src/Doc/IsarImplementation/Logic.thy
changeset 48985 5386df44a037
parent 48938 d468d72a458f
child 50126 3dec88149176
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/Doc/IsarImplementation/Logic.thy	Tue Aug 28 18:57:32 2012 +0200
     1.3 @@ -0,0 +1,1137 @@
     1.4 +theory Logic
     1.5 +imports Base
     1.6 +begin
     1.7 +
     1.8 +chapter {* Primitive logic \label{ch:logic} *}
     1.9 +
    1.10 +text {*
    1.11 +  The logical foundations of Isabelle/Isar are that of the Pure logic,
    1.12 +  which has been introduced as a Natural Deduction framework in
    1.13 +  \cite{paulson700}.  This is essentially the same logic as ``@{text
    1.14 +  "\<lambda>HOL"}'' in the more abstract setting of Pure Type Systems (PTS)
    1.15 +  \cite{Barendregt-Geuvers:2001}, although there are some key
    1.16 +  differences in the specific treatment of simple types in
    1.17 +  Isabelle/Pure.
    1.18 +
    1.19 +  Following type-theoretic parlance, the Pure logic consists of three
    1.20 +  levels of @{text "\<lambda>"}-calculus with corresponding arrows, @{text
    1.21 +  "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
    1.22 +  "\<And>"} for universal quantification (proofs depending on terms), and
    1.23 +  @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
    1.24 +
    1.25 +  Derivations are relative to a logical theory, which declares type
    1.26 +  constructors, constants, and axioms.  Theory declarations support
    1.27 +  schematic polymorphism, which is strictly speaking outside the
    1.28 +  logic.\footnote{This is the deeper logical reason, why the theory
    1.29 +  context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"}
    1.30 +  of the core calculus: type constructors, term constants, and facts
    1.31 +  (proof constants) may involve arbitrary type schemes, but the type
    1.32 +  of a locally fixed term parameter is also fixed!}
    1.33 +*}
    1.34 +
    1.35 +
    1.36 +section {* Types \label{sec:types} *}
    1.37 +
    1.38 +text {*
    1.39 +  The language of types is an uninterpreted order-sorted first-order
    1.40 +  algebra; types are qualified by ordered type classes.
    1.41 +
    1.42 +  \medskip A \emph{type class} is an abstract syntactic entity
    1.43 +  declared in the theory context.  The \emph{subclass relation} @{text
    1.44 +  "c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic
    1.45 +  generating relation; the transitive closure is maintained
    1.46 +  internally.  The resulting relation is an ordering: reflexive,
    1.47 +  transitive, and antisymmetric.
    1.48 +
    1.49 +  A \emph{sort} is a list of type classes written as @{text "s = {c\<^isub>1,
    1.50 +  \<dots>, c\<^isub>m}"}, it represents symbolic intersection.  Notationally, the
    1.51 +  curly braces are omitted for singleton intersections, i.e.\ any
    1.52 +  class @{text "c"} may be read as a sort @{text "{c}"}.  The ordering
    1.53 +  on type classes is extended to sorts according to the meaning of
    1.54 +  intersections: @{text "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff @{text
    1.55 +  "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}.  The empty intersection @{text "{}"} refers to
    1.56 +  the universal sort, which is the largest element wrt.\ the sort
    1.57 +  order.  Thus @{text "{}"} represents the ``full sort'', not the
    1.58 +  empty one!  The intersection of all (finitely many) classes declared
    1.59 +  in the current theory is the least element wrt.\ the sort ordering.
    1.60 +
    1.61 +  \medskip A \emph{fixed type variable} is a pair of a basic name
    1.62 +  (starting with a @{text "'"} character) and a sort constraint, e.g.\
    1.63 +  @{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}.
    1.64 +  A \emph{schematic type variable} is a pair of an indexname and a
    1.65 +  sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually
    1.66 +  printed as @{text "?\<alpha>\<^isub>s"}.
    1.67 +
    1.68 +  Note that \emph{all} syntactic components contribute to the identity
    1.69 +  of type variables: basic name, index, and sort constraint.  The core
    1.70 +  logic handles type variables with the same name but different sorts
    1.71 +  as different, although the type-inference layer (which is outside
    1.72 +  the core) rejects anything like that.
    1.73 +
    1.74 +  A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
    1.75 +  on types declared in the theory.  Type constructor application is
    1.76 +  written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.  For
    1.77 +  @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"}
    1.78 +  instead of @{text "()prop"}.  For @{text "k = 1"} the parentheses
    1.79 +  are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}.
    1.80 +  Further notation is provided for specific constructors, notably the
    1.81 +  right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>,
    1.82 +  \<beta>)fun"}.
    1.83 +  
    1.84 +  The logical category \emph{type} is defined inductively over type
    1.85 +  variables and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
    1.86 +  (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>"}.
    1.87 +
    1.88 +  A \emph{type abbreviation} is a syntactic definition @{text
    1.89 +  "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
    1.90 +  variables @{text "\<^vec>\<alpha>"}.  Type abbreviations appear as type
    1.91 +  constructors in the syntax, but are expanded before entering the
    1.92 +  logical core.
    1.93 +
    1.94 +  A \emph{type arity} declares the image behavior of a type
    1.95 +  constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
    1.96 +  s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)\<kappa>"} is
    1.97 +  of sort @{text "s"} if every argument type @{text "\<tau>\<^isub>i"} is
    1.98 +  of sort @{text "s\<^isub>i"}.  Arity declarations are implicitly
    1.99 +  completed, i.e.\ @{text "\<kappa> :: (\<^vec>s)c"} entails @{text "\<kappa> ::
   1.100 +  (\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}.
   1.101 +
   1.102 +  \medskip The sort algebra is always maintained as \emph{coregular},
   1.103 +  which means that type arities are consistent with the subclass
   1.104 +  relation: for any type constructor @{text "\<kappa>"}, and classes @{text
   1.105 +  "c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> ::
   1.106 +  (\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> ::
   1.107 +  (\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq>
   1.108 +  \<^vec>s\<^isub>2"} component-wise.
   1.109 +
   1.110 +  The key property of a coregular order-sorted algebra is that sort
   1.111 +  constraints can be solved in a most general fashion: for each type
   1.112 +  constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general
   1.113 +  vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such
   1.114 +  that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>,
   1.115 +  \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}.
   1.116 +  Consequently, type unification has most general solutions (modulo
   1.117 +  equivalence of sorts), so type-inference produces primary types as
   1.118 +  expected \cite{nipkow-prehofer}.
   1.119 +*}
   1.120 +
   1.121 +text %mlref {*
   1.122 +  \begin{mldecls}
   1.123 +  @{index_ML_type class: string} \\
   1.124 +  @{index_ML_type sort: "class list"} \\
   1.125 +  @{index_ML_type arity: "string * sort list * sort"} \\
   1.126 +  @{index_ML_type typ} \\
   1.127 +  @{index_ML Term.map_atyps: "(typ -> typ) -> typ -> typ"} \\
   1.128 +  @{index_ML Term.fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
   1.129 +  \end{mldecls}
   1.130 +  \begin{mldecls}
   1.131 +  @{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
   1.132 +  @{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
   1.133 +  @{index_ML Sign.add_type: "Proof.context -> binding * int * mixfix -> theory -> theory"} \\
   1.134 +  @{index_ML Sign.add_type_abbrev: "Proof.context ->
   1.135 +  binding * string list * typ -> theory -> theory"} \\
   1.136 +  @{index_ML Sign.primitive_class: "binding * class list -> theory -> theory"} \\
   1.137 +  @{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
   1.138 +  @{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
   1.139 +  \end{mldecls}
   1.140 +
   1.141 +  \begin{description}
   1.142 +
   1.143 +  \item Type @{ML_type class} represents type classes.
   1.144 +
   1.145 +  \item Type @{ML_type sort} represents sorts, i.e.\ finite
   1.146 +  intersections of classes.  The empty list @{ML "[]: sort"} refers to
   1.147 +  the empty class intersection, i.e.\ the ``full sort''.
