doc-src/IsarImplementation/Thy/logic.thy
changeset 20543 dc294418ff17
parent 20542 a54ca4e90874
child 20547 796ae7fa1049
     1.1 --- a/doc-src/IsarImplementation/Thy/logic.thy	Thu Sep 14 22:48:37 2006 +0200
     1.2 +++ b/doc-src/IsarImplementation/Thy/logic.thy	Fri Sep 15 16:49:41 2006 +0200
     1.3 @@ -15,7 +15,7 @@
     1.4    Isabelle/Pure.
     1.5  
     1.6    Following type-theoretic parlance, the Pure logic consists of three
     1.7 -  levels of @{text "\<lambda>"}-calculus with corresponding arrows: @{text
     1.8 +  levels of @{text "\<lambda>"}-calculus with corresponding arrows, @{text
     1.9    "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
    1.10    "\<And>"} for universal quantification (proofs depending on terms), and
    1.11    @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
    1.12 @@ -80,7 +80,7 @@
    1.13    
    1.14    A \emph{type} is defined inductively over type variables and type
    1.15    constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
    1.16 -  (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
    1.17 +  (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>"}.
    1.18  
    1.19    A \emph{type abbreviation} is a syntactic definition @{text
    1.20    "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
    1.21 @@ -110,10 +110,9 @@
    1.22    vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such
    1.23    that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>,
    1.24    \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}.
    1.25 -  Consequently, unification on the algebra of types has most general
    1.26 -  solutions (modulo equivalence of sorts).  This means that
    1.27 -  type-inference will produce primary types as expected
    1.28 -  \cite{nipkow-prehofer}.
    1.29 +  Consequently, type unification has most general solutions (modulo
    1.30 +  equivalence of sorts), so type-inference produces primary types as
    1.31 +  expected \cite{nipkow-prehofer}.
    1.32  *}
    1.33  
    1.34  text %mlref {*
    1.35 @@ -176,7 +175,7 @@
    1.36    relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
    1.37  
    1.38    \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
    1.39 -  c\<^isub>2)"} declares class relation @{text "c\<^isub>1 \<subseteq>
    1.40 +  c\<^isub>2)"} declares the class relation @{text "c\<^isub>1 \<subseteq>
    1.41    c\<^isub>2"}.
    1.42  
    1.43    \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
    1.44 @@ -201,17 +200,19 @@
    1.45    \medskip A \emph{bound variable} is a natural number @{text "b"},
    1.46    which accounts for the number of intermediate binders between the
    1.47    variable occurrence in the body and its binding position.  For
    1.48 -  example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"}
    1.49 -  would correspond to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a
    1.50 -  named representation.  Note that a bound variable may be represented
    1.51 -  by different de-Bruijn indices at different occurrences, depending
    1.52 -  on the nesting of abstractions.
    1.53 +  example, the de-Bruijn term @{text
    1.54 +  "\<lambda>\<^bsub>nat\<^esub>. \<lambda>\<^bsub>nat\<^esub>. 1 + 0"} would
    1.55 +  correspond to @{text
    1.56 +  "\<lambda>x\<^bsub>nat\<^esub>. \<lambda>y\<^bsub>nat\<^esub>. x + y"} in a named
    1.57 +  representation.  Note that a bound variable may be represented by
    1.58 +  different de-Bruijn indices at different occurrences, depending on
    1.59 +  the nesting of abstractions.
    1.60  
    1.61 -  A \emph{loose variables} is a bound variable that is outside the
    1.62 +  A \emph{loose variable} is a bound variable that is outside the
    1.63    scope of local binders.  The types (and names) for loose variables
    1.64 -  can be managed as a separate context, that is maintained inside-out
    1.65 -  like a stack of hypothetical binders.  The core logic only operates
    1.66 -  on closed terms, without any loose variables.
    1.67 +  can be managed as a separate context, that is maintained as a stack
    1.68 +  of hypothetical binders.  The core logic operates on closed terms,
    1.69 +  without any loose variables.
