--- a/doc-src/IsarImplementation/Thy/logic.thy Mon Feb 16 20:25:21 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,851 +0,0 @@
-theory logic imports base begin
-
-chapter {* Primitive logic \label{ch:logic} *}
-
-text {*
- The logical foundations of Isabelle/Isar are that of the Pure logic,
- which has been introduced as a natural-deduction framework in
- \cite{paulson700}. This is essentially the same logic as ``@{text
- "\<lambda>HOL"}'' in the more abstract setting of Pure Type Systems (PTS)
- \cite{Barendregt-Geuvers:2001}, although there are some key
- differences in the specific treatment of simple types in
- Isabelle/Pure.
-
- Following type-theoretic parlance, the Pure logic consists of three
- levels of @{text "\<lambda>"}-calculus with corresponding arrows, @{text
- "\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text
- "\<And>"} for universal quantification (proofs depending on terms), and
- @{text "\<Longrightarrow>"} for implication (proofs depending on proofs).
-
- Derivations are relative to a logical theory, which declares type
- constructors, constants, and axioms. Theory declarations support
- schematic polymorphism, which is strictly speaking outside the
- logic.\footnote{This is the deeper logical reason, why the theory
- context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"}
- of the core calculus.}
-*}
-
-
-section {* Types \label{sec:types} *}
-
-text {*
- The language of types is an uninterpreted order-sorted first-order
- algebra; types are qualified by ordered type classes.
-
- \medskip A \emph{type class} is an abstract syntactic entity
- declared in the theory context. The \emph{subclass relation} @{text
- "c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic
- generating relation; the transitive closure is maintained
- internally. The resulting relation is an ordering: reflexive,
- transitive, and antisymmetric.
-
- A \emph{sort} is a list of type classes written as @{text "s =
- {c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic
- intersection. Notationally, the curly braces are omitted for
- singleton intersections, i.e.\ any class @{text "c"} may be read as
- a sort @{text "{c}"}. The ordering on type classes is extended to
- sorts according to the meaning of intersections: @{text
- "{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff
- @{text "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}. The empty intersection
- @{text "{}"} refers to the universal sort, which is the largest
- element wrt.\ the sort order. The intersections of all (finitely
- many) classes declared in the current theory are the minimal
- elements wrt.\ the sort order.
-
- \medskip A \emph{fixed type variable} is a pair of a basic name
- (starting with a @{text "'"} character) and a sort constraint, e.g.\
- @{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}.
- A \emph{schematic type variable} is a pair of an indexname and a
- sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually
- printed as @{text "?\<alpha>\<^isub>s"}.
-
- Note that \emph{all} syntactic components contribute to the identity
- of type variables, including the sort constraint. The core logic
- handles type variables with the same name but different sorts as
- different, although some outer layers of the system make it hard to
- produce anything like this.
-
- A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator
- on types declared in the theory. Type constructor application is
- written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}. For
- @{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"}
- instead of @{text "()prop"}. For @{text "k = 1"} the parentheses
- are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}.
- Further notation is provided for specific constructors, notably the
- right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>,
- \<beta>)fun"}.
-
- A \emph{type} is defined inductively over type variables and type
- constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s |
- (\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>"}.
-
- A \emph{type abbreviation} is a syntactic definition @{text
- "(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over
- variables @{text "\<^vec>\<alpha>"}. Type abbreviations appear as type
- constructors in the syntax, but are expanded before entering the
- logical core.
-
- A \emph{type arity} declares the image behavior of a type
- constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>,
- s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)\<kappa>"} is
- of sort @{text "s"} if every argument type @{text "\<tau>\<^isub>i"} is
- of sort @{text "s\<^isub>i"}. Arity declarations are implicitly
- completed, i.e.\ @{text "\<kappa> :: (\<^vec>s)c"} entails @{text "\<kappa> ::
- (\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}.
-
- \medskip The sort algebra is always maintained as \emph{coregular},
- which means that type arities are consistent with the subclass
- relation: for any type constructor @{text "\<kappa>"}, and classes @{text
- "c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> ::
- (\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> ::
- (\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq>
- \<^vec>s\<^isub>2"} component-wise.
