author | haftmann |
Tue, 14 Jul 2009 10:54:04 +0200 | |
changeset 31998 | 2c7a24f74db9 |
parent 31912 | f5bd306f5e9d |
child 33857 | 0cb5002c52db |
permissions | -rw-r--r-- |
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theory HOL_Specific |
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imports Main |
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begin |
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chapter {* Isabelle/HOL \label{ch:hol} *} |
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section {* Primitive types \label{sec:hol-typedef} *} |
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text {* |
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\begin{matharray}{rcl} |
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@{command_def (HOL) "typedecl"} & : & @{text "theory \<rightarrow> theory"} \\ |
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@{command_def (HOL) "typedef"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
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\end{matharray} |
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\begin{rail} |
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'typedecl' typespec infix? |
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; |
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'typedef' altname? abstype '=' repset |
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; |
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altname: '(' (name | 'open' | 'open' name) ')' |
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; |
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abstype: typespec infix? |
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; |
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repset: term ('morphisms' name name)? |
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; |
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\end{rail} |
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\begin{description} |
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\item @{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} is similar |
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to the original @{command "typedecl"} of Isabelle/Pure (see |
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\secref{sec:types-pure}), but also declares type arity @{text "t :: |
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(type, \<dots>, type) type"}, making @{text t} an actual HOL type |
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constructor. %FIXME check, update |
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\item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} sets up |
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a goal stating non-emptiness of the set @{text A}. After finishing |
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the proof, the theory will be augmented by a Gordon/HOL-style type |
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definition, which establishes a bijection between the representing |
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set @{text A} and the new type @{text t}. |
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Technically, @{command (HOL) "typedef"} defines both a type @{text |
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t} and a set (term constant) of the same name (an alternative base |
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name may be given in parentheses). The injection from type to set |
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is called @{text Rep_t}, its inverse @{text Abs_t} (this may be |
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changed via an explicit @{keyword (HOL) "morphisms"} declaration). |
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Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text |
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Abs_t_inverse} provide the most basic characterization as a |
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corresponding injection/surjection pair (in both directions). Rules |
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@{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly |
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more convenient view on the injectivity part, suitable for automated |
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proof tools (e.g.\ in @{attribute simp} or @{attribute iff} |
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declarations). Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and |
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@{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views |
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on surjectivity; these are already declared as set or type rules for |
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the generic @{method cases} and @{method induct} methods. |
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An alternative name may be specified in parentheses; the default is |
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to use @{text t} as indicated before. The ``@{text "(open)"}'' |
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declaration suppresses a separate constant definition for the |
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representing set. |
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\end{description} |
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Note that raw type declarations are rarely used in practice; the |
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main application is with experimental (or even axiomatic!) theory |
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fragments. Instead of primitive HOL type definitions, user-level |
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theories usually refer to higher-level packages such as @{command |
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(HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL) |
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"datatype"} (see \secref{sec:hol-datatype}). |
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*} |
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section {* Adhoc tuples *} |
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text {* |
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\begin{matharray}{rcl} |
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@{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\ |
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\end{matharray} |
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\begin{rail} |
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'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')')) |
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; |
