author | wenzelm |
Sat, 10 May 2008 14:13:03 +0200 | |
changeset 26872 | 336dfd860744 |
parent 24972 | acafb18a47dc |
child 28017 | 4919bd124a58 |
permissions | -rw-r--r-- |
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(* $Id$ *) |
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theory logic imports base begin |
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chapter {* Primitive logic \label{ch:logic} *} |
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text {* |
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The logical foundations of Isabelle/Isar are that of the Pure logic, |
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which has been introduced as a natural-deduction framework in |
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\cite{paulson700}. This is essentially the same logic as ``@{text |
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"\<lambda>HOL"}'' in the more abstract setting of Pure Type Systems (PTS) |
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\cite{Barendregt-Geuvers:2001}, although there are some key |
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differences in the specific treatment of simple types in |
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Isabelle/Pure. |
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Following type-theoretic parlance, the Pure logic consists of three |
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levels of @{text "\<lambda>"}-calculus with corresponding arrows, @{text |
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"\<Rightarrow>"} for syntactic function space (terms depending on terms), @{text |
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"\<And>"} for universal quantification (proofs depending on terms), and |
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@{text "\<Longrightarrow>"} for implication (proofs depending on proofs). |
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Derivations are relative to a logical theory, which declares type |
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constructors, constants, and axioms. Theory declarations support |
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schematic polymorphism, which is strictly speaking outside the |
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logic.\footnote{This is the deeper logical reason, why the theory |
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context @{text "\<Theta>"} is separate from the proof context @{text "\<Gamma>"} |
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of the core calculus.} |
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*} |
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section {* Types \label{sec:types} *} |
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text {* |
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The language of types is an uninterpreted order-sorted first-order |
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algebra; types are qualified by ordered type classes. |
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\medskip A \emph{type class} is an abstract syntactic entity |
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declared in the theory context. The \emph{subclass relation} @{text |
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"c\<^isub>1 \<subseteq> c\<^isub>2"} is specified by stating an acyclic |
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generating relation; the transitive closure is maintained |
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internally. The resulting relation is an ordering: reflexive, |
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transitive, and antisymmetric. |
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A \emph{sort} is a list of type classes written as @{text "s = |
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{c\<^isub>1, \<dots>, c\<^isub>m}"}, which represents symbolic |
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intersection. Notationally, the curly braces are omitted for |
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singleton intersections, i.e.\ any class @{text "c"} may be read as |
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a sort @{text "{c}"}. The ordering on type classes is extended to |
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sorts according to the meaning of intersections: @{text |
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"{c\<^isub>1, \<dots> c\<^isub>m} \<subseteq> {d\<^isub>1, \<dots>, d\<^isub>n}"} iff |
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@{text "\<forall>j. \<exists>i. c\<^isub>i \<subseteq> d\<^isub>j"}. The empty intersection |
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@{text "{}"} refers to the universal sort, which is the largest |
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element wrt.\ the sort order. The intersections of all (finitely |
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many) classes declared in the current theory are the minimal |
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elements wrt.\ the sort order. |
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\medskip A \emph{fixed type variable} is a pair of a basic name |
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(starting with a @{text "'"} character) and a sort constraint, e.g.\ |
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@{text "('a, s)"} which is usually printed as @{text "\<alpha>\<^isub>s"}. |
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A \emph{schematic type variable} is a pair of an indexname and a |
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sort constraint, e.g.\ @{text "(('a, 0), s)"} which is usually |
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printed as @{text "?\<alpha>\<^isub>s"}. |
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Note that \emph{all} syntactic components contribute to the identity |
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of type variables, including the sort constraint. The core logic |
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handles type variables with the same name but different sorts as |
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different, although some outer layers of the system make it hard to |
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produce anything like this. |
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A \emph{type constructor} @{text "\<kappa>"} is a @{text "k"}-ary operator |
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on types declared in the theory. Type constructor application is |
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written postfix as @{text "(\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>k)\<kappa>"}. For |
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@{text "k = 0"} the argument tuple is omitted, e.g.\ @{text "prop"} |
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instead of @{text "()prop"}. For @{text "k = 1"} the parentheses |
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are omitted, e.g.\ @{text "\<alpha> list"} instead of @{text "(\<alpha>)list"}. |
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Further notation is provided for specific constructors, notably the |
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right-associative infix @{text "\<alpha> \<Rightarrow> \<beta>"} instead of @{text "(\<alpha>, |
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\<beta>)fun"}. |
