1.291 -  different type instances of constants of the same basic name are
   1.292 -  commonplace, this rarely happens for variables: type-inference
   1.293 -  always demands ``consistent'' type constraints.
   1.294 +  The identity of atomic terms consists both of the name and the type
   1.295 +  component.  This means that different variables @{text
   1.296 +  "x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text
   1.297 +  "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type
   1.298 +  instantiation.  Some outer layers of the system make it hard to
   1.299 +  produce variables of the same name, but different types.  In
   1.300 +  particular, type-inference always demands ``consistent'' type
   1.301 +  constraints for free variables.  In contrast, mixed instances of
   1.302 +  polymorphic constants occur frequently.
   1.303  
   1.304    \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
   1.305    is the set of type variables occurring in @{text "t"}, but not in
   1.306 -  @{text "\<sigma>"}.  This means that the term implicitly depends on the
   1.307 -  values of various type variables that are not visible in the overall
   1.308 -  type, i.e.\ there are different type instances @{text "t\<vartheta>
   1.309 -  :: \<sigma>"} and @{text "t\<vartheta>' :: \<sigma>"} with the same type.  This
   1.310 -  slightly pathological situation is apt to cause strange effects.
   1.311 +  @{text "\<sigma>"}.  This means that the term implicitly depends on type
   1.312 +  arguments that are not accounted in result type, i.e.\ there are
   1.313 +  different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text
   1.314 +  "t\<vartheta>' :: \<sigma>"} with the same type.  This slightly
   1.315 +  pathological situation demands special care.
   1.316  
   1.317    \medskip A \emph{term abbreviation} is a syntactic definition @{text
   1.318 -  "c\<^isub>\<sigma> \<equiv> t"} of an arbitrary closed term @{text "t"} of type
   1.319 -  @{text "\<sigma>"} without any hidden polymorphism.  A term abbreviation
   1.320 -  looks like a constant at the surface, but is fully expanded before
   1.321 -  entering the logical core.  Abbreviations are usually reverted when
   1.322 -  printing terms, using rules @{text "t \<rightarrow> c\<^isub>\<sigma>"} has a
   1.323 -  higher-order term rewrite system.
   1.324 +  "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"},
   1.325 +  without any hidden polymorphism.  A term abbreviation looks like a
   1.326 +  constant in the syntax, but is fully expanded before entering the
   1.327 +  logical core.  Abbreviations are usually reverted when printing
   1.328 +  terms, using the collective @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for
   1.329 +  higher-order rewriting.
   1.330  
   1.331    \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
   1.332 -  "\<alpha>\<beta>\<eta>"}-conversion. @{text "\<alpha>"}-conversion refers to capture-free
   1.333 +  "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free
   1.334    renaming of bound variables; @{text "\<beta>"}-conversion contracts an
   1.335 -  abstraction applied to some argument term, substituting the argument
   1.336 +  abstraction applied to an argument term, substituting the argument
   1.337    in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
   1.338    "\<eta>"}-conversion contracts vacuous application-abstraction: @{text
   1.339    "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
   1.340 -  @{text "0"} does not occur in @{text "f"}.
   1.341 +  does not occur in @{text "f"}.
   1.342  
   1.343 -  Terms are almost always treated module @{text "\<alpha>"}-conversion, which
   1.344 -  is implicit in the de-Bruijn representation.  The names in
   1.345 -  abstractions of bound variables are maintained only as a comment for
   1.346 -  parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-equivalence is usually
   1.347 -  taken for granted higher rules (\secref{sec:rules}), anything
   1.348 -  depending on higher-order unification or rewriting.
   1.349 +  Terms are normally treated modulo @{text "\<alpha>"}-conversion, which is
   1.350 +  implicit in the de-Bruijn representation.  Names for bound variables
   1.351 +  in abstractions are maintained separately as (meaningless) comments,
   1.352 +  mostly for parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-conversion is
   1.353 +  commonplace in various higher operations (\secref{sec:rules}) that
   1.354 +  are based on higher-order unification and matching.
   1.355  *}
   1.356  
   1.357  text %mlref {*
   1.358 @@ -326,43 +332,43 @@
   1.359  
   1.360    \begin{description}
   1.361  
   1.362 -  \item @{ML_type term} represents de-Bruijn terms with comments in
   1.363 -  abstractions for bound variable names.  This is a datatype with
   1.364 -  constructors @{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}, @{ML
   1.365 -  Abs}, @{ML "op $"}.