-
- The key property of a coregular order-sorted algebra is that sort
- constraints can be solved in a most general fashion: for each type
- constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general
- vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such
- that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>,
- \<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}.
- Consequently, type unification has most general solutions (modulo
- equivalence of sorts), so type-inference produces primary types as
- expected \cite{nipkow-prehofer}.
-*}
-
-text %mlref {*
- \begin{mldecls}
- @{index_ML_type class} \\
- @{index_ML_type sort} \\
- @{index_ML_type arity} \\
- @{index_ML_type typ} \\
- @{index_ML map_atyps: "(typ -> typ) -> typ -> typ"} \\
- @{index_ML fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\
- \end{mldecls}
- \begin{mldecls}
- @{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\
- @{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\
- @{index_ML Sign.add_types: "(string * int * mixfix) list -> theory -> theory"} \\
- @{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 the mapping @{text "f"}
- to all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text
- "\<tau>"}.
-
- \item @{ML fold_atyps}~@{text "f \<tau>"} iterates the operation @{text
- "f"} over all occurrences of atomic types (@{ML TFree}, @{ML TVar})
- in @{text "\<tau>"}; the type structure is traversed from left to right.
-
- \item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"}
- tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}.
-
- \item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether type
- @{text "\<tau>"} is of sort @{text "s"}.
-
- \item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares a new
- type constructors @{text "\<kappa>"} with @{text "k"} arguments and
- optional mixfix syntax.
-
- \item @{ML Sign.add_tyabbrs_i}~@{text "[(\<kappa>, \<^vec>\<alpha>, \<tau>, mx), \<dots>]"}
- defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"} with
- optional mixfix syntax.
-
- \item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>,
- c\<^isub>n])"} declares a new class @{text "c"}, together with class
- relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}.
-
- \item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1,
- c\<^isub>2)"} declares the class relation @{text "c\<^isub>1 \<subseteq>
- c\<^isub>2"}.
-
- \item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares
- the arity @{text "\<kappa> :: (\<^vec>s)s"}.
-
- \end{description}
-*}
-
-
-
-section {* Terms \label{sec:terms} *}
-
-text {*
- \glossary{Term}{FIXME}
-
- The language of terms is that of simply-typed @{text "\<lambda>"}-calculus
- with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72}
- or \cite{paulson-ml2}), with the types being determined determined
- by the corresponding binders. In contrast, free variables and
- constants are have an explicit name and type in each occurrence.
-
- \medskip A \emph{bound variable} is a natural number @{text "b"},
- which accounts for the number of intermediate binders between the
- variable occurrence in the body and its binding position. For
- example, the de-Bruijn term @{text
- "\<lambda>\<^bsub>nat\<^esub>. \<lambda>\<^bsub>nat\<^esub>. 1 + 0"} would
- correspond to @{text
- "\<lambda>x\<^bsub>nat\<^esub>. \<lambda>y\<^bsub>nat\<^esub>. x + y"} in a named
- representation. Note that a bound variable may be represented by
- different de-Bruijn indices at different occurrences, depending on
- the nesting of abstractions.
-
- A \emph{loose variable} is a bound variable that is outside the
- scope of local binders. The types (and names) for loose variables
- can be managed as a separate context, that is maintained as a stack
- of hypothetical binders. The core logic operates on closed terms,
- without any loose variables.
-
- A \emph{fixed variable} is a pair of a basic name and a type, e.g.\
- @{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}. A
- \emph{schematic variable} is a pair of an indexname and a type,
- e.g.\ @{text "((x, 0), \<tau>)"} which is usually printed as @{text
- "?x\<^isub>\<tau>"}.
-
- \medskip A \emph{constant} is a pair of a basic name and a type,
- e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text
- "c\<^isub>\<tau>"}. Constants are declared in the context as polymorphic
- families @{text "c :: \<sigma>"}, meaning that all substitution instances
- @{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid.
-
- The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"}
- wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of
- the matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>,
- ?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in canonical order @{text
- "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}. Within a given theory context,
- there is a one-to-one correspondence between any constant @{text
- "c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1, \<dots>,
- \<tau>\<^isub>n)"} of its type arguments. For example, with @{text "plus
- :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow>
- nat\<^esub>"} corresponds to @{text "plus(nat)"}.
