author wenzelm
Tue Sep 12 17:45:58 2006 +0200 (2006-09-12)
changeset 20520 05fd007bdeb9
parent 20519 d7ad1217c24a
child 20521 189811b39869
permissions -rw-r--r--
     2 (* $Id$ *)
     4 theory logic imports base begin
     6 chapter {* Primitive logic \label{ch:logic} *}
     8 text {*
     9   The logical foundations of Isabelle/Isar are that of the Pure logic,
    10   which has been introduced as a natural-deduction framework in
    11   \cite{paulson700}.  This is essentially the same logic as ``@{text
    12   "\<lambda>HOL"}'' in the more abstract setting of Pure Type Systems (PTS)
    13   \cite{Barendregt-Geuvers:2001}, although there are some key
    14   differences in the specific treatment of simple types in
    15   Isabelle/Pure.
    17   Following type-theoretic parlance, the Pure logic consists of three
    18   levels of @{text "\<lambda>"}-calculus with corresponding arrows: @{text
    19   "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
    20   "\<And>"} for universal quantification (proofs depending on terms), and
    21   @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
    23   Pure derivations are relative to a logical theory, which declares
    24   type constructors, term constants, and axioms.  Theory declarations
    25   support schematic polymorphism, which is strictly speaking outside
    26   the logic.\footnote{Incidently, this is the main logical reason, why
    27   the theory context @{text "\<Theta>"} is separate from the context @{text
    28   "\<Gamma>"} of the core calculus.}
    29 *}
    32 section {* Types \label{sec:types} *}
    34 text {*
    35   The language of types is an uninterpreted order-sorted first-order
    36   algebra; types are qualified by ordered type classes.
    38   \medskip A \emph{type class} is an abstract syntactic entity
    39   declared in the theory context.  The \emph{subclass relation} @{text
    40   "c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic
    41   generating relation; the transitive closure is maintained
    42   internally.  The resulting relation is an ordering: reflexive,
    43   transitive, and antisymmetric.
    45   A \emph{sort} is a list of type classes written as @{text
    46   "{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
    47   intersection.  Notationally, the curly braces are omitted for
    48   singleton intersections, i.e.\ any class @{text "c"} may be read as
    49   a sort @{text "{c}"}.  The ordering on type classes is extended to
    50   sorts according to the meaning of intersections: @{text
    51   "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff
    52   @{text "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}.  The empty intersection
    53   @{text "{}"} refers to the universal sort, which is the largest
    54   element wrt.\ the sort order.  The intersections of all (finitely
    55   many) classes declared in the current theory are the minimal
    56   elements wrt.\ the sort order.
    58   \medskip A \emph{fixed type variable} is a pair of a basic name
    59   (starting with a @{text "'"} character) and a sort constraint.  For
    60   example, @{text "('a, s)"} which is usually printed as @{text
    61   "\<alpha>\<^isub>s"}.  A \emph{schematic type variable} is a pair of an
    62   indexname and a sort constraint.  For example, @{text "(('a, 0),
    63   s)"} which is usually printed as @{text "?\<alpha>\<^isub>s"}.
    65   Note that \emph{all} syntactic components contribute to the identity
    66   of type variables, including the sort constraint.  The core logic
    67   handles type variables with the same name but different sorts as
    68   different, although some outer layers of the system make it hard to
    69   produce anything like this.
    71   A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
    72   on types declared in the theory.  Type constructor application is
    73   usually written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.
    74   For @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text
    75   "prop"} instead of @{text "()prop"}.  For @{text "k = 1"} the
    76   parentheses are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text
    77   "(\<alpha>)list"}.  Further notation is provided for specific constructors,
    78   notably the right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of
    79   @{text "(\<alpha>, \<beta>)fun"}.
    81   A \emph{type} @{text "\<tau>"} is defined inductively over type variables
    82   and type constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s |
    83   ?\<alpha>\<^isub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)k"}.
    85   A \emph{type abbreviation} is a syntactic definition @{text
    86   "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
    87   variables @{text "\<^vec>\<alpha>"}.  Type abbreviations looks like type
    88   constructors at the surface, but are fully expanded before entering
    89   the logical core.
