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doc-src/IsarRef/Thy/HOL_Specific.thy

author | wenzelm |

Mon, 15 Dec 2008 21:54:37 +0100 | |

changeset 29114 | 715178f6ae31 |

parent 29112 | f2b45eea6dac |

child 29560 | fa6c5d62adf5 |

permissions | -rw-r--r-- |

repaired railroad accident;

(* $Id$ *) theory HOL_Specific imports Main begin chapter {* Isabelle/HOL \label{ch:hol} *} section {* Primitive types \label{sec:hol-typedef} *} text {* \begin{matharray}{rcl} @{command_def (HOL) "typedecl"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "typedef"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ \end{matharray} \begin{rail} 'typedecl' typespec infix? ; 'typedef' altname? abstype '=' repset ; altname: '(' (name | 'open' | 'open' name) ')' ; abstype: typespec infix? ; repset: term ('morphisms' name name)? ; \end{rail} \begin{description} \item @{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} is similar to the original @{command "typedecl"} of Isabelle/Pure (see \secref{sec:types-pure}), but also declares type arity @{text "t :: (type, \<dots>, type) type"}, making @{text t} an actual HOL type constructor. %FIXME check, update \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} sets up a goal stating non-emptiness of the set @{text A}. After finishing the proof, the theory will be augmented by a Gordon/HOL-style type definition, which establishes a bijection between the representing set @{text A} and the new type @{text t}. Technically, @{command (HOL) "typedef"} defines both a type @{text t} and a set (term constant) of the same name (an alternative base name may be given in parentheses). The injection from type to set is called @{text Rep_t}, its inverse @{text Abs_t} (this may be changed via an explicit @{keyword (HOL) "morphisms"} declaration). Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text Abs_t_inverse} provide the most basic characterization as a corresponding injection/surjection pair (in both directions). Rules @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly more convenient view on the injectivity part, suitable for automated proof tools (e.g.\ in @{attribute simp} or @{attribute iff} declarations). Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views on surjectivity; these are already declared as set or type rules for the generic @{method cases} and @{method induct} methods. An alternative name may be specified in parentheses; the default is to use @{text t} as indicated before. The ``@{text "(open)"}'' declaration suppresses a separate constant definition for the representing set. \end{description} Note that raw type declarations are rarely used in practice; the main application is with experimental (or even axiomatic!) theory fragments. Instead of primitive HOL type definitions, user-level theories usually refer to higher-level packages such as @{command (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL) "datatype"} (see \secref{sec:hol-datatype}). *} section {* Adhoc tuples *} text {* \begin{matharray}{rcl} @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\ \end{matharray} \begin{rail} 'split\_format' (((name *) + 'and') | ('(' 'complete' ')')) ; \end{rail} \begin{description} \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots> \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into canonical form as specified by the arguments given; the @{text i}-th collection of arguments refers to occurrences in premise @{text i} of the rule. The ``@{text "(complete)"}'' option causes \emph{all} arguments in function applications to be represented canonically according to their tuple type structure. Note that these operations tend to invent funny names for new local parameters to be introduced. \end{description} *} section {* Records \label{sec:hol-record} *} text {* In principle, records merely generalize the concept of tuples, where components may be addressed by labels instead of just position. The logical infrastructure of records in Isabelle/HOL is slightly more advanced, though, supporting truly extensible record schemes. This admits operations that are polymorphic with respect to record extension, yielding ``object-oriented'' effects like (single) inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more details on object-oriented verification and record subtyping in HOL. *} subsection {* Basic concepts *} text {* Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records at the level of terms and types. The notation is as follows: \begin{center} \begin{tabular}{l|l|l} & record terms & record types \\ \hline fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\ schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} & @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\ \end{tabular} \end{center} \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text "(| x = a |)"}. A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value @{text a} and field @{text y} of value @{text b}. The corresponding type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"} and @{text "b :: B"}. A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields @{text x} and @{text y} as before, but also possibly further fields as indicated by the ``@{text "\<dots>"}'' notation (which is actually part of the syntax). The improper field ``@{text "\<dots>"}'' of a record scheme is called the \emph{more part}. Logically it is just a free variable, which is occasionally referred to as ``row variable'' in the literature. The more part of a record scheme may be instantiated by zero or more further components. For example, the previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z = c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part. Fixed records are special instances of record schemes, where ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"} element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}. \medskip Two key observations make extensible records in a simply typed language like HOL work out: \begin{enumerate} \item the more part is internalized, as a free term or type variable, \item field names are externalized, they cannot be accessed within the logic as first-class values. \end{enumerate} \medskip In Isabelle/HOL record types have to be defined explicitly, fixing their field names and types, and their (optional) parent record. Afterwards, records may be formed using above syntax, while obeying the canonical order of fields as given by their declaration. The record package provides several standard operations like selectors and updates. The common setup for various generic proof tools enable succinct reasoning patterns. See also the Isabelle/HOL tutorial \cite{isabelle-hol-book} for further instructions on using records in practice. *} subsection {* Record specifications *} text {* \begin{matharray}{rcl} @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\ \end{matharray} \begin{rail} 'record' typespec '=' (type '+')? (constdecl +) ; \end{rail} \begin{description} \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"}, derived from the optional parent record @{text "\<tau>"} by adding new field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc. The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text \<tau>} needs to specify an instance of an existing record type. At least one new field @{text "c\<^sub>i"} has to be specified. Basically, field names need to belong to a unique record. This is not a real restriction in practice, since fields are qualified by the record name internally. The parent record specification @{text \<tau>} is optional; if omitted @{text t} becomes a root record. The hierarchy of all records declared within a theory context forms a forest structure, i.e.\ a set of trees starting with a root record each. There is no way to merge multiple parent records! For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> :: \<zeta>\<rparr>"}. \end{description} *} subsection {* Record operations *} text {* Any record definition of the form presented above produces certain standard operations. Selectors and updates are provided for any field, including the improper one ``@{text more}''. There are also cumulative record constructor functions. To simplify the presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}. \medskip \textbf{Selectors} and \textbf{updates} are available for any field (including ``@{text more}''): \begin{matharray}{lll} @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ \end{matharray} There is special syntax for application of updates: @{text "r\<lparr>x := a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z := c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that because of postfix notation the order of fields shown here is reverse than in the actual term. Since repeated updates are just function applications, fields may be freely permuted in @{text "\<lparr>x := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned. Thus commutativity of independent updates can be proven within the logic for any two fields, but not as a general theorem. \medskip The \textbf{make} operation provides a cumulative record constructor function: \begin{matharray}{lll} @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ \end{matharray} \medskip We now reconsider the case of non-root records, which are derived of some parent. In general, the latter may depend on another parent as well, resulting in a list of \emph{ancestor records}. Appending the lists of fields of all ancestors results in a certain field prefix. The record package automatically takes care of this by lifting operations over this context of ancestor fields. Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"}, the above record operations will get the following types: \medskip \begin{tabular}{lll} @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ \end{tabular} \medskip \noindent Some further operations address the extension aspect of a derived record scheme specifically: @{text "t.fields"} produces a record fragment consisting of exactly the new fields introduced here (the result may serve as a more part elsewhere); @{text "t.extend"} takes a fixed record and adds a given more part; @{text "t.truncate"} restricts a record scheme to a fixed record. \medskip \begin{tabular}{lll} @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow> \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ @{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ \end{tabular} \medskip \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide for root records. *} subsection {* Derived rules and proof tools *} text {* The record package proves several results internally, declaring these facts to appropriate proof tools. This enables users to reason about record structures quite conveniently. Assume that @{text t} is a record type as specified above. \begin{enumerate} \item Standard conversions for selectors or updates applied to record constructor terms are made part of the default Simplifier context; thus proofs by reduction of basic operations merely require the @{method simp} method without further arguments. These rules are available as @{text "t.simps"}, too. \item Selectors applied to updated records are automatically reduced by an internal simplification procedure, which is also part of the standard Simplifier setup. \item Inject equations of a form analogous to @{prop "(x, y) = (x', y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical Reasoner as @{attribute iff} rules. These rules are available as @{text "t.iffs"}. \item The introduction rule for record equality analogous to @{text "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier, and as the basic rule context as ``@{attribute intro}@{text "?"}''. The rule is called @{text "t.equality"}. \item Representations of arbitrary record expressions as canonical constructor terms are provided both in @{method cases} and @{method induct} format (cf.\ the generic proof methods of the same name, \secref{sec:cases-induct}). Several variations are available, for fixed records, record schemes, more parts etc. The generic proof methods are sufficiently smart to pick the most sensible rule according to the type of the indicated record expression: users just need to apply something like ``@{text "(cases r)"}'' to a certain proof problem. \item The derived record operations @{text "t.make"}, @{text "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not} treated automatically, but usually need to be expanded by hand, using the collective fact @{text "t.defs"}. \end{enumerate} *} section {* Datatypes \label{sec:hol-datatype} *} text {* \begin{matharray}{rcl} @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ \end{matharray} \begin{rail} 'datatype' (dtspec + 'and') ; 'rep\_datatype' ('(' (name +) ')')? (term +) ; dtspec: parname? typespec infix? '=' (cons + '|') ; cons: name (type *) mixfix? \end{rail} \begin{description} \item @{command (HOL) "datatype"} defines inductive datatypes in HOL. \item @{command (HOL) "rep_datatype"} represents existing types as inductive ones, generating the standard infrastructure of derived concepts (primitive recursion etc.). \end{description} The induction and exhaustion theorems generated provide case names according to the constructors involved, while parameters are named after the types (see also \secref{sec:cases-induct}). See \cite{isabelle-HOL} for more details on datatypes, but beware of the old-style theory syntax being used there! Apart from proper proof methods for case-analysis and induction, there are also emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL) induct_tac} available, see \secref{sec:hol-induct-tac}; these admit to refer directly to the internal structure of subgoals (including internally bound parameters). *} section {* Recursive functions \label{sec:recursion} *} text {* \begin{matharray}{rcl} @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ \end{matharray} \begin{rail} 'primrec' target? fixes 'where' equations ; equations: (thmdecl? prop + '|') ; ('fun' | 'function') target? functionopts? fixes 'where' clauses ; clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|') ; functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')' ; 'termination' ( term )? \end{rail} \begin{description} \item @{command (HOL) "primrec"} defines primitive recursive functions over datatypes, see also \cite{isabelle-HOL}. \item @{command (HOL) "function"} defines functions by general wellfounded recursion. A detailed description with examples can be found in \cite{isabelle-function}. The function is specified by a set of (possibly conditional) recursive equations with arbitrary pattern matching. The command generates proof obligations for the completeness and the compatibility of patterns. The defined function is considered partial, and the resulting simplification rules (named @{text "f.psimps"}) and induction rule (named @{text "f.pinduct"}) are guarded by a generated domain predicate @{text "f_dom"}. The @{command (HOL) "termination"} command can then be used to establish that the function is total. \item @{command (HOL) "fun"} is a shorthand notation for ``@{command (HOL) "function"}~@{text "(sequential)"}, followed by automated proof attempts regarding pattern matching and termination. See \cite{isabelle-function} for further details. \item @{command (HOL) "termination"}~@{text f} commences a termination proof for the previously defined function @{text f}. If this is omitted, the command refers to the most recent function definition. After the proof is closed, the recursive equations and the induction principle is established. \end{description} %FIXME check Recursive definitions introduced by the @{command (HOL) "function"} command accommodate reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text "c.induct"} (where @{text c} is the name of the function definition) refers to a specific induction rule, with parameters named according to the user-specified equations. For the @{command (HOL) "primrec"} the induction principle coincides with structural recursion on the datatype the recursion is carried out. Case names of @{command (HOL) "primrec"} are that of the datatypes involved, while those of @{command (HOL) "function"} are numbered (starting from 1). The equations provided by these packages may be referred later as theorem list @{text "f.simps"}, where @{text f} is the (collective) name of the functions defined. Individual equations may be named explicitly as well. The @{command (HOL) "function"} command accepts the following options. \begin{description} \item @{text sequential} enables a preprocessor which disambiguates overlapping patterns by making them mutually disjoint. Earlier equations take precedence over later ones. This allows to give the specification in a format very similar to functional programming. Note that the resulting simplification and induction rules correspond to the transformed specification, not the one given originally. This usually means that each equation given by the user may result in several theroems. Also note that this automatic transformation only works for ML-style datatype patterns. \item @{text domintros} enables the automated generation of introduction rules for the domain predicate. While mostly not needed, they can be helpful in some proofs about partial functions. \item @{text tailrec} generates the unconstrained recursive equations even without a termination proof, provided that the function is tail-recursive. This currently only works \item @{text "default d"} allows to specify a default value for a (partial) function, which will ensure that @{text "f x = d x"} whenever @{text "x \<notin> f_dom"}. \end{description} *} subsection {* Proof methods related to recursive definitions *} text {* \begin{matharray}{rcl} @{method_def (HOL) pat_completeness} & : & @{text method} \\ @{method_def (HOL) relation} & : & @{text method} \\ @{method_def (HOL) lexicographic_order} & : & @{text method} \\ \end{matharray} \begin{rail} 'relation' term ; 'lexicographic\_order' (clasimpmod *) ; \end{rail} \begin{description} \item @{method (HOL) pat_completeness} is a specialized method to solve goals regarding the completeness of pattern matching, as required by the @{command (HOL) "function"} package (cf.\ \cite{isabelle-function}). \item @{method (HOL) relation}~@{text R} introduces a termination proof using the relation @{text R}. The resulting proof state will contain goals expressing that @{text R} is wellfounded, and that the arguments of recursive calls decrease with respect to @{text R}. Usually, this method is used as the initial proof step of manual termination proofs. \item @{method (HOL) "lexicographic_order"} attempts a fully automated termination proof by searching for a lexicographic combination of size measures on the arguments of the function. The method accepts the same arguments as the @{method auto} method, which it uses internally to prove local descents. The same context modifiers as for @{method auto} are accepted, see \secref{sec:clasimp}. In case of failure, extensive information is printed, which can help to analyse the situation (cf.\ \cite{isabelle-function}). \end{description} *} subsection {* Old-style recursive function definitions (TFL) *} text {* The old TFL commands @{command (HOL) "recdef"} and @{command (HOL) "recdef_tc"} for defining recursive are mostly obsolete; @{command (HOL) "function"} or @{command (HOL) "fun"} should be used instead. \begin{matharray}{rcl} @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\ @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ \end{matharray} \begin{rail} 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints? ; recdeftc thmdecl? tc ; hints: '(' 'hints' (recdefmod *) ')' ; recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod ; tc: nameref ('(' nat ')')? ; \end{rail} \begin{description} \item @{command (HOL) "recdef"} defines general well-founded recursive functions (using the TFL package), see also \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells TFL to recover from failed proof attempts, returning unfinished results. The @{text recdef_simp}, @{text recdef_cong}, and @{text recdef_wf} hints refer to auxiliary rules to be used in the internal automated proof process of TFL. Additional @{syntax clasimpmod} declarations (cf.