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\end{rail} |
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\begin{description} |
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\item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots> |
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\<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into |
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canonical form as specified by the arguments given; the @{text i}-th |
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collection of arguments refers to occurrences in premise @{text i} |
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of the rule. The ``@{text "(complete)"}'' option causes \emph{all} |
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arguments in function applications to be represented canonically |
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according to their tuple type structure. |
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Note that these operations tend to invent funny names for new local |
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parameters to be introduced. |
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\end{description} |
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*} |
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section {* Records \label{sec:hol-record} *} |
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text {* |
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In principle, records merely generalize the concept of tuples, where |
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components may be addressed by labels instead of just position. The |
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logical infrastructure of records in Isabelle/HOL is slightly more |
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advanced, though, supporting truly extensible record schemes. This |
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admits operations that are polymorphic with respect to record |
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extension, yielding ``object-oriented'' effects like (single) |
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inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more |
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details on object-oriented verification and record subtyping in HOL. |
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*} |
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subsection {* Basic concepts *} |
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text {* |
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Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records |
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at the level of terms and types. The notation is as follows: |
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\begin{center} |
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\begin{tabular}{l|l|l} |
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& record terms & record types \\ \hline |
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fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\ |
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schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} & |
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@{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\ |
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\end{tabular} |
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\end{center} |
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\noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text |
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"(| x = a |)"}. |
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A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value |
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@{text a} and field @{text y} of value @{text b}. The corresponding |
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type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"} |
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and @{text "b :: B"}. |
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A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields |
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@{text x} and @{text y} as before, but also possibly further fields |
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as indicated by the ``@{text "\<dots>"}'' notation (which is actually part |
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of the syntax). The improper field ``@{text "\<dots>"}'' of a record |
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scheme is called the \emph{more part}. Logically it is just a free |
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variable, which is occasionally referred to as ``row variable'' in |
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the literature. The more part of a record scheme may be |
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instantiated by zero or more further components. For example, the |
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previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z = |
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c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part. |
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Fixed records are special instances of record schemes, where |
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``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"} |
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element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation |
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for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}. |
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\medskip Two key observations make extensible records in a simply |
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typed language like HOL work out: |
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\begin{enumerate} |
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\item the more part is internalized, as a free term or type |
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variable, |
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\item field names are externalized, they cannot be accessed within |
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the logic as first-class values. |
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\end{enumerate} |
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\medskip In Isabelle/HOL record types have to be defined explicitly, |
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fixing their field names and types, and their (optional) parent |