    91   A \emph{type arity} declares the image behavior of a type
    92   constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
    93   s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)\<kappa>"} is
    94   of sort @{text "s"} if every argument type @{text "\<tau>\<^isub>i"} is
    95   of sort @{text "s\<^isub>i"}.  Arity declarations are implicitly
    96   completed, i.e.\ @{text "\<kappa> :: (\<^vec>s)c"} entails @{text "\<kappa> ::
    97   (\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}.
    99   \medskip The sort algebra is always maintained as \emph{coregular},
   100   which means that type arities are consistent with the subclass
   101   relation: for each type constructor @{text "\<kappa>"} and classes @{text
   102   "c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "\<kappa> ::
   103   (\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "\<kappa>
   104   :: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq>
   105   \<^vec>s\<^isub>2"} holds component-wise.
   107   The key property of a coregular order-sorted algebra is that sort
   108   constraints may be always solved in a most general fashion: for each
   109   type constructor @{text "\<kappa>"} and sort @{text "s"} there is a most
   110   general vector of argument sorts @{text "(s\<^isub>1, \<dots>,
   111   s\<^isub>k)"} such that a type scheme @{text
   112   "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is
   113   of sort @{text "s"}.  Consequently, the unification problem on the
   114   algebra of types has most general solutions (modulo renaming and
   115   equivalence of sorts).  Moreover, the usual type-inference algorithm
   116   will produce primary types as expected \cite{nipkow-prehofer}.
   117 *}
   119 text %mlref {*
   120   \begin{mldecls}
   121   @{index_ML_type class} \\
   122   @{index_ML_type sort} \\
   123   @{index_ML_type arity} \\
   124   @{index_ML_type typ} \\
   125   @{index_ML map_atyps: "(typ -> typ) -> typ -> typ"} \\
   126   @{index_ML fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
   127   @{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
   128   @{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
   129   @{index_ML Sign.add_types: "(string * int * mixfix) list -> theory -> theory"} \\
   130   @{index_ML Sign.add_tyabbrs_i: "
   131   (string * string list * typ * mixfix) list -> theory -> theory"} \\
   132   @{index_ML Sign.primitive_class: "string * class list -> theory -> theory"} \\
   133   @{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
   134   @{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
   135   \end{mldecls}
   137   \begin{description}
   139   \item @{ML_type class} represents type classes; this is an alias for
   140   @{ML_type string}.
   142   \item @{ML_type sort} represents sorts; this is an alias for
   143   @{ML_type "class list"}.
   145   \item @{ML_type arity} represents type arities; this is an alias for
   146   triples of the form @{text "(\<kappa>, \<^vec>s, s)"} for @{text "\<kappa> ::
   147   (\<^vec>s)s"} described above.
   149   \item @{ML_type typ} represents types; this is a datatype with
   150   constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
   152   \item @{ML map_atyps}~@{text "f \<tau>"} applies mapping @{text "f"} to
   153   all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text "\<tau>"}.
   155   \item @{ML fold_atyps}~@{text "f \<tau>"} iterates operation @{text "f"}
   156   over all occurrences of atoms (@{ML TFree}, @{ML TVar}) in @{text
   157   "\<tau>"}; the type structure is traversed from left to right.
   159   \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
   160   tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
   162   \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether a type
   163   is of a given sort.
   165   \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares new
   166   type constructors @{text "\<kappa>"} with @{text "k"} arguments and
   167   optional mixfix syntax.
   169   \item @{ML Sign.add_tyabbrs_i}~@{text "[(\<kappa>, \<^vec>\<alpha>, \<tau>, mx), \<dots>]"}
   170   defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"} with
   171   optional mixfix syntax.
   173   \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
   174   c\<^isub>n])"} declares new class @{text "c"}, together with class
   175   relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
   177   \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
   178   c\<^isub>2)"} declares class relation @{text "c\<^isub>1 \<subseteq>
   179   c\<^isub>2"}.
   181   \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
   182   arity @{text "\<kappa> :: (\<^vec>s)s"}.