    1.70  
    1.71    A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
    1.72    @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}.  A
    1.73 @@ -222,8 +223,8 @@
    1.74    \medskip A \emph{constant} is a pair of a basic name and a type,
    1.75    e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text
    1.76    "c\<^isub>\<tau>"}.  Constants are declared in the context as polymorphic
    1.77 -  families @{text "c :: \<sigma>"}, meaning that valid all substitution
    1.78 -  instances @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
    1.79 +  families @{text "c :: \<sigma>"}, meaning that all substitution instances
    1.80 +  @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
    1.81  
    1.82    The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"}
    1.83    wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of
    1.84 @@ -243,13 +244,14 @@
    1.85    polymorphic constants that the user-level type-checker would reject
    1.86    due to violation of type class restrictions.
    1.87  
    1.88 -  \medskip A \emph{term} is defined inductively over variables and
    1.89 -  constants, with abstraction and application as follows: @{text "t =
    1.90 -  b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t |
    1.91 -  t\<^isub>1 t\<^isub>2"}.  Parsing and printing takes care of
    1.92 -  converting between an external representation with named bound
    1.93 -  variables.  Subsequently, we shall use the latter notation instead
    1.94 -  of internal de-Bruijn representation.
    1.95 +  \medskip An \emph{atomic} term is either a variable or constant.  A
    1.96 +  \emph{term} is defined inductively over atomic terms, with
    1.97 +  abstraction and application as follows: @{text "t = b | x\<^isub>\<tau> |
    1.98 +  ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}.
    1.99 +  Parsing and printing takes care of converting between an external
   1.100 +  representation with named bound variables.  Subsequently, we shall
   1.101 +  use the latter notation instead of internal de-Bruijn
   1.102 +  representation.
   1.103  
   1.104    The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a
   1.105    term according to the structure of atomic terms, abstractions, and
   1.106 @@ -275,25 +277,22 @@
   1.107    "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type
   1.108    instantiation.  Some outer layers of the system make it hard to
   1.109    produce variables of the same name, but different types.  In
   1.110 -  particular, type-inference always demands ``consistent'' type
   1.111 -  constraints for free variables.  In contrast, mixed instances of
   1.112 -  polymorphic constants occur frequently.
   1.113 +  contrast, mixed instances of polymorphic constants occur frequently.
   1.114  
   1.115    \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
   1.116    is the set of type variables occurring in @{text "t"}, but not in
   1.117    @{text "\<sigma>"}.  This means that the term implicitly depends on type
   1.118 -  arguments that are not accounted in result type, i.e.\ there are
   1.119 +  arguments that are not accounted in the result type, i.e.\ there are
   1.120    different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text
   1.121    "t\<vartheta>' :: \<sigma>"} with the same type.  This slightly
   1.122 -  pathological situation demands special care.
   1.123 +  pathological situation notoriously demands additional care.
   1.124  
   1.125    \medskip A \emph{term abbreviation} is a syntactic definition @{text
   1.126    "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"},
   1.127    without any hidden polymorphism.  A term abbreviation looks like a
   1.128 -  constant in the syntax, but is fully expanded before entering the
   1.129 -  logical core.  Abbreviations are usually reverted when printing
   1.130 -  terms, using the collective @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for
   1.131 -  higher-order rewriting.
   1.132 +  constant in the syntax, but is expanded before entering the logical
   1.133 +  core.  Abbreviations are usually reverted when printing terms, using
   1.134 +  @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for higher-order rewriting.
   1.135  
   1.136    \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
   1.137    "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free
   1.138 @@ -308,7 +307,7 @@
   1.139    implicit in the de-Bruijn representation.  Names for bound variables
   1.140    in abstractions are maintained separately as (meaningless) comments,
   1.141    mostly for parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-conversion is
   1.142 -  commonplace in various higher operations (\secref{sec:rules}) that
   1.143 +  commonplace in various standard operations (\secref{sec:rules}) that
   1.144    are based on higher-order unification and matching.
   1.145  *}
   1.146  
   1.147 @@ -379,9 +378,8 @@
   1.148  
   1.149    \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
   1.150    Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
   1.151 -  convert between the representations of polymorphic constants: the
   1.152 -  full type instance vs.\ the compact type arguments form (depending
   1.153 -  on the most general declaration given in the context).
   1.154 +  convert between two representations of polymorphic constants: full
   1.155 +  type instance vs.\ compact type arguments form.