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A \emph{type} is defined inductively over type variables and type |
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constructors as follows: @{text "\<tau> = \<alpha>\<^isub>s | ?\<alpha>\<^isub>s | |
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(\<tau>\<^sub>1, \<dots>, \<tau>\<^sub>k)\<kappa>"}. |
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A \emph{type abbreviation} is a syntactic definition @{text |
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"(\<^vec>\<alpha>)\<kappa> = \<tau>"} of an arbitrary type expression @{text "\<tau>"} over |
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variables @{text "\<^vec>\<alpha>"}. Type abbreviations appear as type |
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constructors in the syntax, but are expanded before entering the |
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logical core. |
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A \emph{type arity} declares the image behavior of a type |
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constructor wrt.\ the algebra of sorts: @{text "\<kappa> :: (s\<^isub>1, \<dots>, |
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s\<^isub>k)s"} means that @{text "(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>k)\<kappa>"} is |
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of sort @{text "s"} if every argument type @{text "\<tau>\<^isub>i"} is |
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of sort @{text "s\<^isub>i"}. Arity declarations are implicitly |
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completed, i.e.\ @{text "\<kappa> :: (\<^vec>s)c"} entails @{text "\<kappa> :: |
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(\<^vec>s)c'"} for any @{text "c' \<supseteq> c"}. |
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\medskip The sort algebra is always maintained as \emph{coregular}, |
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which means that type arities are consistent with the subclass |
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relation: for any type constructor @{text "\<kappa>"}, and classes @{text |
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"c\<^isub>1 \<subseteq> c\<^isub>2"}, and arities @{text "\<kappa> :: |
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(\<^vec>s\<^isub>1)c\<^isub>1"} and @{text "\<kappa> :: |
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(\<^vec>s\<^isub>2)c\<^isub>2"} holds @{text "\<^vec>s\<^isub>1 \<subseteq> |
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\<^vec>s\<^isub>2"} component-wise. |
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The key property of a coregular order-sorted algebra is that sort |
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constraints can be solved in a most general fashion: for each type |
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constructor @{text "\<kappa>"} and sort @{text "s"} there is a most general |
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vector of argument sorts @{text "(s\<^isub>1, \<dots>, s\<^isub>k)"} such |
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that a type scheme @{text "(\<alpha>\<^bsub>s\<^isub>1\<^esub>, \<dots>, |
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\<alpha>\<^bsub>s\<^isub>k\<^esub>)\<kappa>"} is of sort @{text "s"}. |
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Consequently, type unification has most general solutions (modulo |
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equivalence of sorts), so type-inference produces primary types as |
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expected \cite{nipkow-prehofer}. |
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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML_type class} \\ |
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@{index_ML_type sort} \\ |
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@{index_ML_type arity} \\ |
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@{index_ML_type typ} \\ |
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@{index_ML map_atyps: "(typ -> typ) -> typ -> typ"} \\ |
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@{index_ML fold_atyps: "(typ -> 'a -> 'a) -> typ -> 'a -> 'a"} \\ |
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\end{mldecls} |
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\begin{mldecls} |
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@{index_ML Sign.subsort: "theory -> sort * sort -> bool"} \\ |
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@{index_ML Sign.of_sort: "theory -> typ * sort -> bool"} \\ |
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@{index_ML Sign.add_types: "(string * int * mixfix) list -> theory -> theory"} \\ |
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@{index_ML Sign.add_tyabbrs_i: " |
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(string * string list * typ * mixfix) list -> theory -> theory"} \\ |
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@{index_ML Sign.primitive_class: "string * class list -> theory -> theory"} \\ |
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@{index_ML Sign.primitive_classrel: "class * class -> theory -> theory"} \\ |
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@{index_ML Sign.primitive_arity: "arity -> theory -> theory"} \\ |
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\end{mldecls} |
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\begin{description} |
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\item @{ML_type class} represents type classes; this is an alias for |
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@{ML_type string}. |
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\item @{ML_type sort} represents sorts; this is an alias for |
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@{ML_type "class list"}. |
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\item @{ML_type arity} represents type arities; this is an alias for |
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triples of the form @{text "(\<kappa>, \<^vec>s, s)"} for @{text "\<kappa> :: |
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(\<^vec>s)s"} described above. |
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\item @{ML_type typ} represents types; this is a datatype with |
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constructors @{ML TFree}, @{ML TVar}, @{ML Type}. |
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\item @{ML map_atyps}~@{text "f \<tau>"} applies the mapping @{text "f"} |
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to all atomic types (@{ML TFree}, @{ML TVar}) occurring in @{text |
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"\<tau>"}. |
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\item @{ML fold_atyps}~@{text "f \<tau>"} iterates the operation @{text |
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"f"} over all occurrences of atomic types (@{ML TFree}, @{ML TVar}) |
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in @{text "\<tau>"}; the type structure is traversed from left to right. |
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\item @{ML Sign.subsort}~@{text "thy (s\<^isub>1, s\<^isub>2)"} |
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tests the subsort relation @{text "s\<^isub>1 \<subseteq> s\<^isub>2"}. |
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\item @{ML Sign.of_sort}~@{text "thy (\<tau>, s)"} tests whether type |