   184   \end{description}
   185 *}
   189 section {* Terms \label{sec:terms} *}
   191 text {*
   192   \glossary{Term}{FIXME}
   194   The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
   195   with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
   196   or \cite{paulson-ml2}), and named free variables and constants.
   197   Terms with loose bound variables are usually considered malformed.
   198   The types of variables and constants is stored explicitly at each
   199   occurrence in the term.
   201   \medskip A \emph{bound variable} is a natural number @{text "b"},
   202   which refers to the next binder that is @{text "b"} steps upwards
   203   from the occurrence of @{text "b"} (counting from zero).  Bindings
   204   may be introduced as abstractions within the term, or as a separate
   205   context (an inside-out list).  This associates each bound variable
   206   with a type.  A \emph{loose variables} is a bound variable that is
   207   outside the current scope of local binders or the context.  For
   208   example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"}
   209   corresponds to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a named
   210   representation.  Also note that the very same bound variable may get
   211   different numbers at different occurrences.
   213   A \emph{fixed variable} is a pair of a basic name and a type.  For
   214   example, @{text "(x, \<tau>)"} which is usually printed @{text
   215   "x\<^isub>\<tau>"}.  A \emph{schematic variable} is a pair of an
   216   indexname and a type.  For example, @{text "((x, 0), \<tau>)"} which is
   217   usually printed as @{text "?x\<^isub>\<tau>"}.
   219   \medskip A \emph{constant} is a atomic terms consisting of a basic
   220   name and a type.  Constants are declared in the context as
   221   polymorphic families @{text "c :: \<sigma>"}, meaning that any @{text
   222   "c\<^isub>\<tau>"} is a valid constant for all substitution instances
   223   @{text "\<tau> \<le> \<sigma>"}.
   225   The list of \emph{type arguments} of @{text "c\<^isub>\<tau>"} wrt.\ the
   226   declaration @{text "c :: \<sigma>"} is the codomain of the type matcher
   227   presented in canonical order (according to the left-to-right
   228   occurrences of type variables in in @{text "\<sigma>"}).  Thus @{text
   229   "c\<^isub>\<tau>"} can be represented more compactly as @{text
   230   "c(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}.  For example, the instance @{text
   231   "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} of some @{text "plus :: \<alpha> \<Rightarrow> \<alpha>
   232   \<Rightarrow> \<alpha>"} has the singleton list @{text "nat"} as type arguments, the
   233   constant may be represented as @{text "plus(nat)"}.
   235   Constant declarations @{text "c :: \<sigma>"} may contain sort constraints
   236   for type variables in @{text "\<sigma>"}.  These are observed by
   237   type-inference as expected, but \emph{ignored} by the core logic.
   238   This means the primitive logic is able to reason with instances of
   239   polymorphic constants that the user-level type-checker would reject.
   241   \medskip A \emph{term} @{text "t"} is defined inductively over
   242   variables and constants, with abstraction and application as
   243   follows: @{text "t = b | x\<^isub>\<tau> | ?x\<^isub>\<tau> | c\<^isub>\<tau> |
   244   \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}.  Parsing and printing takes
   245   care of converting between an external representation with named
   246   bound variables.  Subsequently, we shall use the latter notation
   247   instead of internal de-Bruijn representation.
   249   The subsequent inductive relation @{text "t :: \<tau>"} assigns a
   250   (unique) type to a term, using the special type constructor @{text
   251   "(\<alpha>, \<beta>)fun"}, which is written @{text "\<alpha> \<Rightarrow> \<beta>"}.
   252   \[
   253   \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{}
   254   \qquad
   255   \infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}}
   256   \qquad
   257   \infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}}
   258   \]
   259   A \emph{well-typed term} is a term that can be typed according to these rules.
   261   Typing information can be omitted: type-inference is able to
   262   reconstruct the most general type of a raw term, while assigning
   263   most general types to all of its variables and constants.
   264   Type-inference depends on a context of type constraints for fixed
   265   variables, and declarations for polymorphic constants.