   1.148 +
   1.149 +  \item Type @{ML_type arity} represents type arities.  A triple
   1.150 +  @{text "(\<kappa>, \<^vec>s, s) : arity"} represents @{text "\<kappa> ::
   1.151 +  (\<^vec>s)s"} as described above.
   1.152 +
   1.153 +  \item Type @{ML_type typ} represents types; this is a datatype with
   1.154 +  constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
   1.155 +
   1.156 +  \item @{ML Term.map_atyps}~@{text "f \<tau>"} applies the mapping @{text
   1.157 +  "f"} to all atomic types (@{ML TFree}, @{ML TVar}) occurring in
   1.158 +  @{text "\<tau>"}.
   1.159 +
   1.160 +  \item @{ML Term.fold_atyps}~@{text "f \<tau>"} iterates the operation
   1.161 +  @{text "f"} over all occurrences of atomic types (@{ML TFree}, @{ML
   1.162 +  TVar}) in @{text "\<tau>"}; the type structure is traversed from left to
   1.163 +  right.
   1.164 +
   1.165 +  \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
   1.166 +  tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
   1.167 +
   1.168 +  \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether type
   1.169 +  @{text "\<tau>"} is of sort @{text "s"}.
   1.170 +
   1.171 +  \item @{ML Sign.add_type}~@{text "ctxt (\<kappa>, k, mx)"} declares a
   1.172 +  new type constructors @{text "\<kappa>"} with @{text "k"} arguments and
   1.173 +  optional mixfix syntax.
   1.174 +
   1.175 +  \item @{ML Sign.add_type_abbrev}~@{text "ctxt (\<kappa>, \<^vec>\<alpha>, \<tau>)"}
   1.176 +  defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"}.
   1.177 +
   1.178 +  \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
   1.179 +  c\<^isub>n])"} declares a new class @{text "c"}, together with class
   1.180 +  relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
   1.181 +
   1.182 +  \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
   1.183 +  c\<^isub>2)"} declares the class relation @{text "c\<^isub>1 \<subseteq>
   1.184 +  c\<^isub>2"}.
   1.185 +
   1.186 +  \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
   1.187 +  the arity @{text "\<kappa> :: (\<^vec>s)s"}.
   1.188 +
   1.189 +  \end{description}
   1.190 +*}
   1.191 +
   1.192 +text %mlantiq {*
   1.193 +  \begin{matharray}{rcl}
   1.194 +  @{ML_antiquotation_def "class"} & : & @{text ML_antiquotation} \\
   1.195 +  @{ML_antiquotation_def "sort"} & : & @{text ML_antiquotation} \\
   1.196 +  @{ML_antiquotation_def "type_name"} & : & @{text ML_antiquotation} \\
   1.197 +  @{ML_antiquotation_def "type_abbrev"} & : & @{text ML_antiquotation} \\
   1.198 +  @{ML_antiquotation_def "nonterminal"} & : & @{text ML_antiquotation} \\
   1.199 +  @{ML_antiquotation_def "typ"} & : & @{text ML_antiquotation} \\
   1.200 +  \end{matharray}
   1.201 +
   1.202 +  @{rail "
   1.203 +  @@{ML_antiquotation class} nameref
   1.204 +  ;
   1.205 +  @@{ML_antiquotation sort} sort
   1.206 +  ;
   1.207 +  (@@{ML_antiquotation type_name} |
   1.208 +   @@{ML_antiquotation type_abbrev} |
   1.209 +   @@{ML_antiquotation nonterminal}) nameref
   1.210 +  ;
   1.211 +  @@{ML_antiquotation typ} type
   1.212 +  "}
   1.213 +
   1.214 +  \begin{description}
   1.215 +
   1.216 +  \item @{text "@{class c}"} inlines the internalized class @{text
   1.217 +  "c"} --- as @{ML_type string} literal.
   1.218 +
   1.219 +  \item @{text "@{sort s}"} inlines the internalized sort @{text "s"}
   1.220 +  --- as @{ML_type "string list"} literal.
   1.221 +
   1.222 +  \item @{text "@{type_name c}"} inlines the internalized type
   1.223 +  constructor @{text "c"} --- as @{ML_type string} literal.
   1.224 +
   1.225 +  \item @{text "@{type_abbrev c}"} inlines the internalized type
   1.226 +  abbreviation @{text "c"} --- as @{ML_type string} literal.
   1.227 +
   1.228 +  \item @{text "@{nonterminal c}"} inlines the internalized syntactic
   1.229 +  type~/ grammar nonterminal @{text "c"} --- as @{ML_type string}
   1.230 +  literal.
   1.231 +
   1.232 +  \item @{text "@{typ \<tau>}"} inlines the internalized type @{text "\<tau>"}
   1.233 +  --- as constructor term for datatype @{ML_type typ}.
   1.234 +
   1.235 +  \end{description}
   1.236 +*}
   1.237 +
   1.238 +
   1.239 +section {* Terms \label{sec:terms} *}
   1.240 +
   1.241 +text {*
   1.242 +  The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
   1.243 +  with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
   1.244 +  or \cite{paulson-ml2}), with the types being determined by the
   1.245 +  corresponding binders.  In contrast, free variables and constants
   1.246 +  have an explicit name and type in each occurrence.
   1.247 +
   1.248 +  \medskip A \emph{bound variable} is a natural number @{text "b"},
   1.249 +  which accounts for the number of intermediate binders between the
   1.250 +  variable occurrence in the body and its binding position.  For
   1.251 +  example, the de-Bruijn term @{text "\<lambda>\<^bsub>bool\<^esub>. \<lambda>\<^bsub>bool\<^esub>. 1 \<and> 0"} would
   1.252 +  correspond to @{text "\<lambda>x\<^bsub>bool\<^esub>. \<lambda>y\<^bsub>bool\<^esub>. x \<and> y"} in a named
   1.253 +  representation.  Note that a bound variable may be represented by
   1.254 +  different de-Bruijn indices at different occurrences, depending on
   1.255 +  the nesting of abstractions.
   1.256 +
   1.257 +  A \emph{loose variable} is a bound variable that is outside the
   1.258 +  scope of local binders.  The types (and names) for loose variables
   1.259 +  can be managed as a separate context, that is maintained as a stack
   1.260 +  of hypothetical binders.  The core logic operates on closed terms,
   1.261 +  without any loose variables.
   1.262 +
   1.263 +  A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
   1.264 +  @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"} here.  A
   1.265 +  \emph{schematic variable} is a pair of an indexname and a type,
   1.266 +  e.g.\ @{text "((x, 0), \<tau>)"} which is likewise printed as @{text
   1.267 +  "?x\<^isub>\<tau>"}.
   1.268 +
   1.269 +  \medskip A \emph{constant} is a pair of a basic name and a type,
   1.270 +  e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text "c\<^isub>\<tau>"}
   1.271 +  here.  Constants are declared in the context as polymorphic families
   1.272 +  @{text "c :: \<sigma>"}, meaning that all substitution instances @{text
   1.273 +  "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
   1.274 +
   1.275 +  The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"} wrt.\
   1.276 +  the declaration @{text "c :: \<sigma>"} is defined as the codomain of the
   1.277 +  matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>, ?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in
   1.278 +  canonical order @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}, corresponding to the
   1.279 +  left-to-right occurrences of the @{text "\<alpha>\<^isub>i"} in @{text "\<sigma>"}.
   1.280 +  Within a given theory context, there is a one-to-one correspondence
   1.281 +  between any constant @{text "c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1,
   1.282 +  \<dots>, \<tau>\<^isub>n)"} of its type arguments.  For example, with @{text "plus :: \<alpha>
   1.283 +  \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} corresponds to
   1.284 +  @{text "plus(nat)"}.
   1.285 +
   1.286 +  Constant declarations @{text "c :: \<sigma>"} may contain sort constraints
   1.287 +  for type variables in @{text "\<sigma>"}.  These are observed by
   1.288 +  type-inference as expected, but \emph{ignored} by the core logic.
   1.289 +  This means the primitive logic is able to reason with instances of
   1.290 +  polymorphic constants that the user-level type-checker would reject
   1.291 +  due to violation of type class restrictions.
   1.292 +
   1.293 +  \medskip An \emph{atomic term} is either a variable or constant.