   1.366 +  \item @{ML_type term} represents de-Bruijn terms, with comments in
   1.367 +  abstractions, and explicitly named free variables and constants;
   1.368 +  this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML
   1.369 +  Var}, @{ML Const}, @{ML Abs}, @{ML "op $"}.
   1.370  
   1.371    \item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text
   1.372    "\<alpha>"}-equivalence of two terms.  This is the basic equality relation
   1.373    on type @{ML_type term}; raw datatype equality should only be used
   1.374    for operations related to parsing or printing!
   1.375  
   1.376 -  \item @{ML map_term_types}~@{text "f t"} applies mapping @{text "f"}
   1.377 -  to all types occurring in @{text "t"}.
   1.378 +  \item @{ML map_term_types}~@{text "f t"} applies the mapping @{text
   1.379 +  "f"} to all types occurring in @{text "t"}.
   1.380 +
   1.381 +  \item @{ML fold_types}~@{text "f t"} iterates the operation @{text
   1.382 +  "f"} over all occurrences of types in @{text "t"}; the term
   1.383 +  structure is traversed from left to right.
   1.384  
   1.385 -  \item @{ML fold_types}~@{text "f t"} iterates operation @{text "f"}
   1.386 -  over all occurrences of types in @{text "t"}; the term structure is
   1.387 +  \item @{ML map_aterms}~@{text "f t"} applies the mapping @{text "f"}
   1.388 +  to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
   1.389 +  Const}) occurring in @{text "t"}.
   1.390 +
   1.391 +  \item @{ML fold_aterms}~@{text "f t"} iterates the operation @{text
   1.392 +  "f"} over all occurrences of atomic terms (@{ML Bound}, @{ML Free},
   1.393 +  @{ML Var}, @{ML Const}) in @{text "t"}; the term structure is
   1.394    traversed from left to right.
   1.395  
   1.396 -  \item @{ML map_aterms}~@{text "f t"} applies mapping @{text "f"} to
   1.397 -  all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const})
   1.398 -  occurring in @{text "t"}.
   1.399 -
   1.400 -  \item @{ML fold_aterms}~@{text "f t"} iterates operation @{text "f"}
   1.401 -  over all occurrences of atomic terms in (@{ML Bound}, @{ML Free},
   1.402 -  @{ML Var}, @{ML Const}) @{text "t"}; the term structure is traversed
   1.403 -  from left to right.
   1.404 -
   1.405 -  \item @{ML fastype_of}~@{text "t"} recomputes the type of a
   1.406 -  well-formed term, while omitting any sanity checks.  This operation
   1.407 -  is relatively slow.
   1.408 +  \item @{ML fastype_of}~@{text "t"} determines the type of a
   1.409 +  well-typed term.  This operation is relatively slow, despite the
   1.410 +  omission of any sanity checks.
   1.411  
   1.412    \item @{ML lambda}~@{text "a b"} produces an abstraction @{text
   1.413 -  "\<lambda>a. b"}, where occurrences of the original (atomic) term @{text
   1.414 -  "a"} in the body @{text "b"} are replaced by bound variables.
   1.415 +  "\<lambda>a. b"}, where occurrences of the atomic term @{text "a"} in the
   1.416 +  body @{text "b"} are replaced by bound variables.
   1.417  
   1.418 -  \item @{ML betapply}~@{text "t u"} produces an application @{text "t
   1.419 -  u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} happens to
   1.420 -  be an abstraction.
   1.421 +  \item @{ML betapply}~@{text "(t, u)"} produces an application @{text
   1.422 +  "t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an
   1.423 +  abstraction.
   1.424  
   1.425    \item @{ML Sign.add_consts_i}~@{text "[(c, \<sigma>, mx), \<dots>]"} declares a
   1.426    new constant @{text "c :: \<sigma>"} with optional mixfix syntax.
   1.427 @@ -373,9 +379,9 @@
   1.428  
   1.429    \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
   1.430    Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
   1.431 -  convert between the two representations of constants, namely full
   1.432 -  type instance vs.\ compact type arguments form (depending on the
   1.433 -  most general declaration given in the context).
   1.434 +  convert between the representations of polymorphic constants: the
   1.435 +  full type instance vs.\ the compact type arguments form (depending
   1.436 +  on the most general declaration given in the context).
   1.437  
   1.438    \end{description}
   1.439  *}
   1.440 @@ -424,24 +430,23 @@
   1.441    \emph{theorem} is a proven proposition (depending on a context of
   1.442    hypotheses and the background theory).  Primitive inferences include
   1.443    plain natural deduction rules for the primary connectives @{text
   1.444 -  "\<And>"} and @{text "\<Longrightarrow>"} of the framework.  There are separate (derived)
   1.445 -  rules for equality/equivalence @{text "\<equiv>"} and internal conjunction
   1.446 -  @{text "&"}.