\ \secref{sec:clasimp}) may be given to tune the context of the Simplifier (cf.\ \secref{sec:simplifier}) and Classical reasoner (cf.\ \secref{sec:classical}). \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the proof for leftover termination condition number @{text i} (default 1) as generated by a @{command (HOL) "recdef"} definition of constant @{text c}. Note that in most cases, @{command (HOL) "recdef"} is able to finish its internal proofs without manual intervention. \end{description} \medskip Hints for @{command (HOL) "recdef"} may be also declared globally, using the following attributes. \begin{matharray}{rcl} @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\ @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\ @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\ \end{matharray} \begin{rail} ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ; \end{rail} *} section {* Inductive and coinductive definitions \label{sec:hol-inductive} *} text {* An \textbf{inductive definition} specifies the least predicate (or set) @{text R} closed under given rules: applying a rule to elements of @{text R} yields a result within @{text R}. For example, a structural operational semantics is an inductive definition of an evaluation relation. Dually, a \textbf{coinductive definition} specifies the greatest predicate~/ set @{text R} that is consistent with given rules: every element of @{text R} can be seen as arising by applying a rule to elements of @{text R}. An important example is using bisimulation relations to formalise equivalence of processes and infinite data structures. \medskip The HOL package is related to the ZF one, which is described in a separate paper,\footnote{It appeared in CADE \cite{paulson-CADE}; a longer version is distributed with Isabelle.} which you should refer to in case of difficulties. The package is simpler than that of ZF thanks to implicit type-checking in HOL. The types of the (co)inductive predicates (or sets) determine the domain of the fixedpoint definition, and the package does not have to use inference rules for type-checking. \begin{matharray}{rcl} @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ @{attribute_def (HOL) mono} & : & @{text attribute} \\ \end{matharray} \begin{rail} ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\ ('where' clauses)? ('monos' thmrefs)? ; clauses: (thmdecl? prop + '|') ; 'mono' (() | 'add' | 'del') ; \end{rail} \begin{description} \item @{command (HOL) "inductive"} and @{command (HOL) "coinductive"} define (co)inductive predicates from the introduction rules given in the @{keyword "where"} part. The optional @{keyword "for"} part contains a list of parameters of the (co)inductive predicates that remain fixed throughout the definition. The optional @{keyword "monos"} section contains \emph{monotonicity theorems}, which are required for each operator applied to a recursive set in the introduction rules. There \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for each premise @{text "M R\<^sub>i t"} in an introduction rule! \item @{command (HOL) "inductive_set"} and @{command (HOL) "coinductive_set"} are wrappers for to the previous commands, allowing the definition of (co)inductive sets. \item @{attribute (HOL) mono} declares monotonicity rules. These rule are involved in the automated monotonicity proof of @{command (HOL) "inductive"}. \end{description} *} subsection {* Derived rules *} text {* Each (co)inductive definition @{text R} adds definitions to the theory and also proves some theorems: \begin{description} \item @{text R.intros} is the list of introduction rules as proven theorems, for the recursive predicates (or sets). The rules are also available individually, using the names given them in the theory file; \item @{text R.cases} is the case analysis (or elimination) rule; \item @{text R.induct} or @{text R.coinduct} is the (co)induction rule. \end{description} When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are defined simultaneously, the list of introduction rules is called @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list of mutual induction rules is called @{text "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}. *} subsection {* Monotonicity theorems *} text {* Each theory contains a default set of theorems that are used in monotonicity proofs. New rules can be added to this set via the @{attribute (HOL) mono} attribute. The HOL theory @{text Inductive} shows how this is done. In general, the following monotonicity theorems may be added: \begin{itemize} \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving monotonicity of inductive definitions whose introduction rules have premises involving terms such as @{text "M R\<^sub>i t"}. \item Monotonicity theorems for logical operators, which are of the general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in the case of the operator @{text "\<or>"}, the corresponding theorem is \[ \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}} \] \item De Morgan style equations for reasoning about the ``polarity'' of expressions, e.g. \[ @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"} \] \item Equations for reducing complex operators to more primitive ones whose monotonicity can easily be proved, e.g. \[ @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"} \] \end{itemize} %FIXME: Example of an inductive definition *} section {* Arithmetic proof support *} text {* \begin{matharray}{rcl} @{method_def (HOL) arith} & : & @{text method} \\ @{attribute_def (HOL) arith_split} & : & @{text attribute} \\ \end{matharray} The @{method (HOL) arith} method decides linear arithmetic problems (on types @{text nat}, @{text int}, @{text real}). Any current facts are inserted into the goal before running the procedure. The @{attribute (HOL) arith_split} attribute declares case split rules to be expanded before the arithmetic procedure is invoked. Note that a simpler (but faster) version of arithmetic reasoning is already performed by the Simplifier. *} section {* Invoking automated reasoning tools -- The Sledgehammer *} text {* Isabelle/HOL includes a generic \emph{ATP manager} that allows external automated reasoning tools to crunch a pending goal. Supported provers include E\footnote{\url{http://www.eprover.org}}, SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire. There is also a wrapper to invoke provers remotely via the SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}} web service. The problem passed to external provers consists of the goal together with a smart selection of lemmas from the current theory context. The result of a successful proof search is some source text that usually reconstructs the proof within Isabelle, without requiring external provers again. The Metis prover\footnote{\url{http://www.gilith.com/software/metis/}} that is integrated into Isabelle/HOL is being used here. In this mode of operation, heavy means of automated reasoning are used as a strong relevance filter, while the main proof checking works via explicit inferences going through the Isabelle kernel. Moreover, rechecking Isabelle proof texts with already specified auxiliary facts is much faster than performing fully automated search over and over again. \begin{matharray}{rcl} @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\ @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\ @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\ @{method_def (HOL) metis} & : & @{text method} \\ \end{matharray} \begin{rail} 'sledgehammer' (nameref *) ; 'atp\_messages' ('(' nat ')')? ; 'metis' thmrefs ; \end{rail} \begin{description} \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"} invokes the specified automated theorem provers on the first subgoal. Provers are run in parallel, the first successful result is displayed, and the other attempts are terminated. Provers are defined in the theory context, see also @{command (HOL) print_atps}. If no provers are given as arguments to @{command (HOL) sledgehammer}, the system refers to the default defined as ``ATP provers'' preference by the user interface. There are additional preferences for timeout (default: 60 seconds), and the maximum number of independent prover processes (default: 5); excessive provers are automatically terminated. \item @{command (HOL) print_atps} prints the list of automated theorem provers available to the @{command (HOL) sledgehammer} command. \item @{command (HOL) atp_info} prints information about presently running provers, including elapsed runtime, and the remaining time until timeout. \item @{command (HOL) atp_kill} terminates all presently running provers. \item @{command (HOL) atp_messages} displays recent messages issued by automated theorem provers. This allows to examine results that might have got lost due to the asynchronous nature of default @{command (HOL) sledgehammer} output. An optional message limit may be specified (default 5). \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover with the given facts. Metis is an automated proof tool of medium strength, but is fully integrated into Isabelle/HOL, with explicit inferences going through the kernel. Thus its results are guaranteed to be ``correct by construction''. Note that all facts used with Metis need to be specified as explicit arguments. There are no rule declarations as for other Isabelle provers, like @{method blast} or @{method fast}. \end{description} *} section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *} text {* The following tools of Isabelle/HOL support cases analysis and induction in unstructured tactic scripts; see also \secref{sec:cases-induct} for proper Isar versions of similar ideas. \begin{matharray}{rcl} @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\ @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\ @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\ @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ \end{matharray} \begin{rail} 'case\_tac' goalspec? term rule? ; 'induct\_tac' goalspec? (insts * 'and') rule? ; 'ind\_cases' (prop +) ('for' (name +)) ? ; 'inductive\_cases' (thmdecl? (prop +) + 'and') ; rule: ('rule' ':' thmref) ; \end{rail} \begin{description} \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit to reason about inductive types. Rules are selected according to the declarations by the @{attribute cases} and @{attribute induct} attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL) datatype} package already takes care of this. These unstructured tactics feature both goal addressing and dynamic instantiation. Note that named rule cases are \emph{not} provided as would be by the proper @{method cases} and @{method induct} proof methods (see \secref{sec:cases-induct}). Unlike the @{method induct} method, @{method induct_tac} does not handle structured rule statements, only the compact object-logic conclusion of the subgoal being addressed. \item @{method (HOL) ind_cases} and @{command (HOL) "inductive_cases"} provide an interface to the internal @{ML_text mk_cases} operation. Rules are simplified in an unrestricted forward manner. While @{method (HOL) ind_cases} is a proof method to apply the result immediately as elimination rules, @{command (HOL) "inductive_cases"} provides case split theorems at the theory level for later use. The @{keyword "for"} argument of the @{method (HOL) ind_cases} method allows to specify a list of variables that should be generalized before applying the resulting rule. \end{description} *} section {* Executable code *} text {* Isabelle/Pure provides two generic frameworks to support code generation from executable specifications. Isabelle/HOL instantiates these mechanisms in a way that is amenable to end-user applications. One framework generates code from both functional and relational programs to SML. See \cite{isabelle-HOL} for further information (this actually covers the new-style theory format as well). \begin{matharray}{rcl} @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\ @{attribute_def (HOL) code} & : & @{text attribute} \\ \end{matharray} \begin{rail} 'value' term ; ( 'code\_module' | 'code\_library' ) modespec ? name ? \\ ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\ 'contains' ( ( name '=' term ) + | term + ) ; modespec: '(' ( name * ) ')' ; 'consts\_code' (codespec +) ; codespec: const template attachment ? ; 'types\_code' (tycodespec +) ; tycodespec: name template attachment ? ; const: term ; template: '(' string ')' ; attachment: 'attach' modespec ? verblbrace text verbrbrace ; 'code' (name)? ; \end{rail} \begin{description} \item @{command (HOL) "value"}~@{text t} evaluates and prints a term using the code generator. \end{description} \medskip The other framework generates code from functional programs (including overloading using type classes) to SML \cite{SML}, OCaml \cite{OCaml} and Haskell \cite{haskell-revised-report}. Conceptually, code generation is split up in three steps: \emph{selection} of code theorems, \emph{translation} into an abstract executable view and \emph{serialization} to a specific \emph{target language}. See \cite{isabelle-codegen} for an introduction on how to use it. \begin{matharray}{rcl} @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\ @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ @{attribute_def (HOL) code} & : & @{text attribute} \\ \end{matharray} \begin{rail} 'export\_code' ( constexpr + ) ? \\ ( ( 'in' target ( 'module\_name' string ) ? \\ ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ? ; 'code\_thms' ( constexpr + ) ? ; 'code\_deps' ( constexpr + ) ? ; const: term ; constexpr: ( const | 'name.*' | '*' ) ; typeconstructor: nameref ; class: nameref ; target: 'OCaml' | 'SML' | 'Haskell' ; 'code\_datatype' const + ; 'code\_const' (const + 'and') \\ ( ( '(' target ( syntax ? + 'and' ) ')' ) + ) ; 'code\_type' (typeconstructor + 'and') \\ ( ( '(' target ( syntax ? + 'and' ) ')' ) + ) ; 'code\_class' (class + 'and') \\ ( ( '(' target \\ ( string ? + 'and' ) ')' ) + ) ; 'code\_instance' (( typeconstructor '::' class ) + 'and') \\ ( ( '(' target ( '-' ? + 'and' ) ')' ) + ) ; 'code\_monad' const const target ; 'code\_reserved' target ( string + ) ; 'code\_include' target ( string ( string | '-') ) ; 'code\_modulename' target ( ( string string ) + ) ; 'code\_abort' ( const + ) ; syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string ; 'code' ( 'inline' ) ? ( 'del' ) ? ; \end{rail} \begin{description} \item @{command (HOL) "export_code"} is the canonical interface for generating and serializing code: for a given list of constants, code is generated for the specified target languages. Abstract code is cached incrementally. If no constant is given, the currently cached code is serialized. If no serialization instruction is given, only abstract code is cached. Constants may be specified by giving them literally, referring to all executable contants within a certain theory by giving @{text "name.