   1.294 +  The logical category \emph{term} is defined inductively over atomic
   1.295 +  terms, with abstraction and application as follows: @{text "t = b |
   1.296 +  x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}.  Parsing and printing takes care of
   1.297 +  converting between an external representation with named bound
   1.298 +  variables.  Subsequently, we shall use the latter notation instead
   1.299 +  of internal de-Bruijn representation.
   1.300 +
   1.301 +  The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a
   1.302 +  term according to the structure of atomic terms, abstractions, and
   1.303 +  applicatins:
   1.304 +  \[
   1.305 +  \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{}
   1.306 +  \qquad
   1.307 +  \infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}}
   1.308 +  \qquad
   1.309 +  \infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}}
   1.310 +  \]
   1.311 +  A \emph{well-typed term} is a term that can be typed according to these rules.
   1.312 +
   1.313 +  Typing information can be omitted: type-inference is able to
   1.314 +  reconstruct the most general type of a raw term, while assigning
   1.315 +  most general types to all of its variables and constants.
   1.316 +  Type-inference depends on a context of type constraints for fixed
   1.317 +  variables, and declarations for polymorphic constants.
   1.318 +
   1.319 +  The identity of atomic terms consists both of the name and the type
   1.320 +  component.  This means that different variables @{text
   1.321 +  "x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after
   1.322 +  type instantiation.  Type-inference rejects variables of the same
   1.323 +  name, but different types.  In contrast, mixed instances of
   1.324 +  polymorphic constants occur routinely.
   1.325 +
   1.326 +  \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
   1.327 +  is the set of type variables occurring in @{text "t"}, but not in
   1.328 +  its type @{text "\<sigma>"}.  This means that the term implicitly depends
   1.329 +  on type arguments that are not accounted in the result type, i.e.\
   1.330 +  there are different type instances @{text "t\<vartheta> :: \<sigma>"} and
   1.331 +  @{text "t\<vartheta>' :: \<sigma>"} with the same type.  This slightly
   1.332 +  pathological situation notoriously demands additional care.
   1.333 +
   1.334 +  \medskip A \emph{term abbreviation} is a syntactic definition @{text
   1.335 +  "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"},
   1.336 +  without any hidden polymorphism.  A term abbreviation looks like a
   1.337 +  constant in the syntax, but is expanded before entering the logical
   1.338 +  core.  Abbreviations are usually reverted when printing terms, using
   1.339 +  @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for higher-order rewriting.
   1.340 +
   1.341 +  \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
   1.342 +  "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free
   1.343 +  renaming of bound variables; @{text "\<beta>"}-conversion contracts an
   1.344 +  abstraction applied to an argument term, substituting the argument
   1.345 +  in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
   1.346 +  "\<eta>"}-conversion contracts vacuous application-abstraction: @{text
   1.347 +  "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
   1.348 +  does not occur in @{text "f"}.
   1.349 +
   1.350 +  Terms are normally treated modulo @{text "\<alpha>"}-conversion, which is
   1.351 +  implicit in the de-Bruijn representation.  Names for bound variables
   1.352 +  in abstractions are maintained separately as (meaningless) comments,
   1.353 +  mostly for parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-conversion is
   1.354 +  commonplace in various standard operations (\secref{sec:obj-rules})
   1.355 +  that are based on higher-order unification and matching.
   1.356 +*}
   1.357 +
   1.358 +text %mlref {*
   1.359 +  \begin{mldecls}
   1.360 +  @{index_ML_type term} \\
   1.361 +  @{index_ML_op "aconv": "term * term -> bool"} \\
   1.362 +  @{index_ML Term.map_types: "(typ -> typ) -> term -> term"} \\
   1.363 +  @{index_ML Term.fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
   1.364 +  @{index_ML Term.map_aterms: "(term -> term) -> term -> term"} \\
   1.365 +  @{index_ML Term.fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\
   1.366 +  \end{mldecls}
   1.367 +  \begin{mldecls}
   1.368 +  @{index_ML fastype_of: "term -> typ"} \\
   1.369 +  @{index_ML lambda: "term -> term -> term"} \\
   1.370 +  @{index_ML betapply: "term * term -> term"} \\
   1.371 +  @{index_ML incr_boundvars: "int -> term -> term"} \\
   1.372 +  @{index_ML Sign.declare_const: "Proof.context ->
   1.373 +  (binding * typ) * mixfix -> theory -> term * theory"} \\
   1.374 +  @{index_ML Sign.add_abbrev: "string -> binding * term ->
   1.375 +  theory -> (term * term) * theory"} \\
   1.376 +  @{index_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\
   1.377 +  @{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
   1.378 +  \end{mldecls}
   1.379 +
   1.380 +  \begin{description}
   1.381 +
   1.382 +  \item Type @{ML_type term} represents de-Bruijn terms, with comments
   1.383 +  in abstractions, and explicitly named free variables and constants;
   1.384 +  this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML
   1.385 +  Var}, @{ML Const}, @{ML Abs}, @{ML_op "$"}.
   1.386 +
   1.387 +  \item @{text "t"}~@{ML_text aconv}~@{text "u"} checks @{text
   1.388 +  "\<alpha>"}-equivalence of two terms.  This is the basic equality relation
   1.389 +  on type @{ML_type term}; raw datatype equality should only be used
   1.390 +  for operations related to parsing or printing!
   1.391 +
   1.392 +  \item @{ML Term.map_types}~@{text "f t"} applies the mapping @{text
   1.393 +  "f"} to all types occurring in @{text "t"}.
   1.394 +
   1.395 +  \item @{ML Term.fold_types}~@{text "f t"} iterates the operation
   1.396 +  @{text "f"} over all occurrences of types in @{text "t"}; the term
   1.397 +  structure is traversed from left to right.
   1.398 +
   1.399 +  \item @{ML Term.map_aterms}~@{text "f t"} applies the mapping @{text
   1.400 +  "f"} to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
   1.401 +  Const}) occurring in @{text "t"}.
   1.402 +
   1.403 +  \item @{ML Term.fold_aterms}~@{text "f t"} iterates the operation
   1.404 +  @{text "f"} over all occurrences of atomic terms (@{ML Bound}, @{ML
   1.405 +  Free}, @{ML Var}, @{ML Const}) in @{text "t"}; the term structure is
   1.406 +  traversed from left to right.
   1.407 +
   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 atomic term @{text "a"} in the
   1.414 +  body @{text "b"} are replaced by bound variables.
   1.415 +
   1.416 +  \item @{ML betapply}~@{text "(t, u)"} produces an application @{text
   1.417 +  "t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an
   1.418 +  abstraction.
   1.419 +
   1.420 +  \item @{ML incr_boundvars}~@{text "j"} increments a term's dangling
   1.421 +  bound variables by the offset @{text "j"}.  This is required when
   1.422 +  moving a subterm into a context where it is enclosed by a different
   1.423 +  number of abstractions.  Bound variables with a matching abstraction
   1.424 +  are unaffected.
   1.425 +
   1.426 +  \item @{ML Sign.declare_const}~@{text "ctxt ((c, \<sigma>), mx)"} declares
   1.427 +  a new constant @{text "c :: \<sigma>"} with optional mixfix syntax.
   1.428 +
   1.429 +  \item @{ML Sign.add_abbrev}~@{text "print_mode (c, t)"}
   1.430 +  introduces a new term abbreviation @{text "c \<equiv> t"}.
   1.431 +
   1.432 +  \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
   1.433 +  Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
   1.434 +  convert between two representations of polymorphic constants: full
   1.435 +  type instance vs.\ compact type arguments form.
   1.436 +
   1.437 +  \end{description}
   1.438 +*}
   1.439 +
   1.440 +text %mlantiq {*
   1.441 +  \begin{matharray}{rcl}
   1.442 +  @{ML_antiquotation_def "const_name"} & : & @{text ML_antiquotation} \\
   1.443 +  @{ML_antiquotation_def "const_abbrev"} & : & @{text ML_antiquotation} \\
   1.444 +  @{ML_antiquotation_def "const"} & : & @{text ML_antiquotation} \\
   1.445 +  @{ML_antiquotation_def "term"} & : & @{text ML_antiquotation} \\
   1.446 +  @{ML_antiquotation_def "prop"} & : & @{text ML_antiquotation} \\
   1.447 +  \end{matharray}
   1.448 +
   1.449 +  @{rail "
   1.450 +  (@@{ML_antiquotation const_name} |
   1.451 +   @@{ML_antiquotation const_abbrev}) nameref
   1.452 +  ;
   1.453 +  @@{ML_antiquotation const} ('(' (type + ',') ')')?