   1.156  
   1.157    \end{description}
   1.158  *}
   1.159 @@ -426,7 +424,7 @@
   1.160    \seeglossary{type variable}.  The distinguishing feature of
   1.161    different variables is their binding scope. FIXME}
   1.162  
   1.163 -  A \emph{proposition} is a well-formed term of type @{text "prop"}, a
   1.164 +  A \emph{proposition} is a well-typed term of type @{text "prop"}, a
   1.165    \emph{theorem} is a proven proposition (depending on a context of
   1.166    hypotheses and the background theory).  Primitive inferences include
   1.167    plain natural deduction rules for the primary connectives @{text
   1.168 @@ -437,16 +435,16 @@
   1.169  subsection {* Primitive connectives and rules *}
   1.170  
   1.171  text {*
   1.172 -  The theory @{text "Pure"} contains declarations for the standard
   1.173 -  connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of the logical
   1.174 -  framework, see \figref{fig:pure-connectives}.  The derivability
   1.175 -  judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined
   1.176 -  inductively by the primitive inferences given in
   1.177 -  \figref{fig:prim-rules}, with the global restriction that hypotheses
   1.178 -  @{text "\<Gamma>"} may \emph{not} contain schematic variables.  The builtin
   1.179 -  equality is conceptually axiomatized as shown in
   1.180 +  The theory @{text "Pure"} contains constant declarations for the
   1.181 +  primitive connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of
   1.182 +  the logical framework, see \figref{fig:pure-connectives}.  The
   1.183 +  derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is
   1.184 +  defined inductively by the primitive inferences given in
   1.185 +  \figref{fig:prim-rules}, with the global restriction that the
   1.186 +  hypotheses must \emph{not} contain any schematic variables.  The
   1.187 +  builtin equality is conceptually axiomatized as shown in
   1.188    \figref{fig:pure-equality}, although the implementation works
   1.189 -  directly with derived inference rules.
   1.190 +  directly with derived inferences.
   1.191  
   1.192    \begin{figure}[htb]
   1.193    \begin{center}
   1.194 @@ -496,8 +494,8 @@
   1.195    The introduction and elimination rules for @{text "\<And>"} and @{text
   1.196    "\<Longrightarrow>"} are analogous to formation of dependently typed @{text
   1.197    "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
   1.198 -  are irrelevant in the Pure logic, though, they may never occur
   1.199 -  within propositions.  The system provides a runtime option to record
   1.200 +  are irrelevant in the Pure logic, though; they cannot occur within
   1.201 +  propositions.  The system provides a runtime option to record
   1.202    explicit proof terms for primitive inferences.  Thus all three
   1.203    levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for
   1.204    terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\
   1.205 @@ -505,19 +503,19 @@
   1.206  
   1.207    Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need
   1.208    not be recorded in the hypotheses, because the simple syntactic
   1.209 -  types of Pure are always inhabitable.  Typing ``assumptions'' @{text
   1.210 -  "x :: \<tau>"} are (implicitly) present only with occurrences of @{text
   1.211 -  "x\<^isub>\<tau>"} in the statement body.\footnote{This is the key
   1.212 -  difference ``@{text "\<lambda>HOL"}'' in the PTS framework
   1.213 -  \cite{Barendregt-Geuvers:2001}, where @{text "x : A"} hypotheses are
   1.214 -  treated explicitly for types, in the same way as propositions.}
   1.215 +  types of Pure are always inhabitable.  ``Assumptions'' @{text "x ::
   1.216 +  \<tau>"} for type-membership are only present as long as some @{text
   1.217 +  "x\<^isub>\<tau>"} occurs in the statement body.\footnote{This is the key
   1.218 +  difference to ``@{text "\<lambda>HOL"}'' in the PTS framework
   1.219 +  \cite{Barendregt-Geuvers:2001}, where hypotheses @{text "x : A"} are
   1.220 +  treated uniformly for propositions and types.}
   1.221  
   1.222    \medskip The axiomatization of a theory is implicitly closed by
   1.223    forming all instances of type and term variables: @{text "\<turnstile>
   1.224    A\<vartheta>"} holds for any substitution instance of an axiom
   1.225 -  @{text "\<turnstile> A"}.  By pushing substitution through derivations
   1.226 -  inductively, we get admissible @{text "generalize"} and @{text
   1.227 -  "instance"} rules shown in \figref{fig:subst-rules}.
   1.228 +  @{text "\<turnstile> A"}.  By pushing substitutions through derivations
   1.229 +  inductively, we also get admissible @{text "generalize"} and @{text
   1.230 +  "instance"} rules as shown in \figref{fig:subst-rules}.
