doc-src/IsarImplementation/Thy/Logic.thy
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+theory Logic
+imports Base
+begin
+
+chapter {* Primitive logic \label{ch:logic} *}
+
+text {*
+  The logical foundations of Isabelle/Isar are that of the Pure logic,
+  which has been introduced as a natural-deduction framework in
+  \cite{paulson700}.  This is essentially the same logic as @{text
+  "\<lambda>HOL"}'' in the more abstract setting of Pure Type Systems (PTS)
+  \cite{Barendregt-Geuvers:2001}, although there are some key
+  differences in the specific treatment of simple types in
+  Isabelle/Pure.
+
+  Following type-theoretic parlance, the Pure logic consists of three
+  levels of @{text "\<lambda>"}-calculus with corresponding arrows, @{text
+  "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
+  "\<And>"} for universal quantification (proofs depending on terms), and
+  @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
+
+  Derivations are relative to a logical theory, which declares type
+  constructors, constants, and axioms.  Theory declarations support
+  schematic polymorphism, which is strictly speaking outside the
+  logic.\footnote{This is the deeper logical reason, why the theory
+  context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"}
+  of the core calculus.}
+*}
+
+
+section {* Types \label{sec:types} *}
+
+text {*
+  The language of types is an uninterpreted order-sorted first-order
+  algebra; types are qualified by ordered type classes.
+
+  \medskip A \emph{type class} is an abstract syntactic entity
+  declared in the theory context.  The \emph{subclass relation} @{text
+  "c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic
+  generating relation; the transitive closure is maintained
+  internally.  The resulting relation is an ordering: reflexive,
+  transitive, and antisymmetric.
+
+  A \emph{sort} is a list of type classes written as @{text "s =
+  {c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
+  intersection.  Notationally, the curly braces are omitted for
+  singleton intersections, i.e.\ any class @{text "c"} may be read as
+  a sort @{text "{c}"}.  The ordering on type classes is extended to
+  sorts according to the meaning of intersections: @{text
+  "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff
+  @{text "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}.  The empty intersection
+  @{text "{}"} refers to the universal sort, which is the largest
+  element wrt.\ the sort order.  The intersections of all (finitely
+  many) classes declared in the current theory are the minimal
+  elements wrt.\ the sort order.
+
+  \medskip A \emph{fixed type variable} is a pair of a basic name
+  (starting with a @{text "'"} character) and a sort constraint, e.g.\
+  @{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}.
+  A \emph{schematic type variable} is a pair of an indexname and a
+  sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually
+  printed as @{text "?\<alpha>\<^isub>s"}.
+
+  Note that \emph{all} syntactic components contribute to the identity
+  of type variables, including the sort constraint.  The core logic
+  handles type variables with the same name but different sorts as
+  different, although some outer layers of the system make it hard to
+  produce anything like this.
+
+  A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
+  on types declared in the theory.  Type constructor application is
+  written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}.  For
+  @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"}
+  instead of @{text "()prop"}.  For @{text "k = 1"} the parentheses
+  are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}.
+  Further notation is provided for specific constructors, notably the
+  right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>,
+  \<beta>)fun"}.
+
+  A \emph{type} is defined inductively over type variables and type
+  constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
+  (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>"}.
+
+  A \emph{type abbreviation} is a syntactic definition @{text
+  "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
+  variables @{text "\<^vec>\<alpha>"}.  Type abbreviations appear as type
+  constructors in the syntax, but are expanded before entering the
+  logical core.
+
+  A \emph{type arity} declares the image behavior of a type
+  constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
+  s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)\<kappa>"} is
+  of sort @{text "s"} if every argument type @{text "\<tau>\<^isub>i"} is
+  of sort @{text "s\<^isub>i"}.  Arity declarations are implicitly
+  completed, i.e.\ @{text "\<kappa> :: (\<^vec>s)c"} entails @{text "\<kappa> ::
+  (\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}.
+
+  \medskip The sort algebra is always maintained as \emph{coregular},
+  which means that type arities are consistent with the subclass
+  relation: for any type constructor @{text "\<kappa>"}, and classes @{text
+  "c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> ::
+  (\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> ::
+  (\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq>
+  \<^vec>s\<^isub>2"} component-wise.
