(* $Id$ *)
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).
Pure derivations are relative to a logical theory, which declares
type constructors, term constants, and axioms. Theory declarations
support schematic polymorphism, which is strictly speaking outside
the logic.\footnote{Incidently, this is the main logical reason, why
the theory context @{text "\<Theta>"} is separate from the 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
"{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. For
example, @{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. For example, @{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
usually 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} @{text "\<tau>"} 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)k"}.
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 looks like type
constructors at the surface, but are fully 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 each type constructor @{text "\<kappa>"} and classes @{text
"c\<^isub>1 \<subseteq> c\<^isub>2"}, any arity @{text "\<kappa> ::
(\<^vec>s\<^isub>1)c\<^isub>1"} has a corresponding arity @{text "\<kappa>
:: (\<^vec>s\<^isub>2)c\<^isub>2"} where @{text "\<^vec>s\<^isub>1 \<subseteq>
\<^vec>s\<^isub>2"} holds component-wise.
The key property of a coregular order-sorted algebra is that sort
constraints may be always 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, the unification problem on the
algebra of types has most general solutions (modulo renaming and
equivalence of sorts). Moreover, the usual type-inference algorithm
will produce 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"} \\
@{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"} \\
@{index_ML Sign.add_tyabbrs_i: "
(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 mapping @{text "f"} to
all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text "\<tau>"}.
\item @{ML fold_atyps}~@{text "f \<tau>"} iterates operation @{text "f"}
over all occurrences of atoms (@{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 a type
is of a given sort.
\item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares 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 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 class relation @{text "c\<^isub>1 \<subseteq>
c\<^isub>2"}.
\item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
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}), and named free variables and constants.
Terms with loose bound variables are usually considered malformed.
The types of variables and constants is stored explicitly at each
occurrence in the term.
\medskip A \emph{bound variable} is a natural number @{text "b"},
which refers to the next binder that is @{text "b"} steps upwards
from the occurrence of @{text "b"} (counting from zero). Bindings
may be introduced as abstractions within the term, or as a separate
context (an inside-out list). This associates each bound variable
with a type. A \emph{loose variables} is a bound variable that is
outside the current scope of local binders or the context. For
example, the de-Bruijn term @{text "\<lambda>\<^isub>\<tau>. \<lambda>\<^isub>\<tau>. 1 + 0"}
corresponds to @{text "\<lambda>x\<^isub>\<tau>. \<lambda>y\<^isub>\<tau>. x + y"} in a named
representation. Also note that the very same bound variable may get
different numbers at different occurrences.
A \emph{fixed variable} is a pair of a basic name and a type. For
example, @{text "(x, \<tau>)"} which is usually printed @{text
"x\<^isub>\<tau>"}. A \emph{schematic variable} is a pair of an
indexname and a type. For example, @{text "((x, 0), \<tau>)"} which is
usually printed as @{text "?x\<^isub>\<tau>"}.
\medskip A \emph{constant} is a atomic terms consisting of a basic
name and a type. Constants are declared in the context as
polymorphic families @{text "c :: \<sigma>"}, meaning that any @{text
"c\<^isub>\<tau>"} is a valid constant for all substitution instances
@{text "\<tau> \<le> \<sigma>"}.
The list of \emph{type arguments} of @{text "c\<^isub>\<tau>"} wrt.\ the
declaration @{text "c :: \<sigma>"} is the codomain of the type matcher
presented in canonical order (according to the left-to-right
occurrences of type variables in in @{text "\<sigma>"}). Thus @{text
"c\<^isub>\<tau>"} can be represented more compactly as @{text
"c(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}. For example, the instance @{text
"plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>"} of some @{text "plus :: \<alpha> \<Rightarrow> \<alpha>
\<Rightarrow> \<alpha>"} has the singleton list @{text "nat"} as type arguments, the
constant may be represented as @{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.