   1.231  
   1.232    \begin{figure}[htb]
   1.233    \begin{center}
   1.234 @@ -540,38 +538,38 @@
   1.235    variables.
   1.236  
   1.237    In principle, variables could be substituted in hypotheses as well,
   1.238 -  but this would disrupt monotonicity reasoning: deriving @{text
   1.239 -  "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is correct, but
   1.240 -  @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold --- the result
   1.241 -  belongs to a different proof context.
   1.242 +  but this would disrupt the monotonicity of reasoning: deriving
   1.243 +  @{text "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is
   1.244 +  correct, but @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold:
   1.245 +  the result belongs to a different proof context.
   1.246  
   1.247 -  \medskip The system allows axioms to be stated either as plain
   1.248 -  propositions, or as arbitrary functions (``oracles'') that produce
   1.249 -  propositions depending on given arguments.  It is possible to trace
   1.250 -  the used of axioms (and defined theorems) in derivations.
   1.251 -  Invocations of oracles are recorded invariable.
   1.252 +  \medskip An \emph{oracle} is a function that produces axioms on the
   1.253 +  fly.  Logically, this is an instance of the @{text "axiom"} rule
   1.254 +  (\figref{fig:prim-rules}), but there is an operational difference.
   1.255 +  The system always records oracle invocations within derivations of
   1.256 +  theorems.  Tracing plain axioms (and named theorems) is optional.
   1.257  
   1.258    Axiomatizations should be limited to the bare minimum, typically as
   1.259    part of the initial logical basis of an object-logic formalization.
   1.260 -  Normally, theories will be developed definitionally, by stating
   1.261 -  restricted equalities over constants.
   1.262 +  Later on, theories are usually developed in a strictly definitional
   1.263 +  fashion, by stating only certain equalities over new constants.
   1.264  
   1.265    A \emph{simple definition} consists of a constant declaration @{text
   1.266 -  "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text
   1.267 -  "t"} is a closed term without any hidden polymorphism.  The RHS may
   1.268 -  depend on further defined constants, but not @{text "c"} itself.
   1.269 -  Definitions of constants with function type may be presented
   1.270 -  liberally as @{text "c \<^vec> \<equiv> t"} instead of the puristic @{text
   1.271 -  "c \<equiv> \<lambda>\<^vec>x. t"}.
   1.272 +  "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text "t
   1.273 +  :: \<sigma>"} is a closed term without any hidden polymorphism.  The RHS
   1.274 +  may depend on further defined constants, but not @{text "c"} itself.
   1.275 +  Definitions of functions may be presented as @{text "c \<^vec>x \<equiv>
   1.276 +  t"} instead of the puristic @{text "c \<equiv> \<lambda>\<^vec>x. t"}.
   1.277  
   1.278 -  An \emph{overloaded definition} consists may give zero or one
   1.279 -  equality axioms @{text "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type
   1.280 -  constructor @{text "\<kappa>"}, with distinct variables @{text "\<^vec>\<alpha>"}
   1.281 -  as formal arguments.  The RHS may mention previously defined
   1.282 -  constants as above, or arbitrary constants @{text "d(\<alpha>\<^isub>i)"}
   1.283 -  for some @{text "\<alpha>\<^isub>i"} projected from @{text "\<^vec>\<alpha>"}.
   1.284 -  Thus overloaded definitions essentially work by primitive recursion
   1.285 -  over the syntactic structure of a single type argument.
   1.286 +  An \emph{overloaded definition} consists of a collection of axioms
   1.287 +  for the same constant, with zero or one equations @{text
   1.288 +  "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type constructor @{text "\<kappa>"} (for
   1.289 +  distinct variables @{text "\<^vec>\<alpha>"}).  The RHS may mention
   1.290 +  previously defined constants as above, or arbitrary constants @{text
   1.291 +  "d(\<alpha>\<^isub>i)"} for some @{text "\<alpha>\<^isub>i"} projected from @{text
   1.292 +  "\<^vec>\<alpha>"}.  Thus overloaded definitions essentially work by
   1.293 +  primitive recursion over the syntactic structure of a single type
   1.294 +  argument.
