doc-src/IsarImplementation/Thy/logic.thy
author wenzelm
Thu Sep 07 20:12:08 2006 +0200 (2006-09-07)
changeset 20491 98ba42f19995
parent 20480 4e0522d38968
child 20493 48fea5e99505
permissions -rw-r--r--
tuned;
wenzelm@18537
     1
wenzelm@18537
     2
(* $Id$ *)
wenzelm@18537
     3
wenzelm@18537
     4
theory logic imports base begin
wenzelm@18537
     5
wenzelm@20470
     6
chapter {* Primitive logic \label{ch:logic} *}
wenzelm@18537
     7
wenzelm@20480
     8
text {*
wenzelm@20480
     9
  The logical foundations of Isabelle/Isar are that of the Pure logic,
wenzelm@20480
    10
  which has been introduced as a natural-deduction framework in
wenzelm@20480
    11
  \cite{paulson700}.  This is essentially the same logic as ``@{text
wenzelm@20480
    12
  "\<lambda>HOL"}'' in the more abstract framework of Pure Type Systems (PTS)
wenzelm@20480
    13
  \cite{Barendregt-Geuvers:2001}, although there are some key
wenzelm@20491
    14
  differences in the specific treatment of simple types in
wenzelm@20491
    15
  Isabelle/Pure.
wenzelm@20480
    16
wenzelm@20480
    17
  Following type-theoretic parlance, the Pure logic consists of three
wenzelm@20480
    18
  levels of @{text "\<lambda>"}-calculus with corresponding arrows: @{text
wenzelm@20480
    19
  "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
wenzelm@20480
    20
  "\<And>"} for universal quantification (proofs depending on terms), and
wenzelm@20480
    21
  @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
wenzelm@20480
    22
wenzelm@20480
    23
  Pure derivations are relative to a logical theory, which declares
wenzelm@20491
    24
  type constructors, term constants, and axioms.  Theory declarations
wenzelm@20491
    25
  support schematic polymorphism, which is strictly speaking outside
wenzelm@20491
    26
  the logic.\footnote{Incidently, this is the main logical reason, why
wenzelm@20491
    27
  the theory context @{text "\<Theta>"} is separate from the context @{text
wenzelm@20491
    28
  "\<Gamma>"} of the core calculus.}
wenzelm@20480
    29
*}
wenzelm@20480
    30
wenzelm@20480
    31
wenzelm@20451
    32
section {* Types \label{sec:types} *}
wenzelm@20437
    33
wenzelm@20437
    34
text {*
wenzelm@20480
    35
  The language of types is an uninterpreted order-sorted first-order
wenzelm@20480
    36
  algebra; types are qualified by ordered type classes.
wenzelm@20480
    37
wenzelm@20480
    38
  \medskip A \emph{type class} is an abstract syntactic entity
wenzelm@20480
    39
  declared in the theory context.  The \emph{subclass relation} @{text
wenzelm@20480
    40
  "c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic
wenzelm@20491
    41
  generating relation; the transitive closure is maintained
wenzelm@20491
    42
  internally.  The resulting relation is an ordering: reflexive,
wenzelm@20491
    43
  transitive, and antisymmetric.
wenzelm@20451
    44
wenzelm@20480
    45
  A \emph{sort} is a list of type classes written as @{text
wenzelm@20480
    46
  "{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
wenzelm@20480
    47
  intersection.  Notationally, the curly braces are omitted for
wenzelm@20480
    48
  singleton intersections, i.e.\ any class @{text "c"} may be read as
wenzelm@20480
    49
  a sort @{text "{c}"}.  The ordering on type classes is extended to
wenzelm@20491
    50
  sorts according to the meaning of intersections: @{text
wenzelm@20491
    51
  "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff
wenzelm@20491
    52
  @{text "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}.  The empty intersection
wenzelm@20491
    53
  @{text "{}"} refers to the universal sort, which is the largest
wenzelm@20491
    54
  element wrt.\ the sort order.  The intersections of all (finitely
wenzelm@20491
    55
  many) classes declared in the current theory are the minimal
wenzelm@20491
    56
  elements wrt.\ the sort order.
wenzelm@20480
    57
wenzelm@20491
    58
  \medskip A \emph{fixed type variable} is a pair of a basic name
wenzelm@20480
    59
  (starting with @{text "'"} character) and a sort constraint.  For
wenzelm@20480
    60
  example, @{text "('a, s)"} which is usually printed as @{text
wenzelm@20480
    61
  "\<alpha>\<^isub>s"}.  A \emph{schematic type variable} is a pair of an
wenzelm@20480
    62
  indexname and a sort constraint.  For example, @{text "(('a, 0),
wenzelm@20491
    63
  s)"} which is usually printed as @{text "?\<alpha>\<^isub>s"}.