+
+  The key property of a coregular order-sorted algebra is that sort
+  constraints can be solved in a most general fashion: for each type
+  constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general
+  vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such
+  that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>,
+  \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}.
+  Consequently, type unification has most general solutions (modulo
+  equivalence of sorts), so type-inference produces primary types as
+  expected \cite{nipkow-prehofer}.
+*}
+
+text %mlref {*
+  \begin{mldecls}
+  @{index_ML_type class} \\
+  @{index_ML_type sort} \\
+  @{index_ML_type arity} \\
+  @{index_ML_type typ} \\
+  @{index_ML map_atyps: "(typ -> typ) -> typ -> typ"} \\
+  @{index_ML fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
+  \end{mldecls}
+  \begin{mldecls}
+  @{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
+  @{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
+  @{index_ML Sign.add_types: "(string * int * mixfix) list -> theory -> theory"} \\
+  (string * string list * typ * mixfix) list -> theory -> theory"} \\
+  @{index_ML Sign.primitive_class: "string * class list -> theory -> theory"} \\
+  @{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
+  @{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
+  \end{mldecls}
+
+  \begin{description}
+
+  \item @{ML_type class} represents type classes; this is an alias for
+  @{ML_type string}.
+
+  \item @{ML_type sort} represents sorts; this is an alias for
+  @{ML_type "class list"}.
+
+  \item @{ML_type arity} represents type arities; this is an alias for
+  triples of the form @{text "(\<kappa>, \<^vec>s, s)"} for @{text "\<kappa> ::
+  (\<^vec>s)s"} described above.
+
+  \item @{ML_type typ} represents types; this is a datatype with
+  constructors @{ML TFree}, @{ML TVar}, @{ML Type}.
+
+  \item @{ML map_atyps}~@{text "f \<tau>"} applies the mapping @{text "f"}
+  to all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text
+  "\<tau>"}.
+
+  \item @{ML fold_atyps}~@{text "f \<tau>"} iterates the operation @{text
+  "f"} over all occurrences of atomic types (@{ML TFree}, @{ML TVar})
+  in @{text "\<tau>"}; the type structure is traversed from left to right.
+
+  \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
+  tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
+
+  \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether type
+  @{text "\<tau>"} is of sort @{text "s"}.
+
+  \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares a new
+  type constructors @{text "\<kappa>"} with @{text "k"} arguments and
+  optional mixfix syntax.
+
+  \item @{ML Sign.add_tyabbrs_i}~@{text "[(\<kappa>, \<^vec>\<alpha>, \<tau>, mx), \<dots>]"}
+  defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"} with
+  optional mixfix syntax.
+
+  \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
+  c\<^isub>n])"} declares a new class @{text "c"}, together with class
+  relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
+
+  \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
+  c\<^isub>2)"} declares the class relation @{text "c\<^isub>1 \<subseteq>
+  c\<^isub>2"}.
+
+  \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
+  the arity @{text "\<kappa> :: (\<^vec>s)s"}.
+
+  \end{description}
+*}
+
+
+
+section {* Terms \label{sec:terms} *}
+
+text {*
+  \glossary{Term}{FIXME}
+
+  The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
+  with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
+  or \cite{paulson-ml2}), with the types being determined determined
+  by the corresponding binders.  In contrast, free variables and
+  constants are have an explicit name and type in each occurrence.
+
+  \medskip A \emph{bound variable} is a natural number @{text "b"},
+  which accounts for the number of intermediate binders between the
+  variable occurrence in the body and its binding position.  For
+  example, the de-Bruijn term @{text
+  "\<lambda>\<^bsub>nat\<^esub>. \<lambda>\<^bsub>nat\<^esub>. 1 + 0"} would
+  correspond to @{text
+  "\<lambda>x\<^bsub>nat\<^esub>. \<lambda>y\<^bsub>nat\<^esub>. x + y"} in a named
+  representation.  Note that a bound variable may be represented by
+  different de-Bruijn indices at different occurrences, depending on
+  the nesting of abstractions.
+
+  A \emph{loose variable} is a bound variable that is outside the
+  scope of local binders.  The types (and names) for loose variables
+  can be managed as a separate context, that is maintained as a stack
+  of hypothetical binders.  The core logic operates on closed terms,
+  without any loose variables.
