(*:maxLineLen=78:*)
theory Logic
imports Base
begin
chapter \<open>Primitive logic \label{ch:logic}\<close>
text \<open>
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 ``\<open>\<lambda>HOL\<close>'' 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 \<open>\<lambda>\<close>-calculus with corresponding arrows, \<open>\<Rightarrow>\<close> for syntactic function space
(terms depending on terms), \<open>\<And>\<close> for universal quantification (proofs
depending on terms), and \<open>\<Longrightarrow>\<close> 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>\<open>This is the
deeper logical reason, why the theory context \<open>\<Theta>\<close> is separate from the proof
context \<open>\<Gamma>\<close> of the core calculus: type constructors, term constants, and
facts (proof constants) may involve arbitrary type schemes, but the type of
a locally fixed term parameter is also fixed!\<close>
\<close>
section \<open>Types \label{sec:types}\<close>
text \<open>
The language of types is an uninterpreted order-sorted first-order algebra;
types are qualified by ordered type classes.
\<^medskip>
A \<^emph>\<open>type class\<close> is an abstract syntactic entity declared in the theory
context. The \<^emph>\<open>subclass relation\<close> \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close> 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>\<open>sort\<close> is a list of type classes written as \<open>s = {c\<^sub>1, \<dots>, c\<^sub>m}\<close>, it
represents symbolic intersection. Notationally, the curly braces are omitted
for singleton intersections, i.e.\ any class \<open>c\<close> may be read as a sort
\<open>{c}\<close>. The ordering on type classes is extended to sorts according to the
meaning of intersections: \<open>{c\<^sub>1, \<dots> c\<^sub>m} \<subseteq> {d\<^sub>1, \<dots>, d\<^sub>n}\<close> iff \<open>\<forall>j. \<exists>i. c\<^sub>i \<subseteq>
d\<^sub>j\<close>. The empty intersection \<open>{}\<close> refers to the universal sort, which is the
largest element wrt.\ the sort order. Thus \<open>{}\<close> represents the ``full
sort'', not the empty one! The intersection of all (finitely many) classes
declared in the current theory is the least element wrt.\ the sort ordering.
\<^medskip>
A \<^emph>\<open>fixed type variable\<close> is a pair of a basic name (starting with a \<open>'\<close>
character) and a sort constraint, e.g.\ \<open>('a, s)\<close> which is usually printed
as \<open>\<alpha>\<^sub>s\<close>. A \<^emph>\<open>schematic type variable\<close> is a pair of an indexname and a sort
constraint, e.g.\ \<open>(('a, 0), s)\<close> which is usually printed as \<open>?\<alpha>\<^sub>s\<close>.
Note that \<^emph>\<open>all\<close> syntactic components contribute to the identity of type
variables: basic name, index, and sort constraint. The core logic handles
type variables with the same name but different sorts as different, although
the type-inference layer (which is outside the core) rejects anything like
that.
A \<^emph>\<open>type constructor\<close> \<open>\<kappa>\<close> is a \<open>k\<close>-ary operator on types declared in the
theory. Type constructor application is written postfix as \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>k)\<kappa>\<close>.
For \<open>k = 0\<close> the argument tuple is omitted, e.g.\ \<open>prop\<close> instead of \<open>()prop\<close>.
For \<open>k = 1\<close> the parentheses are omitted, e.g.\ \<open>\<alpha> list\<close> instead of
\<open>(\<alpha>)list\<close>. Further notation is provided for specific constructors, notably
the right-associative infix \<open>\<alpha> \<Rightarrow> \<beta>\<close> instead of \<open>(\<alpha>, \<beta>)fun\<close>.
The logical category \<^emph>\<open>type\<close> is defined inductively over type variables and
type constructors as follows: \<open>\<tau> = \<alpha>\<^sub>s | ?\<alpha>\<^sub>s | (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close>.
A \<^emph>\<open>type abbreviation\<close> is a syntactic definition \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close> of an
arbitrary type expression \<open>\<tau>\<close> over variables \<open>\<^vec>\<alpha>\<close>. Type abbreviations
appear as type constructors in the syntax, but are expanded before entering
the logical core.
A \<^emph>\<open>type arity\<close> declares the image behavior of a type constructor wrt.\ the
algebra of sorts: \<open>\<kappa> :: (s\<^sub>1, \<dots>, s\<^sub>k)s\<close> means that \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>\<close> is of
sort \<open>s\<close> if every argument type \<open>\<tau>\<^sub>i\<close> is of sort \<open>s\<^sub>i\<close>. Arity declarations
are implicitly completed, i.e.\ \<open>\<kappa> :: (\<^vec>s)c\<close> entails \<open>\<kappa> ::
(\<^vec>s)c'\<close> for any \<open>c' \<supseteq> c\<close>.
\<^medskip>
The sort algebra is always maintained as \<^emph>\<open>coregular\<close>, which means that type
arities are consistent with the subclass relation: for any type constructor
\<open>\<kappa>\<close>, and classes \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close>, and arities \<open>\<kappa> :: (\<^vec>s\<^sub>1)c\<^sub>1\<close> and \<open>\<kappa> ::
(\<^vec>s\<^sub>2)c\<^sub>2\<close> holds \<open>\<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2\<close> 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 \<open>\<kappa>\<close> and sort \<open>s\<close> there is a most general vector of argument
sorts \<open>(s\<^sub>1, \<dots>, s\<^sub>k)\<close> such that a type scheme \<open>(\<alpha>\<^bsub>s\<^sub>1\<^esub>, \<dots>, \<alpha>\<^bsub>s\<^sub>k\<^esub>)\<kappa>\<close> is of
sort \<open>s\<close>. Consequently, type unification has most general solutions (modulo
equivalence of sorts), so type-inference produces primary types as expected
@{cite "nipkow-prehofer"}.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML_type class: string} \\
@{index_ML_type sort: "class list"} \\
@{index_ML_type arity: "string * sort list * sort"} \\
@{index_ML_type typ} \\
@{index_ML Term.map_atyps: "(typ -> typ) -> typ -> typ"} \\
@{index_ML Term.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_type: "Proof.context -> binding * int * mixfix -> theory -> theory"} \\
@{index_ML Sign.add_type_abbrev: "Proof.context ->
binding * string list * typ -> theory -> theory"} \\
@{index_ML Sign.primitive_class: "binding * class list -> theory -> theory"} \\
@{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\
@{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\
\end{mldecls}
\<^descr> Type @{ML_type class} represents type classes.
\<^descr> Type @{ML_type sort} represents sorts, i.e.\ finite intersections of
classes. The empty list @{ML "[]: sort"} refers to the empty class
intersection, i.e.\ the ``full sort''.
\<^descr> Type @{ML_type arity} represents type arities. A triple \<open>(\<kappa>, \<^vec>s, s)
: arity\<close> represents \<open>\<kappa> :: (\<^vec>s)s\<close> as described above.
\<^descr> Type @{ML_type typ} represents types; this is a datatype with constructors
@{ML TFree}, @{ML TVar}, @{ML Type}.
\<^descr> @{ML Term.map_atyps}~\<open>f \<tau>\<close> applies the mapping \<open>f\<close> to all atomic types
(@{ML TFree}, @{ML TVar}) occurring in \<open>\<tau>\<close>.
\<^descr> @{ML Term.fold_atyps}~\<open>f \<tau>\<close> iterates the operation \<open>f\<close> over all
occurrences of atomic types (@{ML TFree}, @{ML TVar}) in \<open>\<tau>\<close>; the type
structure is traversed from left to right.
\<^descr> @{ML Sign.subsort}~\<open>thy (s\<^sub>1, s\<^sub>2)\<close> tests the subsort relation \<open>s\<^sub>1 \<subseteq>
s\<^sub>2\<close>.
\<^descr> @{ML Sign.of_sort}~\<open>thy (\<tau>, s)\<close> tests whether type \<open>\<tau>\<close> is of sort \<open>s\<close>.
\<^descr> @{ML Sign.add_type}~\<open>ctxt (\<kappa>, k, mx)\<close> declares a new type constructors \<open>\<kappa>\<close>
with \<open>k\<close> arguments and optional mixfix syntax.