   267   The identity of atomic terms consists both of the name and the type.
   268   Thus different entities @{text "c\<^bsub>\<tau>\<^isub>1\<^esub>"} and
   269   @{text "c\<^bsub>\<tau>\<^isub>2\<^esub>"} may well identified by type
   270   instantiation, by mapping @{text "\<tau>\<^isub>1"} and @{text
   271   "\<tau>\<^isub>2"} to the same @{text "\<tau>"}.  Although,
   272   different type instances of constants of the same basic name are
   273   commonplace, this rarely happens for variables: type-inference
   274   always demands ``consistent'' type constraints.
   276   \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
   277   is the set of type variables occurring in @{text "t"}, but not in
   278   @{text "\<sigma>"}.  This means that the term implicitly depends on the
   279   values of various type variables that are not visible in the overall
   280   type, i.e.\ there are different type instances @{text "t\<vartheta>
   281   :: \<sigma>"} and @{text "t\<vartheta>' :: \<sigma>"} with the same type.  This
   282   slightly pathological situation is apt to cause strange effects.
   284   \medskip A \emph{term abbreviation} is a syntactic definition @{text
   285   "c\<^isub>\<sigma> \<equiv> t"} of an arbitrary closed term @{text "t"} of type
   286   @{text "\<sigma>"} without any hidden polymorphism.  A term abbreviation
   287   looks like a constant at the surface, but is fully expanded before
   288   entering the logical core.  Abbreviations are usually reverted when
   289   printing terms, using rules @{text "t \<rightarrow> c\<^isub>\<sigma>"} has a
   290   higher-order term rewrite system.
   292   \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
   293   "\<alpha>\<beta>\<eta>"}-conversion. @{text "\<alpha>"}-conversion refers to capture-free
   294   renaming of bound variables; @{text "\<beta>"}-conversion contracts an
   295   abstraction applied to some argument term, substituting the argument
   296   in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
   297   "\<eta>"}-conversion contracts vacuous application-abstraction: @{text
   298   "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
   299   @{text "0"} does not occur in @{text "f"}.
   301   Terms are almost always treated module @{text "\<alpha>"}-conversion, which
   302   is implicit in the de-Bruijn representation.  The names in
   303   abstractions of bound variables are maintained only as a comment for
   304   parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-equivalence is usually
   305   taken for granted higher rules (\secref{sec:rules}), anything
   306   depending on higher-order unification or rewriting.
   307 *}
   309 text %mlref {*
   310   \begin{mldecls}
   311   @{index_ML_type term} \\
   312   @{index_ML "op aconv": "term * term -> bool"} \\
   313   @{index_ML map_term_types: "(typ -> typ) -> term -> term"} \\  %FIXME rename map_types
   314   @{index_ML fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
   315   @{index_ML map_aterms: "(term -> term) -> term -> term"} \\
   316   @{index_ML fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\
   317   @{index_ML fastype_of: "term -> typ"} \\
   318   @{index_ML lambda: "term -> term -> term"} \\
   319   @{index_ML betapply: "term * term -> term"} \\
   320   @{index_ML Sign.add_consts_i: "(string * typ * mixfix) list -> theory -> theory"} \\
   321   @{index_ML Sign.add_abbrevs: "string * bool ->
   322   ((string * mixfix) * term) list -> theory -> theory"} \\
   323   @{index_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\
   324   @{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
   325   \end{mldecls}
   327   \begin{description}
   329   \item @{ML_type term} represents de-Bruijn terms with comments in
   330   abstractions for bound variable names.  This is a datatype with
   331   constructors @{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}, @{ML
   332   Abs}, @{ML "op $"}.
   334   \item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text
   335   "\<alpha>"}-equivalence of two terms.  This is the basic equality relation
   336   on type @{ML_type term}; raw datatype equality should only be used
   337   for operations related to parsing or printing!
   339   \item @{ML map_term_types}~@{text "f t"} applies mapping @{text "f"}
   340   to all types occurring in @{text "t"}.
   342   \item @{ML fold_types}~@{text "f t"} iterates operation @{text "f"}
   343   over all occurrences of types in @{text "t"}; the term structure is
   344   traversed from left to right.