+
+  A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
+  @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}.  A
+  \emph{schematic variable} is a pair of an indexname and a type,
+  e.g.\ @{text "((x, 0), \<tau>)"} which is usually printed as @{text
+  "?x\<^isub>\<tau>"}.
+
+  \medskip A \emph{constant} is a pair of a basic name and a type,
+  e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text
+  "c\<^isub>\<tau>"}.  Constants are declared in the context as polymorphic
+  families @{text "c :: \<sigma>"}, meaning that all substitution instances
+  @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
+
+  The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"}
+  wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of
+  the matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>,
+  ?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in canonical order @{text
+  "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}.  Within a given theory context,
+  there is a one-to-one correspondence between any constant @{text
+  "c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1, \<dots>,
+  \<tau>\<^isub>n)"} of its type arguments.  For example, with @{text "plus
+  :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow>
+  nat\<^esub>"} corresponds to @{text "plus(nat)"}.
+
+  Constant declarations @{text "c :: \<sigma>"} may contain sort constraints
+  for type variables in @{text "\<sigma>"}.  These are observed by
+  type-inference as expected, but \emph{ignored} by the core logic.
+  This means the primitive logic is able to reason with instances of
+  polymorphic constants that the user-level type-checker would reject
+  due to violation of type class restrictions.
+
+  \medskip An \emph{atomic} term is either a variable or constant.  A
+  \emph{term} is defined inductively over atomic terms, with
+  abstraction and application as follows: @{text "t = b | 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 converting between an external
+  representation with named bound variables.  Subsequently, we shall
+  use the latter notation instead of internal de-Bruijn
+  representation.
+
+  The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a
+  term according to the structure of atomic terms, abstractions, and
+  applicatins:
+  $+ \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{} + \qquad + \infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}} + \qquad + \infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}} +$
+  A \emph{well-typed term} is a term that can be typed according to these rules.
+
+  Typing information can be omitted: type-inference is able to
+  reconstruct the most general type of a raw term, while assigning
+  most general types to all of its variables and constants.
+  Type-inference depends on a context of type constraints for fixed
+  variables, and declarations for polymorphic constants.
+
+  The identity of atomic terms consists both of the name and the type
+  component.  This means that different variables @{text
+  "x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text
+  "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type
+  instantiation.  Some outer layers of the system make it hard to
+  produce variables of the same name, but different types.  In
+  contrast, mixed instances of polymorphic constants occur frequently.
+
+  \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
+  is the set of type variables occurring in @{text "t"}, but not in
+  @{text "\<sigma>"}.  This means that the term implicitly depends on type
+  arguments that are not accounted in the result type, i.e.\ there are
+  different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text
+  "t\<vartheta>' :: \<sigma>"} with the same type.  This slightly
+  pathological situation notoriously demands additional care.
+
+  \medskip A \emph{term abbreviation} is a syntactic definition @{text
+  "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"},
+  without any hidden polymorphism.  A term abbreviation looks like a
+  constant in the syntax, but is expanded before entering the logical
+  core.  Abbreviations are usually reverted when printing terms, using
+  @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for higher-order rewriting.
+
+  \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
+  "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free
+  renaming of bound variables; @{text "\<beta>"}-conversion contracts an
+  abstraction applied to an argument term, substituting the argument
+  in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
+  "\<eta>"}-conversion contracts vacuous application-abstraction: @{text
+  "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
+  does not occur in @{text "f"}.
+
+  Terms are normally treated modulo @{text "\<alpha>"}-conversion, which is
+  implicit in the de-Bruijn representation.  Names for bound variables
+  in abstractions are maintained separately as (meaningless) comments,
+  mostly for parsing and printing.  Full @{text "\<alpha>\<beta>\<eta>"}-conversion is
+  commonplace in various standard operations (\secref{sec:obj-rules})
+  that are based on higher-order unification and matching.