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record. Afterwards, records may be formed using above syntax, while |
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obeying the canonical order of fields as given by their declaration. |
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The record package provides several standard operations like |
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selectors and updates. The common setup for various generic proof |
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tools enable succinct reasoning patterns. See also the Isabelle/HOL |
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tutorial \cite{isabelle-hol-book} for further instructions on using |
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records in practice. |
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*} |
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subsection {* Record specifications *} |
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text {* |
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\begin{matharray}{rcl} |
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@{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\ |
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\end{matharray} |
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\begin{rail} |
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'record' typespec '=' (type '+')? (constdecl +) |
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; |
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\end{rail} |
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\begin{description} |
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\item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 |
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\<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"}, |
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derived from the optional parent record @{text "\<tau>"} by adding new |
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field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc. |
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The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be |
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covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>, |
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\<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text |
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\<tau>} needs to specify an instance of an existing record type. At |
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least one new field @{text "c\<^sub>i"} has to be specified. |
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Basically, field names need to belong to a unique record. This is |
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not a real restriction in practice, since fields are qualified by |
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the record name internally. |
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The parent record specification @{text \<tau>} is optional; if omitted |
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@{text t} becomes a root record. The hierarchy of all records |
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declared within a theory context forms a forest structure, i.e.\ a |
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set of trees starting with a root record each. There is no way to |
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merge multiple parent records! |
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For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a |
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type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 :: |
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\<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text |
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"(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for |
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@{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> :: |
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\<zeta>\<rparr>"}. |
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\end{description} |
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*} |
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subsection {* Record operations *} |
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text {* |
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Any record definition of the form presented above produces certain |
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standard operations. Selectors and updates are provided for any |
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field, including the improper one ``@{text more}''. There are also |
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cumulative record constructor functions. To simplify the |
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presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>, |
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\<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 :: |
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\<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}. |
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\medskip \textbf{Selectors} and \textbf{updates} are available for |
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any field (including ``@{text more}''): |
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\begin{matharray}{lll} |
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@{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ |
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@{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>"} \\ |
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\end{matharray} |
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There is special syntax for application of updates: @{text "r\<lparr>x := |
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a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for |
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repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z := |
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c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that |
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because of postfix notation the order of fields shown here is |
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reverse than in the actual term. Since repeated updates are just |
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function applications, fields may be freely permuted in @{text "\<lparr>x |
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:= a, y := b, z := c\<rparr>"}, as far as logical equality is concerned. |