   1.454 +  ;
   1.455 +  @@{ML_antiquotation term} term
   1.456 +  ;
   1.457 +  @@{ML_antiquotation prop} prop
   1.458 +  "}
   1.459 +
   1.460 +  \begin{description}
   1.461 +
   1.462 +  \item @{text "@{const_name c}"} inlines the internalized logical
   1.463 +  constant name @{text "c"} --- as @{ML_type string} literal.
   1.464 +
   1.465 +  \item @{text "@{const_abbrev c}"} inlines the internalized
   1.466 +  abbreviated constant name @{text "c"} --- as @{ML_type string}
   1.467 +  literal.
   1.468 +
   1.469 +  \item @{text "@{const c(\<^vec>\<tau>)}"} inlines the internalized
   1.470 +  constant @{text "c"} with precise type instantiation in the sense of
   1.471 +  @{ML Sign.const_instance} --- as @{ML Const} constructor term for
   1.472 +  datatype @{ML_type term}.
   1.473 +
   1.474 +  \item @{text "@{term t}"} inlines the internalized term @{text "t"}
   1.475 +  --- as constructor term for datatype @{ML_type term}.
   1.476 +
   1.477 +  \item @{text "@{prop \<phi>}"} inlines the internalized proposition
   1.478 +  @{text "\<phi>"} --- as constructor term for datatype @{ML_type term}.
   1.479 +
   1.480 +  \end{description}
   1.481 +*}
   1.482 +
   1.483 +
   1.484 +section {* Theorems \label{sec:thms} *}
   1.485 +
   1.486 +text {*
   1.487 +  A \emph{proposition} is a well-typed term of type @{text "prop"}, a
   1.488 +  \emph{theorem} is a proven proposition (depending on a context of
   1.489 +  hypotheses and the background theory).  Primitive inferences include
   1.490 +  plain Natural Deduction rules for the primary connectives @{text
   1.491 +  "\<And>"} and @{text "\<Longrightarrow>"} of the framework.  There is also a builtin
   1.492 +  notion of equality/equivalence @{text "\<equiv>"}.
   1.493 +*}
   1.494 +
   1.495 +
   1.496 +subsection {* Primitive connectives and rules \label{sec:prim-rules} *}
   1.497 +
   1.498 +text {*
   1.499 +  The theory @{text "Pure"} contains constant declarations for the
   1.500 +  primitive connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of
   1.501 +  the logical framework, see \figref{fig:pure-connectives}.  The
   1.502 +  derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is
   1.503 +  defined inductively by the primitive inferences given in
   1.504 +  \figref{fig:prim-rules}, with the global restriction that the
   1.505 +  hypotheses must \emph{not} contain any schematic variables.  The
   1.506 +  builtin equality is conceptually axiomatized as shown in
   1.507 +  \figref{fig:pure-equality}, although the implementation works
   1.508 +  directly with derived inferences.
   1.509 +
   1.510 +  \begin{figure}[htb]
   1.511 +  \begin{center}
   1.512 +  \begin{tabular}{ll}
   1.513 +  @{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\
   1.514 +  @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
   1.515 +  @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
   1.516 +  \end{tabular}
   1.517 +  \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
   1.518 +  \end{center}
   1.519 +  \end{figure}
   1.520 +
   1.521 +  \begin{figure}[htb]
   1.522 +  \begin{center}
   1.523 +  \[
   1.524 +  \infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}}
   1.525 +  \qquad
   1.526 +  \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{}
   1.527 +  \]
   1.528 +  \[
   1.529 +  \infer[@{text "(\<And>\<hyphen>intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}}
   1.530 +  \qquad
   1.531 +  \infer[@{text "(\<And>\<hyphen>elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}
   1.532 +  \]
   1.533 +  \[
   1.534 +  \infer[@{text "(\<Longrightarrow>\<hyphen>intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
   1.535 +  \qquad
   1.536 +  \infer[@{text "(\<Longrightarrow>\<hyphen>elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}}
   1.537 +  \]
   1.538 +  \caption{Primitive inferences of Pure}\label{fig:prim-rules}
   1.539 +  \end{center}
   1.540 +  \end{figure}
   1.541 +
   1.542 +  \begin{figure}[htb]
   1.543 +  \begin{center}
   1.544 +  \begin{tabular}{ll}
   1.545 +  @{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\
   1.546 +  @{text "\<turnstile> x \<equiv> x"} & reflexivity \\
   1.547 +  @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\
   1.548 +  @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
   1.549 +  @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\
   1.550 +  \end{tabular}
   1.551 +  \caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality}
   1.552 +  \end{center}
   1.553 +  \end{figure}
   1.554 +
   1.555 +  The introduction and elimination rules for @{text "\<And>"} and @{text
   1.556 +  "\<Longrightarrow>"} are analogous to formation of dependently typed @{text
   1.557 +  "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
   1.558 +  are irrelevant in the Pure logic, though; they cannot occur within
   1.559 +  propositions.  The system provides a runtime option to record
   1.560 +  explicit proof terms for primitive inferences.  Thus all three
   1.561 +  levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for
   1.562 +  terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\
   1.563 +  \cite{Berghofer-Nipkow:2000:TPHOL}).
   1.564 +
   1.565 +  Observe that locally fixed parameters (as in @{text
   1.566 +  "\<And>\<hyphen>intro"}) need not be recorded in the hypotheses, because
   1.567 +  the simple syntactic types of Pure are always inhabitable.
   1.568 +  ``Assumptions'' @{text "x :: \<tau>"} for type-membership are only
   1.569 +  present as long as some @{text "x\<^isub>\<tau>"} occurs in the statement
   1.570 +  body.\footnote{This is the key difference to ``@{text "\<lambda>HOL"}'' in
   1.571 +  the PTS framework \cite{Barendregt-Geuvers:2001}, where hypotheses
   1.572 +  @{text "x : A"} are treated uniformly for propositions and types.}
   1.573 +
   1.574 +  \medskip The axiomatization of a theory is implicitly closed by
   1.575 +  forming all instances of type and term variables: @{text "\<turnstile>
   1.576 +  A\<vartheta>"} holds for any substitution instance of an axiom
   1.577 +  @{text "\<turnstile> A"}.  By pushing substitutions through derivations
   1.578 +  inductively, we also get admissible @{text "generalize"} and @{text
   1.579 +  "instantiate"} rules as shown in \figref{fig:subst-rules}.
   1.580 +
   1.581 +  \begin{figure}[htb]
   1.582 +  \begin{center}
   1.583 +  \[
   1.584 +  \infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}}
   1.585 +  \quad
   1.586 +  \infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}}
   1.587 +  \]
   1.588 +  \[
   1.589 +  \infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}
   1.590 +  \quad
   1.591 +  \infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}}
   1.592 +  \]
   1.593 +  \caption{Admissible substitution rules}\label{fig:subst-rules}
   1.594 +  \end{center}
   1.595 +  \end{figure}
   1.596 +
   1.597 +  Note that @{text "instantiate"} does not require an explicit
   1.598 +  side-condition, because @{text "\<Gamma>"} may never contain schematic
   1.599 +  variables.
   1.600 +
   1.601 +  In principle, variables could be substituted in hypotheses as well,
   1.602 +  but this would disrupt the monotonicity of reasoning: deriving
   1.603 +  @{text "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is
   1.604 +  correct, but @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold:
   1.605 +  the result belongs to a different proof context.
   1.606 +
   1.607 +  \medskip An \emph{oracle} is a function that produces axioms on the
   1.608 +  fly.  Logically, this is an instance of the @{text "axiom"} rule
   1.609 +  (\figref{fig:prim-rules}), but there is an operational difference.
   1.610 +  The system always records oracle invocations within derivations of
   1.611 +  theorems by a unique tag.