+*}
+
+text %mlref {*
+  \begin{mldecls}
+  @{index_ML_type term} \\
+  @{index_ML "op aconv": "term * term -> bool"} \\
+  @{index_ML map_types: "(typ -> typ) -> term -> term"} \\
+  @{index_ML fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
+  @{index_ML map_aterms: "(term -> term) -> term -> term"} \\
+  @{index_ML fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\
+  \end{mldecls}
+  \begin{mldecls}
+  @{index_ML fastype_of: "term -> typ"} \\
+  @{index_ML lambda: "term -> term -> term"} \\
+  @{index_ML betapply: "term * term -> term"} \\
+  @{index_ML Sign.declare_const: "Properties.T -> (binding * typ) * mixfix ->
+  theory -> term * theory"} \\
+  @{index_ML Sign.add_abbrev: "string -> Properties.T -> binding * term ->
+  theory -> (term * term) * theory"} \\
+  @{index_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\
+  @{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
+  \end{mldecls}
+
+  \begin{description}
+
+  \item @{ML_type term} represents de-Bruijn terms, with comments in
+  abstractions, and explicitly named free variables and constants;
+  this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML
+  Var}, @{ML Const}, @{ML Abs}, @{ML "op \$"}.
+
+  \item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text
+  "\<alpha>"}-equivalence of two terms.  This is the basic equality relation
+  on type @{ML_type term}; raw datatype equality should only be used
+  for operations related to parsing or printing!
+
+  \item @{ML map_types}~@{text "f t"} applies the mapping @{text
+  "f"} to all types occurring in @{text "t"}.
+
+  \item @{ML fold_types}~@{text "f t"} iterates the operation @{text
+  "f"} over all occurrences of types in @{text "t"}; the term
+  structure is traversed from left to right.
+
+  \item @{ML map_aterms}~@{text "f t"} applies the mapping @{text "f"}
+  to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
+  Const}) occurring in @{text "t"}.
+
+  \item @{ML fold_aterms}~@{text "f t"} iterates the operation @{text
+  "f"} over all occurrences of atomic terms (@{ML Bound}, @{ML Free},
+  @{ML Var}, @{ML Const}) in @{text "t"}; the term structure is
+  traversed from left to right.
+
+  \item @{ML fastype_of}~@{text "t"} determines the type of a
+  well-typed term.  This operation is relatively slow, despite the
+  omission of any sanity checks.
+
+  \item @{ML lambda}~@{text "a b"} produces an abstraction @{text
+  "\<lambda>a. b"}, where occurrences of the atomic term @{text "a"} in the
+  body @{text "b"} are replaced by bound variables.
+
+  \item @{ML betapply}~@{text "(t, u)"} produces an application @{text
+  "t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an
+  abstraction.
+
+  \item @{ML Sign.declare_const}~@{text "properties ((c, \<sigma>), mx)"}
+  declares a new constant @{text "c :: \<sigma>"} with optional mixfix
+  syntax.
+
+  \item @{ML Sign.add_abbrev}~@{text "print_mode properties (c, t)"}
+  introduces a new term abbreviation @{text "c \<equiv> t"}.
+
+  \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
+  Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
+  convert between two representations of polymorphic constants: full
+  type instance vs.\ compact type arguments form.
+
+  \end{description}
+*}
+
+
+section {* Theorems \label{sec:thms} *}
+
+text {*
+  \glossary{Proposition}{FIXME A \seeglossary{term} of
+  \seeglossary{type} @{text "prop"}.  Internally, there is nothing
+  special about propositions apart from their type, but the concrete
+  syntax enforces a clear distinction.  Propositions are structured
+  via implication @{text "A \<Longrightarrow> B"} or universal quantification @{text
+  "\<And>x. B x"} --- anything else is considered atomic.  The canonical
+  form for propositions is that of a \seeglossary{Hereditary Harrop
+  Formula}. FIXME}
+
+  \glossary{Theorem}{A proven proposition within a certain theory and
+  proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are
+  rarely spelled out explicitly.  Theorems are usually normalized
+  according to the \seeglossary{HHF} format. FIXME}
+
+  \glossary{Fact}{Sometimes used interchangeably for
+  \seeglossary{theorem}.  Strictly speaking, a list of theorems,
+  essentially an extra-logical conjunction.  Facts emerge either as
+  local assumptions, or as results of local goal statements --- both
+  may be simultaneous, hence the list representation. FIXME}
+
+  \glossary{Schematic variable}{FIXME}
+
+  \glossary{Fixed variable}{A variable that is bound within a certain
+  proof context; an arbitrary-but-fixed entity within a portion of
+  proof text. FIXME}
+
+  \glossary{Free variable}{Synonymous for \seeglossary{fixed
+  variable}. FIXME}
+
+  \glossary{Bound variable}{FIXME}
+
+  \glossary{Variable}{See \seeglossary{schematic variable},
+  \seeglossary{fixed variable}, \seeglossary{bound variable}, or
+  \seeglossary{type variable}.  The distinguishing feature of
+  different variables is their binding scope. FIXME}
+
+  A \emph{proposition} is a well-typed term of type @{text "prop"}, a
+  \emph{theorem} is a proven proposition (depending on a context of
+  hypotheses and the background theory).  Primitive inferences include
+  plain natural deduction rules for the primary connectives @{text
+  "\<And>"} and @{text "\<Longrightarrow>"} of the framework.  There is also a builtin
+  notion of equality/equivalence @{text "\<equiv>"}.