   346   \item @{ML map_aterms}~@{text "f t"} applies mapping @{text "f"} to
   347   all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const})
   348   occurring in @{text "t"}.
   350   \item @{ML fold_aterms}~@{text "f t"} iterates operation @{text "f"}
   351   over all occurrences of atomic terms in (@{ML Bound}, @{ML Free},
   352   @{ML Var}, @{ML Const}) @{text "t"}; the term structure is traversed
   353   from left to right.
   355   \item @{ML fastype_of}~@{text "t"} recomputes the type of a
   356   well-formed term, while omitting any sanity checks.  This operation
   357   is relatively slow.
   359   \item @{ML lambda}~@{text "a b"} produces an abstraction @{text
   360   "\<lambda>a. b"}, where occurrences of the original (atomic) term @{text
   361   "a"} in the body @{text "b"} are replaced by bound variables.
   363   \item @{ML betapply}~@{text "t u"} produces an application @{text "t
   364   u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} happens to
   365   be an abstraction.
   367   \item @{ML Sign.add_consts_i}~@{text "[(c, \<sigma>, mx), \<dots>]"} declares a
   368   new constant @{text "c :: \<sigma>"} with optional mixfix syntax.
   370   \item @{ML Sign.add_abbrevs}~@{text "print_mode [((c, t), mx), \<dots>]"}
   371   declares a new term abbreviation @{text "c \<equiv> t"} with optional
   372   mixfix syntax.
   374   \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
   375   Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
   376   convert between the two representations of constants, namely full
   377   type instance vs.\ compact type arguments form (depending on the
   378   most general declaration given in the context).
   380   \end{description}
   381 *}
   384 section {* Theorems \label{sec:thms} *}
   386 text {*
   387   \glossary{Proposition}{A \seeglossary{term} of \seeglossary{type}
   388   @{text "prop"}.  Internally, there is nothing special about
   389   propositions apart from their type, but the concrete syntax enforces
   390   a clear distinction.  Propositions are structured via implication
   391   @{text "A \<Longrightarrow> B"} or universal quantification @{text "\<And>x. B x"} ---
   392   anything else is considered atomic.  The canonical form for
   393   propositions is that of a \seeglossary{Hereditary Harrop Formula}. FIXME}
   395   \glossary{Theorem}{A proven proposition within a certain theory and
   396   proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are
   397   rarely spelled out explicitly.  Theorems are usually normalized
   398   according to the \seeglossary{HHF} format. FIXME}
   400   \glossary{Fact}{Sometimes used interchangeably for
   401   \seeglossary{theorem}.  Strictly speaking, a list of theorems,
   402   essentially an extra-logical conjunction.  Facts emerge either as
   403   local assumptions, or as results of local goal statements --- both
   404   may be simultaneous, hence the list representation. FIXME}
   406   \glossary{Schematic variable}{FIXME}
   408   \glossary{Fixed variable}{A variable that is bound within a certain
   409   proof context; an arbitrary-but-fixed entity within a portion of
   410   proof text. FIXME}
   412   \glossary{Free variable}{Synonymous for \seeglossary{fixed
   413   variable}. FIXME}
   415   \glossary{Bound variable}{FIXME}
   417   \glossary{Variable}{See \seeglossary{schematic variable},
   418   \seeglossary{fixed variable}, \seeglossary{bound variable}, or
   419   \seeglossary{type variable}.  The distinguishing feature of
   420   different variables is their binding scope. FIXME}
   422   A \emph{proposition} is a well-formed term of type @{text "prop"}.
   423   The connectives of minimal logic are declared as constants of the
   424   basic theory:
   426   \smallskip
   427   \begin{tabular}{ll}
   428   @{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\
   429   @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
   430   \end{tabular}
   432   \medskip A \emph{theorem} is a proven proposition, depending on a
   433   collection of assumptions, and axioms from the theory context.  The
   434   judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined
   435   inductively by the primitive inferences given in
   436   \figref{fig:prim-rules}; there is a global syntactic restriction
   437   that the hypotheses may not contain schematic variables.