-
- Constant declarations @{text "c :: \<sigma>"} may contain sort constraints
- for type variables in @{text "\<sigma>"}. These are observed by
- type-inference as expected, but \emph{ignored} by the core logic.
- This means the primitive logic is able to reason with instances of
- polymorphic constants that the user-level type-checker would reject
- due to violation of type class restrictions.
-
- \medskip An \emph{atomic} term is either a variable or constant. A
- \emph{term} is defined inductively over atomic terms, with
- abstraction and application as follows: @{text "t = b | x\<^isub>\<tau> |
- ?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}.
- Parsing and printing takes care of converting between an external
- representation with named bound variables. Subsequently, we shall
- use the latter notation instead of internal de-Bruijn
- representation.
-
- The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a
- term according to the structure of atomic terms, abstractions, and
- applicatins:
- \[
- \infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{}
- \qquad
- \infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}}
- \qquad
- \infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}}
- \]
- A \emph{well-typed term} is a term that can be typed according to these rules.
-
- Typing information can be omitted: type-inference is able to
- reconstruct the most general type of a raw term, while assigning
- most general types to all of its variables and constants.
- Type-inference depends on a context of type constraints for fixed
- variables, and declarations for polymorphic constants.
-
- The identity of atomic terms consists both of the name and the type
- component. This means that different variables @{text
- "x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text
- "x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type
- instantiation. Some outer layers of the system make it hard to
- produce variables of the same name, but different types. In
- contrast, mixed instances of polymorphic constants occur frequently.
-
- \medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"}
- is the set of type variables occurring in @{text "t"}, but not in
- @{text "\<sigma>"}. This means that the term implicitly depends on type
- arguments that are not accounted in the result type, i.e.\ there are
- different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text
- "t\<vartheta>' :: \<sigma>"} with the same type. This slightly
- pathological situation notoriously demands additional care.
-
- \medskip A \emph{term abbreviation} is a syntactic definition @{text
- "c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"},
- without any hidden polymorphism. A term abbreviation looks like a
- constant in the syntax, but is expanded before entering the logical
- core. Abbreviations are usually reverted when printing terms, using
- @{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for higher-order rewriting.
-
- \medskip Canonical operations on @{text "\<lambda>"}-terms include @{text
- "\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free
- renaming of bound variables; @{text "\<beta>"}-conversion contracts an
- abstraction applied to an argument term, substituting the argument
- in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text
- "\<eta>"}-conversion contracts vacuous application-abstraction: @{text
- "\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable
- does not occur in @{text "f"}.
-
- Terms are normally treated modulo @{text "\<alpha>"}-conversion, which is
- implicit in the de-Bruijn representation. Names for bound variables
- in abstractions are maintained separately as (meaningless) comments,
- mostly for parsing and printing. Full @{text "\<alpha>\<beta>\<eta>"}-conversion is
- commonplace in various standard operations (\secref{sec:obj-rules})
- that are based on higher-order unification and matching.
-*}
-
-text %mlref {*
- \begin{mldecls}
- @{index_ML_type term} \\
- @{index_ML "op aconv": "term * term -> bool"} \\
- @{index_ML map_types: "(typ -> typ) -> term -> term"} \\
- @{index_ML fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\
- @{index_ML map_aterms: "(term -> term) -> term -> term"} \\
- @{index_ML fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\
- \end{mldecls}
- \begin{mldecls}
- @{index_ML fastype_of: "term -> typ"} \\
- @{index_ML lambda: "term -> term -> term"} \\
- @{index_ML betapply: "term * term -> term"} \\
- @{index_ML Sign.declare_const: "Properties.T -> (binding * typ) * mixfix ->
- theory -> term * theory"} \\
- @{index_ML Sign.add_abbrev: "string -> Properties.T -> binding * term ->
- theory -> (term * term) * theory"} \\
- @{index_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\
- @{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\
- \end{mldecls}
-
- \begin{description}
-
- \item @{ML_type term} represents de-Bruijn terms, with comments in
- abstractions, and explicitly named free variables and constants;
- this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML
- Var}, @{ML Const}, @{ML Abs}, @{ML "op $"}.