+*}
+
+subsection {* Primitive connectives and rules \label{sec:prim-rules} *}
+
+text {*
+  The theory @{text "Pure"} contains constant declarations for the
+  primitive connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of
+  the logical framework, see \figref{fig:pure-connectives}.  The
+  derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is
+  defined inductively by the primitive inferences given in
+  \figref{fig:prim-rules}, with the global restriction that the
+  hypotheses must \emph{not} contain any schematic variables.  The
+  builtin equality is conceptually axiomatized as shown in
+  \figref{fig:pure-equality}, although the implementation works
+  directly with derived inferences.
+
+  \begin{figure}[htb]
+  \begin{center}
+  \begin{tabular}{ll}
+  @{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\
+  @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
+  @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
+  \end{tabular}
+  \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
+  \end{center}
+  \end{figure}
+
+  \begin{figure}[htb]
+  \begin{center}
+  $+ \infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}} + \qquad + \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{} +$
+  $+ \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}} + \qquad + \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}} +$
+  $+ \infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}} + \qquad + \infer[@{text "(\<Longrightarrow>_elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}} +$
+  \caption{Primitive inferences of Pure}\label{fig:prim-rules}
+  \end{center}
+  \end{figure}
+
+  \begin{figure}[htb]
+  \begin{center}
+  \begin{tabular}{ll}
+  @{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\
+  @{text "\<turnstile> x \<equiv> x"} & reflexivity \\
+  @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\
+  @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
+  @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\
+  \end{tabular}
+  \caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality}
+  \end{center}
+  \end{figure}
+
+  The introduction and elimination rules for @{text "\<And>"} and @{text
+  "\<Longrightarrow>"} are analogous to formation of dependently typed @{text
+  "\<lambda>"}-terms representing the underlying proof objects.  Proof terms
+  are irrelevant in the Pure logic, though; they cannot occur within
+  propositions.  The system provides a runtime option to record
+  explicit proof terms for primitive inferences.  Thus all three
+  levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for
+  terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\
+  \cite{Berghofer-Nipkow:2000:TPHOL}).
+
+  Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need
+  not be recorded in the hypotheses, because the simple syntactic
+  types of Pure are always inhabitable.  Assumptions'' @{text "x ::
+  \<tau>"} for type-membership are only present as long as some @{text
+  "x\<^isub>\<tau>"} occurs in the statement body.\footnote{This is the key
+  difference to @{text "\<lambda>HOL"}'' in the PTS framework
+  \cite{Barendregt-Geuvers:2001}, where hypotheses @{text "x : A"} are
+  treated uniformly for propositions and types.}
+
+  \medskip The axiomatization of a theory is implicitly closed by
+  forming all instances of type and term variables: @{text "\<turnstile>
+  A\<vartheta>"} holds for any substitution instance of an axiom
+  @{text "\<turnstile> A"}.  By pushing substitutions through derivations
+  inductively, we also get admissible @{text "generalize"} and @{text
+  "instance"} rules as shown in \figref{fig:subst-rules}.
+
+  \begin{figure}[htb]
+  \begin{center}
+  $+ \infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}} + \quad + \infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}} +$
+  $+ \infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}} + \quad + \infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}} +$
+  \end{center}
+  \end{figure}
+
+  Note that @{text "instantiate"} does not require an explicit
+  side-condition, because @{text "\<Gamma>"} may never contain schematic
+  variables.
+
+  In principle, variables could be substituted in hypotheses as well,
+  but this would disrupt the monotonicity of reasoning: deriving
+  @{text "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is
+  correct, but @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold:
+  the result belongs to a different proof context.