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Thus commutativity of independent updates can be proven within the |
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logic for any two fields, but not as a general theorem. |
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\medskip The \textbf{make} operation provides a cumulative record |
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constructor function: |
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\begin{matharray}{lll} |
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@{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
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\end{matharray} |
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\medskip We now reconsider the case of non-root records, which are |
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derived of some parent. In general, the latter may depend on |
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another parent as well, resulting in a list of \emph{ancestor |
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records}. Appending the lists of fields of all ancestors results in |
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a certain field prefix. The record package automatically takes care |
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of this by lifting operations over this context of ancestor fields. |
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Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor |
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fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"}, |
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the above record operations will get the following types: |
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\medskip |
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\begin{tabular}{lll} |
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@{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ |
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@{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> |
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\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> |
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\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
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@{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> |
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\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
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\end{tabular} |
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\medskip |
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\noindent Some further operations address the extension aspect of a |
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derived record scheme specifically: @{text "t.fields"} produces a |
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record fragment consisting of exactly the new fields introduced here |
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(the result may serve as a more part elsewhere); @{text "t.extend"} |
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takes a fixed record and adds a given more part; @{text |
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"t.truncate"} restricts a record scheme to a fixed record. |
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\medskip |
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\begin{tabular}{lll} |
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@{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
295 |
@{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow> |
|
296 |
\<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
|
297 |
@{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>"} \\ |
|
298 |
\end{tabular} |
|
299 |
\medskip |
|
26849 | 300 |
|
301 |
\noindent Note that @{text "t.make"} and @{text "t.fields"} coincide |
|
302 |
for root records. |
|
303 |
*} |
|
304 |
||
305 |
||
306 |
subsection {* Derived rules and proof tools *} |
|
307 |
||
308 |
text {* |
|
309 |
The record package proves several results internally, declaring |
|
310 |
these facts to appropriate proof tools. This enables users to |
|
311 |
reason about record structures quite conveniently. Assume that |
|
312 |
@{text t} is a record type as specified above. |
|
313 |
||
314 |
\begin{enumerate} |
|
315 |
||
316 |
\item Standard conversions for selectors or updates applied to |
|
317 |
record constructor terms are made part of the default Simplifier |
|
318 |
context; thus proofs by reduction of basic operations merely require |
|
319 |
the @{method simp} method without further arguments. These rules |
|
320 |
are available as @{text "t.simps"}, too. |
|
321 |
||
322 |
\item Selectors applied to updated records are automatically reduced |
|
323 |
by an internal simplification procedure, which is also part of the |
|
324 |
standard Simplifier setup. |
|
325 |
||
326 |
\item Inject equations of a form analogous to @{prop "(x, y) = (x', |
|
327 |
y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical |
|
328 |
Reasoner as @{attribute iff} rules. These rules are available as |
|
329 |
@{text "t.iffs"}. |
|
330 |
||
331 |
\item The introduction rule for record equality analogous to @{text |
|
332 |
"x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier, |
|
333 |
and as the basic rule context as ``@{attribute intro}@{text "?"}''. |
|
334 |
The rule is called @{text "t.equality"}. |
|
335 |
||
336 |
\item Representations of arbitrary record expressions as canonical |
|
337 |
constructor terms are provided both in @{method cases} and @{method |
|
338 |
induct} format (cf.\ the generic proof methods of the same name, |
|
339 |
\secref{sec:cases-induct}). Several variations are available, for |
|
340 |
fixed records, record schemes, more parts etc. |
|
341 |
||
342 |
The generic proof methods are sufficiently smart to pick the most |
|
343 |
sensible rule according to the type of the indicated record |
|
344 |
expression: users just need to apply something like ``@{text "(cases |
|
345 |
r)"}'' to a certain proof problem. |
|
346 |
||
347 |
\item The derived record operations @{text "t.make"}, @{text |
|
348 |
"t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not} |
|
349 |
treated automatically, but usually need to be expanded by hand, |
|
350 |
using the collective fact @{text "t.defs"}. |
|
351 |
||
352 |
\end{enumerate} |
|
353 |
*} |
|
354 |
||
355 |
||
356 |
section {* Datatypes \label{sec:hol-datatype} *} |
|
357 |
||
358 |
text {* |
|
359 |
\begin{matharray}{rcl} |
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@{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\ |