-
- \item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text
- "\<alpha>"}-equivalence of two terms. This is the basic equality relation
- on type @{ML_type term}; raw datatype equality should only be used
- for operations related to parsing or printing!
-
- \item @{ML map_types}~@{text "f t"} applies the mapping @{text
- "f"} to all types occurring in @{text "t"}.
-
- \item @{ML fold_types}~@{text "f t"} iterates the operation @{text
- "f"} over all occurrences of types in @{text "t"}; the term
- structure is traversed from left to right.
-
- \item @{ML map_aterms}~@{text "f t"} applies the mapping @{text "f"}
- to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML
- Const}) occurring in @{text "t"}.
-
- \item @{ML fold_aterms}~@{text "f t"} iterates the operation @{text
- "f"} over all occurrences of atomic terms (@{ML Bound}, @{ML Free},
- @{ML Var}, @{ML Const}) in @{text "t"}; the term structure is
- traversed from left to right.
-
- \item @{ML fastype_of}~@{text "t"} determines the type of a
- well-typed term. This operation is relatively slow, despite the
- omission of any sanity checks.
-
- \item @{ML lambda}~@{text "a b"} produces an abstraction @{text
- "\<lambda>a. b"}, where occurrences of the atomic term @{text "a"} in the
- body @{text "b"} are replaced by bound variables.
-
- \item @{ML betapply}~@{text "(t, u)"} produces an application @{text
- "t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an
- abstraction.
-
- \item @{ML Sign.declare_const}~@{text "properties ((c, \<sigma>), mx)"}
- declares a new constant @{text "c :: \<sigma>"} with optional mixfix
- syntax.
-
- \item @{ML Sign.add_abbrev}~@{text "print_mode properties (c, t)"}
- introduces a new term abbreviation @{text "c \<equiv> t"}.
-
- \item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML
- Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"}
- convert between two representations of polymorphic constants: full
- type instance vs.\ compact type arguments form.
-
- \end{description}
-*}
-
-
-section {* Theorems \label{sec:thms} *}
-
-text {*
- \glossary{Proposition}{FIXME A \seeglossary{term} of
- \seeglossary{type} @{text "prop"}. Internally, there is nothing
- special about propositions apart from their type, but the concrete
- syntax enforces a clear distinction. Propositions are structured
- via implication @{text "A \<Longrightarrow> B"} or universal quantification @{text
- "\<And>x. B x"} --- anything else is considered atomic. The canonical
- form for propositions is that of a \seeglossary{Hereditary Harrop
- Formula}. FIXME}
-
- \glossary{Theorem}{A proven proposition within a certain theory and
- proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are
- rarely spelled out explicitly. Theorems are usually normalized
- according to the \seeglossary{HHF} format. FIXME}
-
- \glossary{Fact}{Sometimes used interchangeably for
- \seeglossary{theorem}. Strictly speaking, a list of theorems,
- essentially an extra-logical conjunction. Facts emerge either as
- local assumptions, or as results of local goal statements --- both
- may be simultaneous, hence the list representation. FIXME}
-
- \glossary{Schematic variable}{FIXME}
-
- \glossary{Fixed variable}{A variable that is bound within a certain
- proof context; an arbitrary-but-fixed entity within a portion of
- proof text. FIXME}
-
- \glossary{Free variable}{Synonymous for \seeglossary{fixed
- variable}. FIXME}
-
- \glossary{Bound variable}{FIXME}
-
- \glossary{Variable}{See \seeglossary{schematic variable},
- \seeglossary{fixed variable}, \seeglossary{bound variable}, or
- \seeglossary{type variable}. The distinguishing feature of
- different variables is their binding scope. FIXME}
-
- A \emph{proposition} is a well-typed term of type @{text "prop"}, a
- \emph{theorem} is a proven proposition (depending on a context of
- hypotheses and the background theory). Primitive inferences include
- plain natural deduction rules for the primary connectives @{text
- "\<And>"} and @{text "\<Longrightarrow>"} of the framework. There is also a builtin
- notion of equality/equivalence @{text "\<equiv>"}.