+
+  \medskip An \emph{oracle} is a function that produces axioms on the
+  fly.  Logically, this is an instance of the @{text "axiom"} rule
+  (\figref{fig:prim-rules}), but there is an operational difference.
+  The system always records oracle invocations within derivations of
+  theorems.  Tracing plain axioms (and named theorems) is optional.
+
+  Axiomatizations should be limited to the bare minimum, typically as
+  part of the initial logical basis of an object-logic formalization.
+  Later on, theories are usually developed in a strictly definitional
+  fashion, by stating only certain equalities over new constants.
+
+  A \emph{simple definition} consists of a constant declaration @{text
+  "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text "t
+  :: \<sigma>"} is a closed term without any hidden polymorphism.  The RHS
+  may depend on further defined constants, but not @{text "c"} itself.
+  Definitions of functions may be presented as @{text "c \<^vec>x \<equiv>
+  t"} instead of the puristic @{text "c \<equiv> \<lambda>\<^vec>x. t"}.
+
+  An \emph{overloaded definition} consists of a collection of axioms
+  for the same constant, with zero or one equations @{text
+  "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type constructor @{text "\<kappa>"} (for
+  distinct variables @{text "\<^vec>\<alpha>"}).  The RHS may mention
+  previously defined constants as above, or arbitrary constants @{text
+  "d(\<alpha>\<^isub>i)"} for some @{text "\<alpha>\<^isub>i"} projected from @{text
+  "\<^vec>\<alpha>"}.  Thus overloaded definitions essentially work by
+  primitive recursion over the syntactic structure of a single type
+  argument.
+*}
+
+text %mlref {*
+  \begin{mldecls}
+  @{index_ML_type ctyp} \\
+  @{index_ML_type cterm} \\
+  @{index_ML Thm.ctyp_of: "theory -> typ -> ctyp"} \\
+  @{index_ML Thm.cterm_of: "theory -> term -> cterm"} \\
+  \end{mldecls}
+  \begin{mldecls}
+  @{index_ML_type thm} \\
+  @{index_ML proofs: "int ref"} \\
+  @{index_ML Thm.assume: "cterm -> thm"} \\
+  @{index_ML Thm.forall_intr: "cterm -> thm -> thm"} \\
+  @{index_ML Thm.forall_elim: "cterm -> thm -> thm"} \\
+  @{index_ML Thm.implies_intr: "cterm -> thm -> thm"} \\
+  @{index_ML Thm.implies_elim: "thm -> thm -> thm"} \\
+  @{index_ML Thm.generalize: "string list * string list -> int -> thm -> thm"} \\
+  @{index_ML Thm.instantiate: "(ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm"} \\
+  @{index_ML Thm.axiom: "theory -> string -> thm"} \\
+  @{index_ML Thm.add_oracle: "bstring * ('a -> cterm) -> theory
+  -> (string * ('a -> thm)) * theory"} \\
+  \end{mldecls}
+  \begin{mldecls}
+  @{index_ML Theory.add_axioms_i: "(binding * term) list -> theory -> theory"} \\
+  @{index_ML Theory.add_deps: "string -> string * typ -> (string * typ) list -> theory -> theory"} \\
+  @{index_ML Theory.add_defs_i: "bool -> bool -> (binding * term) list -> theory -> theory"} \\
+  \end{mldecls}
+
+  \begin{description}
+
+  \item @{ML_type ctyp} and @{ML_type cterm} represent certified types
+  and terms, respectively.  These are abstract datatypes that
+  guarantee that its values have passed the full well-formedness (and
+  well-typedness) checks, relative to the declarations of type
+  constructors, constants etc. in the theory.
+
+  \item @{ML ctyp_of}~@{text "thy \<tau>"} and @{ML cterm_of}~@{text "thy
+  t"} explicitly checks types and terms, respectively.  This also
+  involves some basic normalizations, such expansion of type and term
+  abbreviations from the theory context.
+
+  Re-certification is relatively slow and should be avoided in tight
+  reasoning loops.  There are separate operations to decompose
+  certified entities (including actual theorems).
+
+  \item @{ML_type thm} represents proven propositions.  This is an
+  abstract datatype that guarantees that its values have been
+  constructed by basic principles of the @{ML_struct Thm} module.