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@{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
26849 | 362 |
\end{matharray} |
363 |
||
364 |
\begin{rail} |
|
365 |
'datatype' (dtspec + 'and') |
|
366 |
; |
|
27452 | 367 |
'rep\_datatype' ('(' (name +) ')')? (term +) |
26849 | 368 |
; |
369 |
||
370 |
dtspec: parname? typespec infix? '=' (cons + '|') |
|
371 |
; |
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cons: name ( type * ) mixfix? |
26849 | 373 |
\end{rail} |
374 |
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\begin{description} |
26849 | 376 |
|
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\item @{command (HOL) "datatype"} defines inductive datatypes in |
26849 | 378 |
HOL. |
379 |
||
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\item @{command (HOL) "rep_datatype"} represents existing types as |
26849 | 381 |
inductive ones, generating the standard infrastructure of derived |
382 |
concepts (primitive recursion etc.). |
|
383 |
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\end{description} |
26849 | 385 |
|
386 |
The induction and exhaustion theorems generated provide case names |
|
387 |
according to the constructors involved, while parameters are named |
|
388 |
after the types (see also \secref{sec:cases-induct}). |
|
389 |
||
390 |
See \cite{isabelle-HOL} for more details on datatypes, but beware of |
|
391 |
the old-style theory syntax being used there! Apart from proper |
|
392 |
proof methods for case-analysis and induction, there are also |
|
393 |
emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL) |
|
394 |
induct_tac} available, see \secref{sec:hol-induct-tac}; these admit |
|
395 |
to refer directly to the internal structure of subgoals (including |
|
396 |
internally bound parameters). |
|
397 |
*} |
|
398 |
||
399 |
||
400 |
section {* Recursive functions \label{sec:recursion} *} |
|
401 |
||
402 |
text {* |
|
403 |
\begin{matharray}{rcl} |
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@{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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@{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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@{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
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@{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
26849 | 408 |
\end{matharray} |
409 |
||
410 |
\begin{rail} |
|
411 |
'primrec' target? fixes 'where' equations |
|
412 |
; |
|
413 |
equations: (thmdecl? prop + '|') |
|
414 |
; |
|
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('fun' | 'function') target? functionopts? fixes 'where' clauses |
26849 | 416 |
; |
417 |
clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|') |
|
418 |
; |
|
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functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')' |
26849 | 420 |
; |
421 |
'termination' ( term )? |
|
422 |
\end{rail} |
|
423 |
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\begin{description} |
26849 | 425 |
|
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\item @{command (HOL) "primrec"} defines primitive recursive |
26849 | 427 |
functions over datatypes, see also \cite{isabelle-HOL}. |
428 |
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\item @{command (HOL) "function"} defines functions by general |
26849 | 430 |
wellfounded recursion. A detailed description with examples can be |
431 |
found in \cite{isabelle-function}. The function is specified by a |
|
432 |
set of (possibly conditional) recursive equations with arbitrary |
|
433 |
pattern matching. The command generates proof obligations for the |
|
434 |
completeness and the compatibility of patterns. |
|
435 |
||
436 |
The defined function is considered partial, and the resulting |
|
437 |
simplification rules (named @{text "f.psimps"}) and induction rule |
|
438 |
(named @{text "f.pinduct"}) are guarded by a generated domain |
|
439 |
predicate @{text "f_dom"}. The @{command (HOL) "termination"} |
|
440 |
command can then be used to establish that the function is total. |
|
441 |
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\item @{command (HOL) "fun"} is a shorthand notation for ``@{command |
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(HOL) "function"}~@{text "(sequential)"}, followed by automated |
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proof attempts regarding pattern matching and termination. See |
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\cite{isabelle-function} for further details. |
26849 | 446 |
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\item @{command (HOL) "termination"}~@{text f} commences a |
26849 | 448 |
termination proof for the previously defined function @{text f}. If |
449 |
this is omitted, the command refers to the most recent function |
|
450 |
definition. After the proof is closed, the recursive equations and |
|
451 |
the induction principle is established. |
|
452 |
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\end{description} |
26849 | 454 |
|
455 |
%FIXME check |
|
456 |
||
27452 | 457 |
Recursive definitions introduced by the @{command (HOL) "function"} |
458 |
command accommodate |
|
26849 | 459 |
reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text |
460 |
"c.induct"} (where @{text c} is the name of the function definition) |
|
461 |
refers to a specific induction rule, with parameters named according |
|
27452 | 462 |
to the user-specified equations. |
463 |
For the @{command (HOL) "primrec"} the induction principle coincides |
|
464 |
with structural recursion on the datatype the recursion is carried |
|
465 |
out. |
|
466 |
Case names of @{command (HOL) |
|
26849 | 467 |
"primrec"} are that of the datatypes involved, while those of |
468 |
@{command (HOL) "function"} are numbered (starting from 1). |
|
469 |
||
470 |
The equations provided by these packages may be referred later as |
|
471 |
theorem list @{text "f.simps"}, where @{text f} is the (collective) |
|
472 |
name of the functions defined. Individual equations may be named |
|
473 |
explicitly as well. |
|
474 |
||
475 |
The @{command (HOL) "function"} command accepts the following |
|
476 |
options. |
|
477 |
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\begin{description} |
26849 | 479 |
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\item @{text sequential} enables a preprocessor which disambiguates |
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481 |
overlapping patterns by making them mutually disjoint. Earlier |
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|
482 |
equations take precedence over later ones. This allows to give the |
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|
483 |
specification in a format very similar to functional programming. |