-*}
-
-subsection {* Primitive connectives and rules \label{sec:prim-rules} *}
-
-text {*
- The theory @{text "Pure"} contains constant declarations for the
- primitive connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of
- the logical framework, see \figref{fig:pure-connectives}. The
- derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is
- defined inductively by the primitive inferences given in
- \figref{fig:prim-rules}, with the global restriction that the
- hypotheses must \emph{not} contain any schematic variables. The
- builtin equality is conceptually axiomatized as shown in
- \figref{fig:pure-equality}, although the implementation works
- directly with derived inferences.
-
- \begin{figure}[htb]
- \begin{center}
- \begin{tabular}{ll}
- @{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\
- @{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\
- @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\
- \end{tabular}
- \caption{Primitive connectives of Pure}\label{fig:pure-connectives}
- \end{center}
- \end{figure}
-
- \begin{figure}[htb]
- \begin{center}
- \[
- \infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}}
- \qquad
- \infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{}
- \]
- \[
- \infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}}
- \qquad
- \infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}
- \]
- \[
- \infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
- \qquad
- \infer[@{text "(\<Longrightarrow>_elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}}
- \]
- \caption{Primitive inferences of Pure}\label{fig:prim-rules}
- \end{center}
- \end{figure}
-
- \begin{figure}[htb]
- \begin{center}
- \begin{tabular}{ll}
- @{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\
- @{text "\<turnstile> x \<equiv> x"} & reflexivity \\
- @{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\
- @{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\
- @{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\
- \end{tabular}
- \caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality}
- \end{center}
- \end{figure}
-
- The introduction and elimination rules for @{text "\<And>"} and @{text
- "\<Longrightarrow>"} are analogous to formation of dependently typed @{text
- "\<lambda>"}-terms representing the underlying proof objects. Proof terms
- are irrelevant in the Pure logic, though; they cannot occur within
- propositions. The system provides a runtime option to record
- explicit proof terms for primitive inferences. Thus all three
- levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for
- terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\
- \cite{Berghofer-Nipkow:2000:TPHOL}).
-
- Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need
- not be recorded in the hypotheses, because the simple syntactic
- types of Pure are always inhabitable. ``Assumptions'' @{text "x ::
- \<tau>"} for type-membership are only present as long as some @{text
- "x\<^isub>\<tau>"} occurs in the statement body.\footnote{This is the key
- difference to ``@{text "\<lambda>HOL"}'' in the PTS framework
- \cite{Barendregt-Geuvers:2001}, where hypotheses @{text "x : A"} are
- treated uniformly for propositions and types.}
-
- \medskip The axiomatization of a theory is implicitly closed by
- forming all instances of type and term variables: @{text "\<turnstile>
- A\<vartheta>"} holds for any substitution instance of an axiom
- @{text "\<turnstile> A"}. By pushing substitutions through derivations
- inductively, we also get admissible @{text "generalize"} and @{text
- "instance"} rules as shown in \figref{fig:subst-rules}.
-
- \begin{figure}[htb]
- \begin{center}
- \[
- \infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}}
- \quad
- \infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}}
- \]
- \[
- \infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}
- \quad
- \infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}}
- \]
- \caption{Admissible substitution rules}\label{fig:subst-rules}
- \end{center}
- \end{figure}
-
- Note that @{text "instantiate"} does not require an explicit
- side-condition, because @{text "\<Gamma>"} may never contain schematic
- variables.
-
- In principle, variables could be substituted in hypotheses as well,
- but this would disrupt the monotonicity of reasoning: deriving
- @{text "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is
- correct, but @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold:
- the result belongs to a different proof context.
-
- \medskip An \emph{oracle} is a function that produces axioms on the
- fly. Logically, this is an instance of the @{text "axiom"} rule
- (\figref{fig:prim-rules}), but there is an operational difference.
- The system always records oracle invocations within derivations of
- theorems. Tracing plain axioms (and named theorems) is optional.
-
- Axiomatizations should be limited to the bare minimum, typically as
- part of the initial logical basis of an object-logic formalization.