   1.612 +
   1.613 +  Axiomatizations should be limited to the bare minimum, typically as
   1.614 +  part of the initial logical basis of an object-logic formalization.
   1.615 +  Later on, theories are usually developed in a strictly definitional
   1.616 +  fashion, by stating only certain equalities over new constants.
   1.617 +
   1.618 +  A \emph{simple definition} consists of a constant declaration @{text
   1.619 +  "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text "t
   1.620 +  :: \<sigma>"} is a closed term without any hidden polymorphism.  The RHS
   1.621 +  may depend on further defined constants, but not @{text "c"} itself.
   1.622 +  Definitions of functions may be presented as @{text "c \<^vec>x \<equiv>
   1.623 +  t"} instead of the puristic @{text "c \<equiv> \<lambda>\<^vec>x. t"}.
   1.624 +
   1.625 +  An \emph{overloaded definition} consists of a collection of axioms
   1.626 +  for the same constant, with zero or one equations @{text
   1.627 +  "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type constructor @{text "\<kappa>"} (for
   1.628 +  distinct variables @{text "\<^vec>\<alpha>"}).  The RHS may mention
   1.629 +  previously defined constants as above, or arbitrary constants @{text
   1.630 +  "d(\<alpha>\<^isub>i)"} for some @{text "\<alpha>\<^isub>i"} projected from @{text
   1.631 +  "\<^vec>\<alpha>"}.  Thus overloaded definitions essentially work by
   1.632 +  primitive recursion over the syntactic structure of a single type
   1.633 +  argument.  See also \cite[\S4.3]{Haftmann-Wenzel:2006:classes}.
   1.634 +*}
   1.635 +
   1.636 +text %mlref {*
   1.637 +  \begin{mldecls}
   1.638 +  @{index_ML Logic.all: "term -> term -> term"} \\
   1.639 +  @{index_ML Logic.mk_implies: "term * term -> term"} \\
   1.640 +  \end{mldecls}
   1.641 +  \begin{mldecls}
   1.642 +  @{index_ML_type ctyp} \\
   1.643 +  @{index_ML_type cterm} \\
   1.644 +  @{index_ML Thm.ctyp_of: "theory -> typ -> ctyp"} \\
   1.645 +  @{index_ML Thm.cterm_of: "theory -> term -> cterm"} \\
   1.646 +  @{index_ML Thm.apply: "cterm -> cterm -> cterm"} \\
   1.647 +  @{index_ML Thm.lambda: "cterm -> cterm -> cterm"} \\
   1.648 +  @{index_ML Thm.all: "cterm -> cterm -> cterm"} \\
   1.649 +  @{index_ML Drule.mk_implies: "cterm * cterm -> cterm"} \\
   1.650 +  \end{mldecls}
   1.651 +  \begin{mldecls}
   1.652 +  @{index_ML_type thm} \\
   1.653 +  @{index_ML proofs: "int Unsynchronized.ref"} \\
   1.654 +  @{index_ML Thm.transfer: "theory -> thm -> thm"} \\
   1.655 +  @{index_ML Thm.assume: "cterm -> thm"} \\
   1.656 +  @{index_ML Thm.forall_intr: "cterm -> thm -> thm"} \\
   1.657 +  @{index_ML Thm.forall_elim: "cterm -> thm -> thm"} \\
   1.658 +  @{index_ML Thm.implies_intr: "cterm -> thm -> thm"} \\
   1.659 +  @{index_ML Thm.implies_elim: "thm -> thm -> thm"} \\
   1.660 +  @{index_ML Thm.generalize: "string list * string list -> int -> thm -> thm"} \\
   1.661 +  @{index_ML Thm.instantiate: "(ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm"} \\
   1.662 +  @{index_ML Thm.add_axiom: "Proof.context ->
   1.663 +  binding * term -> theory -> (string * thm) * theory"} \\
   1.664 +  @{index_ML Thm.add_oracle: "binding * ('a -> cterm) -> theory ->
   1.665 +  (string * ('a -> thm)) * theory"} \\
   1.666 +  @{index_ML Thm.add_def: "Proof.context -> bool -> bool ->
   1.667 +  binding * term -> theory -> (string * thm) * theory"} \\
   1.668 +  \end{mldecls}
   1.669 +  \begin{mldecls}
   1.670 +  @{index_ML Theory.add_deps: "Proof.context -> string ->
   1.671 +  string * typ -> (string * typ) list -> theory -> theory"} \\
   1.672 +  \end{mldecls}
   1.673 +
   1.674 +  \begin{description}
   1.675 +
   1.676 +  \item @{ML Logic.all}~@{text "a B"} produces a Pure quantification
   1.677 +  @{text "\<And>a. B"}, where occurrences of the atomic term @{text "a"} in
   1.678 +  the body proposition @{text "B"} are replaced by bound variables.
   1.679 +  (See also @{ML lambda} on terms.)
   1.680 +
   1.681 +  \item @{ML Logic.mk_implies}~@{text "(A, B)"} produces a Pure
   1.682 +  implication @{text "A \<Longrightarrow> B"}.
   1.683 +
   1.684 +  \item Types @{ML_type ctyp} and @{ML_type cterm} represent certified
   1.685 +  types and terms, respectively.  These are abstract datatypes that
   1.686 +  guarantee that its values have passed the full well-formedness (and
   1.687 +  well-typedness) checks, relative to the declarations of type
   1.688 +  constructors, constants etc.\ in the background theory.  The
   1.689 +  abstract types @{ML_type ctyp} and @{ML_type cterm} are part of the
   1.690 +  same inference kernel that is mainly responsible for @{ML_type thm}.
   1.691 +  Thus syntactic operations on @{ML_type ctyp} and @{ML_type cterm}
   1.692 +  are located in the @{ML_struct Thm} module, even though theorems are
   1.693 +  not yet involved at that stage.
   1.694 +
   1.695 +  \item @{ML Thm.ctyp_of}~@{text "thy \<tau>"} and @{ML
   1.696 +  Thm.cterm_of}~@{text "thy t"} explicitly checks types and terms,
   1.697 +  respectively.  This also involves some basic normalizations, such
   1.698 +  expansion of type and term abbreviations from the theory context.
   1.699 +  Full re-certification is relatively slow and should be avoided in
   1.700 +  tight reasoning loops.
   1.701 +
   1.702 +  \item @{ML Thm.apply}, @{ML Thm.lambda}, @{ML Thm.all}, @{ML
   1.703 +  Drule.mk_implies} etc.\ compose certified terms (or propositions)
   1.704 +  incrementally.  This is equivalent to @{ML Thm.cterm_of} after
   1.705 +  unchecked @{ML_op "$"}, @{ML lambda}, @{ML Logic.all}, @{ML
   1.706 +  Logic.mk_implies} etc., but there can be a big difference in
   1.707 +  performance when large existing entities are composed by a few extra
   1.708 +  constructions on top.  There are separate operations to decompose
   1.709 +  certified terms and theorems to produce certified terms again.
   1.710 +
   1.711 +  \item Type @{ML_type thm} represents proven propositions.  This is
   1.712 +  an abstract datatype that guarantees that its values have been
   1.713 +  constructed by basic principles of the @{ML_struct Thm} module.
   1.714 +  Every @{ML_type thm} value contains a sliding back-reference to the
   1.715 +  enclosing theory, cf.\ \secref{sec:context-theory}.
   1.716 +
   1.717 +  \item @{ML proofs} specifies the detail of proof recording within
   1.718 +  @{ML_type thm} values: @{ML 0} records only the names of oracles,
   1.719 +  @{ML 1} records oracle names and propositions, @{ML 2} additionally
   1.720 +  records full proof terms.  Officially named theorems that contribute
   1.721 +  to a result are recorded in any case.
   1.722 +
   1.723 +  \item @{ML Thm.transfer}~@{text "thy thm"} transfers the given
   1.724 +  theorem to a \emph{larger} theory, see also \secref{sec:context}.
   1.725 +  This formal adjustment of the background context has no logical
   1.726 +  significance, but is occasionally required for formal reasons, e.g.\
   1.727 +  when theorems that are imported from more basic theories are used in
   1.728 +  the current situation.