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|
484 |
Note that the resulting simplification and induction rules |
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|
485 |
correspond to the transformed specification, not the one given |
26849 | 486 |
originally. This usually means that each equation given by the user |
487 |
may result in several theroems. Also note that this automatic |
|
488 |
transformation only works for ML-style datatype patterns. |
|
489 |
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490 |
\item @{text domintros} enables the automated generation of |
26849 | 491 |
introduction rules for the domain predicate. While mostly not |
492 |
needed, they can be helpful in some proofs about partial functions. |
|
493 |
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494 |
\item @{text tailrec} generates the unconstrained recursive |
26849 | 495 |
equations even without a termination proof, provided that the |
496 |
function is tail-recursive. This currently only works |
|
497 |
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\item @{text "default d"} allows to specify a default value for a |
26849 | 499 |
(partial) function, which will ensure that @{text "f x = d x"} |
500 |
whenever @{text "x \<notin> f_dom"}. |
|
501 |
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502 |
\end{description} |
26849 | 503 |
*} |
504 |
||
505 |
||
506 |
subsection {* Proof methods related to recursive definitions *} |
|
507 |
||
508 |
text {* |
|
509 |
\begin{matharray}{rcl} |
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@{method_def (HOL) pat_completeness} & : & @{text method} \\ |
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@{method_def (HOL) relation} & : & @{text method} \\ |
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512 |
@{method_def (HOL) lexicographic_order} & : & @{text method} \\ |
26849 | 513 |
\end{matharray} |
514 |
||
515 |
\begin{rail} |
|
516 |
'relation' term |
|
517 |
; |
|
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|
518 |
'lexicographic\_order' ( clasimpmod * ) |
26849 | 519 |
; |
520 |
\end{rail} |
|
521 |
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\begin{description} |
26849 | 523 |
|
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|
524 |
\item @{method (HOL) pat_completeness} is a specialized method to |
26849 | 525 |
solve goals regarding the completeness of pattern matching, as |
526 |
required by the @{command (HOL) "function"} package (cf.\ |
|
527 |
\cite{isabelle-function}). |
|
528 |
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|
529 |
\item @{method (HOL) relation}~@{text R} introduces a termination |
26849 | 530 |
proof using the relation @{text R}. The resulting proof state will |
531 |
contain goals expressing that @{text R} is wellfounded, and that the |
|
532 |
arguments of recursive calls decrease with respect to @{text R}. |
|
533 |
Usually, this method is used as the initial proof step of manual |
|
534 |
termination proofs. |
|
535 |
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|
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\item @{method (HOL) "lexicographic_order"} attempts a fully |
26849 | 537 |
automated termination proof by searching for a lexicographic |
538 |
combination of size measures on the arguments of the function. The |
|
539 |
method accepts the same arguments as the @{method auto} method, |
|
540 |
which it uses internally to prove local descents. The same context |
|
541 |
modifiers as for @{method auto} are accepted, see |
|
542 |
\secref{sec:clasimp}. |
|
543 |
||
544 |
In case of failure, extensive information is printed, which can help |
|
545 |
to analyse the situation (cf.\ \cite{isabelle-function}). |
|
546 |
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|
547 |
\end{description} |
26849 | 548 |
*} |
549 |
||
550 |
||
551 |
subsection {* Old-style recursive function definitions (TFL) *} |
|
552 |
||
553 |
text {* |
|
554 |
The old TFL commands @{command (HOL) "recdef"} and @{command (HOL) |
|
555 |
"recdef_tc"} for defining recursive are mostly obsolete; @{command |
|
556 |
(HOL) "function"} or @{command (HOL) "fun"} should be used instead. |
|
557 |
||
558 |
\begin{matharray}{rcl} |
|
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@{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\ |
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@{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
26849 | 561 |
\end{matharray} |
562 |
||
563 |
\begin{rail} |
|
564 |
'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints? |
|
565 |
; |
|
566 |
recdeftc thmdecl? tc |
|
567 |
; |
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|
568 |
hints: '(' 'hints' ( recdefmod * ) ')' |
26849 | 569 |
; |
570 |
recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod |
|
571 |
; |
|
572 |
tc: nameref ('(' nat ')')? |
|
573 |
; |
|
574 |
\end{rail} |
|
575 |
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\begin{description} |
26849 | 577 |
|
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|
578 |
\item @{command (HOL) "recdef"} defines general well-founded |
26849 | 579 |
recursive functions (using the TFL package), see also |
580 |
\cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells |
|
581 |
TFL to recover from failed proof attempts, returning unfinished |
|
582 |
results. The @{text recdef_simp}, @{text recdef_cong}, and @{text |
|
583 |
recdef_wf} hints refer to auxiliary rules to be used in the internal |
|
584 |
automated proof process of TFL. Additional @{syntax clasimpmod} |
|
585 |
declarations (cf.\ \secref{sec:clasimp}) may be given to tune the |
|
586 |
context of the Simplifier (cf.\ \secref{sec:simplifier}) and |
|
587 |
Classical reasoner (cf.\ \secref{sec:classical}). |
|
588 |
||
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589 |
\item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the |
26849 | 590 |
proof for leftover termination condition number @{text i} (default |
591 |
1) as generated by a @{command (HOL) "recdef"} definition of |
|
592 |
constant @{text c}. |
|
593 |
||
594 |
Note that in most cases, @{command (HOL) "recdef"} is able to finish |
|
595 |
its internal proofs without manual intervention. |
|
596 |
||
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|
597 |
\end{description} |
26849 | 598 |
|
599 |
\medskip Hints for @{command (HOL) "recdef"} may be also declared |
|
600 |
globally, using the following attributes. |
|
601 |
||
602 |
\begin{matharray}{rcl} |
|
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|
603 |
@{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\ |
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|
604 |
@{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\ |
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|
605 |
@{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\ |