\<^descr> @{ML Sign.add_type_abbrev}~\<open>ctxt (\<kappa>, \<^vec>\<alpha>, \<tau>)\<close> defines a new type
abbreviation \<open>(\<^vec>\<alpha>)\<kappa> = \<tau>\<close>.
\<^descr> @{ML Sign.primitive_class}~\<open>(c, [c\<^sub>1, \<dots>, c\<^sub>n])\<close> declares a new class \<open>c\<close>,
together with class relations \<open>c \<subseteq> c\<^sub>i\<close>, for \<open>i = 1, \<dots>, n\<close>.
\<^descr> @{ML Sign.primitive_classrel}~\<open>(c\<^sub>1, c\<^sub>2)\<close> declares the class relation
\<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close>.
\<^descr> @{ML Sign.primitive_arity}~\<open>(\<kappa>, \<^vec>s, s)\<close> declares the arity \<open>\<kappa> ::
(\<^vec>s)s\<close>.
\<close>
text %mlantiq \<open>
\begin{matharray}{rcl}
@{ML_antiquotation_def "class"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "sort"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "type_name"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "type_abbrev"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "nonterminal"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "typ"} & : & \<open>ML_antiquotation\<close> \\
\end{matharray}
@{rail \<open>
@@{ML_antiquotation class} embedded
;
@@{ML_antiquotation sort} sort
;
(@@{ML_antiquotation type_name} |
@@{ML_antiquotation type_abbrev} |
@@{ML_antiquotation nonterminal}) embedded
;
@@{ML_antiquotation typ} type
\<close>}
\<^descr> \<open>@{class c}\<close> inlines the internalized class \<open>c\<close> --- as @{ML_type string}
literal.
\<^descr> \<open>@{sort s}\<close> inlines the internalized sort \<open>s\<close> --- as @{ML_type "string
list"} literal.
\<^descr> \<open>@{type_name c}\<close> inlines the internalized type constructor \<open>c\<close> --- as
@{ML_type string} literal.
\<^descr> \<open>@{type_abbrev c}\<close> inlines the internalized type abbreviation \<open>c\<close> --- as
@{ML_type string} literal.
\<^descr> \<open>@{nonterminal c}\<close> inlines the internalized syntactic type~/ grammar
nonterminal \<open>c\<close> --- as @{ML_type string} literal.
\<^descr> \<open>@{typ \<tau>}\<close> inlines the internalized type \<open>\<tau>\<close> --- as constructor term for
datatype @{ML_type typ}.
\<close>
section \<open>Terms \label{sec:terms}\<close>
text \<open>
The language of terms is that of simply-typed \<open>\<lambda>\<close>-calculus with de-Bruijn
indices for bound variables (cf.\ @{cite debruijn72} or @{cite
"paulson-ml2"}), with the types being determined by the corresponding
binders. In contrast, free variables and constants have an explicit name and
type in each occurrence.
\<^medskip>
A \<^emph>\<open>bound variable\<close> is a natural number \<open>b\<close>, 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 \<open>\<lambda>\<^bsub>bool\<^esub>. \<lambda>\<^bsub>bool\<^esub>. 1 \<and> 0\<close>
would correspond to \<open>\<lambda>x\<^bsub>bool\<^esub>. \<lambda>y\<^bsub>bool\<^esub>. x \<and> y\<close> 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>\<open>loose variable\<close> 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>\<open>fixed variable\<close> is a pair of a basic name and a type, e.g.\ \<open>(x, \<tau>)\<close>
which is usually printed \<open>x\<^sub>\<tau>\<close> here. A \<^emph>\<open>schematic variable\<close> is a pair of an
indexname and a type, e.g.\ \<open>((x, 0), \<tau>)\<close> which is likewise printed as
\<open>?x\<^sub>\<tau>\<close>.
\<^medskip>
A \<^emph>\<open>constant\<close> is a pair of a basic name and a type, e.g.\ \<open>(c, \<tau>)\<close> which is
usually printed as \<open>c\<^sub>\<tau>\<close> here. Constants are declared in the context as
polymorphic families \<open>c :: \<sigma>\<close>, meaning that all substitution instances \<open>c\<^sub>\<tau>\<close>
for \<open>\<tau> = \<sigma>\<vartheta>\<close> are valid.
The vector of \<^emph>\<open>type arguments\<close> of constant \<open>c\<^sub>\<tau>\<close> wrt.\ the declaration \<open>c
:: \<sigma>\<close> is defined as the codomain of the matcher \<open>\<vartheta> = {?\<alpha>\<^sub>1 \<mapsto> \<tau>\<^sub>1,
\<dots>, ?\<alpha>\<^sub>n \<mapsto> \<tau>\<^sub>n}\<close> presented in canonical order \<open>(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n)\<close>, corresponding
to the left-to-right occurrences of the \<open>\<alpha>\<^sub>i\<close> in \<open>\<sigma>\<close>. Within a given theory
context, there is a one-to-one correspondence between any constant \<open>c\<^sub>\<tau>\<close> and
the application \<open>c(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n)\<close> of its type arguments. For example, with
\<open>plus :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>\<close>, the instance \<open>plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> nat\<^esub>\<close> corresponds to
\<open>plus(nat)\<close>.
Constant declarations \<open>c :: \<sigma>\<close> may contain sort constraints for type
variables in \<open>\<sigma>\<close>. These are observed by type-inference as expected, but
\<^emph>\<open>ignored\<close> 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>\<open>atomic term\<close> is either a variable or constant. The logical category
\<^emph>\<open>term\<close> is defined inductively over atomic terms, with abstraction and
application as follows: \<open>t = b | x\<^sub>\<tau> | ?x\<^sub>\<tau> | c\<^sub>\<tau> | \<lambda>\<^sub>\<tau>. t | t\<^sub>1 t\<^sub>2\<close>.
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 \<open>t :: \<tau>\<close> assigns a (unique) type to a term according
to the structure of atomic terms, abstractions, and applications:
\[
\infer{\<open>a\<^sub>\<tau> :: \<tau>\<close>}{}
\qquad
\infer{\<open>(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>\<close>}{\<open>t :: \<sigma>\<close>}
\qquad
\infer{\<open>t u :: \<sigma>\<close>}{\<open>t :: \<tau> \<Rightarrow> \<sigma>\<close> & \<open>u :: \<tau>\<close>}
\]
A \<^emph>\<open>well-typed term\<close> 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 \<open>x\<^bsub>\<tau>\<^sub>1\<^esub>\<close> and \<open>x\<^bsub>\<tau>\<^sub>2\<^esub>\<close> may
become the same after type instantiation. Type-inference rejects variables
of the same name, but different types. In contrast, mixed instances of
polymorphic constants occur routinely.
\<^medskip>
The \<^emph>\<open>hidden polymorphism\<close> of a term \<open>t :: \<sigma>\<close> is the set of type variables
occurring in \<open>t\<close>, but not in its type \<open>\<sigma>\<close>. 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 \<open>t\<vartheta> :: \<sigma>\<close> and
\<open>t\<vartheta>' :: \<sigma>\<close> with the same type. This slightly pathological
situation notoriously demands additional care.
\<^medskip>
A \<^emph>\<open>term abbreviation\<close> is a syntactic definition \<open>c\<^sub>\<sigma> \<equiv> t\<close> of a closed term
\<open>t\<close> of type \<open>\<sigma>\<close>, 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 \<open>t \<rightarrow>
c\<^sub>\<sigma>\<close> as rules for higher-order rewriting.
\<^medskip>
Canonical operations on \<open>\<lambda>\<close>-terms include \<open>\<alpha>\<beta>\<eta>\<close>-conversion: \<open>\<alpha>\<close>-conversion
refers to capture-free renaming of bound variables; \<open>\<beta>\<close>-conversion contracts
an abstraction applied to an argument term, substituting the argument in the
body: \<open>(\<lambda>x. b)a\<close> becomes \<open>b[a/x]\<close>; \<open>\<eta>\<close>-conversion contracts vacuous
application-abstraction: \<open>\<lambda>x. f x\<close> becomes \<open>f\<close>, provided that the bound
variable does not occur in \<open>f\<close>.