+  Every @{ML thm} value contains a sliding back-reference to the
+  enclosing theory, cf.\ \secref{sec:context-theory}.
+
+  \item @{ML proofs} determines the detail of proof recording within
+  @{ML_type thm} values: @{ML 0} records only oracles, @{ML 1} records
+  oracles, axioms and named theorems, @{ML 2} records full proof
+  terms.
+
+  \item @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML
+  Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim}
+  correspond to the primitive inferences of \figref{fig:prim-rules}.
+
+  \item @{ML Thm.generalize}~@{text "(\<^vec>\<alpha>, \<^vec>x)"}
+  corresponds to the @{text "generalize"} rules of
+  \figref{fig:subst-rules}.  Here collections of type and term
+  variables are generalized simultaneously, specified by the given
+  basic names.
+
+  \item @{ML Thm.instantiate}~@{text "(\<^vec>\<alpha>\<^isub>s,
+  \<^vec>x\<^isub>\<tau>)"} corresponds to the @{text "instantiate"} rules
+  of \figref{fig:subst-rules}.  Type variables are substituted before
+  term variables.  Note that the types in @{text "\<^vec>x\<^isub>\<tau>"}
+  refer to the instantiated versions.
+
+  \item @{ML Thm.axiom}~@{text "thy name"} retrieves a named
+  axiom, cf.\ @{text "axiom"} in \figref{fig:prim-rules}.
+
+  \item @{ML Thm.add_oracle}~@{text "(name, oracle)"} produces a named
+  oracle rule, essentially generating arbitrary axioms on the fly,
+  cf.\ @{text "axiom"} in \figref{fig:prim-rules}.
+
+  \item @{ML Theory.add_axioms_i}~@{text "[(name, A), \<dots>]"} declares
+  arbitrary propositions as axioms.
+
+  \item @{ML Theory.add_deps}~@{text "name c\<^isub>\<tau>
+  \<^vec>d\<^isub>\<sigma>"} declares dependencies of a named specification
+  for constant @{text "c\<^isub>\<tau>"}, relative to existing
+  specifications for constants @{text "\<^vec>d\<^isub>\<sigma>"}.
+
+  \<^vec>x \<equiv> t), \<dots>]"} states a definitional axiom for an existing
+  constant @{text "c"}.  Dependencies are recorded (cf.\ @{ML
+  Theory.add_deps}), unless the @{text "unchecked"} option is set.
+
+  \end{description}
+*}
+
+
+subsection {* Auxiliary definitions *}
+
+text {*
+  Theory @{text "Pure"} provides a few auxiliary definitions, see
+  \figref{fig:pure-aux}.  These special constants are normally not
+  exposed to the user, but appear in internal encodings.
+
+  \begin{figure}[htb]
+  \begin{center}
+  \begin{tabular}{ll}
+  @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\
+  @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \$1ex] + @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, suppressed) \\ + @{text "#A \<equiv> A"} \\[1ex] + @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\ + @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex] + @{text "TYPE :: \<alpha> itself"} & (prefix @{text "TYPE"}) \\ + @{text "(unspecified)"} \\ + \end{tabular} + \caption{Definitions of auxiliary connectives}\label{fig:pure-aux} + \end{center} + \end{figure} + + Derived conjunction rules include introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A & + B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}. + Conjunction allows to treat simultaneous assumptions and conclusions + uniformly. For example, multiple claims are intermediately + represented as explicit conjunction, but this is refined into + separate sub-goals before the user continues the proof; the final + result is projected into a list of theorems (cf.\ + \secref{sec:tactical-goals}). + + The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex + propositions appear as atomic, without changing the meaning: @{text + "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable. See + \secref{sec:tactical-goals} for specific operations. + + The @{text "term"} marker turns any well-typed term into a derivable + proposition: @{text "\<turnstile> TERM t"} holds unconditionally. Although + this is logically vacuous, it allows to treat terms and proofs + uniformly, similar to a type-theoretic framework. + + The @{text "TYPE"} constructor is the canonical representative of + the unspecified type @{text "\<alpha> itself"}; it essentially injects the + language of types into that of terms. There is specific notation + @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau> + itself\<^esub>"}. + Although being devoid of any particular meaning, the @{text + "TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term + language. In particular, @{text "TYPE(\<alpha>)"} may be used as formal + argument in primitive definitions, in order to circumvent hidden + polymorphism (cf.