   1.447 +  "\<And>"} and @{text "\<Longrightarrow>"} of the framework.  There is also a builtin
   1.448 +  notion of equality/equivalence @{text "\<equiv>"}.
   1.449  *}
   1.450  
   1.451 -subsection {* Standard connectives and rules *}
   1.452 +subsection {* Primitive connectives and rules *}
   1.453  
   1.454  text {*
   1.455 -  The basic theory is called @{text "Pure"}, it contains declarations
   1.456 -  for the standard logical connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and
   1.457 -  @{text "\<equiv>"} of the framework, see \figref{fig:pure-connectives}.
   1.458 -  The derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is
   1.459 -  defined inductively by the primitive inferences given in
   1.460 -  \figref{fig:prim-rules}, with the global syntactic restriction that
   1.461 -  hypotheses may never contain schematic variables.  The builtin
   1.462 -  equality is conceptually axiomatized shown in
   1.463 +  The theory @{text "Pure"} contains declarations for the standard
   1.464 +  connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of the logical
   1.465 +  framework, see \figref{fig:pure-connectives}.  The derivability
   1.466 +  judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined
   1.467 +  inductively by the primitive inferences given in
   1.468 +  \figref{fig:prim-rules}, with the global restriction that hypotheses
   1.469 +  @{text "\<Gamma>"} may \emph{not} contain schematic variables.  The builtin
   1.470 +  equality is conceptually axiomatized as shown in
   1.471    \figref{fig:pure-equality}, although the implementation works
   1.472 -  directly with (derived) inference rules.
   1.473 +  directly with derived inference rules.
   1.474  
   1.475    \begin{figure}[htb]
   1.476    \begin{center}
   1.477 @@ -450,7 +455,7 @@
   1.478    @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
   1.479    @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
   1.480    \end{tabular}
   1.481 -  \caption{Standard connectives of Pure}\label{fig:pure-connectives}
   1.482 +  \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
   1.483    \end{center}
   1.484    \end{figure}
   1.485  
   1.486 @@ -462,9 +467,9 @@
   1.487    \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{}
   1.488    \]
   1.489    \[
   1.490 -  \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}{@{text "\<Gamma> \<turnstile> b x"} & @{text "x \<notin> \<Gamma>"}}
   1.491 +  \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}}
   1.492    \qquad
   1.493 -  \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b a"}}{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}
   1.494 +  \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}
   1.495    \]
   1.496    \[
   1.497    \infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
   1.498 @@ -478,34 +483,34 @@
   1.499    \begin{figure}[htb]
   1.500    \begin{center}
   1.501    \begin{tabular}{ll}
   1.502 -  @{text "\<turnstile> (\<lambda>x. b x) a \<equiv> b a"} & @{text "\<beta>"}-conversion \\
   1.503 +  @{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\
   1.504    @{text "\<turnstile> x \<equiv> x"} & reflexivity \\
   1.505    @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\
   1.506    @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
   1.507 -  @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & coincidence with equivalence \\
   1.508 +  @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\
   1.509    \end{tabular}
   1.510 -  \caption{Conceptual axiomatization of builtin equality}\label{fig:pure-equality}
   1.511 +  \caption{Conceptual axiomatization of @{text "\<equiv>"}}\label{fig:pure-equality}
   1.512    \end{center}
   1.513    \end{figure}
   1.514  
   1.515    The introduction and elimination rules for @{text "\<And>"} and @{text
   1.516 -  "\<Longrightarrow>"} are analogous to formation of (dependently typed) @{text
   1.517 +  "\<Longrightarrow>"} are analogous to formation of dependently typed @{text
   1.518    "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
   1.519 -  are \emph{irrelevant} in the Pure logic, they may never occur within
   1.520 -  propositions, i.e.\ the @{text "\<Longrightarrow>"} arrow is non-dependent.  The
   1.521 -  system provides a runtime option to record explicit proof terms for
   1.522 -  primitive inferences, cf.\ \cite{Berghofer-Nipkow:2000:TPHOL}.  Thus
   1.523 -  the three-fold @{text "\<lambda>"}-structure can be made explicit.
   1.524 +  are irrelevant in the Pure logic, though, they may never occur
   1.525 +  within propositions.  The system provides a runtime option to record
   1.526 +  explicit proof terms for primitive inferences.  Thus all three
   1.527 +  levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for
   1.528 +  terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\
   1.529 +  \cite{Berghofer-Nipkow:2000:TPHOL}).