- Later on, theories are usually developed in a strictly definitional
- fashion, by stating only certain equalities over new constants.
-
- A \emph{simple definition} consists of a constant declaration @{text
- "c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text "t
- :: \<sigma>"} is a closed term without any hidden polymorphism. The RHS
- may depend on further defined constants, but not @{text "c"} itself.
- Definitions of functions may be presented as @{text "c \<^vec>x \<equiv>
- t"} instead of the puristic @{text "c \<equiv> \<lambda>\<^vec>x. t"}.
-
- An \emph{overloaded definition} consists of a collection of axioms
- for the same constant, with zero or one equations @{text
- "c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type constructor @{text "\<kappa>"} (for
- distinct variables @{text "\<^vec>\<alpha>"}). The RHS may mention
- previously defined constants as above, or arbitrary constants @{text
- "d(\<alpha>\<^isub>i)"} for some @{text "\<alpha>\<^isub>i"} projected from @{text
- "\<^vec>\<alpha>"}. Thus overloaded definitions essentially work by
- primitive recursion over the syntactic structure of a single type
- argument.
-*}
-
-text %mlref {*
- \begin{mldecls}
- @{index_ML_type ctyp} \\
- @{index_ML_type cterm} \\
- @{index_ML Thm.ctyp_of: "theory -> typ -> ctyp"} \\
- @{index_ML Thm.cterm_of: "theory -> term -> cterm"} \\
- \end{mldecls}
- \begin{mldecls}
- @{index_ML_type thm} \\
- @{index_ML proofs: "int ref"} \\
- @{index_ML Thm.assume: "cterm -> thm"} \\
- @{index_ML Thm.forall_intr: "cterm -> thm -> thm"} \\
- @{index_ML Thm.forall_elim: "cterm -> thm -> thm"} \\
- @{index_ML Thm.implies_intr: "cterm -> thm -> thm"} \\
- @{index_ML Thm.implies_elim: "thm -> thm -> thm"} \\
- @{index_ML Thm.generalize: "string list * string list -> int -> thm -> thm"} \\
- @{index_ML Thm.instantiate: "(ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm"} \\
- @{index_ML Thm.axiom: "theory -> string -> thm"} \\
- @{index_ML Thm.add_oracle: "bstring * ('a -> cterm) -> theory
- -> (string * ('a -> thm)) * theory"} \\
- \end{mldecls}
- \begin{mldecls}
- @{index_ML Theory.add_axioms_i: "(binding * term) list -> theory -> theory"} \\
- @{index_ML Theory.add_deps: "string -> string * typ -> (string * typ) list -> theory -> theory"} \\
- @{index_ML Theory.add_defs_i: "bool -> bool -> (binding * term) list -> theory -> theory"} \\
- \end{mldecls}
-
- \begin{description}
-
- \item @{ML_type ctyp} and @{ML_type cterm} represent certified types
- and terms, respectively. These are abstract datatypes that
- guarantee that its values have passed the full well-formedness (and
- well-typedness) checks, relative to the declarations of type
- constructors, constants etc. in the theory.
-
- \item @{ML ctyp_of}~@{text "thy \<tau>"} and @{ML cterm_of}~@{text "thy
- t"} explicitly checks types and terms, respectively. This also
- involves some basic normalizations, such expansion of type and term
- abbreviations from the theory context.
-
- Re-certification is relatively slow and should be avoided in tight
- reasoning loops. There are separate operations to decompose
- certified entities (including actual theorems).
-
- \item @{ML_type thm} represents proven propositions. This is an
- abstract datatype that guarantees that its values have been
- constructed by basic principles of the @{ML_struct Thm} module.
- Every @{ML thm} value contains a sliding back-reference to the
- enclosing theory, cf.\ \secref{sec:context-theory}.
-
- \item @{ML proofs} determines the detail of proof recording within
- @{ML_type thm} values: @{ML 0} records only oracles, @{ML 1} records
- oracles, axioms and named theorems, @{ML 2} records full proof
- terms.
-
- \item @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML
- Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim}
- correspond to the primitive inferences of \figref{fig:prim-rules}.