   1.295  *}
   1.296  
   1.297  text %mlref {*
   1.298 @@ -612,10 +610,13 @@
   1.299    \item @{ML_type thm} represents proven propositions.  This is an
   1.300    abstract datatype that guarantees that its values have been
   1.301    constructed by basic principles of the @{ML_struct Thm} module.
   1.302 +  Every @{ML thm} value contains a sliding back-reference to the
   1.303 +  enclosing theory, cf.\ \secref{sec:context-theory}.
   1.304  
   1.305 -  \item @{ML proofs} determines the detail of proof recording: @{ML 0}
   1.306 -  records only oracles, @{ML 1} records oracles, axioms and named
   1.307 -  theorems, @{ML 2} records full proof terms.
   1.308 +  \item @{ML proofs} determines the detail of proof recording within
   1.309 +  @{ML_type thm} values: @{ML 0} records only oracles, @{ML 1} records
   1.310 +  oracles, axioms and named theorems, @{ML 2} records full proof
   1.311 +  terms.
   1.312  
   1.313    \item @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML
   1.314    Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim}
   1.315 @@ -623,8 +624,9 @@
   1.316  
   1.317    \item @{ML Thm.generalize}~@{text "(\<^vec>\<alpha>, \<^vec>x)"}
   1.318    corresponds to the @{text "generalize"} rules of
   1.319 -  \figref{fig:subst-rules}.  A collection of type and term variables
   1.320 -  is generalized simultaneously, according to the given basic names.
   1.321 +  \figref{fig:subst-rules}.  Here collections of type and term
   1.322 +  variables are generalized simultaneously, specified by the given
   1.323 +  basic names.
   1.324  
   1.325    \item @{ML Thm.instantiate}~@{text "(\<^vec>\<alpha>\<^isub>s,
   1.326    \<^vec>x\<^isub>\<tau>)"} corresponds to the @{text "instantiate"} rules
   1.327 @@ -635,45 +637,43 @@
   1.328    \item @{ML Thm.get_axiom_i}~@{text "thy name"} retrieves a named
   1.329    axiom, cf.\ @{text "axiom"} in \figref{fig:prim-rules}.
   1.330  
   1.331 -  \item @{ML Thm.invoke_oracle_i}~@{text "thy name arg"} invokes the
   1.332 -  oracle function @{text "name"} of the theory.  Logically, this is
   1.333 -  just another instance of @{text "axiom"} in \figref{fig:prim-rules},
   1.334 -  but the system records an explicit trace of oracle invocations with
   1.335 -  the @{text "thm"} value.
   1.336 +  \item @{ML Thm.invoke_oracle_i}~@{text "thy name arg"} invokes a
   1.337 +  named oracle function, cf.\ @{text "axiom"} in
   1.338 +  \figref{fig:prim-rules}.
   1.339  
   1.340 -  \item @{ML Theory.add_axioms_i}~@{text "[(name, A), \<dots>]"} adds
   1.341 -  arbitrary axioms, without any checking of the proposition.
   1.342 +  \item @{ML Theory.add_axioms_i}~@{text "[(name, A), \<dots>]"} declares
   1.343 +  arbitrary propositions as axioms.
   1.344  
   1.345 -  \item @{ML Theory.add_oracle}~@{text "(name, f)"} declares an
   1.346 -  oracle, i.e.\ a function for generating arbitrary axioms.
   1.347 +  \item @{ML Theory.add_oracle}~@{text "(name, f)"} declares an oracle
   1.348 +  function for generating arbitrary axioms on the fly.
   1.349  
   1.350    \item @{ML Theory.add_deps}~@{text "name c\<^isub>\<tau>
   1.351 -  \<^vec>d\<^isub>\<sigma>"} declares dependencies of a new specification for
   1.352 -  constant @{text "c\<^isub>\<tau>"} from relative to existing ones for
   1.353 -  constants @{text "\<^vec>d\<^isub>\<sigma>"}.
   1.354 +  \<^vec>d\<^isub>\<sigma>"} declares dependencies of a named specification
   1.355 +  for constant @{text "c\<^isub>\<tau>"}, relative to existing
   1.356 +  specifications for constants @{text "\<^vec>d\<^isub>\<sigma>"}.
   1.357  
   1.358    \item @{ML Theory.add_defs_i}~@{text "unchecked overloaded [(name, c
   1.359 -  \<^vec>x \<equiv> t), \<dots>]"} states a definitional axiom for an already
   1.360 -  declared constant @{text "c"}.  Dependencies are recorded using @{ML
   1.361 -  Theory.add_deps}, unless the @{text "unchecked"} option is set.