*"}, or referring to \emph{all} executable constants currently available by giving @{text "*"}. By default, for each involved theory one corresponding name space module is generated. Alternativly, a module name may be specified after the @{keyword "module_name"} keyword; then \emph{all} code is placed in this module. For \emph{SML} and \emph{OCaml}, the file specification refers to a single file; for \emph{Haskell}, it refers to a whole directory, where code is generated in multiple files reflecting the module hierarchy. The file specification ``@{text "-"}'' denotes standard output. For \emph{SML}, omitting the file specification compiles code internally in the context of the current ML session. Serializers take an optional list of arguments in parentheses. For \emph{Haskell} a module name prefix may be given using the ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate datatype declaration. \item @{command (HOL) "code_thms"} prints a list of theorems representing the corresponding program containing all given constants; if no constants are given, the currently cached code theorems are printed. \item @{command (HOL) "code_deps"} visualizes dependencies of theorems representing the corresponding program containing all given constants; if no constants are given, the currently cached code theorems are visualized. \item @{command (HOL) "code_datatype"} specifies a constructor set for a logical type. \item @{command (HOL) "code_const"} associates a list of constants with target-specific serializations; omitting a serialization deletes an existing serialization. \item @{command (HOL) "code_type"} associates a list of type constructors with target-specific serializations; omitting a serialization deletes an existing serialization. \item @{command (HOL) "code_class"} associates a list of classes with target-specific class names; omitting a serialization deletes an existing serialization. This applies only to \emph{Haskell}. \item @{command (HOL) "code_instance"} declares a list of type constructor / class instance relations as ``already present'' for a given target. Omitting a ``@{text "-"}'' deletes an existing ``already present'' declaration. This applies only to \emph{Haskell}. \item @{command (HOL) "code_monad"} provides an auxiliary mechanism to generate monadic code for Haskell. \item @{command (HOL) "code_reserved"} declares a list of names as reserved for a given target, preventing it to be shadowed by any generated code. \item @{command (HOL) "code_include"} adds arbitrary named content (``include'') to generated code. A ``@{text "-"}'' as last argument will remove an already added ``include''. \item @{command (HOL) "code_modulename"} declares aliasings from one module name onto another. \item @{command (HOL) "code_abort"} declares constants which are not required to have a definition by means of defining equations; if needed these are implemented by program abort instead. \item @{attribute (HOL) code} explicitly selects (or with option ``@{text "del"}'' deselects) a defining equation for code generation. Usually packages introducing defining equations provide a reasonable default setup for selection. \item @{attribute (HOL) code}~@{text inline} declares (or with option ``@{text "del"}'' removes) inlining theorems which are applied as rewrite rules to any defining equation during preprocessing. \item @{command (HOL) "print_codesetup"} gives an overview on selected defining equations, code generator datatypes and preprocessor setup. \end{description} *} section {* Definition by specification \label{sec:hol-specification} *} text {* \begin{matharray}{rcl} @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ \end{matharray} \begin{rail} ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +) ; decl: ((name ':')? term '(' 'overloaded' ')'?) \end{rail} \begin{description} \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a goal stating the existence of terms with the properties specified to hold for the constants given in @{text decls}. After finishing the proof, the theory will be augmented with definitions for the given constants, as well as with theorems stating the properties for these constants. \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up a goal stating the existence of terms with the properties specified to hold for the constants given in @{text decls}. After finishing the proof, the theory will be augmented with axioms expressing the properties given in the first place. \item @{text decl} declares a constant to be defined by the specification given. The definition for the constant @{text c} is bound to the name @{text c_def} unless a theorem name is given in the declaration. Overloaded constants should be declared as such. \end{description} Whether to use @{command (HOL) "specification"} or @{command (HOL) "ax_specification"} is to some extent a matter of style. @{command (HOL) "specification"} introduces no new axioms, and so by construction cannot introduce inconsistencies, whereas @{command (HOL) "ax_specification"} does introduce axioms, but only after the user has explicitly proven it to be safe. A practical issue must be considered, though: After introducing two constants with the same properties using @{command (HOL) "specification"}, one can prove that the two constants are, in fact, equal. If this might be a problem, one should use @{command (HOL) "ax_specification"}. *} end