Terms are normally treated modulo \<open>\<alpha>\<close>-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 \<open>\<alpha>\<beta>\<eta>\<close>-conversion is commonplace in various standard
operations (\secref{sec:obj-rules}) that are based on higher-order
unification and matching.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML_type term} \\
@{index_ML_op "aconv": "term * term -> bool"} \\
@{index_ML Term.map_types: "(typ -> typ) -> term -> term"} \\
@{index_ML Term.fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
@{index_ML Term.map_aterms: "(term -> term) -> term -> term"} \\
@{index_ML Term.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 incr_boundvars: "int -> term -> term"} \\
@{index_ML Sign.declare_const: "Proof.context ->
(binding * typ) * mixfix -> theory -> term * theory"} \\
@{index_ML Sign.add_abbrev: "string -> 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}
\<^descr> Type @{ML_type term} represents de-Bruijn terms, with comments in
abstractions, and explicitly named free variables and constants; this is a
datatype with constructors @{index_ML Bound}, @{index_ML Free}, @{index_ML
Var}, @{index_ML Const}, @{index_ML Abs}, @{index_ML_op "$"}.
\<^descr> \<open>t\<close>~@{ML_text aconv}~\<open>u\<close> checks \<open>\<alpha>\<close>-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!
\<^descr> @{ML Term.map_types}~\<open>f t\<close> applies the mapping \<open>f\<close> to all types occurring
in \<open>t\<close>.
\<^descr> @{ML Term.fold_types}~\<open>f t\<close> iterates the operation \<open>f\<close> over all
occurrences of types in \<open>t\<close>; the term structure is traversed from left to
right.
\<^descr> @{ML Term.map_aterms}~\<open>f t\<close> applies the mapping \<open>f\<close> to all atomic terms
(@{ML Bound}, @{ML Free}, @{ML Var}, @{ML Const}) occurring in \<open>t\<close>.
\<^descr> @{ML Term.fold_aterms}~\<open>f t\<close> iterates the operation \<open>f\<close> over all
occurrences of atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
Const}) in \<open>t\<close>; the term structure is traversed from left to right.
\<^descr> @{ML fastype_of}~\<open>t\<close> determines the type of a well-typed term. This
operation is relatively slow, despite the omission of any sanity checks.
\<^descr> @{ML lambda}~\<open>a b\<close> produces an abstraction \<open>\<lambda>a. b\<close>, where occurrences of
the atomic term \<open>a\<close> in the body \<open>b\<close> are replaced by bound variables.
\<^descr> @{ML betapply}~\<open>(t, u)\<close> produces an application \<open>t u\<close>, with topmost
\<open>\<beta>\<close>-conversion if \<open>t\<close> is an abstraction.
\<^descr> @{ML incr_boundvars}~\<open>j\<close> increments a term's dangling bound variables by
the offset \<open>j\<close>. This is required when moving a subterm into a context where
it is enclosed by a different number of abstractions. Bound variables with a
matching abstraction are unaffected.
\<^descr> @{ML Sign.declare_const}~\<open>ctxt ((c, \<sigma>), mx)\<close> declares a new constant \<open>c ::
\<sigma>\<close> with optional mixfix syntax.
\<^descr> @{ML Sign.add_abbrev}~\<open>print_mode (c, t)\<close> introduces a new term
abbreviation \<open>c \<equiv> t\<close>.
\<^descr> @{ML Sign.const_typargs}~\<open>thy (c, \<tau>)\<close> and @{ML Sign.const_instance}~\<open>thy
(c, [\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>n])\<close> convert between two representations of polymorphic
constants: full type instance vs.\ compact type arguments form.
\<close>
text %mlantiq \<open>
\begin{matharray}{rcl}
@{ML_antiquotation_def "const_name"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "const_abbrev"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "const"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "term"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "prop"} & : & \<open>ML_antiquotation\<close> \\
\end{matharray}
@{rail \<open>
(@@{ML_antiquotation const_name} |
@@{ML_antiquotation const_abbrev}) embedded
;
@@{ML_antiquotation const} ('(' (type + ',') ')')?
;
@@{ML_antiquotation term} term
;
@@{ML_antiquotation prop} prop
\<close>}
\<^descr> \<open>@{const_name c}\<close> inlines the internalized logical constant name \<open>c\<close> ---
as @{ML_type string} literal.
\<^descr> \<open>@{const_abbrev c}\<close> inlines the internalized abbreviated constant name \<open>c\<close>
--- as @{ML_type string} literal.
\<^descr> \<open>@{const c(\<^vec>\<tau>)}\<close> inlines the internalized constant \<open>c\<close> with precise
type instantiation in the sense of @{ML Sign.const_instance} --- as @{ML
Const} constructor term for datatype @{ML_type term}.
\<^descr> \<open>@{term t}\<close> inlines the internalized term \<open>t\<close> --- as constructor term for
datatype @{ML_type term}.
\<^descr> \<open>@{prop \<phi>}\<close> inlines the internalized proposition \<open>\<phi>\<close> --- as constructor
term for datatype @{ML_type term}.
\<close>
section \<open>Theorems \label{sec:thms}\<close>
text \<open>
A \<^emph>\<open>proposition\<close> is a well-typed term of type \<open>prop\<close>, a \<^emph>\<open>theorem\<close> 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 \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> of the framework. There is also a builtin
notion of equality/equivalence \<open>\<equiv>\<close>.
\<close>
subsection \<open>Primitive connectives and rules \label{sec:prim-rules}\<close>
text \<open>
The theory \<open>Pure\<close> contains constant declarations for the primitive
connectives \<open>\<And>\<close>, \<open>\<Longrightarrow>\<close>, and \<open>\<equiv>\<close> of the logical framework, see
\figref{fig:pure-connectives}. The derivability judgment \<open>A\<^sub>1, \<dots>, A\<^sub>n \<turnstile> B\<close>
is defined inductively by the primitive inferences given in
\figref{fig:prim-rules}, with the global restriction that the hypotheses
must \<^emph>\<open>not\<close> 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}
\<open>all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop\<close> & universal quantification (binder \<open>\<And>\<close>) \\
\<open>\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop\<close> & implication (right associative infix) \\
\<open>\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop\<close> & equality relation (infix) \\
\end{tabular}
\caption{Primitive connectives of Pure}\label{fig:pure-connectives}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\[
\infer[\<open>(axiom)\<close>]{\<open>\<turnstile> A\<close>}{\<open>A \<in> \<Theta>\<close>}
\qquad
\infer[\<open>(assume)\<close>]{\<open>A \<turnstile> A\<close>}{}
\]
\[
\infer[\<open>(\<And>\<hyphen>intro)\<close>]{\<open>\<Gamma> \<turnstile> \<And>x. B[x]\<close>}{\<open>\<Gamma> \<turnstile> B[x]\<close> & \<open>x \<notin> \<Gamma>\<close>}
\qquad
\infer[\<open>(\<And>\<hyphen>elim)\<close>]{\<open>\<Gamma> \<turnstile> B[a]\<close>}{\<open>\<Gamma> \<turnstile> \<And>x. B[x]\<close>}
\]
\[
\infer[\<open>(\<Longrightarrow>\<hyphen>intro)\<close>]{\<open>\<Gamma> - A \<turnstile> A \<Longrightarrow> B\<close>}{\<open>\<Gamma> \<turnstile> B\<close>}
\qquad
\infer[\<open>(\<Longrightarrow>\<hyphen>elim)\<close>]{\<open>\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B\<close>}{\<open>\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B\<close> & \<open>\<Gamma>\<^sub>2 \<turnstile> A\<close>}
\]
\caption{Primitive inferences of Pure}\label{fig:prim-rules}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
\<open>\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]\<close> & \<open>\<beta>\<close>-conversion \\
\<open>\<turnstile> x \<equiv> x\<close> & reflexivity \\
\<open>\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y\<close> & substitution \\
\<open>\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g\<close> & extensionality \\
\<open>\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B\<close> & logical equivalence \\
\end{tabular}
\caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality}
\end{center}
\end{figure}
The introduction and elimination rules for \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> are analogous to
formation of dependently typed \<open>\<lambda>\<close>-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, see also
\secref{sec:proof-terms}. Thus all three levels of \<open>\<lambda>\<close>-calculus become
explicit: \<open>\<Rightarrow>\<close> for terms, and \<open>\<And>/\<Longrightarrow>\<close> for proofs (cf.\ @{cite
"Berghofer-Nipkow:2000:TPHOL"}).