26849 | 606 |
\end{matharray} |
607 |
||
608 |
\begin{rail} |
|
609 |
('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') |
|
610 |
; |
|
611 |
\end{rail} |
|
612 |
*} |
|
613 |
||
614 |
||
615 |
section {* Inductive and coinductive definitions \label{sec:hol-inductive} *} |
|
616 |
||
617 |
text {* |
|
618 |
An \textbf{inductive definition} specifies the least predicate (or |
|
619 |
set) @{text R} closed under given rules: applying a rule to elements |
|
620 |
of @{text R} yields a result within @{text R}. For example, a |
|
621 |
structural operational semantics is an inductive definition of an |
|
622 |
evaluation relation. |
|
623 |
||
624 |
Dually, a \textbf{coinductive definition} specifies the greatest |
|
625 |
predicate~/ set @{text R} that is consistent with given rules: every |
|
626 |
element of @{text R} can be seen as arising by applying a rule to |
|
627 |
elements of @{text R}. An important example is using bisimulation |
|
628 |
relations to formalise equivalence of processes and infinite data |
|
629 |
structures. |
|
630 |
||
631 |
\medskip The HOL package is related to the ZF one, which is |
|
632 |
described in a separate paper,\footnote{It appeared in CADE |
|
633 |
\cite{paulson-CADE}; a longer version is distributed with Isabelle.} |
|
634 |
which you should refer to in case of difficulties. The package is |
|
635 |
simpler than that of ZF thanks to implicit type-checking in HOL. |
|
636 |
The types of the (co)inductive predicates (or sets) determine the |
|
637 |
domain of the fixedpoint definition, and the package does not have |
|
638 |
to use inference rules for type-checking. |
|
639 |
||
640 |
\begin{matharray}{rcl} |
|
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|
641 |
@{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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|
642 |
@{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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|
643 |
@{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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|
644 |
@{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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|
645 |
@{attribute_def (HOL) mono} & : & @{text attribute} \\ |
26849 | 646 |
\end{matharray} |
647 |
||
648 |
\begin{rail} |
|
649 |
('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\ |
|
650 |
('where' clauses)? ('monos' thmrefs)? |
|
651 |
; |
|
652 |
clauses: (thmdecl? prop + '|') |
|
653 |
; |
|
654 |
'mono' (() | 'add' | 'del') |
|
655 |
; |
|
656 |
\end{rail} |
|
657 |
||
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|
658 |
\begin{description} |
26849 | 659 |
|
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|
660 |
\item @{command (HOL) "inductive"} and @{command (HOL) |
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|
661 |
"coinductive"} define (co)inductive predicates from the |
26849 | 662 |
introduction rules given in the @{keyword "where"} part. The |
663 |
optional @{keyword "for"} part contains a list of parameters of the |
|
664 |
(co)inductive predicates that remain fixed throughout the |
|
665 |
definition. The optional @{keyword "monos"} section contains |
|
666 |
\emph{monotonicity theorems}, which are required for each operator |
|
667 |
applied to a recursive set in the introduction rules. There |
|
668 |
\emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, |
|
669 |
for each premise @{text "M R\<^sub>i t"} in an introduction rule! |
|
670 |
||
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|
671 |
\item @{command (HOL) "inductive_set"} and @{command (HOL) |
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|
672 |
"coinductive_set"} are wrappers for to the previous commands, |
26849 | 673 |
allowing the definition of (co)inductive sets. |
674 |
||
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|
675 |
\item @{attribute (HOL) mono} declares monotonicity rules. These |
26849 | 676 |
rule are involved in the automated monotonicity proof of @{command |
677 |
(HOL) "inductive"}. |
|
678 |
||
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|
679 |
\end{description} |
26849 | 680 |
*} |
681 |
||
682 |
||
683 |
subsection {* Derived rules *} |
|
684 |
||
685 |
text {* |
|
686 |
Each (co)inductive definition @{text R} adds definitions to the |
|
687 |
theory and also proves some theorems: |
|
688 |
||
689 |
\begin{description} |
|
690 |
||
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|
691 |
\item @{text R.intros} is the list of introduction rules as proven |
26849 | 692 |
theorems, for the recursive predicates (or sets). The rules are |
693 |
also available individually, using the names given them in the |
|
694 |
theory file; |
|
695 |
||
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|
696 |
\item @{text R.cases} is the case analysis (or elimination) rule; |
26849 | 697 |
|
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|
698 |
\item @{text R.induct} or @{text R.coinduct} is the (co)induction |
26849 | 699 |
rule. |
700 |
||
701 |
\end{description} |
|
702 |
||
703 |
When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are |
|
704 |
defined simultaneously, the list of introduction rules is called |
|
705 |
@{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are |
|
706 |
called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list |
|
707 |
of mutual induction rules is called @{text |
|
708 |
"R\<^sub>1_\<dots>_R\<^sub>n.inducts"}. |
|
709 |
*} |
|
710 |
||
711 |
||
712 |
subsection {* Monotonicity theorems *} |
|
713 |
||
714 |
text {* |
|
715 |
Each theory contains a default set of theorems that are used in |
|
716 |
monotonicity proofs. New rules can be added to this set via the |
|
717 |
@{attribute (HOL) mono} attribute. The HOL theory @{text Inductive} |
|
718 |
shows how this is done. In general, the following monotonicity |
|
719 |
theorems may be added: |
|
720 |
||
721 |
\begin{itemize} |
|
722 |
||
723 |
\item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving |
|
724 |
monotonicity of inductive definitions whose introduction rules have |
|
725 |
premises involving terms such as @{text "M R\<^sub>i t"}. |
|
726 |
||
727 |
\item Monotonicity theorems for logical operators, which are of the |
|
728 |
general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in |
|
729 |
the case of the operator @{text "\<or>"}, the corresponding theorem is |
|
730 |
\[ |
|
731 |
\infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}} |
|
732 |
\] |
|
733 |
||
734 |
\item De Morgan style equations for reasoning about the ``polarity'' |
|
735 |
of expressions, e.g. |
|
736 |
\[ |
|
737 |
@{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad |
|
738 |
@{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"} |
|
739 |
\] |
|
740 |
||
741 |