wenzelm@20451
    64
wenzelm@20480
    65
  Note that \emph{all} syntactic components contribute to the identity
wenzelm@20491
    66
  of type variables, including the literal sort constraint.  The core
wenzelm@20491
    67
  logic handles type variables with the same name but different sorts
wenzelm@20491
    68
  as different, although some outer layers of the system make it hard
wenzelm@20491
    69
  to produce anything like this.
wenzelm@20451
    70
wenzelm@20491
    71
  A \emph{type constructor} is a @{text "k"}-ary operator on types
wenzelm@20491
    72
  declared in the theory.  Type constructor application is usually
wenzelm@20491
    73
  written postfix.  For @{text "k = 0"} the argument tuple is omitted,
wenzelm@20491
    74
  e.g.\ @{text "prop"} instead of @{text "()prop"}.  For @{text "k =
wenzelm@20491
    75
  1"} the parentheses are omitted, e.g.\ @{text "\<alpha> list"} instead of
wenzelm@20491
    76
  @{text "(\<alpha>)list"}.  Further notation is provided for specific
wenzelm@20491
    77
  constructors, notably right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"}
wenzelm@20491
    78
  instead of @{text "(\<alpha>, \<beta>)fun"} constructor.
wenzelm@20480
    79
  
wenzelm@20480
    80
  A \emph{type} is defined inductively over type variables and type
wenzelm@20491
    81
  constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
wenzelm@20491
    82
  (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)c"}.
wenzelm@20451
    83
wenzelm@20480
    84
  A \emph{type abbreviation} is a syntactic abbreviation of an
wenzelm@20480
    85
  arbitrary type expression of the theory.  Type abbreviations looks
wenzelm@20480
    86
  like type constructors at the surface, but are expanded before the
wenzelm@20480
    87
  core logic encounters them.
wenzelm@20480
    88
wenzelm@20480
    89
  A \emph{type arity} declares the image behavior of a type
wenzelm@20480
    90
  constructor wrt.\ the algebra of sorts: @{text "c :: (s\<^isub>1, \<dots>,
wenzelm@20491
    91
  s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)c"} is
wenzelm@20480
    92
  of sort @{text "s"} if each argument type @{text "\<tau>\<^isub>i"} is of
wenzelm@20491
    93
  sort @{text "s\<^isub>i"}.  Arity declarations are implicitly
wenzelm@20491
    94
  completed, i.e.\ @{text "c :: (\<^vec>s)c"} entails @{text "c ::
wenzelm@20491
    95
  (\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}.
wenzelm@20491
    96
wenzelm@20491
    97
  \medskip The sort algebra is always maintained as \emph{coregular},
wenzelm@20491
    98
  which means that type arities are consistent with the subclass
wenzelm@20491
    99
  relation: for each type constructor @{text "c"} and classes @{text
wenzelm@20491
   100
  "c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "c ::
wenzelm@20480
   101
  (\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "c
wenzelm@20480
   102
  :: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq>
wenzelm@20480
   103
  \<^vec>s\<^isub>2"} holds pointwise for all argument sorts.
wenzelm@20451
   104
wenzelm@20491
   105
  The key property of a coregular order-sorted algebra is that sort
wenzelm@20480
   106
  constraints may be always fulfilled in a most general fashion: for
wenzelm@20480
   107
  each type constructor @{text "c"} and sort @{text "s"} there is a
wenzelm@20480
   108
  most general vector of argument sorts @{text "(s\<^isub>1, \<dots>,
wenzelm@20491
   109
  s\<^isub>k)"} such that a type scheme @{text
wenzelm@20491
   110
  "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)c"} is
wenzelm@20491
   111
  of sort @{text "s"}.  Consequently, the unification problem on the
wenzelm@20491
   112
  algebra of types has most general solutions (modulo renaming and
wenzelm@20491
   113
  equivalence of sorts).  Moreover, the usual type-inference algorithm
wenzelm@20491
   114
  will produce primary types as expected \cite{nipkow-prehofer}.