Observe that locally fixed parameters (as in \<open>\<And>\<hyphen>intro\<close>) need not be recorded
in the hypotheses, because the simple syntactic types of Pure are always
inhabitable. ``Assumptions'' \<open>x :: \<tau>\<close> for type-membership are only present
as long as some \<open>x\<^sub>\<tau>\<close> occurs in the statement body.\<^footnote>\<open>This is the key
difference to ``\<open>\<lambda>HOL\<close>'' in the PTS framework @{cite
"Barendregt-Geuvers:2001"}, where hypotheses \<open>x : A\<close> are treated uniformly
for propositions and types.\<close>
\<^medskip>
The axiomatization of a theory is implicitly closed by forming all instances
of type and term variables: \<open>\<turnstile> A\<vartheta>\<close> holds for any substitution
instance of an axiom \<open>\<turnstile> A\<close>. By pushing substitutions through derivations
inductively, we also get admissible \<open>generalize\<close> and \<open>instantiate\<close> rules as
shown in \figref{fig:subst-rules}.
\begin{figure}[htb]
\begin{center}
\[
\infer{\<open>\<Gamma> \<turnstile> B[?\<alpha>]\<close>}{\<open>\<Gamma> \<turnstile> B[\<alpha>]\<close> & \<open>\<alpha> \<notin> \<Gamma>\<close>}
\quad
\infer[\quad\<open>(generalize)\<close>]{\<open>\<Gamma> \<turnstile> B[?x]\<close>}{\<open>\<Gamma> \<turnstile> B[x]\<close> & \<open>x \<notin> \<Gamma>\<close>}
\]
\[
\infer{\<open>\<Gamma> \<turnstile> B[\<tau>]\<close>}{\<open>\<Gamma> \<turnstile> B[?\<alpha>]\<close>}
\quad
\infer[\quad\<open>(instantiate)\<close>]{\<open>\<Gamma> \<turnstile> B[t]\<close>}{\<open>\<Gamma> \<turnstile> B[?x]\<close>}
\]
\caption{Admissible substitution rules}\label{fig:subst-rules}
\end{center}
\end{figure}
Note that \<open>instantiate\<close> does not require an explicit side-condition, because
\<open>\<Gamma>\<close> may never contain schematic variables.
In principle, variables could be substituted in hypotheses as well, but this
would disrupt the monotonicity of reasoning: deriving \<open>\<Gamma>\<vartheta> \<turnstile>
B\<vartheta>\<close> from \<open>\<Gamma> \<turnstile> B\<close> is correct, but \<open>\<Gamma>\<vartheta> \<supseteq> \<Gamma>\<close> does not
necessarily hold: the result belongs to a different proof context.
\<^medskip>
An \<^emph>\<open>oracle\<close> is a function that produces axioms on the fly. Logically, this
is an instance of the \<open>axiom\<close> rule (\figref{fig:prim-rules}), but there is
an operational difference. The system always records oracle invocations
within derivations of theorems by a unique tag.
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>\<open>simple definition\<close> consists of a constant declaration \<open>c :: \<sigma>\<close> together
with an axiom \<open>\<turnstile> c \<equiv> t\<close>, where \<open>t :: \<sigma>\<close> is a closed term without any hidden
polymorphism. The RHS may depend on further defined constants, but not \<open>c\<close>
itself. Definitions of functions may be presented as \<open>c \<^vec>x \<equiv> t\<close>
instead of the puristic \<open>c \<equiv> \<lambda>\<^vec>x. t\<close>.
An \<^emph>\<open>overloaded definition\<close> consists of a collection of axioms for the same
constant, with zero or one equations \<open>c((\<^vec>\<alpha>)\<kappa>) \<equiv> t\<close> for each type
constructor \<open>\<kappa>\<close> (for distinct variables \<open>\<^vec>\<alpha>\<close>). The RHS may mention
previously defined constants as above, or arbitrary constants \<open>d(\<alpha>\<^sub>i)\<close> for
some \<open>\<alpha>\<^sub>i\<close> projected from \<open>\<^vec>\<alpha>\<close>. Thus overloaded definitions
essentially work by primitive recursion over the syntactic structure of a
single type argument. See also @{cite \<open>\S4.3\<close>
"Haftmann-Wenzel:2006:classes"}.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML Logic.all: "term -> term -> term"} \\
@{index_ML Logic.mk_implies: "term * term -> term"} \\
\end{mldecls}
\begin{mldecls}
@{index_ML_type ctyp} \\
@{index_ML_type cterm} \\
@{index_ML Thm.ctyp_of: "Proof.context -> typ -> ctyp"} \\
@{index_ML Thm.cterm_of: "Proof.context -> term -> cterm"} \\
@{index_ML Thm.apply: "cterm -> cterm -> cterm"} \\
@{index_ML Thm.lambda: "cterm -> cterm -> cterm"} \\
@{index_ML Thm.all: "Proof.context -> cterm -> cterm -> cterm"} \\
@{index_ML Drule.mk_implies: "cterm * cterm -> cterm"} \\
\end{mldecls}
\begin{mldecls}
@{index_ML_type thm} \\
@{index_ML Thm.peek_status: "thm -> {oracle: bool, unfinished: bool, failed: bool}"} \\
@{index_ML Thm.transfer: "theory -> thm -> thm"} \\
@{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: "((indexname * sort) * ctyp) list * ((indexname * typ) * cterm) list
-> thm -> thm"} \\
@{index_ML Thm.add_axiom: "Proof.context ->
binding * term -> theory -> (string * thm) * theory"} \\
@{index_ML Thm.add_oracle: "binding * ('a -> cterm) -> theory ->
(string * ('a -> thm)) * theory"} \\
@{index_ML Thm.add_def: "Defs.context -> bool -> bool ->
binding * term -> theory -> (string * thm) * theory"} \\
\end{mldecls}
\begin{mldecls}
@{index_ML Theory.add_deps: "Defs.context -> string ->
Defs.entry -> Defs.entry list -> theory -> theory"} \\
\end{mldecls}
\<^descr> @{ML Thm.peek_status}~\<open>thm\<close> informs about the current status of the
derivation object behind the given theorem. This is a snapshot of a
potentially ongoing (parallel) evaluation of proofs. The three Boolean
values indicate the following: \<^verbatim>\<open>oracle\<close> if the finished part contains some
oracle invocation; \<^verbatim>\<open>unfinished\<close> if some future proofs are still pending;
\<^verbatim>\<open>failed\<close> if some future proof has failed, rendering the theorem invalid!
\<^descr> @{ML Logic.all}~\<open>a B\<close> produces a Pure quantification \<open>\<And>a. B\<close>, where
occurrences of the atomic term \<open>a\<close> in the body proposition \<open>B\<close> are replaced
by bound variables. (See also @{ML lambda} on terms.)
\<^descr> @{ML Logic.mk_implies}~\<open>(A, B)\<close> produces a Pure implication \<open>A \<Longrightarrow> B\<close>.
\<^descr> Types @{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
background theory. The abstract types @{ML_type ctyp} and @{ML_type cterm}
are part of the same inference kernel that is mainly responsible for
@{ML_type thm}. Thus syntactic operations on @{ML_type ctyp} and @{ML_type
cterm} are located in the @{ML_structure Thm} module, even though theorems
are not yet involved at that stage.
\<^descr> @{ML Thm.ctyp_of}~\<open>ctxt \<tau>\<close> and @{ML Thm.cterm_of}~\<open>ctxt t\<close> explicitly
check types and terms, respectively. This also involves some basic
normalizations, such expansion of type and term abbreviations from the
underlying theory context. Full re-certification is relatively slow and
should be avoided in tight reasoning loops.