-
- \item @{ML Thm.generalize}~@{text "(\<^vec>\<alpha>, \<^vec>x)"}
- corresponds to the @{text "generalize"} rules of
- \figref{fig:subst-rules}. Here collections of type and term
- variables are generalized simultaneously, specified by the given
- basic names.
-
- \item @{ML Thm.instantiate}~@{text "(\<^vec>\<alpha>\<^isub>s,
- \<^vec>x\<^isub>\<tau>)"} corresponds to the @{text "instantiate"} rules
- of \figref{fig:subst-rules}. Type variables are substituted before
- term variables. Note that the types in @{text "\<^vec>x\<^isub>\<tau>"}
- refer to the instantiated versions.
-
- \item @{ML Thm.axiom}~@{text "thy name"} retrieves a named
- axiom, cf.\ @{text "axiom"} in \figref{fig:prim-rules}.
-
- \item @{ML Thm.add_oracle}~@{text "(name, oracle)"} produces a named
- oracle rule, essentially generating arbitrary axioms on the fly,
- cf.\ @{text "axiom"} in \figref{fig:prim-rules}.
-
- \item @{ML Theory.add_axioms_i}~@{text "[(name, A), \<dots>]"} declares
- arbitrary propositions as axioms.
-
- \item @{ML Theory.add_deps}~@{text "name c\<^isub>\<tau>
- \<^vec>d\<^isub>\<sigma>"} declares dependencies of a named specification
- for constant @{text "c\<^isub>\<tau>"}, relative to existing
- specifications for constants @{text "\<^vec>d\<^isub>\<sigma>"}.
-
- \item @{ML Theory.add_defs_i}~@{text "unchecked overloaded [(name, c
- \<^vec>x \<equiv> t), \<dots>]"} states a definitional axiom for an existing
- constant @{text "c"}. Dependencies are recorded (cf.\ @{ML
- Theory.add_deps}), unless the @{text "unchecked"} option is set.
-
- \end{description}
-*}
-
-
-subsection {* Auxiliary definitions *}
-
-text {*
- Theory @{text "Pure"} provides a few auxiliary definitions, see
- \figref{fig:pure-aux}. These special constants are normally not
- exposed to the user, but appear in internal encodings.
-
- \begin{figure}[htb]
- \begin{center}
- \begin{tabular}{ll}
- @{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\
- @{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex]
- @{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, suppressed) \\
- @{text "#A \<equiv> A"} \\[1ex]
- @{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\
- @{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex]
- @{text "TYPE :: \<alpha> itself"} & (prefix @{text "TYPE"}) \\
- @{text "(unspecified)"} \\
- \end{tabular}
- \caption{Definitions of auxiliary connectives}\label{fig:pure-aux}
- \end{center}
- \end{figure}
-
- Derived conjunction rules include introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A &
- B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}.
- Conjunction allows to treat simultaneous assumptions and conclusions
- uniformly. For example, multiple claims are intermediately
- represented as explicit conjunction, but this is refined into
- separate sub-goals before the user continues the proof; the final
- result is projected into a list of theorems (cf.\
- \secref{sec:tactical-goals}).
-
- The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex
- propositions appear as atomic, without changing the meaning: @{text
- "\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable. See
- \secref{sec:tactical-goals} for specific operations.
-
- The @{text "term"} marker turns any well-typed term into a derivable
- proposition: @{text "\<turnstile> TERM t"} holds unconditionally. Although
- this is logically vacuous, it allows to treat terms and proofs
- uniformly, similar to a type-theoretic framework.
-
- The @{text "TYPE"} constructor is the canonical representative of
- the unspecified type @{text "\<alpha> itself"}; it essentially injects the
- language of types into that of terms. There is specific notation
- @{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau>
- itself\<^esub>"}.
- Although being devoid of any particular meaning, the @{text
- "TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term
- language. In particular, @{text "TYPE(\<alpha>)"} may be used as formal
- argument in primitive definitions, in order to circumvent hidden
- polymorphism (cf.\ \secref{sec:terms}). For example, @{text "c
- TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of
- a proposition @{text "A"} that depends on an additional type
- argument, which is essentially a predicate on types.