\medskip A \emph{term} @{text "t"} is defined inductively over
variables and constants, 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 subsequent inductive relation @{text "t :: \<tau>"} assigns a
(unique) type to a term, using the special type constructor @{text
"(\<alpha>, \<beta>)fun"}, which is written @{text "\<alpha> \<Rightarrow> \<beta>"}.
\[
\infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{}
\qquad
\infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}}
\qquad
\infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}}
\]
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.
Thus different entities @{text "c\<^bsub>\<tau>\<^isub>1\<^esub>"} and
@{text "c\<^bsub>\<tau>\<^isub>2\<^esub>"} may well identified by type
instantiation, by mapping @{text "\<tau>\<^isub>1"} and @{text
"\<tau>\<^isub>2"} to the same @{text "\<tau>"}. Although,
different type instances of constants of the same basic name are
commonplace, this rarely happens for variables: type-inference
always demands ``consistent'' type constraints.
\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 the
values of various type variables that are not visible in the overall
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 is apt to cause strange effects.
\medskip A \emph{term abbreviation} is a syntactic definition @{text
"c\<^isub>\<sigma> \<equiv> t"} of an arbitrary closed term @{text "t"} of type
@{text "\<sigma>"} without any hidden polymorphism. A term abbreviation
looks like a constant at the surface, but is fully expanded before
entering the logical core. Abbreviations are usually reverted when
printing terms, using rules @{text "t \<rightarrow> c\<^isub>\<sigma>"} has a
higher-order term rewrite system.
\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 some 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
@{text "0"} does not occur in @{text "f"}.
Terms are almost always treated module @{text "\<alpha>"}-conversion, which
is implicit in the de-Bruijn representation. The names in
abstractions of bound variables are maintained only as a comment for
parsing and printing. Full @{text "\<alpha>\<beta>\<eta>"}-equivalence is usually
taken for granted higher rules (\secref{sec:rules}), anything
depending on higher-order unification or rewriting.
*}
text %mlref {*
\begin{mldecls}
@{index_ML_type term} \\
@{index_ML "op aconv": "term * term -> bool"} \\
@{index_ML map_term_types: "(typ -> typ) -> term -> term"} \\ %FIXME rename map_types
@{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"} \\
@{index_ML fastype_of: "term -> typ"} \\
@{index_ML lambda: "term -> term -> term"} \\
@{index_ML betapply: "term * term -> term"} \\
@{index_ML Sign.add_consts_i: "(string * typ * mixfix) list -> theory -> theory"} \\
@{index_ML Sign.add_abbrevs: "string * bool ->
((string * mixfix) * term) list -> theory -> 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 for bound variable names. 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_term_types}~@{text "f t"} applies mapping @{text "f"}
to all types occurring in @{text "t"}.
\item @{ML fold_types}~@{text "f t"} iterates 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 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 operation @{text "f"}
over all occurrences of atomic terms in (@{ML Bound}, @{ML Free},
@{ML Var}, @{ML Const}) @{text "t"}; the term structure is traversed
from left to right.
\item @{ML fastype_of}~@{text "t"} recomputes the type of a
well-formed term, while omitting any sanity checks. This operation
is relatively slow.
\item @{ML lambda}~@{text "a b"} produces an abstraction @{text
"\<lambda>a. b"}, where occurrences of the original (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"} happens to
be an abstraction.
\item @{ML Sign.add_consts_i}~@{text "[(c, \<sigma>, mx), \<dots>]"} declares a
new constant @{text "c :: \<sigma>"} with optional mixfix syntax.
\item @{ML Sign.add_abbrevs}~@{text "print_mode [((c, t), mx), \<dots>]"}
declares a new term abbreviation @{text "c \<equiv> t"} with optional
mixfix syntax.
\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 the two representations of constants, namely full
type instance vs.\ compact type arguments form (depending on the
most general declaration given in the context).
\end{description}
*}
section {* Theorems \label{sec:thms} *}
text {*
\glossary{Proposition}{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-formed term of type @{text "prop"}.