\<^descr> @{ML Thm.apply}, @{ML Thm.lambda}, @{ML Thm.all}, @{ML Drule.mk_implies}
etc.\ compose certified terms (or propositions) incrementally. This is
equivalent to @{ML Thm.cterm_of} after unchecked @{ML_op "$"}, @{ML lambda},
@{ML Logic.all}, @{ML Logic.mk_implies} etc., but there can be a big
difference in performance when large existing entities are composed by a few
extra constructions on top. There are separate operations to decompose
certified terms and theorems to produce certified terms again.
\<^descr> Type @{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_structure Thm} module. Every @{ML_type thm} value
refers its background theory, cf.\ \secref{sec:context-theory}.
\<^descr> @{ML Thm.transfer}~\<open>thy thm\<close> transfers the given theorem to a \<^emph>\<open>larger\<close>
theory, see also \secref{sec:context}. This formal adjustment of the
background context has no logical significance, but is occasionally required
for formal reasons, e.g.\ when theorems that are imported from more basic
theories are used in the current situation.
\<^descr> @{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}.
\<^descr> @{ML Thm.generalize}~\<open>(\<^vec>\<alpha>, \<^vec>x)\<close> corresponds to the
\<open>generalize\<close> rules of \figref{fig:subst-rules}. Here collections of type and
term variables are generalized simultaneously, specified by the given basic
names.
\<^descr> @{ML Thm.instantiate}~\<open>(\<^vec>\<alpha>\<^sub>s, \<^vec>x\<^sub>\<tau>)\<close> corresponds to the
\<open>instantiate\<close> rules of \figref{fig:subst-rules}. Type variables are
substituted before term variables. Note that the types in \<open>\<^vec>x\<^sub>\<tau>\<close> refer
to the instantiated versions.
\<^descr> @{ML Thm.add_axiom}~\<open>ctxt (name, A)\<close> declares an arbitrary proposition as
axiom, and retrieves it as a theorem from the resulting theory, cf.\ \<open>axiom\<close>
in \figref{fig:prim-rules}. Note that the low-level representation in the
axiom table may differ slightly from the returned theorem.
\<^descr> @{ML Thm.add_oracle}~\<open>(binding, oracle)\<close> produces a named oracle rule,
essentially generating arbitrary axioms on the fly, cf.\ \<open>axiom\<close> in
\figref{fig:prim-rules}.
\<^descr> @{ML Thm.add_def}~\<open>ctxt unchecked overloaded (name, c \<^vec>x \<equiv> t)\<close>
states a definitional axiom for an existing constant \<open>c\<close>. Dependencies are
recorded via @{ML Theory.add_deps}, unless the \<open>unchecked\<close> option is set.
Note that the low-level representation in the axiom table may differ
slightly from the returned theorem.
\<^descr> @{ML Theory.add_deps}~\<open>ctxt name c\<^sub>\<tau> \<^vec>d\<^sub>\<sigma>\<close> declares dependencies of
a named specification for constant \<open>c\<^sub>\<tau>\<close>, relative to existing
specifications for constants \<open>\<^vec>d\<^sub>\<sigma>\<close>. This also works for type
constructors.
\<close>
text %mlantiq \<open>
\begin{matharray}{rcl}
@{ML_antiquotation_def "ctyp"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "cterm"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "cprop"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "thm"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "thms"} & : & \<open>ML_antiquotation\<close> \\
@{ML_antiquotation_def "lemma"} & : & \<open>ML_antiquotation\<close> \\
\end{matharray}
@{rail \<open>
@@{ML_antiquotation ctyp} typ
;
@@{ML_antiquotation cterm} term
;
@@{ML_antiquotation cprop} prop
;
@@{ML_antiquotation thm} thm
;
@@{ML_antiquotation thms} thms
;
@@{ML_antiquotation lemma} ('(' @'open' ')')? ((prop +) + @'and') \<newline>
@'by' method method?
\<close>}
\<^descr> \<open>@{ctyp \<tau>}\<close> produces a certified type wrt.\ the current background theory
--- as abstract value of type @{ML_type ctyp}.
\<^descr> \<open>@{cterm t}\<close> and \<open>@{cprop \<phi>}\<close> produce a certified term wrt.\ the current
background theory --- as abstract value of type @{ML_type cterm}.
\<^descr> \<open>@{thm a}\<close> produces a singleton fact --- as abstract value of type
@{ML_type thm}.
\<^descr> \<open>@{thms a}\<close> produces a general fact --- as abstract value of type
@{ML_type "thm list"}.
\<^descr> \<open>@{lemma \<phi> by meth}\<close> produces a fact that is proven on the spot according
to the minimal proof, which imitates a terminal Isar proof. The result is an
abstract value of type @{ML_type thm} or @{ML_type "thm list"}, depending on
the number of propositions given here.
The internal derivation object lacks a proper theorem name, but it is
formally closed, unless the \<open>(open)\<close> option is specified (this may impact
performance of applications with proof terms).
Since ML antiquotations are always evaluated at compile-time, there is no
run-time overhead even for non-trivial proofs. Nonetheless, the
justification is syntactically limited to a single @{command "by"} step.
More complex Isar proofs should be done in regular theory source, before
compiling the corresponding ML text that uses the result.
\<close>
subsection \<open>Auxiliary connectives \label{sec:logic-aux}\<close>
text \<open>
Theory \<open>Pure\<close> provides a few auxiliary connectives that are defined on top
of the primitive ones, see \figref{fig:pure-aux}. These special constants
are useful in certain internal encodings, and are normally not directly
exposed to the user.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
\<open>conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop\<close> & (infix \<open>&&&\<close>) \\
\<open>\<turnstile> A &&& B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)\<close> \\[1ex]
\<open>prop :: prop \<Rightarrow> prop\<close> & (prefix \<open>#\<close>, suppressed) \\
\<open>#A \<equiv> A\<close> \\[1ex]
\<open>term :: \<alpha> \<Rightarrow> prop\<close> & (prefix \<open>TERM\<close>) \\
\<open>term x \<equiv> (\<And>A. A \<Longrightarrow> A)\<close> \\[1ex]
\<open>type :: \<alpha> itself\<close> & (prefix \<open>TYPE\<close>) \\
\<open>(unspecified)\<close> \\
\end{tabular}
\caption{Definitions of auxiliary connectives}\label{fig:pure-aux}
\end{center}
\end{figure}
The introduction \<open>A \<Longrightarrow> B \<Longrightarrow> A &&& B\<close>, and eliminations (projections) \<open>A &&& B
\<Longrightarrow> A\<close> and \<open>A &&& B \<Longrightarrow> B\<close> are available as derived rules. Conjunction allows to
treat simultaneous assumptions and conclusions uniformly, e.g.\ consider \<open>A
\<Longrightarrow> B \<Longrightarrow> C &&& D\<close>. In particular, the goal mechanism represents multiple claims
as explicit conjunction internally, but this is refined (via backwards
introduction) into separate sub-goals before the user commences the proof;
the final result is projected into a list of theorems using eliminations
(cf.\ \secref{sec:tactical-goals}).
The \<open>prop\<close> marker (\<open>#\<close>) makes arbitrarily complex propositions appear as
atomic, without changing the meaning: \<open>\<Gamma> \<turnstile> A\<close> and \<open>\<Gamma> \<turnstile> #A\<close> are
interchangeable. See \secref{sec:tactical-goals} for specific operations.
The \<open>term\<close> marker turns any well-typed term into a derivable proposition: \<open>\<turnstile>
TERM t\<close> holds unconditionally. Although this is logically vacuous, it allows
to treat terms and proofs uniformly, similar to a type-theoretic framework.
The \<open>TYPE\<close> constructor is the canonical representative of the unspecified
type \<open>\<alpha> itself\<close>; it essentially injects the language of types into that of
terms. There is specific notation \<open>TYPE(\<tau>)\<close> for \<open>TYPE\<^bsub>\<tau> itself\<^esub>\<close>. Although
being devoid of any particular meaning, the term \<open>TYPE(\<tau>)\<close> accounts for the
type \<open>\<tau>\<close> within the term language. In particular, \<open>TYPE(\<alpha>)\<close> may be used as
formal argument in primitive definitions, in order to circumvent hidden
polymorphism (cf.\ \secref{sec:terms}). For example, \<open>c TYPE(\<alpha>) \<equiv> A[\<alpha>]\<close>
defines \<open>c :: \<alpha> itself \<Rightarrow> prop\<close> in terms of a proposition \<open>A\<close> that depends on
an additional type argument, which is essentially a predicate on types.