   1.530  
   1.531 -  Observe that locally fixed parameters (as used in rule @{text
   1.532 -  "\<And>_intro"}) need not be recorded in the hypotheses, because the
   1.533 -  simple syntactic types of Pure are always inhabitable.  The typing
   1.534 -  ``assumption'' @{text "x :: \<tau>"} is logically vacuous, it disappears
   1.535 -  automatically whenever the statement body ceases to mention variable
   1.536 -  @{text "x\<^isub>\<tau>"}.\footnote{This greatly simplifies many basic
   1.537 -  reasoning steps, and is the key difference to the formulation of
   1.538 -  this logic as ``@{text "\<lambda>HOL"}'' in the PTS framework
   1.539 -  \cite{Barendregt-Geuvers:2001}.}
   1.540 +  Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need
   1.541 +  not be recorded in the hypotheses, because the simple syntactic
   1.542 +  types of Pure are always inhabitable.  Typing ``assumptions'' @{text
   1.543 +  "x :: \<tau>"} are (implicitly) present only with occurrences of @{text
   1.544 +  "x\<^isub>\<tau>"} in the statement body.\footnote{This is the key
   1.545 +  difference ``@{text "\<lambda>HOL"}'' in the PTS framework
   1.546 +  \cite{Barendregt-Geuvers:2001}, where @{text "x : A"} hypotheses are
   1.547 +  treated explicitly for types, in the same way as propositions.}
   1.548  
   1.549    \medskip FIXME @{text "\<alpha>\<beta>\<eta>"}-equivalence and primitive definitions
   1.550  
   1.551 @@ -514,13 +519,11 @@
   1.552    "\<alpha>\<beta>\<eta>"}-equivalence on terms, while coinciding with bi-implication.
   1.553  
   1.554    \medskip The axiomatization of a theory is implicitly closed by
   1.555 -  forming all instances of type and term variables: @{text "\<turnstile> A\<vartheta>"} for
   1.556 -  any substitution instance of axiom @{text "\<turnstile> A"}.  By pushing
   1.557 -  substitution through derivations inductively, we get admissible
   1.558 -  substitution rules for theorems shown in \figref{fig:subst-rules}.
   1.559 -  Alternatively, the term substitution rules could be derived from
   1.560 -  @{text "\<And>_intro/elim"}.  The versions for types are genuine
   1.561 -  admissible rules, due to the lack of true polymorphism in the logic.
   1.562 +  forming all instances of type and term variables: @{text "\<turnstile>
   1.563 +  A\<vartheta>"} holds for any substitution instance of an axiom
   1.564 +  @{text "\<turnstile> A"}.  By pushing substitution through derivations
   1.565 +  inductively, we get admissible @{text "generalize"} and @{text
   1.566 +  "instance"} rules shown in \figref{fig:subst-rules}.
   1.567  
   1.568    \begin{figure}[htb]
   1.569    \begin{center}
   1.570 @@ -538,11 +541,15 @@
   1.571    \end{center}
   1.572    \end{figure}
   1.573  
   1.574 -  Since @{text "\<Gamma>"} may never contain any schematic variables, the
   1.575 -  @{text "instantiate"} do not require an explicit side-condition.  In
   1.576 -  principle, variables could be substituted in hypotheses as well, but
   1.577 -  this could disrupt monotonicity of the basic calculus: derivations
   1.578 -  could leave the current proof context.
   1.579 +  Note that @{text "instantiate"} does not require an explicit
   1.580 +  side-condition, because @{text "\<Gamma>"} may never contain schematic
   1.581 +  variables.
   1.582 +
   1.583 +  In principle, variables could be substituted in hypotheses as well,
   1.584 +  but this would disrupt monotonicity reasoning: deriving @{text
   1.585 +  "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is correct, but
   1.586 +  @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold --- the result
   1.587 +  belongs to a different proof context.
   1.588  *}
   1.589  
   1.590  text %mlref {*
   1.591 @@ -567,16 +574,16 @@
   1.592  subsection {* Auxiliary connectives *}
   1.593  
   1.594  text {*
   1.595 -  Pure also provides various auxiliary connectives based on primitive
   1.596 -  definitions, see \figref{fig:pure-aux}.  These are normally not
   1.597 -  exposed to the user, but appear in internal encodings only.
   1.598 +  Theory @{text "Pure"} also defines a few auxiliary connectives, see
   1.599 +  \figref{fig:pure-aux}.  These are normally not exposed to the user,
   1.600 +  but appear in internal encodings only.