The connectives of minimal logic are declared as constants of the
basic theory:
\smallskip
\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) \\
\end{tabular}
\medskip A \emph{theorem} is a proven proposition, depending on a
collection of assumptions, and axioms from the theory context. The
judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is defined
inductively by the primitive inferences given in
\figref{fig:prim-rules}; there is a global syntactic restriction
that the hypotheses may not contain schematic variables.
\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 the Pure logic}\label{fig:prim-rules}
\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 \emph{irrelevant} in the Pure logic, they may never occur within
propositions, i.e.\ the @{text "\<Longrightarrow>"} arrow of the framework is a
non-dependent one.
Also note that fixed parameters as in @{text "\<And>_intro"} need not be
recorded in the context @{text "\<Gamma>"}, since syntactic types are
always inhabitable. An ``assumption'' @{text "x :: \<tau>"} is logically
vacuous, because @{text "\<tau>"} is always non-empty. This is the deeper
reason why @{text "\<Gamma>"} only consists of hypothetical proofs, but no
hypothetical terms.
The corresponding proof terms are left implicit in the classic
``LCF-approach'', although they could be exploited separately
\cite{Berghofer-Nipkow:2000}. The implementation provides a runtime
option to control the generation of full proof terms.
\medskip The axiomatization of a theory is implicitly closed by
forming all instances of type and term variables: @{text "\<turnstile> A\<vartheta>"} for
any substitution instance of axiom @{text "\<turnstile> A"}. By pushing
substitution through derivations inductively, we get admissible
substitution rules for theorems shown in \figref{fig:subst-rules}.
\begin{figure}[htb]
\begin{center}
\[
\infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}}
\quad
\infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}}
\]
\[
\infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}
\quad
\infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}}
\]
\caption{Admissible substitution rules}\label{fig:subst-rules}
\end{center}
\end{figure}
Note that @{text "instantiate_term"} could be derived using @{text
"\<And>_intro/elim"}, but this is not how it is implemented. The type
instantiation rule is a genuine admissible one, due to the lack of
true polymorphism in the logic.
Since @{text "\<Gamma>"} may never contain any schematic variables, the
@{text "instantiate"} do not require an explicit side-condition. In
principle, variables could be substituted in hypotheses as well, but
this could disrupt monotonicity of the basic calculus: derivations
could leave the current proof context.
\medskip The framework also provides builtin equality @{text "\<equiv>"},
which is conceptually axiomatized shown in \figref{fig:equality},
although the implementation provides derived rules directly:
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
@{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
@{text "\<turnstile> (\<lambda>x. b x) a \<equiv> b a"} & @{text "\<beta>"}-conversion \\
@{text "\<turnstile> x \<equiv> x"} & reflexivity law \\
@{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution law \\
@{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"} & coincidence with equivalence \\
\end{tabular}
\caption{Conceptual axiomatization of equality.}\label{fig:equality}
\end{center}
\end{figure}
Since the basic representation of terms already accounts for @{text
"\<alpha>"}-conversion, Pure equality essentially acts like @{text
"\<alpha>\<beta>\<eta>"}-equivalence on terms, while coinciding with bi-implication.
\medskip Conjunction is defined in Pure as a derived connective, see
\figref{fig:conjunction}. This is occasionally useful to represent
simultaneous statements behind the scenes --- framework conjunction
is usually not exposed to the user.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
@{text "& :: prop \<Rightarrow> prop \<Rightarrow> prop"} & conjunction (hidden) \\
@{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\
\end{tabular}
\caption{Definition of conjunction.}\label{fig:equality}
\end{center}
\end{figure}
The definition allows to derive the usual introduction @{text "\<turnstile> A \<Longrightarrow>
B \<Longrightarrow> A & B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B
\<Longrightarrow> B"}.
*}
section {* Rules \label{sec:rules} *}
text {*
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