\<close>
text %mlref \<open>
\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}
\<^descr> @{ML Conjunction.intr} derives \<open>A &&& B\<close> from \<open>A\<close> and \<open>B\<close>.
\<^descr> @{ML Conjunction.elim} derives \<open>A\<close> and \<open>B\<close> from \<open>A &&& B\<close>.
\<^descr> @{ML Drule.mk_term} derives \<open>TERM t\<close>.
\<^descr> @{ML Drule.dest_term} recovers term \<open>t\<close> from \<open>TERM t\<close>.
\<^descr> @{ML Logic.mk_type}~\<open>\<tau>\<close> produces the term \<open>TYPE(\<tau>)\<close>.
\<^descr> @{ML Logic.dest_type}~\<open>TYPE(\<tau>)\<close> recovers the type \<open>\<tau>\<close>.
\<close>
subsection \<open>Sort hypotheses\<close>
text \<open>
Type variables are decorated with sorts, as explained in \secref{sec:types}.
This constrains type instantiation to certain ranges of types: variable
\<open>\<alpha>\<^sub>s\<close> may only be assigned to types \<open>\<tau>\<close> that belong to sort \<open>s\<close>. Within the
logic, sort constraints act like implicit preconditions on the result \<open>\<lparr>\<alpha>\<^sub>1
: s\<^sub>1\<rparr>, \<dots>, \<lparr>\<alpha>\<^sub>n : s\<^sub>n\<rparr>, \<Gamma> \<turnstile> \<phi>\<close> where the type variables \<open>\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n\<close> cover
the propositions \<open>\<Gamma>\<close>, \<open>\<phi>\<close>, as well as the proof of \<open>\<Gamma> \<turnstile> \<phi>\<close>.
These \<^emph>\<open>sort hypothesis\<close> of a theorem are passed monotonically through
further derivations. They are redundant, as long as the statement of a
theorem still contains the type variables that are accounted here. The
logical significance of sort hypotheses is limited to the boundary case
where type variables disappear from the proposition, e.g.\ \<open>\<lparr>\<alpha>\<^sub>s : s\<rparr> \<turnstile> \<phi>\<close>.
Since such dangling type variables can be renamed arbitrarily without
changing the proposition \<open>\<phi>\<close>, the inference kernel maintains sort hypotheses
in anonymous form \<open>s \<turnstile> \<phi>\<close>.
In most practical situations, such extra sort hypotheses may be stripped in
a final bookkeeping step, e.g.\ at the end of a proof: they are typically
left over from intermediate reasoning with type classes that can be
satisfied by some concrete type \<open>\<tau>\<close> of sort \<open>s\<close> to replace the hypothetical
type variable \<open>\<alpha>\<^sub>s\<close>.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML Thm.extra_shyps: "thm -> sort list"} \\
@{index_ML Thm.strip_shyps: "thm -> thm"} \\
\end{mldecls}
\<^descr> @{ML Thm.extra_shyps}~\<open>thm\<close> determines the extraneous sort hypotheses of
the given theorem, i.e.\ the sorts that are not present within type
variables of the statement.
\<^descr> @{ML Thm.strip_shyps}~\<open>thm\<close> removes any extraneous sort hypotheses that
can be witnessed from the type signature.
\<close>
text %mlex \<open>
The following artificial example demonstrates the derivation of @{prop
False} with a pending sort hypothesis involving a logically empty sort.
\<close>
class empty =
assumes bad: "\<And>(x::'a) y. x \<noteq> y"
theorem (in empty) false: False
using bad by blast
ML_val \<open>@{assert} (Thm.extra_shyps @{thm false} = [@{sort empty}])\<close>
text \<open>
Thanks to the inference kernel managing sort hypothesis according to their
logical significance, this example is merely an instance of \<^emph>\<open>ex falso
quodlibet consequitur\<close> --- not a collapse of the logical framework!
\<close>
section \<open>Object-level rules \label{sec:obj-rules}\<close>
text \<open>
The primitive inferences covered so far mostly serve foundational purposes.
User-level reasoning usually works via object-level rules that are
represented as theorems of Pure. Composition of rules involves
\<^emph>\<open>backchaining\<close>, \<^emph>\<open>higher-order unification\<close> modulo \<open>\<alpha>\<beta>\<eta>\<close>-conversion of
\<open>\<lambda>\<close>-terms, and so-called \<^emph>\<open>lifting\<close> of rules into a context of \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close>
connectives. Thus the full power of higher-order Natural Deduction in
Isabelle/Pure becomes readily available.
\<close>
subsection \<open>Hereditary Harrop Formulae\<close>
text \<open>
The idea of object-level rules is to model Natural Deduction inferences in
the style of Gentzen @{cite "Gentzen:1935"}, but we allow arbitrary nesting
similar to @{cite extensions91}. The most basic rule format is that of a
\<^emph>\<open>Horn Clause\<close>:
\[
\infer{\<open>A\<close>}{\<open>A\<^sub>1\<close> & \<open>\<dots>\<close> & \<open>A\<^sub>n\<close>}
\]
where \<open>A, A\<^sub>1, \<dots>, A\<^sub>n\<close> are atomic propositions of the framework, usually of
the form \<open>Trueprop B\<close>, where \<open>B\<close> is a (compound) object-level statement.
This object-level inference corresponds to an iterated implication in Pure
like this:
\[
\<open>A\<^sub>1 \<Longrightarrow> \<dots> A\<^sub>n \<Longrightarrow> A\<close>
\]
As an example consider conjunction introduction: \<open>A \<Longrightarrow> B \<Longrightarrow> A \<and> B\<close>. Any
parameters occurring in such rule statements are conceptionally treated as
arbitrary:
\[
\<open>\<And>x\<^sub>1 \<dots> x\<^sub>m. A\<^sub>1 x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> \<dots> A\<^sub>n x\<^sub>1 \<dots> x\<^sub>m \<Longrightarrow> A x\<^sub>1 \<dots> x\<^sub>m\<close>
\]
Nesting of rules means that the positions of \<open>A\<^sub>i\<close> may again hold compound
rules, not just atomic propositions. Propositions of this format are called
\<^emph>\<open>Hereditary Harrop Formulae\<close> in the literature @{cite "Miller:1991"}. Here
we give an inductive characterization as follows:
\<^medskip>
\begin{tabular}{ll}
\<open>\<^bold>x\<close> & set of variables \\
\<open>\<^bold>A\<close> & set of atomic propositions \\
\<open>\<^bold>H = \<And>\<^bold>x\<^sup>*. \<^bold>H\<^sup>* \<Longrightarrow> \<^bold>A\<close> & set of Hereditary Harrop Formulas \\
\end{tabular}
\<^medskip>
Thus we essentially impose nesting levels on propositions formed from \<open>\<And>\<close>
and \<open>\<Longrightarrow>\<close>. At each level there is a prefix of parameters and compound
premises, concluding an atomic proposition. Typical examples are
\<open>\<longrightarrow>\<close>-introduction \<open>(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B\<close> or mathematical induction \<open>P 0 \<Longrightarrow> (\<And>n. P n
\<Longrightarrow> P (Suc n)) \<Longrightarrow> P n\<close>. Even deeper nesting occurs in well-founded induction
\<open>(\<And>x. (\<And>y. y \<prec> x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x\<close>, but this already marks the limit of
rule complexity that is usually seen in practice.
\<^medskip>
Regular user-level inferences in Isabelle/Pure always maintain the following
canonical form of results:
\<^item> Normalization by \<open>(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)\<close>, which is a theorem of
Pure, means that quantifiers are pushed in front of implication at each
level of nesting. The normal form is a Hereditary Harrop Formula.