-*}
-
-text %mlref {*
- \begin{mldecls}
- @{index_ML Conjunction.intr: "thm -> thm -> thm"} \\
- @{index_ML Conjunction.elim: "thm -> thm * thm"} \\
- @{index_ML Drule.mk_term: "cterm -> thm"} \\
- @{index_ML Drule.dest_term: "thm -> cterm"} \\
- @{index_ML Logic.mk_type: "typ -> term"} \\
- @{index_ML Logic.dest_type: "term -> typ"} \\
- \end{mldecls}
-
- \begin{description}
-
- \item @{ML Conjunction.intr} derives @{text "A & B"} from @{text
- "A"} and @{text "B"}.
-
- \item @{ML Conjunction.elim} derives @{text "A"} and @{text "B"}
- from @{text "A & B"}.
-
- \item @{ML Drule.mk_term} derives @{text "TERM t"}.
-
- \item @{ML Drule.dest_term} recovers term @{text "t"} from @{text
- "TERM t"}.
-
- \item @{ML Logic.mk_type}~@{text "\<tau>"} produces the term @{text
- "TYPE(\<tau>)"}.
-
- \item @{ML Logic.dest_type}~@{text "TYPE(\<tau>)"} recovers the type
- @{text "\<tau>"}.
-
- \end{description}
-*}
-
-
-section {* Object-level rules \label{sec:obj-rules} *}
-
-text %FIXME {*
-
-FIXME
-
- A \emph{rule} is any Pure theorem in HHF normal form; there is a
- separate calculus for rule composition, which is modeled after
- Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows
- rules to be nested arbitrarily, similar to \cite{extensions91}.
-
- Normally, all theorems accessible to the user are proper rules.
- Low-level inferences are occasional required internally, but the
- result should be always presented in canonical form. The higher
- interfaces of Isabelle/Isar will always produce proper rules. It is
- important to maintain this invariant in add-on applications!
-
- There are two main principles of rule composition: @{text
- "resolution"} (i.e.\ backchaining of rules) and @{text
- "by-assumption"} (i.e.\ closing a branch); both principles are
- combined in the variants of @{text "elim-resolution"} and @{text
- "dest-resolution"}. Raw @{text "composition"} is occasionally
- useful as well, also it is strictly speaking outside of the proper
- rule calculus.
-
- Rules are treated modulo general higher-order unification, which is
- unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion
- on @{text "\<lambda>"}-terms. Moreover, propositions are understood modulo
- the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}.
-
- This means that any operations within the rule calculus may be
- subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions. It is common
- practice not to contract or expand unnecessarily. Some mechanisms
- prefer an one form, others the opposite, so there is a potential
- danger to produce some oscillation!
-
- Only few operations really work \emph{modulo} HHF conversion, but
- expect a normal form: quantifiers @{text "\<And>"} before implications
- @{text "\<Longrightarrow>"} at each level of nesting.
-
-\glossary{Hereditary Harrop Formula}{The set of propositions in HHF
-format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow>
-A)"}, for variables @{text "x"} and atomic propositions @{text "A"}.
-Any proposition may be put into HHF form by normalizing with the rule
-@{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}. In Isabelle, the outermost
-quantifier prefix is represented via \seeglossary{schematic
-variables}, such that the top-level structure is merely that of a
-\seeglossary{Horn Clause}}.
-
-\glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.}
-
-
- \[
- \infer[@{text "(assumption)"}]{@{text "C\<vartheta>"}}
- {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text i})}}
- \]
-
-
- \[
- \infer[@{text "(compose)"}]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}}
- {@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}}
- \]
-
-
- \[
- \infer[@{text "(\<And>_lift)"}]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}}
- \]
- \[
- \infer[@{text "(\<Longrightarrow>_lift)"}]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}}
- \]
-
- The @{text resolve} scheme is now acquired from @{text "\<And>_lift"},
- @{text "\<Longrightarrow>_lift"}, and @{text compose}.
-
- \[
- \infer[@{text "(resolution)"}]
- {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}}
- {\begin{tabular}{l}
- @{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\
- @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\
- @{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\
- \end{tabular}}
- \]
-
-
- FIXME @{text "elim_resolution"}, @{text "dest_resolution"}
-*}
-
-
-end