   1.729 +
   1.730 +  \item @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML
   1.731 +  Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim}
   1.732 +  correspond to the primitive inferences of \figref{fig:prim-rules}.
   1.733 +
   1.734 +  \item @{ML Thm.generalize}~@{text "(\<^vec>\<alpha>, \<^vec>x)"}
   1.735 +  corresponds to the @{text "generalize"} rules of
   1.736 +  \figref{fig:subst-rules}.  Here collections of type and term
   1.737 +  variables are generalized simultaneously, specified by the given
   1.738 +  basic names.
   1.739 +
   1.740 +  \item @{ML Thm.instantiate}~@{text "(\<^vec>\<alpha>\<^isub>s,
   1.741 +  \<^vec>x\<^isub>\<tau>)"} corresponds to the @{text "instantiate"} rules
   1.742 +  of \figref{fig:subst-rules}.  Type variables are substituted before
   1.743 +  term variables.  Note that the types in @{text "\<^vec>x\<^isub>\<tau>"}
   1.744 +  refer to the instantiated versions.
   1.745 +
   1.746 +  \item @{ML Thm.add_axiom}~@{text "ctxt (name, A)"} declares an
   1.747 +  arbitrary proposition as axiom, and retrieves it as a theorem from
   1.748 +  the resulting theory, cf.\ @{text "axiom"} in
   1.749 +  \figref{fig:prim-rules}.  Note that the low-level representation in
   1.750 +  the axiom table may differ slightly from the returned theorem.
   1.751 +
   1.752 +  \item @{ML Thm.add_oracle}~@{text "(binding, oracle)"} produces a named
   1.753 +  oracle rule, essentially generating arbitrary axioms on the fly,
   1.754 +  cf.\ @{text "axiom"} in \figref{fig:prim-rules}.
   1.755 +
   1.756 +  \item @{ML Thm.add_def}~@{text "ctxt unchecked overloaded (name, c
   1.757 +  \<^vec>x \<equiv> t)"} states a definitional axiom for an existing constant
   1.758 +  @{text "c"}.  Dependencies are recorded via @{ML Theory.add_deps},
   1.759 +  unless the @{text "unchecked"} option is set.  Note that the
   1.760 +  low-level representation in the axiom table may differ slightly from
   1.761 +  the returned theorem.
   1.762 +
   1.763 +  \item @{ML Theory.add_deps}~@{text "ctxt name c\<^isub>\<tau> \<^vec>d\<^isub>\<sigma>"}
   1.764 +  declares dependencies of a named specification for constant @{text
   1.765 +  "c\<^isub>\<tau>"}, relative to existing specifications for constants @{text
   1.766 +  "\<^vec>d\<^isub>\<sigma>"}.
   1.767 +
   1.768 +  \end{description}
   1.769 +*}
   1.770 +
   1.771 +
   1.772 +text %mlantiq {*
   1.773 +  \begin{matharray}{rcl}
   1.774 +  @{ML_antiquotation_def "ctyp"} & : & @{text ML_antiquotation} \\
   1.775 +  @{ML_antiquotation_def "cterm"} & : & @{text ML_antiquotation} \\
   1.776 +  @{ML_antiquotation_def "cprop"} & : & @{text ML_antiquotation} \\
   1.777 +  @{ML_antiquotation_def "thm"} & : & @{text ML_antiquotation} \\
   1.778 +  @{ML_antiquotation_def "thms"} & : & @{text ML_antiquotation} \\
   1.779 +  @{ML_antiquotation_def "lemma"} & : & @{text ML_antiquotation} \\
   1.780 +  \end{matharray}
   1.781 +
   1.782 +  @{rail "
   1.783 +  @@{ML_antiquotation ctyp} typ
   1.784 +  ;
   1.785 +  @@{ML_antiquotation cterm} term
   1.786 +  ;
   1.787 +  @@{ML_antiquotation cprop} prop
   1.788 +  ;
   1.789 +  @@{ML_antiquotation thm} thmref
   1.790 +  ;
   1.791 +  @@{ML_antiquotation thms} thmrefs
   1.792 +  ;
   1.793 +  @@{ML_antiquotation lemma} ('(' @'open' ')')? ((prop +) + @'and') \\
   1.794 +    @'by' method method?
   1.795 +  "}
   1.796 +
   1.797 +  \begin{description}
   1.798 +
   1.799 +  \item @{text "@{ctyp \<tau>}"} produces a certified type wrt.\ the
   1.800 +  current background theory --- as abstract value of type @{ML_type
   1.801 +  ctyp}.
   1.802 +
   1.803 +  \item @{text "@{cterm t}"} and @{text "@{cprop \<phi>}"} produce a
   1.804 +  certified term wrt.\ the current background theory --- as abstract
   1.805 +  value of type @{ML_type cterm}.
   1.806 +
   1.807 +  \item @{text "@{thm a}"} produces a singleton fact --- as abstract
   1.808 +  value of type @{ML_type thm}.
   1.809 +
   1.810 +  \item @{text "@{thms a}"} produces a general fact --- as abstract
   1.811 +  value of type @{ML_type "thm list"}.
   1.812 +
   1.813 +  \item @{text "@{lemma \<phi> by meth}"} produces a fact that is proven on
   1.814 +  the spot according to the minimal proof, which imitates a terminal
   1.815 +  Isar proof.  The result is an abstract value of type @{ML_type thm}
   1.816 +  or @{ML_type "thm list"}, depending on the number of propositions
   1.817 +  given here.
   1.818 +
   1.819 +  The internal derivation object lacks a proper theorem name, but it
   1.820 +  is formally closed, unless the @{text "(open)"} option is specified
   1.821 +  (this may impact performance of applications with proof terms).
   1.822 +
   1.823 +  Since ML antiquotations are always evaluated at compile-time, there
   1.824 +  is no run-time overhead even for non-trivial proofs.  Nonetheless,
   1.825 +  the justification is syntactically limited to a single @{command
   1.826 +  "by"} step.  More complex Isar proofs should be done in regular
   1.827 +  theory source, before compiling the corresponding ML text that uses
   1.828 +  the result.
   1.829 +
   1.830 +  \end{description}
   1.831 +
   1.832 +*}
   1.833 +
   1.834 +
   1.835 +subsection {* Auxiliary connectives \label{sec:logic-aux} *}
   1.836 +
   1.837 +text {* Theory @{text "Pure"} provides a few auxiliary connectives
   1.838 +  that are defined on top of the primitive ones, see
   1.839 +  \figref{fig:pure-aux}.  These special constants are useful in
   1.840 +  certain internal encodings, and are normally not directly exposed to
   1.841 +  the user.
   1.842 +
   1.843 +  \begin{figure}[htb]
   1.844 +  \begin{center}
   1.845 +  \begin{tabular}{ll}
   1.846 +  @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&&&"}) \\
   1.847 +  @{text "\<turnstile> A &&& B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex]
   1.848 +  @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, suppressed) \\
   1.849 +  @{text "#A \<equiv> A"} \\[1ex]
   1.850 +  @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\
   1.851 +  @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex]
   1.852 +  @{text "TYPE :: \<alpha> itself"} & (prefix @{text "TYPE"}) \\
   1.853 +  @{text "(unspecified)"} \\
   1.854 +  \end{tabular}
   1.855 +  \caption{Definitions of auxiliary connectives}\label{fig:pure-aux}
   1.856 +  \end{center}
   1.857 +  \end{figure}
   1.858 +
   1.859 +  The introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A &&& B"}, and eliminations
   1.860 +  (projections) @{text "A &&& B \<Longrightarrow> A"} and @{text "A &&& B \<Longrightarrow> B"} are
   1.861 +  available as derived rules.  Conjunction allows to treat
   1.862 +  simultaneous assumptions and conclusions uniformly, e.g.\ consider
   1.863 +  @{text "A \<Longrightarrow> B \<Longrightarrow> C &&& D"}.  In particular, the goal mechanism
   1.864 +  represents multiple claims as explicit conjunction internally, but
   1.865 +  this is refined (via backwards introduction) into separate sub-goals
   1.866 +  before the user commences the proof; the final result is projected
   1.867 +  into a list of theorems using eliminations (cf.\
   1.868 +  \secref{sec:tactical-goals}).