\<^item> The outermost prefix of parameters is represented via schematic variables:
instead of \<open>\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x\<close> we have \<open>\<^vec>H
?\<^vec>x \<Longrightarrow> A ?\<^vec>x\<close>. Note that this representation looses information
about the order of parameters, and vacuous quantifiers vanish automatically.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML Simplifier.norm_hhf: "Proof.context -> thm -> thm"} \\
\end{mldecls}
\<^descr> @{ML Simplifier.norm_hhf}~\<open>ctxt thm\<close> normalizes the given theorem
according to the canonical form specified above. This is occasionally
helpful to repair some low-level tools that do not handle Hereditary Harrop
Formulae properly.
\<close>
subsection \<open>Rule composition\<close>
text \<open>
The rule calculus of Isabelle/Pure provides two main inferences: @{inference
resolution} (i.e.\ back-chaining of rules) and @{inference assumption}
(i.e.\ closing a branch), both modulo higher-order unification. There are
also combined variants, notably @{inference elim_resolution} and @{inference
dest_resolution}.
To understand the all-important @{inference resolution} principle, we first
consider raw @{inference_def composition} (modulo higher-order unification
with substitution \<open>\<vartheta>\<close>):
\[
\infer[(@{inference_def composition})]{\<open>\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>\<close>}
{\<open>\<^vec>A \<Longrightarrow> B\<close> & \<open>B' \<Longrightarrow> C\<close> & \<open>B\<vartheta> = B'\<vartheta>\<close>}
\]
Here the conclusion of the first rule is unified with the premise of the
second; the resulting rule instance inherits the premises of the first and
conclusion of the second. Note that \<open>C\<close> can again consist of iterated
implications. We can also permute the premises of the second rule
back-and-forth in order to compose with \<open>B'\<close> in any position (subsequently
we shall always refer to position 1 w.l.o.g.).
In @{inference composition} the internal structure of the common part \<open>B\<close>
and \<open>B'\<close> is not taken into account. For proper @{inference resolution} we
require \<open>B\<close> to be atomic, and explicitly observe the structure \<open>\<And>\<^vec>x.
\<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x\<close> of the premise of the second rule. The idea
is to adapt the first rule by ``lifting'' it into this context, by means of
iterated application of the following inferences:
\[
\infer[(@{inference_def imp_lift})]{\<open>(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)\<close>}{\<open>\<^vec>A \<Longrightarrow> B\<close>}
\]
\[
\infer[(@{inference_def all_lift})]{\<open>(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))\<close>}{\<open>\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a\<close>}
\]
By combining raw composition with lifting, we get full @{inference
resolution} as follows:
\[
\infer[(@{inference_def resolution})]
{\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>\<close>}
{\begin{tabular}{l}
\<open>\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a\<close> \\
\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C\<close> \\
\<open>(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>\<close> \\
\end{tabular}}
\]
Continued resolution of rules allows to back-chain a problem towards more
and sub-problems. Branches are closed either by resolving with a rule of 0
premises, or by producing a ``short-circuit'' within a solved situation
(again modulo unification):
\[
\infer[(@{inference_def assumption})]{\<open>C\<vartheta>\<close>}
{\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C\<close> & \<open>A\<vartheta> = H\<^sub>i\<vartheta>\<close>~~\mbox{(for some~\<open>i\<close>)}}
\]
%FIXME @{inference_def elim_resolution}, @{inference_def dest_resolution}
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML_op "RSN": "thm * (int * thm) -> thm"} \\
@{index_ML_op "RS": "thm * thm -> thm"} \\
@{index_ML_op "RLN": "thm list * (int * thm list) -> thm list"} \\
@{index_ML_op "RL": "thm list * thm list -> thm list"} \\
@{index_ML_op "MRS": "thm list * thm -> thm"} \\
@{index_ML_op "OF": "thm * thm list -> thm"} \\
\end{mldecls}
\<^descr> \<open>rule\<^sub>1 RSN (i, rule\<^sub>2)\<close> resolves the conclusion of \<open>rule\<^sub>1\<close> with the
\<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>, according to the @{inference resolution}
principle explained above. Unless there is precisely one resolvent it raises
exception @{ML THM}.
This corresponds to the rule attribute @{attribute THEN} in Isar source
language.
\<^descr> \<open>rule\<^sub>1 RS rule\<^sub>2\<close> abbreviates \<open>rule\<^sub>1 RSN (1, rule\<^sub>2)\<close>.
\<^descr> \<open>rules\<^sub>1 RLN (i, rules\<^sub>2)\<close> joins lists of rules. For every \<open>rule\<^sub>1\<close> in
\<open>rules\<^sub>1\<close> and \<open>rule\<^sub>2\<close> in \<open>rules\<^sub>2\<close>, it resolves the conclusion of \<open>rule\<^sub>1\<close>
with the \<open>i\<close>-th premise of \<open>rule\<^sub>2\<close>, accumulating multiple results in one
big list. Note that such strict enumerations of higher-order unifications
can be inefficient compared to the lazy variant seen in elementary tactics
like @{ML resolve_tac}.
\<^descr> \<open>rules\<^sub>1 RL rules\<^sub>2\<close> abbreviates \<open>rules\<^sub>1 RLN (1, rules\<^sub>2)\<close>.
\<^descr> \<open>[rule\<^sub>1, \<dots>, rule\<^sub>n] MRS rule\<close> resolves \<open>rule\<^sub>i\<close> against premise \<open>i\<close> of
\<open>rule\<close>, for \<open>i = n, \<dots>, 1\<close>. By working from right to left, newly emerging
premises are concatenated in the result, without interfering.
\<^descr> \<open>rule OF rules\<close> is an alternative notation for \<open>rules MRS rule\<close>, which
makes rule composition look more like function application. Note that the
argument \<open>rules\<close> need not be atomic.
This corresponds to the rule attribute @{attribute OF} in Isar source
language.
\<close>
section \<open>Proof terms \label{sec:proof-terms}\<close>
text \<open>
The Isabelle/Pure inference kernel can record the proof of each theorem as a
proof term that contains all logical inferences in detail. Rule composition
by resolution (\secref{sec:obj-rules}) and type-class reasoning is broken
down to primitive rules of the logical framework. The proof term can be
inspected by a separate proof-checker, for example.
According to the well-known \<^emph>\<open>Curry-Howard isomorphism\<close>, a proof can be
viewed as a \<open>\<lambda>\<close>-term. Following this idea, proofs in Isabelle are internally
represented by a datatype similar to the one for terms described in
\secref{sec:terms}. On top of these syntactic terms, two more layers of
\<open>\<lambda>\<close>-calculus are added, which correspond to \<open>\<And>x :: \<alpha>. B x\<close> and \<open>A \<Longrightarrow> B\<close>
according to the propositions-as-types principle. The resulting 3-level
\<open>\<lambda>\<close>-calculus resembles ``\<open>\<lambda>HOL\<close>'' in the more abstract setting of Pure Type
Systems (PTS) @{cite "Barendregt-Geuvers:2001"}, if some fine points like
schematic polymorphism and type classes are ignored.
\<^medskip>
\<^emph>\<open>Proof abstractions\<close> of the form \<open>\<^bold>\<lambda>x :: \<alpha>. prf\<close> or \<open>\<^bold>\<lambda>p : A. prf\<close>
correspond to introduction of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>, and \<^emph>\<open>proof applications\<close> of the form
\<open>p \<cdot> t\<close> or \<open>p \<bullet> q\<close> correspond to elimination of \<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>. Actual types \<open>\<alpha>\<close>,
propositions \<open>A\<close>, and terms \<open>t\<close> might be suppressed and reconstructed from
the overall proof term.
\<^medskip>
Various atomic proofs indicate special situations within the proof
construction as follows.
A \<^emph>\<open>bound proof variable\<close> is a natural number \<open>b\<close> that acts as de-Bruijn
index for proof term abstractions.
A \<^emph>\<open>minimal proof\<close> ``\<open>?\<close>'' is a dummy proof term. This indicates some
unrecorded part of the proof.
\<open>Hyp A\<close> refers to some pending hypothesis by giving its proposition. This
indicates an open context of implicit hypotheses, similar to loose bound
variables or free variables within a term (\secref{sec:terms}).
An \<^emph>\<open>axiom\<close> or \<^emph>\<open>oracle\<close> \<open>a : A[\<^vec>\<tau>]\<close> refers some postulated \<open>proof
constant\<close>, which is subject to schematic polymorphism of theory content, and
the particular type instantiation may be given explicitly. The vector of
types \<open>\<^vec>\<tau>\<close> refers to the schematic type variables in the generic
proposition \<open>A\<close> in canonical order.