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@{text "\<tau>"} is of sort @{text "s"}. |
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\item @{ML Sign.add_types}~@{text "[(\<kappa>, k, mx), \<dots>]"} declares a new |
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type constructors @{text "\<kappa>"} with @{text "k"} arguments and |
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optional mixfix syntax. |
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\item @{ML Sign.add_tyabbrs_i}~@{text "[(\<kappa>, \<^vec>\<alpha>, \<tau>, mx), \<dots>]"} |
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defines a new type abbreviation @{text "(\<^vec>\<alpha>)\<kappa> = \<tau>"} with |
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optional mixfix syntax. |
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\item @{ML Sign.primitive_class}~@{text "(c, [c\<^isub>1, \<dots>, |
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c\<^isub>n])"} declares a new class @{text "c"}, together with class |
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relations @{text "c \<subseteq> c\<^isub>i"}, for @{text "i = 1, \<dots>, n"}. |
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\item @{ML Sign.primitive_classrel}~@{text "(c\<^isub>1, |
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c\<^isub>2)"} declares the class relation @{text "c\<^isub>1 \<subseteq> |
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c\<^isub>2"}. |
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\item @{ML Sign.primitive_arity}~@{text "(\<kappa>, \<^vec>s, s)"} declares |
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the arity @{text "\<kappa> :: (\<^vec>s)s"}. |
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\end{description} |
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*} |
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section {* Terms \label{sec:terms} *} |
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text {* |
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\glossary{Term}{FIXME} |
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The language of terms is that of simply-typed @{text "\<lambda>"}-calculus |
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with de-Bruijn indices for bound variables (cf.\ \cite{debruijn72} |
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or \cite{paulson-ml2}), with the types being determined determined |
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by the corresponding binders. In contrast, free variables and |
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constants are have an explicit name and type in each occurrence. |
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\medskip A \emph{bound variable} is a natural number @{text "b"}, |
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which accounts for the number of intermediate binders between the |
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variable occurrence in the body and its binding position. For |
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example, the de-Bruijn term @{text |
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"\<lambda>\<^bsub>nat\<^esub>. \<lambda>\<^bsub>nat\<^esub>. 1 + 0"} would |
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correspond to @{text |
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"\<lambda>x\<^bsub>nat\<^esub>. \<lambda>y\<^bsub>nat\<^esub>. x + y"} in a named |
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representation. Note that a bound variable may be represented by |
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different de-Bruijn indices at different occurrences, depending on |
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the nesting of abstractions. |
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A \emph{loose variable} is a bound variable that is outside the |
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scope of local binders. The types (and names) for loose variables |
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can be managed as a separate context, that is maintained as a stack |
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of hypothetical binders. The core logic operates on closed terms, |
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without any loose variables. |
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A \emph{fixed variable} is a pair of a basic name and a type, e.g.\ |
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@{text "(x, \<tau>)"} which is usually printed @{text "x\<^isub>\<tau>"}. A |
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\emph{schematic variable} is a pair of an indexname and a type, |
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e.g.\ @{text "((x, 0), \<tau>)"} which is usually printed as @{text |
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"?x\<^isub>\<tau>"}. |
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\medskip A \emph{constant} is a pair of a basic name and a type, |
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e.g.\ @{text "(c, \<tau>)"} which is usually printed as @{text |
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"c\<^isub>\<tau>"}. Constants are declared in the context as polymorphic |
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families @{text "c :: \<sigma>"}, meaning that all substitution instances |
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@{text "c\<^isub>\<tau>"} for @{text "\<tau> = \<sigma>\<vartheta>"} are valid. |
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The vector of \emph{type arguments} of constant @{text "c\<^isub>\<tau>"} |
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wrt.\ the declaration @{text "c :: \<sigma>"} is defined as the codomain of |
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the matcher @{text "\<vartheta> = {?\<alpha>\<^isub>1 \<mapsto> \<tau>\<^isub>1, \<dots>, |
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?\<alpha>\<^isub>n \<mapsto> \<tau>\<^isub>n}"} presented in canonical order @{text |
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"(\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n)"}. Within a given theory context, |
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there is a one-to-one correspondence between any constant @{text |
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"c\<^isub>\<tau>"} and the application @{text "c(\<tau>\<^isub>1, \<dots>, |
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\<tau>\<^isub>n)"} of its type arguments. For example, with @{text "plus |
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:: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"}, the instance @{text "plus\<^bsub>nat \<Rightarrow> nat \<Rightarrow> |
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nat\<^esub>"} corresponds to @{text "plus(nat)"}. |
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Constant declarations @{text "c :: \<sigma>"} may contain sort constraints |
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for type variables in @{text "\<sigma>"}. These are observed by |
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type-inference as expected, but \emph{ignored} by the core logic. |
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This means the primitive logic is able to reason with instances of |
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polymorphic constants that the user-level type-checker would reject |
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due to violation of type class restrictions. |