   439   \begin{figure}[htb]
   440   \begin{center}
   441   \[
   442   \infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}}
   443   \qquad
   444   \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{}
   445   \]
   446   \[
   447   \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}{@{text "\<Gamma> \<turnstile> b x"} & @{text "x \<notin> \<Gamma>"}}
   448   \qquad
   449   \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b a"}}{@{text "\<Gamma> \<turnstile> \<And>x. b x"}}
   450   \]
   451   \[
   452   \infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
   453   \qquad
   454   \infer[@{text "(\<Longrightarrow>_elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}}
   455   \]
   456   \caption{Primitive inferences of the Pure logic}\label{fig:prim-rules}
   457   \end{center}
   458   \end{figure}
   460   The introduction and elimination rules for @{text "\<And>"} and @{text
   461   "\<Longrightarrow>"} are analogous to formation of (dependently typed) @{text
   462   "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
   463   are \emph{irrelevant} in the Pure logic, they may never occur within
   464   propositions, i.e.\ the @{text "\<Longrightarrow>"} arrow of the framework is a
   465   non-dependent one.
   467   Also note that fixed parameters as in @{text "\<And>_intro"} need not be
   468   recorded in the context @{text "\<Gamma>"}, since syntactic types are
   469   always inhabitable.  An ``assumption'' @{text "x :: \<tau>"} is logically
   470   vacuous, because @{text "\<tau>"} is always non-empty.  This is the deeper
   471   reason why @{text "\<Gamma>"} only consists of hypothetical proofs, but no
   472   hypothetical terms.
   474   The corresponding proof terms are left implicit in the classic
   475   ``LCF-approach'', although they could be exploited separately
   476   \cite{Berghofer-Nipkow:2000}.  The implementation provides a runtime
   477   option to control the generation of full proof terms.
   479   \medskip The axiomatization of a theory is implicitly closed by
   480   forming all instances of type and term variables: @{text "\<turnstile> A\<vartheta>"} for
   481   any substitution instance of axiom @{text "\<turnstile> A"}.  By pushing
   482   substitution through derivations inductively, we get admissible
   483   substitution rules for theorems shown in \figref{fig:subst-rules}.
   485   \begin{figure}[htb]
   486   \begin{center}
   487   \[
   488   \infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}}
   489   \quad
   490   \infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}}
   491   \]
   492   \[
   493   \infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}
   494   \quad
   495   \infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}}
   496   \]
   497   \caption{Admissible substitution rules}\label{fig:subst-rules}
   498   \end{center}
   499   \end{figure}
   501   Note that @{text "instantiate_term"} could be derived using @{text
   502   "\<And>_intro/elim"}, but this is not how it is implemented.  The type
   503   instantiation rule is a genuine admissible one, due to the lack of
   504   true polymorphism in the logic.
   506   Since @{text "\<Gamma>"} may never contain any schematic variables, the
   507   @{text "instantiate"} do not require an explicit side-condition.  In
   508   principle, variables could be substituted in hypotheses as well, but
   509   this could disrupt monotonicity of the basic calculus: derivations
   510   could leave the current proof context.
   512   \medskip The framework also provides builtin equality @{text "\<equiv>"},
   513   which is conceptually axiomatized shown in \figref{fig:equality},
   514   although the implementation provides derived rules directly:
   516   \begin{figure}[htb]
   517   \begin{center}
   518   \begin{tabular}{ll}
   519   @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
   520   @{text "\<turnstile> (\<lambda>x. b x) a \<equiv> b a"} & @{text "\<beta>"}-conversion \\
   521   @{text "\<turnstile> x \<equiv> x"} & reflexivity law \\
   522   @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution law \\
   523   @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
   524   @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & coincidence with equivalence \\
   525   \end{tabular}
   526   \caption{Conceptual axiomatization of equality.}\label{fig:equality}
   527   \end{center}
   528   \end{figure}
   530   Since the basic representation of terms already accounts for @{text
   531   "\<alpha>"}-conversion, Pure equality essentially acts like @{text
   532   "\<alpha>\<beta>\<eta>"}-equivalence on terms, while coinciding with bi-implication.