   1.869 +
   1.870 +  The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex
   1.871 +  propositions appear as atomic, without changing the meaning: @{text
   1.872 +  "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable.  See
   1.873 +  \secref{sec:tactical-goals} for specific operations.
   1.874 +
   1.875 +  The @{text "term"} marker turns any well-typed term into a derivable
   1.876 +  proposition: @{text "\<turnstile> TERM t"} holds unconditionally.  Although
   1.877 +  this is logically vacuous, it allows to treat terms and proofs
   1.878 +  uniformly, similar to a type-theoretic framework.
   1.879 +
   1.880 +  The @{text "TYPE"} constructor is the canonical representative of
   1.881 +  the unspecified type @{text "\<alpha> itself"}; it essentially injects the
   1.882 +  language of types into that of terms.  There is specific notation
   1.883 +  @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau>
   1.884 + itself\<^esub>"}.
   1.885 +  Although being devoid of any particular meaning, the term @{text
   1.886 +  "TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term
   1.887 +  language.  In particular, @{text "TYPE(\<alpha>)"} may be used as formal
   1.888 +  argument in primitive definitions, in order to circumvent hidden
   1.889 +  polymorphism (cf.\ \secref{sec:terms}).  For example, @{text "c
   1.890 +  TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of
   1.891 +  a proposition @{text "A"} that depends on an additional type
   1.892 +  argument, which is essentially a predicate on types.
   1.893 +*}
   1.894 +
   1.895 +text %mlref {*
   1.896 +  \begin{mldecls}
   1.897 +  @{index_ML Conjunction.intr: "thm -> thm -> thm"} \\
   1.898 +  @{index_ML Conjunction.elim: "thm -> thm * thm"} \\
   1.899 +  @{index_ML Drule.mk_term: "cterm -> thm"} \\
   1.900 +  @{index_ML Drule.dest_term: "thm -> cterm"} \\
   1.901 +  @{index_ML Logic.mk_type: "typ -> term"} \\
   1.902 +  @{index_ML Logic.dest_type: "term -> typ"} \\
   1.903 +  \end{mldecls}
   1.904 +
   1.905 +  \begin{description}
   1.906 +
   1.907 +  \item @{ML Conjunction.intr} derives @{text "A &&& B"} from @{text
   1.908 +  "A"} and @{text "B"}.
   1.909 +
   1.910 +  \item @{ML Conjunction.elim} derives @{text "A"} and @{text "B"}
   1.911 +  from @{text "A &&& B"}.
   1.912 +
   1.913 +  \item @{ML Drule.mk_term} derives @{text "TERM t"}.
   1.914 +
   1.915 +  \item @{ML Drule.dest_term} recovers term @{text "t"} from @{text
   1.916 +  "TERM t"}.
   1.917 +
   1.918 +  \item @{ML Logic.mk_type}~@{text "\<tau>"} produces the term @{text
   1.919 +  "TYPE(\<tau>)"}.
   1.920 +
   1.921 +  \item @{ML Logic.dest_type}~@{text "TYPE(\<tau>)"} recovers the type
   1.922 +  @{text "\<tau>"}.
   1.923 +
   1.924 +  \end{description}
   1.925 +*}
   1.926 +
   1.927 +
   1.928 +section {* Object-level rules \label{sec:obj-rules} *}
   1.929 +
   1.930 +text {*
   1.931 +  The primitive inferences covered so far mostly serve foundational
   1.932 +  purposes.  User-level reasoning usually works via object-level rules
   1.933 +  that are represented as theorems of Pure.  Composition of rules
   1.934 +  involves \emph{backchaining}, \emph{higher-order unification} modulo
   1.935 +  @{text "\<alpha>\<beta>\<eta>"}-conversion of @{text "\<lambda>"}-terms, and so-called
   1.936 +  \emph{lifting} of rules into a context of @{text "\<And>"} and @{text
   1.937 +  "\<Longrightarrow>"} connectives.  Thus the full power of higher-order Natural
   1.938 +  Deduction in Isabelle/Pure becomes readily available.
   1.939 +*}
   1.940 +
   1.941 +
   1.942 +subsection {* Hereditary Harrop Formulae *}
   1.943 +
   1.944 +text {*
   1.945 +  The idea of object-level rules is to model Natural Deduction
   1.946 +  inferences in the style of Gentzen \cite{Gentzen:1935}, but we allow
   1.947 +  arbitrary nesting similar to \cite{extensions91}.  The most basic
   1.948 +  rule format is that of a \emph{Horn Clause}:
   1.949 +  \[
   1.950 +  \infer{@{text "A"}}{@{text "A\<^sub>1"} & @{text "\<dots>"} & @{text "A\<^sub>n"}}
   1.951 +  \]
   1.952 +  where @{text "A, A\<^sub>1, \<dots>, A\<^sub>n"} are atomic propositions
   1.953 +  of the framework, usually of the form @{text "Trueprop B"}, where
   1.954 +  @{text "B"} is a (compound) object-level statement.  This
   1.955 +  object-level inference corresponds to an iterated implication in
   1.956 +  Pure like this:
   1.957 +  \[
   1.958 +  @{text "A\<^sub>1 \<Longrightarrow> \<dots> A\<^sub>n \<Longrightarrow> A"}
   1.959 +  \]
   1.960 +  As an example consider conjunction introduction: @{text "A \<Longrightarrow> B \<Longrightarrow> A \<and>
   1.961 +  B"}.  Any parameters occurring in such rule statements are
   1.962 +  conceptionally treated as arbitrary:
   1.963 +  \[
   1.964 +  @{text "\<And>x\<^sub>1 \<dots> x\<^sub>m. A\<^sub>1 x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> \<dots> A\<^sub>n x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> A x\<^sub>1 \<dots> x\<^sub>m"}
   1.965 +  \]
   1.966 +
   1.967 +  Nesting of rules means that the positions of @{text "A\<^sub>i"} may
   1.968 +  again hold compound rules, not just atomic propositions.
   1.969 +  Propositions of this format are called \emph{Hereditary Harrop
   1.970 +  Formulae} in the literature \cite{Miller:1991}.  Here we give an
   1.971 +  inductive characterization as follows:
   1.972 +
   1.973 +  \medskip
   1.974 +  \begin{tabular}{ll}
   1.975 +  @{text "\<^bold>x"} & set of variables \\
   1.976 +  @{text "\<^bold>A"} & set of atomic propositions \\
   1.977 +  @{text "\<^bold>H  =  \<And>\<^bold>x\<^sup>*. \<^bold>H\<^sup>* \<Longrightarrow> \<^bold>A"} & set of Hereditary Harrop Formulas \\
   1.978 +  \end{tabular}
   1.979 +  \medskip
   1.980 +
   1.981 +  Thus we essentially impose nesting levels on propositions formed
   1.982 +  from @{text "\<And>"} and @{text "\<Longrightarrow>"}.  At each level there is a prefix
   1.983 +  of parameters and compound premises, concluding an atomic
   1.984 +  proposition.  Typical examples are @{text "\<longrightarrow>"}-introduction @{text
   1.985 +  "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B"} or mathematical induction @{text "P 0 \<Longrightarrow> (\<And>n. P n
   1.986 +  \<Longrightarrow> P (Suc n)) \<Longrightarrow> P n"}.  Even deeper nesting occurs in well-founded
   1.987 +  induction @{text "(\<And>x. (\<And>y. y \<prec> x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x"}, but this
   1.988 +  already marks the limit of rule complexity that is usually seen in
   1.989 +  practice.
   1.990 +
   1.991 +  \medskip Regular user-level inferences in Isabelle/Pure always
   1.992 +  maintain the following canonical form of results:
   1.993 +
   1.994 +  \begin{itemize}
   1.995 +
   1.996 +  \item Normalization by @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"},
   1.997 +  which is a theorem of Pure, means that quantifiers are pushed in
   1.998 +  front of implication at each level of nesting.  The normal form is a
   1.999 +  Hereditary Harrop Formula.