A \<^emph>\<open>proof promise\<close> \<open>a : A[\<^vec>\<tau>]\<close> is a placeholder for some proof of
polymorphic proposition \<open>A\<close>, with explicit type instantiation as given by
the vector \<open>\<^vec>\<tau>\<close>, as above. Unlike axioms or oracles, proof promises
may be \<^emph>\<open>fulfilled\<close> eventually, by substituting \<open>a\<close> by some particular proof
\<open>q\<close> at the corresponding type instance. This acts like Hindley-Milner
\<open>let\<close>-polymorphism: a generic local proof definition may get used at
different type instances, and is replaced by the concrete instance
eventually.
A \<^emph>\<open>named theorem\<close> wraps up some concrete proof as a closed formal entity,
in the manner of constant definitions for proof terms. The \<^emph>\<open>proof body\<close> of
such boxed theorems involves some digest about oracles and promises
occurring in the original proof. This allows the inference kernel to manage
this critical information without the full overhead of explicit proof terms.
\<close>
subsection \<open>Reconstructing and checking proof terms\<close>
text \<open>
Fully explicit proof terms can be large, but most of this information is
redundant and can be reconstructed from the context. Therefore, the
Isabelle/Pure inference kernel records only \<^emph>\<open>implicit\<close> proof terms, by
omitting all typing information in terms, all term and type labels of proof
abstractions, and some argument terms of applications \<open>p \<cdot> t\<close> (if possible).
There are separate operations to reconstruct the full proof term later on,
using \<^emph>\<open>higher-order pattern unification\<close> @{cite "nipkow-patterns" and
"Berghofer-Nipkow:2000:TPHOL"}.
The \<^emph>\<open>proof checker\<close> expects a fully reconstructed proof term, and can turn
it into a theorem by replaying its primitive inferences within the kernel.
\<close>
subsection \<open>Concrete syntax of proof terms\<close>
text \<open>
The concrete syntax of proof terms is a slight extension of the regular
inner syntax of Isabelle/Pure @{cite "isabelle-isar-ref"}. Its main
syntactic category @{syntax (inner) proof} is defined as follows:
\begin{center}
\begin{supertabular}{rclr}
@{syntax_def (inner) proof} & = & \<^verbatim>\<open>Lam\<close> \<open>params\<close> \<^verbatim>\<open>.\<close> \<open>proof\<close> \\
& \<open>|\<close> & \<open>\<^bold>\<lambda>\<close> \<open>params\<close> \<^verbatim>\<open>.\<close> \<open>proof\<close> \\
& \<open>|\<close> & \<open>proof\<close> \<^verbatim>\<open>%\<close> \<open>any\<close> \\
& \<open>|\<close> & \<open>proof\<close> \<open>\<cdot>\<close> \<open>any\<close> \\
& \<open>|\<close> & \<open>proof\<close> \<^verbatim>\<open>%%\<close> \<open>proof\<close> \\
& \<open>|\<close> & \<open>proof\<close> \<open>\<bullet>\<close> \<open>proof\<close> \\
& \<open>|\<close> & \<open>id | longid\<close> \\
\\
\<open>param\<close> & = & \<open>idt\<close> \\
& \<open>|\<close> & \<open>idt\<close> \<^verbatim>\<open>:\<close> \<open>prop\<close> \\
& \<open>|\<close> & \<^verbatim>\<open>(\<close> \<open>param\<close> \<^verbatim>\<open>)\<close> \\
\\
\<open>params\<close> & = & \<open>param\<close> \\
& \<open>|\<close> & \<open>param\<close> \<open>params\<close> \\
\end{supertabular}
\end{center}
Implicit term arguments in partial proofs are indicated by ``\<open>_\<close>''. Type
arguments for theorems and axioms may be specified using \<open>p \<cdot> TYPE(type)\<close>
(they must appear before any other term argument of a theorem or axiom, but
may be omitted altogether).
\<^medskip>
There are separate read and print operations for proof terms, in order to
avoid conflicts with the regular term language.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML_type proof} \\
@{index_ML_type proof_body} \\
@{index_ML Proofterm.proofs: "int Unsynchronized.ref"} \\
@{index_ML Reconstruct.reconstruct_proof:
"Proof.context -> term -> proof -> proof"} \\
@{index_ML Reconstruct.expand_proof: "Proof.context ->
(string * term option) list -> proof -> proof"} \\
@{index_ML Proof_Checker.thm_of_proof: "theory -> proof -> thm"} \\
@{index_ML Proof_Syntax.read_proof: "theory -> bool -> bool -> string -> proof"} \\
@{index_ML Proof_Syntax.pretty_proof: "Proof.context -> proof -> Pretty.T"} \\
\end{mldecls}
\<^descr> Type @{ML_type proof} represents proof terms; this is a datatype with
constructors @{index_ML Abst}, @{index_ML AbsP}, @{index_ML_op "%"},
@{index_ML_op "%%"}, @{index_ML PBound}, @{index_ML MinProof}, @{index_ML
Hyp}, @{index_ML PAxm}, @{index_ML Oracle}, @{index_ML Promise}, @{index_ML
PThm} as explained above. %FIXME OfClass (!?)
\<^descr> Type @{ML_type proof_body} represents the nested proof information of a
named theorem, consisting of a digest of oracles and named theorem over some
proof term. The digest only covers the directly visible part of the proof:
in order to get the full information, the implicit graph of nested theorems
needs to be traversed (e.g.\ using @{ML Proofterm.fold_body_thms}).
\<^descr> @{ML Thm.proof_of}~\<open>thm\<close> and @{ML Thm.proof_body_of}~\<open>thm\<close> produce the
proof term or proof body (with digest of oracles and theorems) from a given
theorem. Note that this involves a full join of internal futures that
fulfill pending proof promises, and thus disrupts the natural bottom-up
construction of proofs by introducing dynamic ad-hoc dependencies. Parallel
performance may suffer by inspecting proof terms at run-time.
\<^descr> @{ML Proofterm.proofs} specifies the detail of proof recording within
@{ML_type thm} values produced by the inference kernel: @{ML 0} records only
the names of oracles, @{ML 1} records oracle names and propositions, @{ML 2}
additionally records full proof terms. Officially named theorems that
contribute to a result are recorded in any case.
\<^descr> @{ML Reconstruct.reconstruct_proof}~\<open>ctxt prop prf\<close> turns the implicit
proof term \<open>prf\<close> into a full proof of the given proposition.
Reconstruction may fail if \<open>prf\<close> is not a proof of \<open>prop\<close>, or if it does not
contain sufficient information for reconstruction. Failure may only happen
for proofs that are constructed manually, but not for those produced
automatically by the inference kernel.
\<^descr> @{ML Reconstruct.expand_proof}~\<open>ctxt [thm\<^sub>1, \<dots>, thm\<^sub>n] prf\<close> expands and
reconstructs the proofs of all specified theorems, with the given (full)
proof. Theorems that are not unique specified via their name may be
disambiguated by giving their proposition.
\<^descr> @{ML Proof_Checker.thm_of_proof}~\<open>thy prf\<close> turns the given (full) proof
into a theorem, by replaying it using only primitive rules of the inference
kernel.
\<^descr> @{ML Proof_Syntax.read_proof}~\<open>thy b\<^sub>1 b\<^sub>2 s\<close> reads in a proof term. The
Boolean flags indicate the use of sort and type information. Usually, typing
information is left implicit and is inferred during proof reconstruction.
%FIXME eliminate flags!?
\<^descr> @{ML Proof_Syntax.pretty_proof}~\<open>ctxt prf\<close> pretty-prints the given proof
term.
\<close>
text %mlex \<open>
\<^item> \<^file>\<open>~~/src/HOL/Proofs/ex/Proof_Terms.thy\<close> provides basic examples involving
proof terms.
\<^item> \<^file>\<open>~~/src/HOL/Proofs/ex/XML_Data.thy\<close> demonstrates export and import of
proof terms via XML/ML data representation.
\<close>
end