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\medskip An \emph{atomic} term is either a variable or constant. A |
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\emph{term} is defined inductively over atomic terms, with |
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abstraction and application as follows: @{text "t = b | x\<^isub>\<tau> | |
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?x\<^isub>\<tau> | c\<^isub>\<tau> | \<lambda>\<^isub>\<tau>. t | t\<^isub>1 t\<^isub>2"}. |
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Parsing and printing takes care of converting between an external |
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representation with named bound variables. Subsequently, we shall |
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use the latter notation instead of internal de-Bruijn |
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representation. |
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The inductive relation @{text "t :: \<tau>"} assigns a (unique) type to a |
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term according to the structure of atomic terms, abstractions, and |
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applicatins: |
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\[ |
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\infer{@{text "a\<^isub>\<tau> :: \<tau>"}}{} |
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\qquad |
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\infer{@{text "(\<lambda>x\<^sub>\<tau>. t) :: \<tau> \<Rightarrow> \<sigma>"}}{@{text "t :: \<sigma>"}} |
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\qquad |
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\infer{@{text "t u :: \<sigma>"}}{@{text "t :: \<tau> \<Rightarrow> \<sigma>"} & @{text "u :: \<tau>"}} |
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\] |
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A \emph{well-typed term} is a term that can be typed according to these rules. |
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Typing information can be omitted: type-inference is able to |
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reconstruct the most general type of a raw term, while assigning |
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most general types to all of its variables and constants. |
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Type-inference depends on a context of type constraints for fixed |
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variables, and declarations for polymorphic constants. |
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The identity of atomic terms consists both of the name and the type |
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component. This means that different variables @{text |
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"x\<^bsub>\<tau>\<^isub>1\<^esub>"} and @{text |
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"x\<^bsub>\<tau>\<^isub>2\<^esub>"} may become the same after type |
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instantiation. Some outer layers of the system make it hard to |
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produce variables of the same name, but different types. In |
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contrast, mixed instances of polymorphic constants occur frequently. |
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\medskip The \emph{hidden polymorphism} of a term @{text "t :: \<sigma>"} |
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is the set of type variables occurring in @{text "t"}, but not in |
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@{text "\<sigma>"}. This means that the term implicitly depends on type |
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arguments that are not accounted in the result type, i.e.\ there are |
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different type instances @{text "t\<vartheta> :: \<sigma>"} and @{text |
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"t\<vartheta>' :: \<sigma>"} with the same type. This slightly |
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pathological situation notoriously demands additional care. |
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\medskip A \emph{term abbreviation} is a syntactic definition @{text |
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"c\<^isub>\<sigma> \<equiv> t"} of a closed term @{text "t"} of type @{text "\<sigma>"}, |
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without any hidden polymorphism. A term abbreviation looks like a |
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constant in the syntax, but is expanded before entering the logical |
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core. Abbreviations are usually reverted when printing terms, using |
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@{text "t \<rightarrow> c\<^isub>\<sigma>"} as rules for higher-order rewriting. |
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\medskip Canonical operations on @{text "\<lambda>"}-terms include @{text |
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"\<alpha>\<beta>\<eta>"}-conversion: @{text "\<alpha>"}-conversion refers to capture-free |
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renaming of bound variables; @{text "\<beta>"}-conversion contracts an |
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abstraction applied to an argument term, substituting the argument |
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in the body: @{text "(\<lambda>x. b)a"} becomes @{text "b[a/x]"}; @{text |
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"\<eta>"}-conversion contracts vacuous application-abstraction: @{text |
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"\<lambda>x. f x"} becomes @{text "f"}, provided that the bound variable |
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does not occur in @{text "f"}. |
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Terms are normally treated modulo @{text "\<alpha>"}-conversion, which is |
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implicit in the de-Bruijn representation. Names for bound variables |
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in abstractions are maintained separately as (meaningless) comments, |
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mostly for parsing and printing. Full @{text "\<alpha>\<beta>\<eta>"}-conversion is |
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commonplace in various standard operations (\secref{sec:rules}) that |
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are based on higher-order unification and matching. |
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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML_type term} \\ |
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@{index_ML "op aconv": "term * term -> bool"} \\ |
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@{index_ML map_types: "(typ -> typ) -> term -> term"} \\ |
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@{index_ML fold_types: "(typ -> 'a -> 'a) -> term -> 'a -> 'a"} \\ |
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@{index_ML map_aterms: "(term -> term) -> term -> term"} \\ |
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@{index_ML fold_aterms: "(term -> 'a -> 'a) -> term -> 'a -> 'a"} \\ |
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\end{mldecls} |
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\begin{mldecls} |