\ \secref{sec:terms}). For example, @{text "c + TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of + a proposition @{text "A"} that depends on an additional type + argument, which is essentially a predicate on types. +*} + +text %mlref {* + \begin{mldecls} + @{index_ML Conjunction.intr: "thm -> thm -> thm"} \\ + @{index_ML Conjunction.elim: "thm -> thm * thm"} \\ + @{index_ML Drule.mk_term: "cterm -> thm"} \\ + @{index_ML Drule.dest_term: "thm -> cterm"} \\ + @{index_ML Logic.mk_type: "typ -> term"} \\ + @{index_ML Logic.dest_type: "term -> typ"} \\ + \end{mldecls} + + \begin{description} + + \item @{ML Conjunction.intr} derives @{text "A & B"} from @{text + "A"} and @{text "B"}. + + \item @{ML Conjunction.elim} derives @{text "A"} and @{text "B"} + from @{text "A & B"}. + + \item @{ML Drule.mk_term} derives @{text "TERM t"}. + + \item @{ML Drule.dest_term} recovers term @{text "t"} from @{text + "TERM t"}. + + \item @{ML Logic.mk_type}~@{text "\<tau>"} produces the term @{text + "TYPE(\<tau>)"}. + + \item @{ML Logic.dest_type}~@{text "TYPE(\<tau>)"} recovers the type + @{text "\<tau>"}. + + \end{description} +*} + + +section {* Object-level rules \label{sec:obj-rules} *} + +text %FIXME {* + +FIXME + + A \emph{rule} is any Pure theorem in HHF normal form; there is a + separate calculus for rule composition, which is modeled after + Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows + rules to be nested arbitrarily, similar to \cite{extensions91}. + + Normally, all theorems accessible to the user are proper rules. + Low-level inferences are occasional required internally, but the + result should be always presented in canonical form. The higher + interfaces of Isabelle/Isar will always produce proper rules. It is + important to maintain this invariant in add-on applications! + + There are two main principles of rule composition: @{text + "resolution"} (i.e.\ backchaining of rules) and @{text + "by-assumption"} (i.e.\ closing a branch); both principles are + combined in the variants of @{text "elim-resolution"} and @{text + "dest-resolution"}. Raw @{text "composition"} is occasionally + useful as well, also it is strictly speaking outside of the proper + rule calculus. + + Rules are treated modulo general higher-order unification, which is + unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion + on @{text "\<lambda>"}-terms. Moreover, propositions are understood modulo + the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}. + + This means that any operations within the rule calculus may be + subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions. It is common + practice not to contract or expand unnecessarily. Some mechanisms + prefer an one form, others the opposite, so there is a potential + danger to produce some oscillation! + + Only few operations really work \emph{modulo} HHF conversion, but + expect a normal form: quantifiers @{text "\<And>"} before implications + @{text "\<Longrightarrow>"} at each level of nesting. + +\glossary{Hereditary Harrop Formula}{The set of propositions in HHF +format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow> +A)"}, for variables @{text "x"} and atomic propositions @{text "A"}. +Any proposition may be put into HHF form by normalizing with the rule +@{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}. In Isabelle, the outermost +quantifier prefix is represented via \seeglossary{schematic +variables}, such that the top-level structure is merely that of a +\seeglossary{Horn Clause}}. + +\glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.} + + + \[ + \infer[@{text "(assumption)"}]{@{text "C\<vartheta>"}} + {@{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})}} +$
+
+
+  $+ \infer[@{text "(compose)"}]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}} + {@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}} +$
+
+
+  $+ \infer[@{text "(\<And>_lift)"}]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}} +$
+  $+ \infer[@{text "(\<Longrightarrow>_lift)"}]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}} +$
+
+  The @{text resolve} scheme is now acquired from @{text "\<And>_lift"},
+  @{text "\<Longrightarrow>_lift"}, and @{text compose}.
+
+  $+ \infer[@{text "(resolution)"}] + {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}} + {\begin{tabular}{l} + @{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\ + @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\ + @{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\ + \end{tabular}} +$
+
+
+  FIXME @{text "elim_resolution"}, @{text "dest_resolution"}
+*}
+
+end