   1.362 +  \<^vec>x \<equiv> t), \<dots>]"} states a definitional axiom for an existing
   1.363 +  constant @{text "c"}.  Dependencies are recorded (cf.\ @{ML
   1.364 +  Theory.add_deps}), unless the @{text "unchecked"} option is set.
   1.365  
   1.366    \end{description}
   1.367  *}
   1.368  
   1.369  
   1.370 -subsection {* Auxiliary connectives *}
   1.371 +subsection {* Auxiliary definitions *}
   1.372  
   1.373  text {*
   1.374 -  Theory @{text "Pure"} also defines a few auxiliary connectives, see
   1.375 -  \figref{fig:pure-aux}.  These are normally not exposed to the user,
   1.376 -  but appear in internal encodings only.
   1.377 +  Theory @{text "Pure"} provides a few auxiliary definitions, see
   1.378 +  \figref{fig:pure-aux}.  These special constants are normally not
   1.379 +  exposed to the user, but appear in internal encodings.
   1.380  
   1.381    \begin{figure}[htb]
   1.382    \begin{center}
   1.383    \begin{tabular}{ll}
   1.384    @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\
   1.385    @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex]
   1.386 -  @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, hidden) \\
   1.387 +  @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, suppressed) \\
   1.388    @{text "#A \<equiv> A"} \\[1ex]
   1.389    @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\
   1.390    @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex]
   1.391 @@ -688,9 +688,9 @@
   1.392    B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}.
   1.393    Conjunction allows to treat simultaneous assumptions and conclusions
   1.394    uniformly.  For example, multiple claims are intermediately
   1.395 -  represented as explicit conjunction, but this is usually refined
   1.396 -  into separate sub-goals before the user continues the proof; the
   1.397 -  final result is projected into a list of theorems (cf.\
   1.398 +  represented as explicit conjunction, but this is refined into
   1.399 +  separate sub-goals before the user continues the proof; the final
   1.400 +  result is projected into a list of theorems (cf.\
   1.401    \secref{sec:tactical-goals}).
   1.402  
   1.403    The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex
   1.404 @@ -698,10 +698,10 @@
   1.405    "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable.  See
   1.406    \secref{sec:tactical-goals} for specific operations.
   1.407  
   1.408 -  The @{text "term"} marker turns any well-formed term into a
   1.409 -  derivable proposition: @{text "\<turnstile> TERM t"} holds unconditionally.
   1.410 -  Although this is logically vacuous, it allows to treat terms and
   1.411 -  proofs uniformly, similar to a type-theoretic framework.
   1.412 +  The @{text "term"} marker turns any well-typed term into a derivable
   1.413 +  proposition: @{text "\<turnstile> TERM t"} holds unconditionally.  Although
   1.414 +  this is logically vacuous, it allows to treat terms and proofs
   1.415 +  uniformly, similar to a type-theoretic framework.
   1.416  
   1.417    The @{text "TYPE"} constructor is the canonical representative of
   1.418    the unspecified type @{text "\<alpha> itself"}; it essentially injects the
   1.419 @@ -733,13 +733,13 @@
   1.420    \item @{ML Conjunction.intr} derives @{text "A & B"} from @{text
   1.421    "A"} and @{text "B"}.
   1.422  
   1.423 -  \item @{ML Conjunction.intr} derives @{text "A"} and @{text "B"}
   1.424 +  \item @{ML Conjunction.elim} derives @{text "A"} and @{text "B"}
   1.425    from @{text "A & B"}.
   1.426  
   1.427 -  \item @{ML Drule.mk_term}~@{text "t"} derives @{text "TERM t"}.
   1.428 +  \item @{ML Drule.mk_term} derives @{text "TERM t"}.
   1.429  
   1.430 -  \item @{ML Drule.dest_term}~@{text "TERM t"} recovers term @{text
   1.431 -  "t"}.
   1.432 +  \item @{ML Drule.dest_term} recovers term @{text "t"} from @{text
   1.433 +  "TERM t"}.
   1.434  
   1.435    \item @{ML Logic.mk_type}~@{text "\<tau>"} produces the term @{text
   1.436    "TYPE(\<tau>)"}.