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@{index_ML fastype_of: "term -> typ"} \\ |
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@{index_ML lambda: "term -> term -> term"} \\ |
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@{index_ML betapply: "term * term -> term"} \\ |
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changeset
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@{index_ML Sign.declare_const: "Markup.property list -> bstring * typ * mixfix -> |
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|
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theory -> term * theory"} \\ |
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|
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@{index_ML Sign.add_abbrev: "string -> Markup.property list -> bstring * term -> |
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|
332 |
theory -> (term * term) * theory"} \\ |
20519 | 333 |
@{index_ML Sign.const_typargs: "theory -> string * typ -> typ list"} \\ |
334 |
@{index_ML Sign.const_instance: "theory -> string * typ list -> typ"} \\ |
|
20514 | 335 |
\end{mldecls} |
18537 | 336 |
|
20514 | 337 |
\begin{description} |
18537 | 338 |
|
20537 | 339 |
\item @{ML_type term} represents de-Bruijn terms, with comments in |
340 |
abstractions, and explicitly named free variables and constants; |
|
341 |
this is a datatype with constructors @{ML Bound}, @{ML Free}, @{ML |
|
342 |
Var}, @{ML Const}, @{ML Abs}, @{ML "op $"}. |
|
20519 | 343 |
|
344 |
\item @{text "t"}~@{ML aconv}~@{text "u"} checks @{text |
|
345 |
"\<alpha>"}-equivalence of two terms. This is the basic equality relation |
|
346 |
on type @{ML_type term}; raw datatype equality should only be used |
|
347 |
for operations related to parsing or printing! |
|
348 |
||
20547 | 349 |
\item @{ML map_types}~@{text "f t"} applies the mapping @{text |
20537 | 350 |
"f"} to all types occurring in @{text "t"}. |
351 |
||
352 |
\item @{ML fold_types}~@{text "f t"} iterates the operation @{text |
|
353 |
"f"} over all occurrences of types in @{text "t"}; the term |
|
354 |
structure is traversed from left to right. |
|
20519 | 355 |
|
20537 | 356 |
\item @{ML map_aterms}~@{text "f t"} applies the mapping @{text "f"} |
357 |
to all atomic terms (@{ML Bound}, @{ML Free}, @{ML Var}, @{ML |
|
358 |
Const}) occurring in @{text "t"}. |
|
359 |
||
360 |
\item @{ML fold_aterms}~@{text "f t"} iterates the operation @{text |
|
361 |
"f"} over all occurrences of atomic terms (@{ML Bound}, @{ML Free}, |
|
362 |
@{ML Var}, @{ML Const}) in @{text "t"}; the term structure is |
|
20519 | 363 |
traversed from left to right. |
364 |
||
20537 | 365 |
\item @{ML fastype_of}~@{text "t"} determines the type of a |
366 |
well-typed term. This operation is relatively slow, despite the |
|
367 |
omission of any sanity checks. |
|
20519 | 368 |
|
369 |
\item @{ML lambda}~@{text "a b"} produces an abstraction @{text |
|
20537 | 370 |
"\<lambda>a. b"}, where occurrences of the atomic term @{text "a"} in the |
371 |
body @{text "b"} are replaced by bound variables. |
|
20519 | 372 |
|
20537 | 373 |
\item @{ML betapply}~@{text "(t, u)"} produces an application @{text |
374 |
"t u"}, with topmost @{text "\<beta>"}-conversion if @{text "t"} is an |
|
375 |
abstraction. |
|
20519 | 376 |
|
24972
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\item @{ML Sign.declare_const}~@{text "properties (c, \<sigma>, mx)"} |
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declares a new constant @{text "c :: \<sigma>"} with optional mixfix |
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|
379 |
syntax. |
20519 | 380 |
|
24828 | 381 |
\item @{ML Sign.add_abbrev}~@{text "print_mode properties (c, t)"} |
21827 | 382 |
introduces a new term abbreviation @{text "c \<equiv> t"}. |
20519 | 383 |
|
20520 | 384 |
\item @{ML Sign.const_typargs}~@{text "thy (c, \<tau>)"} and @{ML |
385 |
Sign.const_instance}~@{text "thy (c, [\<tau>\<^isub>1, \<dots>, \<tau>\<^isub>n])"} |
|
20543 | 386 |
convert between two representations of polymorphic constants: full |
387 |
type instance vs.\ compact type arguments form. |
|
18537 | 388 |
|
20514 | 389 |
\end{description} |
18537 | 390 |
*} |
391 |
||
392 |
||
20451 | 393 |
section {* Theorems \label{sec:thms} *} |
18537 | 394 |
|
395 |
text {* |
|
20521 | 396 |
\glossary{Proposition}{FIXME A \seeglossary{term} of |
397 |
\seeglossary{type} @{text "prop"}. Internally, there is nothing |
|
398 |
special about propositions apart from their type, but the concrete |
|
399 |
syntax enforces a clear distinction. Propositions are structured |
|
400 |
via implication @{text "A \<Longrightarrow> B"} or universal quantification @{text |
|
401 |
"\<And>x. B x"} --- anything else is considered atomic. The canonical |
|
402 |
form for propositions is that of a \seeglossary{Hereditary Harrop |
|
403 |
Formula}. FIXME} |
|
20480 | 404 |
|
20501 | 405 |
\glossary{Theorem}{A proven proposition within a certain theory and |
406 |
proof context, formally @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}; both contexts are |
|
407 |
rarely spelled out explicitly. Theorems are usually normalized |
|
408 |
according to the \seeglossary{HHF} format. FIXME} |
|
20480 | 409 |
|
20519 | 410 |
\glossary{Fact}{Sometimes used interchangeably for |
20501 | 411 |
\seeglossary{theorem}. Strictly speaking, a list of theorems, |
412 |
essentially an extra-logical conjunction. Facts emerge either as |
|
413 |
local assumptions, or as results of local goal statements --- both |
|
414 |
may be simultaneous, hence the list representation. FIXME} |
|
18537 | 415 |
|
20501 | 416 |
\glossary{Schematic variable}{FIXME} |
417 |
||
418 |
\glossary{Fixed variable}{A variable that is bound within a certain |
|
419 |
proof context; an arbitrary-but-fixed entity within a portion of |
|
420 |
proof text. FIXME} |
|
18537 | 421 |
|
20501 | 422 |
\glossary{Free variable}{Synonymous for \seeglossary{fixed |
423 |
variable}. FIXME} |
|
424 |
||
425 |
\glossary{Bound variable}{FIXME} |
|
18537 | 426 |
|
20501 | 427 |
\glossary{Variable}{See \seeglossary{schematic variable}, |
428 |
\seeglossary{fixed variable}, \seeglossary{bound variable}, or |
|
429 |
\seeglossary{type variable}. The distinguishing feature of |
|
430 |
different variables is their binding scope. FIXME} |
|
18537 | 431 |
|
20543 | 432 |
A \emph{proposition} is a well-typed term of type @{text "prop"}, a |
20521 | 433 |
\emph{theorem} is a proven proposition (depending on a context of |
434 |
hypotheses and the background theory). Primitive inferences include |
|
435 |
plain natural deduction rules for the primary connectives @{text |
|
20537 | 436 |
"\<And>"} and @{text "\<Longrightarrow>"} of the framework. There is also a builtin |
437 |
notion of equality/equivalence @{text "\<equiv>"}. |
|
20521 | 438 |
*} |
439 |
||
26872 | 440 |
subsection {* Primitive connectives and rules \label{sec:prim-rules} *} |
18537 | 441 |
|
20521 | 442 |
text {* |
20543 | 443 |
The theory @{text "Pure"} contains constant declarations for the |
444 |
primitive connectives @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"} of |
|
445 |
the logical framework, see \figref{fig:pure-connectives}. The |
|
446 |
derivability judgment @{text "A\<^isub>1, \<dots>, A\<^isub>n \<turnstile> B"} is |
|
447 |
defined inductively by the primitive inferences given in |
|
448 |
\figref{fig:prim-rules}, with the global restriction that the |
|
449 |
hypotheses must \emph{not} contain any schematic variables. The |
|
450 |
builtin equality is conceptually axiomatized as shown in |