wenzelm@20480
   115
*}
wenzelm@20451
   116
wenzelm@20480
   117
text %mlref {*
wenzelm@20480
   118
  \begin{mldecls}
wenzelm@20480
   119
  @{index_ML_type class} \\
wenzelm@20480
   120
  @{index_ML_type sort} \\
wenzelm@20480
   121
  @{index_ML_type typ} \\
wenzelm@20480
   122
  @{index_ML_type arity} \\
wenzelm@20480
   123
  @{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
wenzelm@20480
   124
  @{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
wenzelm@20480
   125
  @{index_ML Sign.add_types: "(bstring * int * mixfix) list -> theory -> theory"} \\
wenzelm@20480
   126
  @{index_ML Sign.add_tyabbrs_i: "
wenzelm@20480
   127
  (bstring * string list * typ * mixfix) list -> theory -> theory"} \\
wenzelm@20480
   128
  @{index_ML Sign.primitive_class: "string * class list -> theory -> theory"} \\
wenzelm@20480
   129
  @{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
wenzelm@20480
   130
  @{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
wenzelm@20480
   131
  \end{mldecls}
wenzelm@20480
   132
wenzelm@20480
   133
  \begin{description}
wenzelm@20480
   134
wenzelm@20480
   135
  \item @{ML_type class} represents type classes; this is an alias for
wenzelm@20480
   136
  @{ML_type string}.
wenzelm@20480
   137
wenzelm@20480
   138
  \item @{ML_type sort} represents sorts; this is an alias for
wenzelm@20480
   139
  @{ML_type "class list"}.
wenzelm@20451
   140
wenzelm@20480
   141
  \item @{ML_type arity} represents type arities; this is an alias for
wenzelm@20480
   142
  triples of the form @{text "(c, \<^vec>s, s)"} for @{text "c ::
wenzelm@20480
   143
  (\<^vec>s)s"} described above.
wenzelm@20480
   144
wenzelm@20480
   145
  \item @{ML_type typ} represents types; this is a datatype with
wenzelm@20480
   146
  constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
wenzelm@20480
   147
wenzelm@20480
   148
  \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
wenzelm@20480
   149
  tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
wenzelm@20480
   150
wenzelm@20480
   151
  \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether a type
wenzelm@20491
   152
  is of a given sort.
wenzelm@20480
   153
wenzelm@20480
   154
  \item @{ML Sign.add_types}~@{text "[(c, k, mx), \<dots>]"} declares new
wenzelm@20491
   155
  type constructors @{text "c"} with @{text "k"} arguments and
wenzelm@20480
   156
  optional mixfix syntax.
wenzelm@20451
   157
wenzelm@20480
   158
  \item @{ML Sign.add_tyabbrs_i}~@{text "[(c, \<^vec>\<alpha>, \<tau>, mx), \<dots>]"}
wenzelm@20491
   159
  defines a new type abbreviation @{text "(\<^vec>\<alpha>)c = \<tau>"} with
wenzelm@20491
   160
  optional mixfix syntax.
wenzelm@20480
   161
wenzelm@20480
   162
  \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
wenzelm@20480
   163
  c\<^isub>n])"} declares new class @{text "c"} derived together with
wenzelm@20491
   164
  class relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
wenzelm@20480
   165
wenzelm@20480
   166
  \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
wenzelm@20480
   167
  c\<^isub>2)"} declares class relation @{text "c\<^isub>1 \<subseteq>
wenzelm@20480
   168
  c\<^isub>2"}.
wenzelm@20480
   169
wenzelm@20480
   170
  \item @{ML Sign.primitive_arity}~@{text "(c, \<^vec>s, s)"} declares
wenzelm@20480
   171
  arity @{text "c :: (\<^vec>s)s"}.
wenzelm@20480
   172
wenzelm@20480
   173
  \end{description}
wenzelm@20437
   174
*}
wenzelm@20437
   175
wenzelm@20437
   176
wenzelm@20480
   177
wenzelm@20451
   178
section {* Terms \label{sec:terms} *}
wenzelm@18537
   179
wenzelm@18537
   180
text {*
wenzelm@20451
   181
  \glossary{Term}{FIXME}
wenzelm@18537
   182
wenzelm@20491
   183
  The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
wenzelm@20491
   184
  with de-Bruijn indices for bound variables, and named free
wenzelm@20491
   185
  variables, and constants.  Terms with loose bound variables are
wenzelm@20491
   186
  usually considered malformed.  The types of variables and constants
wenzelm@20491
   187
  is stored explicitly at each occurrence in the term (which is a
wenzelm@20491
   188
  known performance issue).
wenzelm@20491
   189
wenzelm@20451
   190
  FIXME de-Bruijn representation of lambda terms
wenzelm@20480
   191
wenzelm@20480
   192
  Term syntax provides explicit abstraction @{text "\<lambda>x :: \<alpha>. b(x)"}
wenzelm@20480
   193
  and application @{text "t u"}, while types are usually implicit
wenzelm@20480
   194
  thanks to type-inference.