  1.1000 +
  1.1001 +  \item The outermost prefix of parameters is represented via
  1.1002 +  schematic variables: instead of @{text "\<And>\<^vec>x. \<^vec>H \<^vec>x
  1.1003 +  \<Longrightarrow> A \<^vec>x"} we have @{text "\<^vec>H ?\<^vec>x \<Longrightarrow> A ?\<^vec>x"}.
  1.1004 +  Note that this representation looses information about the order of
  1.1005 +  parameters, and vacuous quantifiers vanish automatically.
  1.1006 +
  1.1007 +  \end{itemize}
  1.1008 +*}
  1.1009 +
  1.1010 +text %mlref {*
  1.1011 +  \begin{mldecls}
  1.1012 +  @{index_ML Simplifier.norm_hhf: "thm -> thm"} \\
  1.1013 +  \end{mldecls}
  1.1014 +
  1.1015 +  \begin{description}
  1.1016 +
  1.1017 +  \item @{ML Simplifier.norm_hhf}~@{text thm} normalizes the given
  1.1018 +  theorem according to the canonical form specified above.  This is
  1.1019 +  occasionally helpful to repair some low-level tools that do not
  1.1020 +  handle Hereditary Harrop Formulae properly.
  1.1021 +
  1.1022 +  \end{description}
  1.1023 +*}
  1.1024 +
  1.1025 +
  1.1026 +subsection {* Rule composition *}
  1.1027 +
  1.1028 +text {*
  1.1029 +  The rule calculus of Isabelle/Pure provides two main inferences:
  1.1030 +  @{inference resolution} (i.e.\ back-chaining of rules) and
  1.1031 +  @{inference assumption} (i.e.\ closing a branch), both modulo
  1.1032 +  higher-order unification.  There are also combined variants, notably
  1.1033 +  @{inference elim_resolution} and @{inference dest_resolution}.
  1.1034 +
  1.1035 +  To understand the all-important @{inference resolution} principle,
  1.1036 +  we first consider raw @{inference_def composition} (modulo
  1.1037 +  higher-order unification with substitution @{text "\<vartheta>"}):
  1.1038 +  \[
  1.1039 +  \infer[(@{inference_def composition})]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}}
  1.1040 +  {@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}}
  1.1041 +  \]
  1.1042 +  Here the conclusion of the first rule is unified with the premise of
  1.1043 +  the second; the resulting rule instance inherits the premises of the
  1.1044 +  first and conclusion of the second.  Note that @{text "C"} can again
  1.1045 +  consist of iterated implications.  We can also permute the premises
  1.1046 +  of the second rule back-and-forth in order to compose with @{text
  1.1047 +  "B'"} in any position (subsequently we shall always refer to
  1.1048 +  position 1 w.l.o.g.).
  1.1049 +
  1.1050 +  In @{inference composition} the internal structure of the common
  1.1051 +  part @{text "B"} and @{text "B'"} is not taken into account.  For
  1.1052 +  proper @{inference resolution} we require @{text "B"} to be atomic,
  1.1053 +  and explicitly observe the structure @{text "\<And>\<^vec>x. \<^vec>H
  1.1054 +  \<^vec>x \<Longrightarrow> B' \<^vec>x"} of the premise of the second rule.  The
  1.1055 +  idea is to adapt the first rule by ``lifting'' it into this context,
  1.1056 +  by means of iterated application of the following inferences:
  1.1057 +  \[
  1.1058 +  \infer[(@{inference_def imp_lift})]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}}
  1.1059 +  \]
  1.1060 +  \[
  1.1061 +  \infer[(@{inference_def all_lift})]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}}
  1.1062 +  \]
  1.1063 +  By combining raw composition with lifting, we get full @{inference
  1.1064 +  resolution} as follows:
  1.1065 +  \[
  1.1066 +  \infer[(@{inference_def resolution})]
  1.1067 +  {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}}
  1.1068 +  {\begin{tabular}{l}
  1.1069 +    @{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\
  1.1070 +    @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\
  1.1071 +    @{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\
  1.1072 +   \end{tabular}}
  1.1073 +  \]
  1.1074 +
  1.1075 +  Continued resolution of rules allows to back-chain a problem towards
  1.1076 +  more and sub-problems.  Branches are closed either by resolving with
  1.1077 +  a rule of 0 premises, or by producing a ``short-circuit'' within a
  1.1078 +  solved situation (again modulo unification):
  1.1079 +  \[
  1.1080 +  \infer[(@{inference_def assumption})]{@{text "C\<vartheta>"}}
  1.1081 +  {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text i})}}
  1.1082 +  \]
  1.1083 +
  1.1084 +  FIXME @{inference_def elim_resolution}, @{inference_def dest_resolution}
  1.1085 +*}
  1.1086 +
  1.1087 +text %mlref {*
  1.1088 +  \begin{mldecls}
  1.1089 +  @{index_ML_op "RSN": "thm * (int * thm) -> thm"} \\
  1.1090 +  @{index_ML_op "RS": "thm * thm -> thm"} \\
  1.1091 +
  1.1092 +  @{index_ML_op "RLN": "thm list * (int * thm list) -> thm list"} \\
  1.1093 +  @{index_ML_op "RL": "thm list * thm list -> thm list"} \\
  1.1094 +
  1.1095 +  @{index_ML_op "MRS": "thm list * thm -> thm"} \\
  1.1096 +  @{index_ML_op "OF": "thm * thm list -> thm"} \\
  1.1097 +  \end{mldecls}
  1.1098 +
  1.1099 +  \begin{description}
  1.1100 +
  1.1101 +  \item @{text "rule\<^sub>1 RSN (i, rule\<^sub>2)"} resolves the conclusion of
  1.1102 +  @{text "rule\<^sub>1"} with the @{text i}-th premise of @{text "rule\<^sub>2"},
  1.1103 +  according to the @{inference resolution} principle explained above.
  1.1104 +  Unless there is precisely one resolvent it raises exception @{ML
  1.1105 +  THM}.
  1.1106 +
  1.1107 +  This corresponds to the rule attribute @{attribute THEN} in Isar
  1.1108 +  source language.
  1.1109 +
  1.1110 +  \item @{text "rule\<^sub>1 RS rule\<^sub>2"} abbreviates @{text "rule\<^sub>1 RS (1,
  1.1111 +  rule\<^sub>2)"}.
  1.1112 +
  1.1113 +  \item @{text "rules\<^sub>1 RLN (i, rules\<^sub>2)"} joins lists of rules.  For
  1.1114 +  every @{text "rule\<^sub>1"} in @{text "rules\<^sub>1"} and @{text "rule\<^sub>2"} in
  1.1115 +  @{text "rules\<^sub>2"}, it resolves the conclusion of @{text "rule\<^sub>1"} with
  1.1116 +  the @{text "i"}-th premise of @{text "rule\<^sub>2"}, accumulating multiple
  1.1117 +  results in one big list.  Note that such strict enumerations of
  1.1118 +  higher-order unifications can be inefficient compared to the lazy
  1.1119 +  variant seen in elementary tactics like @{ML resolve_tac}.
  1.1120 +
  1.1121 +  \item @{text "rules\<^sub>1 RL rules\<^sub>2"} abbreviates @{text "rules\<^sub>1 RLN (1,
  1.1122 +  rules\<^sub>2)"}.
  1.1123 +
  1.1124 +  \item @{text "[rule\<^sub>1, \<dots>, rule\<^sub>n] MRS rule"} resolves @{text "rule\<^isub>i"}
  1.1125 +  against premise @{text "i"} of @{text "rule"}, for @{text "i = n, \<dots>,
  1.1126 +  1"}.  By working from right to left, newly emerging premises are
  1.1127 +  concatenated in the result, without interfering.
  1.1128 +
  1.1129 +  \item @{text "rule OF rules"} is an alternative notation for @{text
  1.1130 +  "rules MRS rule"}, which makes rule composition look more like
  1.1131 +  function application.  Note that the argument @{text "rules"} need
  1.1132 +  not be atomic.
  1.1133 +
  1.1134 +  This corresponds to the rule attribute @{attribute OF} in Isar
  1.1135 +  source language.
  1.1136 +
  1.1137 +  \end{description}
  1.1138 +*}
  1.1139 +
  1.1140 +end