|
20521 | 451 |
\figref{fig:pure-equality}, although the implementation works |
20543 | 452 |
directly with derived inferences. |
20521 | 453 |
|
454 |
\begin{figure}[htb] |
|
455 |
\begin{center} |
|
20501 | 456 |
\begin{tabular}{ll} |
457 |
@{text "all :: (\<alpha> \<Rightarrow> prop) \<Rightarrow> prop"} & universal quantification (binder @{text "\<And>"}) \\ |
|
458 |
@{text "\<Longrightarrow> :: prop \<Rightarrow> prop \<Rightarrow> prop"} & implication (right associative infix) \\ |
|
20521 | 459 |
@{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} & equality relation (infix) \\ |
20501 | 460 |
\end{tabular} |
20537 | 461 |
\caption{Primitive connectives of Pure}\label{fig:pure-connectives} |
20521 | 462 |
\end{center} |
463 |
\end{figure} |
|
18537 | 464 |
|
20501 | 465 |
\begin{figure}[htb] |
466 |
\begin{center} |
|
20498 | 467 |
\[ |
468 |
\infer[@{text "(axiom)"}]{@{text "\<turnstile> A"}}{@{text "A \<in> \<Theta>"}} |
|
469 |
\qquad |
|
470 |
\infer[@{text "(assume)"}]{@{text "A \<turnstile> A"}}{} |
|
471 |
\] |
|
472 |
\[ |
|
20537 | 473 |
\infer[@{text "(\<And>_intro)"}]{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}}{@{text "\<Gamma> \<turnstile> b[x]"} & @{text "x \<notin> \<Gamma>"}} |
20498 | 474 |
\qquad |
20537 | 475 |
\infer[@{text "(\<And>_elim)"}]{@{text "\<Gamma> \<turnstile> b[a]"}}{@{text "\<Gamma> \<turnstile> \<And>x. b[x]"}} |
20498 | 476 |
\] |
477 |
\[ |
|
478 |
\infer[@{text "(\<Longrightarrow>_intro)"}]{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}} |
|
479 |
\qquad |
|
480 |
\infer[@{text "(\<Longrightarrow>_elim)"}]{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}} |
|
481 |
\] |
|
20521 | 482 |
\caption{Primitive inferences of Pure}\label{fig:prim-rules} |
483 |
\end{center} |
|
484 |
\end{figure} |
|
485 |
||
486 |
\begin{figure}[htb] |
|
487 |
\begin{center} |
|
488 |
\begin{tabular}{ll} |
|
20537 | 489 |
@{text "\<turnstile> (\<lambda>x. b[x]) a \<equiv> b[a]"} & @{text "\<beta>"}-conversion \\ |
20521 | 490 |
@{text "\<turnstile> x \<equiv> x"} & reflexivity \\ |
491 |
@{text "\<turnstile> x \<equiv> y \<Longrightarrow> P x \<Longrightarrow> P y"} & substitution \\ |
|
492 |
@{text "\<turnstile> (\<And>x. f x \<equiv> g x) \<Longrightarrow> f \<equiv> g"} & extensionality \\ |
|
20537 | 493 |
@{text "\<turnstile> (A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A \<equiv> B"} & logical equivalence \\ |
20521 | 494 |
\end{tabular} |
20542 | 495 |
\caption{Conceptual axiomatization of Pure equality}\label{fig:pure-equality} |
20501 | 496 |
\end{center} |
497 |
\end{figure} |
|
18537 | 498 |
|
20501 | 499 |
The introduction and elimination rules for @{text "\<And>"} and @{text |
20537 | 500 |
"\<Longrightarrow>"} are analogous to formation of dependently typed @{text |
20501 | 501 |
"\<lambda>"}-terms representing the underlying proof objects. Proof terms |
20543 | 502 |
are irrelevant in the Pure logic, though; they cannot occur within |
503 |
propositions. The system provides a runtime option to record |
|
20537 | 504 |
explicit proof terms for primitive inferences. Thus all three |
505 |
levels of @{text "\<lambda>"}-calculus become explicit: @{text "\<Rightarrow>"} for |
|
506 |
terms, and @{text "\<And>/\<Longrightarrow>"} for proofs (cf.\ |
|
507 |
\cite{Berghofer-Nipkow:2000:TPHOL}). |
|
20491 | 508 |
|
20537 | 509 |
Observe that locally fixed parameters (as in @{text "\<And>_intro"}) need |
510 |
not be recorded in the hypotheses, because the simple syntactic |
|
20543 | 511 |
types of Pure are always inhabitable. ``Assumptions'' @{text "x :: |
512 |
\<tau>"} for type-membership are only present as long as some @{text |
|
513 |
"x\<^isub>\<tau>"} occurs in the statement body.\footnote{This is the key |
|
514 |
difference to ``@{text "\<lambda>HOL"}'' in the PTS framework |
|
515 |
\cite{Barendregt-Geuvers:2001}, where hypotheses @{text "x : A"} are |
|
516 |
treated uniformly for propositions and types.} |
|
20501 | 517 |
|
518 |
\medskip The axiomatization of a theory is implicitly closed by |
|
20537 | 519 |
forming all instances of type and term variables: @{text "\<turnstile> |
520 |
A\<vartheta>"} holds for any substitution instance of an axiom |
|
20543 | 521 |
@{text "\<turnstile> A"}. By pushing substitutions through derivations |
522 |
inductively, we also get admissible @{text "generalize"} and @{text |
|
523 |
"instance"} rules as shown in \figref{fig:subst-rules}. |
|
20501 | 524 |
|
525 |
\begin{figure}[htb] |
|
526 |
\begin{center} |
|
20498 | 527 |
\[ |
20501 | 528 |
\infer{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}}{@{text "\<Gamma> \<turnstile> B[\<alpha>]"} & @{text "\<alpha> \<notin> \<Gamma>"}} |
529 |
\quad |
|
530 |
\infer[\quad@{text "(generalize)"}]{@{text "\<Gamma> \<turnstile> B[?x]"}}{@{text "\<Gamma> \<turnstile> B[x]"} & @{text "x \<notin> \<Gamma>"}} |
|
20498 | 531 |
\] |
532 |
\[ |
|
20501 | 533 |
\infer{@{text "\<Gamma> \<turnstile> B[\<tau>]"}}{@{text "\<Gamma> \<turnstile> B[?\<alpha>]"}} |
534 |
\quad |
|
535 |
\infer[\quad@{text "(instantiate)"}]{@{text "\<Gamma> \<turnstile> B[t]"}}{@{text "\<Gamma> \<turnstile> B[?x]"}} |
|
20498 | 536 |
\] |
20501 | 537 |
\caption{Admissible substitution rules}\label{fig:subst-rules} |
538 |
\end{center} |
|
539 |
\end{figure} |
|
18537 | 540 |
|
20537 | 541 |
Note that @{text "instantiate"} does not require an explicit |
542 |
side-condition, because @{text "\<Gamma>"} may never contain schematic |
|
543 |
variables. |
|
544 |
||
545 |
In principle, variables could be substituted in hypotheses as well, |
|
20543 | 546 |
but this would disrupt the monotonicity of reasoning: deriving |
547 |
@{text "\<Gamma>\<vartheta> \<turnstile> B\<vartheta>"} from @{text "\<Gamma> \<turnstile> B"} is |
|
548 |
correct, but @{text "\<Gamma>\<vartheta> \<supseteq> \<Gamma>"} does not necessarily hold: |
|
549 |
the result belongs to a different proof context. |
|
20542 | 550 |
|
20543 | 551 |
\medskip An \emph{oracle} is a function that produces axioms on the |
552 |
fly. Logically, this is an instance of the @{text "axiom"} rule |
|
553 |
(\figref{fig:prim-rules}), but there is an operational difference. |
|
554 |
The system always records oracle invocations within derivations of |
|
555 |
theorems. Tracing plain axioms (and named theorems) is optional. |
|
20542 | 556 |
|
557 |
Axiomatizations should be limited to the bare minimum, typically as |
|
558 |
part of the initial logical basis of an object-logic formalization. |
|
20543 | 559 |
Later on, theories are usually developed in a strictly definitional |
560 |
fashion, by stating only certain equalities over new constants. |
|
20542 | 561 |
|
562 |
A \emph{simple definition} consists of a constant declaration @{text |
|
20543 | 563 |
"c :: \<sigma>"} together with an axiom @{text "\<turnstile> c \<equiv> t"}, where @{text "t |
564 |
:: \<sigma>"} is a closed term without any hidden polymorphism. The RHS |
|
565 |
may depend on further defined constants, but not @{text "c"} itself. |
|
566 |
Definitions of functions may be presented as @{text "c \<^vec>x \<equiv> |
|
567 |
t"} instead of the puristic @{text "c \<equiv> \<lambda>\<^vec>x. t"}. |
|
20542 | 568 |
|
20543 | 569 |
An \emph{overloaded definition} consists of a collection of axioms |
570 |
for the same constant, with zero or one equations @{text |
|
571 |
"c((\<^vec>\<alpha>)\<kappa>) \<equiv> t"} for each type constructor @{text "\<kappa>"} (for |
|
572 |
distinct variables @{text "\<^vec>\<alpha>"}). The RHS may mention |
|
573 |
previously defined constants as above, or arbitrary constants @{text |
|
574 |