wenzelm@20480
   195
wenzelm@20480
   196
  Terms of type @{text "prop"} are called
wenzelm@20480
   197
  propositions.  Logical statements are composed via @{text "\<And>x ::
wenzelm@20480
   198
  \<alpha>. B(x)"} and @{text "A \<Longrightarrow> B"}.
wenzelm@18537
   199
*}
wenzelm@18537
   200
wenzelm@18537
   201
wenzelm@18537
   202
text {*
wenzelm@18537
   203
wenzelm@18537
   204
FIXME
wenzelm@18537
   205
wenzelm@18537
   206
\glossary{Schematic polymorphism}{FIXME}
wenzelm@18537
   207
wenzelm@18537
   208
\glossary{Type variable}{FIXME}
wenzelm@18537
   209
wenzelm@18537
   210
*}
wenzelm@18537
   211
wenzelm@18537
   212
wenzelm@20451
   213
section {* Theorems \label{sec:thms} *}
wenzelm@18537
   214
wenzelm@18537
   215
text {*
wenzelm@18537
   216
wenzelm@20480
   217
  Primitive reasoning operates on judgments of the form @{text "\<Gamma> \<turnstile>
wenzelm@20480
   218
  \<phi>"}, with standard introduction and elimination rules for @{text
wenzelm@20480
   219
  "\<And>"} and @{text "\<Longrightarrow>"} that refer to fixed parameters @{text "x"} and
wenzelm@20480
   220
  hypotheses @{text "A"} from the context @{text "\<Gamma>"}.  The
wenzelm@20480
   221
  corresponding proof terms are left implicit in the classic
wenzelm@20480
   222
  ``LCF-approach'', although they could be exploited separately
wenzelm@20480
   223
  \cite{Berghofer-Nipkow:2000}.
wenzelm@20480
   224
wenzelm@20480
   225
  The framework also provides definitional equality @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha>
wenzelm@20480
   226
  \<Rightarrow> prop"}, with @{text "\<alpha>\<beta>\<eta>"}-conversion rules.  The internal
wenzelm@20480
   227
  conjunction @{text "& :: prop \<Rightarrow> prop \<Rightarrow> prop"} enables the view of
wenzelm@20480
   228
  assumptions and conclusions emerging uniformly as simultaneous
wenzelm@20480
   229
  statements.
wenzelm@20480
   230
wenzelm@20480
   231
wenzelm@20480
   232
wenzelm@18537
   233
  FIXME
wenzelm@18537
   234
wenzelm@18537
   235
\glossary{Proposition}{A \seeglossary{term} of \seeglossary{type}
wenzelm@18537
   236
@{text "prop"}.  Internally, there is nothing special about
wenzelm@18537
   237
propositions apart from their type, but the concrete syntax enforces a
wenzelm@18537
   238
clear distinction.  Propositions are structured via implication @{text
wenzelm@18537
   239
"A \<Longrightarrow> B"} or universal quantification @{text "\<And>x. B x"} --- anything
wenzelm@18537
   240
else is considered atomic.  The canonical form for propositions is
wenzelm@18537
   241
that of a \seeglossary{Hereditary Harrop Formula}.}
wenzelm@18537
   242
wenzelm@18537
   243
\glossary{Theorem}{A proven proposition within a certain theory and
wenzelm@18537
   244
proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are
wenzelm@18537
   245
rarely spelled out explicitly.  Theorems are usually normalized
wenzelm@18537
   246
according to the \seeglossary{HHF} format.}
wenzelm@18537
   247
wenzelm@18537
   248
\glossary{Fact}{Sometimes used interchangably for
wenzelm@18537
   249
\seeglossary{theorem}.  Strictly speaking, a list of theorems,
wenzelm@18537
   250
essentially an extra-logical conjunction.  Facts emerge either as
wenzelm@18537
   251
local assumptions, or as results of local goal statements --- both may
wenzelm@18537
   252
be simultaneous, hence the list representation.}
wenzelm@18537
   253
wenzelm@18537
   254
\glossary{Schematic variable}{FIXME}
wenzelm@18537
   255
wenzelm@18537
   256
\glossary{Fixed variable}{A variable that is bound within a certain
wenzelm@18537
   257
proof context; an arbitrary-but-fixed entity within a portion of proof
wenzelm@18537
   258
text.}
wenzelm@18537
   259
wenzelm@18537
   260
\glossary{Free variable}{Synonymous for \seeglossary{fixed variable}.}
wenzelm@18537
   261
wenzelm@18537
   262
\glossary{Bound variable}{FIXME}
wenzelm@18537
   263
wenzelm@18537
   264
\glossary{Variable}{See \seeglossary{schematic variable},
wenzelm@18537
   265
\seeglossary{fixed variable}, \seeglossary{bound variable}, or
wenzelm@18537
   266
\seeglossary{type variable}.  The distinguishing feature of different
wenzelm@18537
   267
variables is their binding scope.}
wenzelm@18537
   268
wenzelm@18537
   269
*}
wenzelm@18537
   270
wenzelm@20491
   271
wenzelm@20491
   272
section {* Proof terms *}
wenzelm@18537
   273
wenzelm@20491
   274
text {*
wenzelm@20491
   275
  FIXME !?