   535   \medskip Conjunction is defined in Pure as a derived connective, see
   536   \figref{fig:conjunction}.  This is occasionally useful to represent
   537   simultaneous statements behind the scenes --- framework conjunction
   538   is usually not exposed to the user.
   540   \begin{figure}[htb]
   541   \begin{center}
   542   \begin{tabular}{ll}
   543   @{text "& :: prop \<Rightarrow> prop \<Rightarrow> prop"} & conjunction (hidden) \\
   544   @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\
   545   \end{tabular}
   546   \caption{Definition of conjunction.}\label{fig:equality}
   547   \end{center}
   548   \end{figure}
   550   The definition allows to derive the usual introduction @{text "\<turnstile> A \<Longrightarrow>
   551   B \<Longrightarrow> A & B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B
   552   \<Longrightarrow> B"}.
   553 *}
   556 section {* Rules \label{sec:rules} *}
   558 text {*
   560 FIXME
   562   A \emph{rule} is any Pure theorem in HHF normal form; there is a
   563   separate calculus for rule composition, which is modeled after
   564   Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
   565   rules to be nested arbitrarily, similar to \cite{extensions91}.
   567   Normally, all theorems accessible to the user are proper rules.
   568   Low-level inferences are occasional required internally, but the
   569   result should be always presented in canonical form.  The higher
   570   interfaces of Isabelle/Isar will always produce proper rules.  It is
   571   important to maintain this invariant in add-on applications!
   573   There are two main principles of rule composition: @{text
   574   "resolution"} (i.e.\ backchaining of rules) and @{text
   575   "by-assumption"} (i.e.\ closing a branch); both principles are
   576   combined in the variants of @{text "elim-resolution"} and @{text
   577   "dest-resolution"}.  Raw @{text "composition"} is occasionally
   578   useful as well, also it is strictly speaking outside of the proper
   579   rule calculus.
   581   Rules are treated modulo general higher-order unification, which is
   582   unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion
   583   on @{text "\<lambda>"}-terms.  Moreover, propositions are understood modulo
   584   the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.
   586   This means that any operations within the rule calculus may be
   587   subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions.  It is common
   588   practice not to contract or expand unnecessarily.  Some mechanisms
   589   prefer an one form, others the opposite, so there is a potential
   590   danger to produce some oscillation!
   592   Only few operations really work \emph{modulo} HHF conversion, but
   593   expect a normal form: quantifiers @{text "\<And>"} before implications
   594   @{text "\<Longrightarrow>"} at each level of nesting.
   596 \glossary{Hereditary Harrop Formula}{The set of propositions in HHF
   597 format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow>
   598 A)"}, for variables @{text "x"} and atomic propositions @{text "A"}.
   599 Any proposition may be put into HHF form by normalizing with the rule
   600 @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.  In Isabelle, the outermost
   601 quantifier prefix is represented via \seeglossary{schematic
   602 variables}, such that the top-level structure is merely that of a
   603 \seeglossary{Horn Clause}}.
   605 \glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
   608   \[
   609   \infer[@{text "(assumption)"}]{@{text "C\<vartheta>"}}
   610   {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text i})}}
   611   \]
   614   \[
   615   \infer[@{text "(compose)"}]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}}
   616   {@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}}
   617   \]
   620   \[
   621   \infer[@{text "(\<And>_lift)"}]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}}
   622   \]
   623   \[
   624   \infer[@{text "(\<Longrightarrow>_lift)"}]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}}
   625   \]
   627   The @{text resolve} scheme is now acquired from @{text "\<And>_lift"},
   628   @{text "\<Longrightarrow>_lift"}, and @{text compose}.
   630   \[
   631   \infer[@{text "(resolution)"}]
   632   {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}}
   633   {\begin{tabular}{l}
   634     @{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\
   635     @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\
   636     @{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\
   637    \end{tabular}}
   638   \]
   641   FIXME @{text "elim_resolution"}, @{text "dest_resolution"}
   642 *}
   645 end