"d(\<alpha>\<^isub>i)"} for some @{text "\<alpha>\<^isub>i"} projected from @{text |
|
575 |
"\<^vec>\<alpha>"}. Thus overloaded definitions essentially work by |
|
576 |
primitive recursion over the syntactic structure of a single type |
|
577 |
argument. |
|
20521 | 578 |
*} |
20498 | 579 |
|
20521 | 580 |
text %mlref {* |
581 |
\begin{mldecls} |
|
582 |
@{index_ML_type ctyp} \\ |
|
583 |
@{index_ML_type cterm} \\ |
|
20547 | 584 |
@{index_ML Thm.ctyp_of: "theory -> typ -> ctyp"} \\ |
585 |
@{index_ML Thm.cterm_of: "theory -> term -> cterm"} \\ |
|
586 |
\end{mldecls} |
|
587 |
\begin{mldecls} |
|
20521 | 588 |
@{index_ML_type thm} \\ |
20542 | 589 |
@{index_ML proofs: "int ref"} \\ |
590 |
@{index_ML Thm.assume: "cterm -> thm"} \\ |
|
591 |
@{index_ML Thm.forall_intr: "cterm -> thm -> thm"} \\ |
|
592 |
@{index_ML Thm.forall_elim: "cterm -> thm -> thm"} \\ |
|
593 |
@{index_ML Thm.implies_intr: "cterm -> thm -> thm"} \\ |
|
594 |
@{index_ML Thm.implies_elim: "thm -> thm -> thm"} \\ |
|
595 |
@{index_ML Thm.generalize: "string list * string list -> int -> thm -> thm"} \\ |
|
596 |
@{index_ML Thm.instantiate: "(ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm"} \\ |
|
597 |
@{index_ML Thm.get_axiom_i: "theory -> string -> thm"} \\ |
|
598 |
@{index_ML Thm.invoke_oracle_i: "theory -> string -> theory * Object.T -> thm"} \\ |
|
20547 | 599 |
\end{mldecls} |
600 |
\begin{mldecls} |
|
20542 | 601 |
@{index_ML Theory.add_axioms_i: "(string * term) list -> theory -> theory"} \\ |
602 |
@{index_ML Theory.add_deps: "string -> string * typ -> (string * typ) list -> theory -> theory"} \\ |
|
603 |
@{index_ML Theory.add_oracle: "string * (theory * Object.T -> term) -> theory -> theory"} \\ |
|
604 |
@{index_ML Theory.add_defs_i: "bool -> bool -> (bstring * term) list -> theory -> theory"} \\ |
|
20521 | 605 |
\end{mldecls} |
606 |
||
607 |
\begin{description} |
|
608 |
||
20542 | 609 |
\item @{ML_type ctyp} and @{ML_type cterm} represent certified types |
610 |
and terms, respectively. These are abstract datatypes that |
|
611 |
guarantee that its values have passed the full well-formedness (and |
|
612 |
well-typedness) checks, relative to the declarations of type |
|
613 |
constructors, constants etc. in the theory. |
|
614 |
||
20547 | 615 |
\item @{ML ctyp_of}~@{text "thy \<tau>"} and @{ML cterm_of}~@{text "thy |
616 |
t"} explicitly checks types and terms, respectively. This also |
|
617 |
involves some basic normalizations, such expansion of type and term |
|
618 |
abbreviations from the theory context. |
|
619 |
||
620 |
Re-certification is relatively slow and should be avoided in tight |
|
621 |
reasoning loops. There are separate operations to decompose |
|
622 |
certified entities (including actual theorems). |
|
20542 | 623 |
|
624 |
\item @{ML_type thm} represents proven propositions. This is an |
|
625 |
abstract datatype that guarantees that its values have been |
|
626 |
constructed by basic principles of the @{ML_struct Thm} module. |
|
20543 | 627 |
Every @{ML thm} value contains a sliding back-reference to the |
628 |
enclosing theory, cf.\ \secref{sec:context-theory}. |
|
20542 | 629 |
|
20543 | 630 |
\item @{ML proofs} determines the detail of proof recording within |
631 |
@{ML_type thm} values: @{ML 0} records only oracles, @{ML 1} records |
|
632 |
oracles, axioms and named theorems, @{ML 2} records full proof |
|
633 |
terms. |
|
20542 | 634 |
|
635 |
\item @{ML Thm.assume}, @{ML Thm.forall_intr}, @{ML |
|
636 |
Thm.forall_elim}, @{ML Thm.implies_intr}, and @{ML Thm.implies_elim} |
|
637 |
correspond to the primitive inferences of \figref{fig:prim-rules}. |
|
638 |
||
639 |
\item @{ML Thm.generalize}~@{text "(\<^vec>\<alpha>, \<^vec>x)"} |
|
640 |
corresponds to the @{text "generalize"} rules of |
|
20543 | 641 |
\figref{fig:subst-rules}. Here collections of type and term |
642 |
variables are generalized simultaneously, specified by the given |
|
643 |
basic names. |
|
20521 | 644 |
|
20542 | 645 |
\item @{ML Thm.instantiate}~@{text "(\<^vec>\<alpha>\<^isub>s, |
646 |
\<^vec>x\<^isub>\<tau>)"} corresponds to the @{text "instantiate"} rules |
|
647 |
of \figref{fig:subst-rules}. Type variables are substituted before |
|
648 |
term variables. Note that the types in @{text "\<^vec>x\<^isub>\<tau>"} |
|
649 |
refer to the instantiated versions. |
|
650 |
||
651 |
\item @{ML Thm.get_axiom_i}~@{text "thy name"} retrieves a named |
|
652 |
axiom, cf.\ @{text "axiom"} in \figref{fig:prim-rules}. |
|
653 |
||
20543 | 654 |
\item @{ML Thm.invoke_oracle_i}~@{text "thy name arg"} invokes a |
655 |
named oracle function, cf.\ @{text "axiom"} in |
|
656 |
\figref{fig:prim-rules}. |
|
20521 | 657 |
|
20543 | 658 |
\item @{ML Theory.add_axioms_i}~@{text "[(name, A), \<dots>]"} declares |
659 |
arbitrary propositions as axioms. |
|
20542 | 660 |
|
20543 | 661 |
\item @{ML Theory.add_oracle}~@{text "(name, f)"} declares an oracle |
662 |
function for generating arbitrary axioms on the fly. |
|
20542 | 663 |
|
664 |
\item @{ML Theory.add_deps}~@{text "name c\<^isub>\<tau> |
|
20543 | 665 |
\<^vec>d\<^isub>\<sigma>"} declares dependencies of a named specification |
666 |
for constant @{text "c\<^isub>\<tau>"}, relative to existing |
|
667 |
specifications for constants @{text "\<^vec>d\<^isub>\<sigma>"}. |
|
20542 | 668 |
|
669 |
\item @{ML Theory.add_defs_i}~@{text "unchecked overloaded [(name, c |
|
20543 | 670 |
\<^vec>x \<equiv> t), \<dots>]"} states a definitional axiom for an existing |
671 |
constant @{text "c"}. Dependencies are recorded (cf.\ @{ML |
|
672 |
Theory.add_deps}), unless the @{text "unchecked"} option is set. |
|
20521 | 673 |
|
674 |
\end{description} |
|
675 |
*} |
|
676 |
||
677 |
||
20543 | 678 |
subsection {* Auxiliary definitions *} |
20521 | 679 |
|
680 |
text {* |
|
20543 | 681 |
Theory @{text "Pure"} provides a few auxiliary definitions, see |
682 |
\figref{fig:pure-aux}. These special constants are normally not |
|
683 |
exposed to the user, but appear in internal encodings. |
|
20501 | 684 |
|
685 |
\begin{figure}[htb] |
|
686 |
\begin{center} |
|
20498 | 687 |
\begin{tabular}{ll} |
20521 | 688 |
@{text "conjunction :: prop \<Rightarrow> prop \<Rightarrow> prop"} & (infix @{text "&"}) \\ |
689 |
@{text "\<turnstile> A & B \<equiv> (\<And>C. (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C)"} \\[1ex] |
|
20543 | 690 |
@{text "prop :: prop \<Rightarrow> prop"} & (prefix @{text "#"}, suppressed) \\ |
20521 | 691 |
@{text "#A \<equiv> A"} \\[1ex] |
692 |
@{text "term :: \<alpha> \<Rightarrow> prop"} & (prefix @{text "TERM"}) \\ |
|
693 |
@{text "term x \<equiv> (\<And>A. A \<Longrightarrow> A)"} \\[1ex] |
|
694 |
@{text "TYPE :: \<alpha> itself"} & (prefix @{text "TYPE"}) \\ |
|
695 |
@{text "(unspecified)"} \\ |
|
20498 | 696 |
\end{tabular} |
20521 | 697 |
\caption{Definitions of auxiliary connectives}\label{fig:pure-aux} |
20501 | 698 |
\end{center} |
699 |
\end{figure} |
|
700 |
||
20537 | 701 |
Derived conjunction rules include introduction @{text "A \<Longrightarrow> B \<Longrightarrow> A & |
702 |
B"}, and destructions @{text "A & B \<Longrightarrow> A"} and @{text "A & B \<Longrightarrow> B"}. |
|
703 |
Conjunction allows to treat simultaneous assumptions and conclusions |
|
704 |
uniformly. For example, multiple claims are intermediately |
|
20543 | 705 |
represented as explicit conjunction, but this is refined into |
706 |
separate sub-goals before the user continues the proof; the final |
|
707 |
result is projected into a list of theorems (cf.\ |
|
20537 | 708 |
\secref{sec:tactical-goals}). |
20498 | 709 |
|
20537 | 710 |