wenzelm@20491
   276
*}
wenzelm@18537
   277
wenzelm@20480
   278
wenzelm@20491
   279
section {* Rules \label{sec:rules} *}
wenzelm@18537
   280
wenzelm@18537
   281
text {*
wenzelm@18537
   282
wenzelm@18537
   283
FIXME
wenzelm@18537
   284
wenzelm@20491
   285
  A \emph{rule} is any Pure theorem in HHF normal form; there is a
wenzelm@20491
   286
  separate calculus for rule composition, which is modeled after
wenzelm@20491
   287
  Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
wenzelm@20491
   288
  rules to be nested arbitrarily, similar to \cite{extensions91}.
wenzelm@20491
   289
wenzelm@20491
   290
  Normally, all theorems accessible to the user are proper rules.
wenzelm@20491
   291
  Low-level inferences are occasional required internally, but the
wenzelm@20491
   292
  result should be always presented in canonical form.  The higher
wenzelm@20491
   293
  interfaces of Isabelle/Isar will always produce proper rules.  It is
wenzelm@20491
   294
  important to maintain this invariant in add-on applications!
wenzelm@20491
   295
wenzelm@20491
   296
  There are two main principles of rule composition: @{text
wenzelm@20491
   297
  "resolution"} (i.e.\ backchaining of rules) and @{text
wenzelm@20491
   298
  "by-assumption"} (i.e.\ closing a branch); both principles are
wenzelm@20491
   299
  combined in the variants of @{text "elim-resosultion"} and @{text
wenzelm@20491
   300
  "dest-resolution"}.  Raw @{text "composition"} is occasionally
wenzelm@20491
   301
  useful as well, also it is strictly speaking outside of the proper
wenzelm@20491
   302
  rule calculus.
wenzelm@20491
   303
wenzelm@20491
   304
  Rules are treated modulo general higher-order unification, which is
wenzelm@20491
   305
  unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion
wenzelm@20491
   306
  on @{text "\<lambda>"}-terms.  Moreover, propositions are understood modulo
wenzelm@20491
   307
  the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.
wenzelm@20491
   308
wenzelm@20491
   309
  This means that any operations within the rule calculus may be
wenzelm@20491
   310
  subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions.  It is common
wenzelm@20491
   311
  practice not to contract or expand unnecessarily.  Some mechanisms
wenzelm@20491
   312
  prefer an one form, others the opposite, so there is a potential
wenzelm@20491
   313
  danger to produce some oscillation!
wenzelm@20491
   314
wenzelm@20491
   315
  Only few operations really work \emph{modulo} HHF conversion, but
wenzelm@20491
   316
  expect a normal form: quantifiers @{text "\<And>"} before implications
wenzelm@20491
   317
  @{text "\<Longrightarrow>"} at each level of nesting.
wenzelm@20491
   318
wenzelm@18537
   319
\glossary{Hereditary Harrop Formula}{The set of propositions in HHF
wenzelm@18537
   320
format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow>
wenzelm@18537
   321
A)"}, for variables @{text "x"} and atomic propositions @{text "A"}.
wenzelm@18537
   322
Any proposition may be put into HHF form by normalizing with the rule
wenzelm@18537
   323
@{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.  In Isabelle, the outermost
wenzelm@18537
   324
quantifier prefix is represented via \seeglossary{schematic
wenzelm@18537
   325
variables}, such that the top-level structure is merely that of a
wenzelm@18537
   326
\seeglossary{Horn Clause}}.
wenzelm@18537
   327
wenzelm@18537
   328
\glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
wenzelm@18537
   329
wenzelm@18537
   330
*}
wenzelm@18537
   331
wenzelm@18537
   332
end