   1.601  
   1.602    \begin{figure}[htb]
   1.603    \begin{center}
   1.604    \begin{tabular}{ll}
   1.605    @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\
   1.606    @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex]
   1.607 -  @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}) \\
   1.608 +  @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, hidden) \\
   1.609    @{text "#A \<equiv> A"} \\[1ex]
   1.610    @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\
   1.611    @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex]
   1.612 @@ -587,39 +594,38 @@
   1.613    \end{center}
   1.614    \end{figure}
   1.615  
   1.616 -  Conjunction as an explicit connective allows to treat both
   1.617 -  simultaneous assumptions and conclusions uniformly.  The definition
   1.618 -  allows to derive the usual introduction @{text "\<turnstile> A \<Longrightarrow> B \<Longrightarrow> A & B"},
   1.619 -  and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}.  For
   1.620 -  example, several claims may be stated at the same time, which is
   1.621 -  intermediately represented as an assumption, but the user only
   1.622 -  encounters several sub-goals, and several resulting facts in the
   1.623 -  very end (cf.\ \secref{sec:tactical-goals}).
   1.624 +  Derived conjunction rules include introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A &
   1.625 +  B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}.
   1.626 +  Conjunction allows to treat simultaneous assumptions and conclusions
   1.627 +  uniformly.  For example, multiple claims are intermediately
   1.628 +  represented as explicit conjunction, but this is usually refined
   1.629 +  into separate sub-goals before the user continues the proof; the
   1.630 +  final result is projected into a list of theorems (cf.\
   1.631 +  \secref{sec:tactical-goals}).
   1.632  
   1.633 -  The @{text "#"} marker allows complex propositions (nested @{text
   1.634 -  "\<And>"} and @{text "\<Longrightarrow>"}) to appear formally as atomic, without changing
   1.635 -  the meaning: @{text "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are
   1.636 -  interchangeable.  See \secref{sec:tactical-goals} for specific
   1.637 -  operations.
   1.638 +  The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex
   1.639 +  propositions appear as atomic, without changing the meaning: @{text
   1.640 +  "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable.  See
   1.641 +  \secref{sec:tactical-goals} for specific operations.
   1.642  
   1.643 -  The @{text "TERM"} marker turns any well-formed term into a
   1.644 -  derivable proposition: @{text "\<turnstile> TERM t"} holds
   1.645 -  unconditionally.  Despite its logically vacous meaning, this is
   1.646 -  occasionally useful to treat syntactic terms and proven propositions
   1.647 -  uniformly, as in a type-theoretic framework.
   1.648 +  The @{text "term"} marker turns any well-formed term into a
   1.649 +  derivable proposition: @{text "\<turnstile> TERM t"} holds unconditionally.
   1.650 +  Although this is logically vacuous, it allows to treat terms and
   1.651 +  proofs uniformly, similar to a type-theoretic framework.
   1.652  
   1.653 -  The @{text "TYPE"} constructor (which is the canonical
   1.654 -  representative of the unspecified type @{text "\<alpha> itself"}) injects
   1.655 -  the language of types into that of terms.  There is specific
   1.656 -  notation @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau>
   1.657 +  The @{text "TYPE"} constructor is the canonical representative of
   1.658 +  the unspecified type @{text "\<alpha> itself"}; it essentially injects the
   1.659 +  language of types into that of terms.  There is specific notation
   1.660 +  @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau>
   1.661   itself\<^esub>"}.
   1.662 -  Although being devoid of any particular meaning, the term @{text
   1.663 -  "TYPE(\<tau>)"} is able to carry the type @{text "\<tau>"} formally.  @{text
   1.664 -  "TYPE(\<alpha>)"} may be used as an additional formal argument in primitive
   1.665 -  definitions, in order to avoid hidden polymorphism (cf.\
   1.666 -  \secref{sec:terms}).  For example, @{text "c TYPE(\<alpha>) \<equiv> A[\<alpha>]"} turns
   1.667 -  out as a formally correct definition of some proposition @{text "A"}
   1.668 -  that depends on an additional type argument.
   1.669 +  Although being devoid of any particular meaning, the @{text
   1.670 +  "TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term
   1.671 +  language.  In particular, @{text "TYPE(\<alpha>)"} may be used as formal
   1.672 +  argument in primitive definitions, in order to circumvent hidden
   1.673 +  polymorphism (cf.\ \secref{sec:terms}).  For example, @{text "c
   1.674 +  TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of
   1.675 +  a proposition @{text "A"} that depends on an additional type
   1.676 +  argument, which is essentially a predicate on types.
   1.677  *}
   1.678  
   1.679  text %mlref {*