The @{text "prop"} marker (@{text "#"}) makes arbitrarily complex |
711 |
propositions appear as atomic, without changing the meaning: @{text |
|
712 |
"\<Gamma> \<turnstile> A"} and @{text "\<Gamma> \<turnstile> #A"} are interchangeable. See |
|
713 |
\secref{sec:tactical-goals} for specific operations. |
|
20521 | 714 |
|
20543 | 715 |
The @{text "term"} marker turns any well-typed term into a derivable |
716 |
proposition: @{text "\<turnstile> TERM t"} holds unconditionally. Although |
|
717 |
this is logically vacuous, it allows to treat terms and proofs |
|
718 |
uniformly, similar to a type-theoretic framework. |
|
20498 | 719 |
|
20537 | 720 |
The @{text "TYPE"} constructor is the canonical representative of |
721 |
the unspecified type @{text "\<alpha> itself"}; it essentially injects the |
|
722 |
language of types into that of terms. There is specific notation |
|
723 |
@{text "TYPE(\<tau>)"} for @{text "TYPE\<^bsub>\<tau> |
|
20521 | 724 |
itself\<^esub>"}. |
20537 | 725 |
Although being devoid of any particular meaning, the @{text |
726 |
"TYPE(\<tau>)"} accounts for the type @{text "\<tau>"} within the term |
|
727 |
language. In particular, @{text "TYPE(\<alpha>)"} may be used as formal |
|
728 |
argument in primitive definitions, in order to circumvent hidden |
|
729 |
polymorphism (cf.\ \secref{sec:terms}). For example, @{text "c |
|
730 |
TYPE(\<alpha>) \<equiv> A[\<alpha>]"} defines @{text "c :: \<alpha> itself \<Rightarrow> prop"} in terms of |
|
731 |
a proposition @{text "A"} that depends on an additional type |
|
732 |
argument, which is essentially a predicate on types. |
|
20521 | 733 |
*} |
20501 | 734 |
|
20521 | 735 |
text %mlref {* |
736 |
\begin{mldecls} |
|
737 |
@{index_ML Conjunction.intr: "thm -> thm -> thm"} \\ |
|
738 |
@{index_ML Conjunction.elim: "thm -> thm * thm"} \\ |
|
739 |
@{index_ML Drule.mk_term: "cterm -> thm"} \\ |
|
740 |
@{index_ML Drule.dest_term: "thm -> cterm"} \\ |
|
741 |
@{index_ML Logic.mk_type: "typ -> term"} \\ |
|
742 |
@{index_ML Logic.dest_type: "term -> typ"} \\ |
|
743 |
\end{mldecls} |
|
744 |
||
745 |
\begin{description} |
|
746 |
||
20542 | 747 |
\item @{ML Conjunction.intr} derives @{text "A & B"} from @{text |
748 |
"A"} and @{text "B"}. |
|
749 |
||
20543 | 750 |
\item @{ML Conjunction.elim} derives @{text "A"} and @{text "B"} |
20542 | 751 |
from @{text "A & B"}. |
752 |
||
20543 | 753 |
\item @{ML Drule.mk_term} derives @{text "TERM t"}. |
20542 | 754 |
|
20543 | 755 |
\item @{ML Drule.dest_term} recovers term @{text "t"} from @{text |
756 |
"TERM t"}. |
|
20542 | 757 |
|
758 |
\item @{ML Logic.mk_type}~@{text "\<tau>"} produces the term @{text |
|
759 |
"TYPE(\<tau>)"}. |
|
760 |
||
761 |
\item @{ML Logic.dest_type}~@{text "TYPE(\<tau>)"} recovers the type |
|
762 |
@{text "\<tau>"}. |
|
20521 | 763 |
|
764 |
\end{description} |
|
20491 | 765 |
*} |
18537 | 766 |
|
20480 | 767 |
|
20491 | 768 |
section {* Rules \label{sec:rules} *} |
18537 | 769 |
|
20929 | 770 |
text %FIXME {* |
18537 | 771 |
|
772 |
FIXME |
|
773 |
||
20491 | 774 |
A \emph{rule} is any Pure theorem in HHF normal form; there is a |
775 |
separate calculus for rule composition, which is modeled after |
|
776 |
Gentzen's Natural Deduction \cite{Gentzen:1935}, but allows |
|
777 |
rules to be nested arbitrarily, similar to \cite{extensions91}. |
|
778 |
||
779 |
Normally, all theorems accessible to the user are proper rules. |
|
780 |
Low-level inferences are occasional required internally, but the |
|
781 |
result should be always presented in canonical form. The higher |
|
782 |
interfaces of Isabelle/Isar will always produce proper rules. It is |
|
783 |
important to maintain this invariant in add-on applications! |
|
784 |
||
785 |
There are two main principles of rule composition: @{text |
|
786 |
"resolution"} (i.e.\ backchaining of rules) and @{text |
|
787 |
"by-assumption"} (i.e.\ closing a branch); both principles are |
|
20519 | 788 |
combined in the variants of @{text "elim-resolution"} and @{text |
20491 | 789 |
"dest-resolution"}. Raw @{text "composition"} is occasionally |
790 |
useful as well, also it is strictly speaking outside of the proper |
|
791 |
rule calculus. |
|
792 |
||
793 |
Rules are treated modulo general higher-order unification, which is |
|
794 |
unification modulo the equational theory of @{text "\<alpha>\<beta>\<eta>"}-conversion |
|
795 |
on @{text "\<lambda>"}-terms. Moreover, propositions are understood modulo |
|
796 |
the (derived) equivalence @{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}. |
|
797 |
||
798 |
This means that any operations within the rule calculus may be |
|
799 |
subject to spontaneous @{text "\<alpha>\<beta>\<eta>"}-HHF conversions. It is common |
|
800 |
practice not to contract or expand unnecessarily. Some mechanisms |
|
801 |
prefer an one form, others the opposite, so there is a potential |
|
802 |
danger to produce some oscillation! |
|
803 |
||
804 |
Only few operations really work \emph{modulo} HHF conversion, but |
|
805 |
expect a normal form: quantifiers @{text "\<And>"} before implications |
|
806 |
@{text "\<Longrightarrow>"} at each level of nesting. |
|
807 |
||
18537 | 808 |
\glossary{Hereditary Harrop Formula}{The set of propositions in HHF |
809 |
format is defined inductively as @{text "H = (\<And>x\<^sup>*. H\<^sup>* \<Longrightarrow> |
|
810 |
A)"}, for variables @{text "x"} and atomic propositions @{text "A"}. |
|
811 |
Any proposition may be put into HHF form by normalizing with the rule |
|
812 |
@{text "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}. In Isabelle, the outermost |
|
813 |
quantifier prefix is represented via \seeglossary{schematic |
|
814 |
variables}, such that the top-level structure is merely that of a |
|
815 |
\seeglossary{Horn Clause}}. |
|
816 |
||
817 |
\glossary{HHF}{See \seeglossary{Hereditary Harrop Formula}.} |
|
818 |
||
20498 | 819 |
|
820 |
\[ |
|
821 |
\infer[@{text "(assumption)"}]{@{text "C\<vartheta>"}} |
|
822 |
{@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text i})}} |
|
823 |
\] |
|
824 |
||
825 |
||
826 |
\[ |
|
827 |
\infer[@{text "(compose)"}]{@{text "\<^vec>A\<vartheta> \<Longrightarrow> C\<vartheta>"}} |
|
828 |
{@{text "\<^vec>A \<Longrightarrow> B"} & @{text "B' \<Longrightarrow> C"} & @{text "B\<vartheta> = B'\<vartheta>"}} |
|
829 |
\] |
|
830 |
||
831 |
||
832 |
\[ |
|
833 |
\infer[@{text "(\<And>_lift)"}]{@{text "(\<And>\<^vec>x. \<^vec>A (?\<^vec>a \<^vec>x)) \<Longrightarrow> (\<And>\<^vec>x. B (?\<^vec>a \<^vec>x))"}}{@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"}} |
|
834 |
\] |
|
835 |
\[ |
|
836 |
\infer[@{text "(\<Longrightarrow>_lift)"}]{@{text "(\<^vec>H \<Longrightarrow> \<^vec>A) \<Longrightarrow> (\<^vec>H \<Longrightarrow> B)"}}{@{text "\<^vec>A \<Longrightarrow> B"}} |
|
837 |
\] |
|
838 |
||
839 |
The @{text resolve} scheme is now acquired from @{text "\<And>_lift"}, |
|
840 |
@{text "\<Longrightarrow>_lift"}, and @{text compose}. |
|
841 |
||
842 |
\[ |
|
843 |
\infer[@{text "(resolution)"}] |
|
844 |
{@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (?\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}} |
|
845 |
{\begin{tabular}{l} |
|
846 |
@{text "\<^vec>A ?\<^vec>a \<Longrightarrow> B ?\<^vec>a"} \\ |
|
847 |
@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\ |
|
848 |
@{text "(\<lambda>\<^vec>x. B (?\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\ |
|
849 |
\end{tabular}} |
|
850 |
\] |
|
851 |
||
852 |
||
853 |
FIXME @{text "elim_resolution"}, @{text "dest_resolution"} |
|
18537 | 854 |
*} |
855 |
||
20498 | 856 |
|
18537 | 857 |
end |