\item Equations for reducing complex operators to more primitive |
|
742 |
ones whose monotonicity can easily be proved, e.g. |
|
743 |
\[ |
|
744 |
@{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad |
|
745 |
@{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"} |
|
746 |
\] |
|
747 |
||
748 |
\end{itemize} |
|
749 |
||
750 |
%FIXME: Example of an inductive definition |
|
751 |
*} |
|
752 |
||
753 |
||
754 |
section {* Arithmetic proof support *} |
|
755 |
||
756 |
text {* |
|
757 |
\begin{matharray}{rcl} |
|
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|
758 |
@{method_def (HOL) arith} & : & @{text method} \\ |
30863 | 759 |
@{attribute_def (HOL) arith} & : & @{text attribute} \\ |
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|
760 |
@{attribute_def (HOL) arith_split} & : & @{text attribute} \\ |
26849 | 761 |
\end{matharray} |
762 |
||
763 |
The @{method (HOL) arith} method decides linear arithmetic problems |
|
764 |
(on types @{text nat}, @{text int}, @{text real}). Any current |
|
765 |
facts are inserted into the goal before running the procedure. |
|
766 |
||
30863 | 767 |
The @{attribute (HOL) arith} attribute declares facts that are |
768 |
always supplied to the arithmetic provers implicitly. |
|
26849 | 769 |
|
30863 | 770 |
The @{attribute (HOL) arith_split} attribute declares case split |
30865 | 771 |
rules to be expanded before @{method (HOL) arith} is invoked. |
30863 | 772 |
|
773 |
Note that a simpler (but faster) arithmetic prover is |
|
774 |
already invoked by the Simplifier. |
|
26849 | 775 |
*} |
776 |
||
777 |
||
30169 | 778 |
section {* Intuitionistic proof search *} |
779 |
||
780 |
text {* |
|
781 |
\begin{matharray}{rcl} |
|
30171 | 782 |
@{method_def (HOL) iprover} & : & @{text method} \\ |
30169 | 783 |
\end{matharray} |
784 |
||
785 |
\begin{rail} |
|
31912
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changeset
|
786 |
'iprover' ('!' ?) ( rulemod * ) |
30169 | 787 |
; |
788 |
\end{rail} |
|
789 |
||
30171 | 790 |
The @{method (HOL) iprover} method performs intuitionistic proof |
791 |
search, depending on specifically declared rules from the context, |
|
792 |
or given as explicit arguments. Chained facts are inserted into the |
|
793 |
goal before commencing proof search; ``@{method (HOL) iprover}@{text |
|
794 |
"!"}'' means to include the current @{fact prems} as well. |
|
30169 | 795 |
|
796 |
Rules need to be classified as @{attribute (Pure) intro}, |
|
797 |
@{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the |
|
798 |
``@{text "!"}'' indicator refers to ``safe'' rules, which may be |
|
799 |
applied aggressively (without considering back-tracking later). |
|
800 |
Rules declared with ``@{text "?"}'' are ignored in proof search (the |
|
801 |
single-step @{method rule} method still observes these). An |
|
802 |
explicit weight annotation may be given as well; otherwise the |
|
803 |
number of rule premises will be taken into account here. |
|
804 |
*} |
|
805 |
||
806 |
||
30171 | 807 |
section {* Coherent Logic *} |
808 |
||
809 |
text {* |
|
810 |
\begin{matharray}{rcl} |
|
811 |
@{method_def (HOL) "coherent"} & : & @{text method} \\ |
|
812 |
\end{matharray} |
|
813 |
||
814 |
\begin{rail} |
|
815 |
'coherent' thmrefs? |
|
816 |
; |
|
817 |
\end{rail} |
|
818 |
||
819 |
The @{method (HOL) coherent} method solves problems of |
|
820 |
\emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers |
|
821 |
applications in confluence theory, lattice theory and projective |
|
822 |
geometry. See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some |
|
823 |
examples. |
|
824 |
*} |
|
825 |
||
826 |
||
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|
827 |
section {* Checking and refuting propositions *} |
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changeset
|
828 |
|
f5bd306f5e9d
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changeset
|
829 |
text {* |
f5bd306f5e9d
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haftmann
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changeset
|
830 |
Identifying incorrect propositions usually involves evaluation of |
f5bd306f5e9d
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changeset
|
831 |
particular assignments and systematic counter example search. This |
f5bd306f5e9d
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haftmann
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diff
changeset
|
832 |
is supported by the following commands. |
f5bd306f5e9d
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diff
changeset
|
833 |
|
f5bd306f5e9d
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diff
changeset
|
834 |
\begin{matharray}{rcl} |
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diff
changeset
|
835 |
@{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
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changeset
|
836 |
@{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
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changeset
|
837 |
@{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"} |
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changeset
|
838 |
\end{matharray} |
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diff
changeset
|
839 |
|
f5bd306f5e9d
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changeset
|
840 |
\begin{rail} |
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changeset
|
841 |
'value' ( ( '[' name ']' ) ? ) modes? term |
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changeset
|
842 |
; |
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changeset
|
843 |
|
f5bd306f5e9d
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changeset
|
844 |
'quickcheck' ( ( '[' args ']' ) ? ) nat? |
f5bd306f5e9d
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changeset
|
845 |
; |
f5bd306f5e9d
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changeset
|
846 |
|
f5bd306f5e9d
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changeset
|
847 |
'quickcheck_params' ( ( '[' args ']' ) ? ) |
f5bd306f5e9d
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changeset
|
848 |
; |
f5bd306f5e9d
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diff
changeset
|
849 |
|
f5bd306f5e9d
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changeset
|
850 |
modes: '(' (name + ) ')' |
f5bd306f5e9d
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haftmann
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diff
changeset
|
851 |
; |
f5bd306f5e9d
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changeset
|
852 |
|
f5bd306f5e9d
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changeset
|
853 |
args: ( name '=' value + ',' ) |
f5bd306f5e9d
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haftmann
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diff
changeset
|
854 |
; |
f5bd306f5e9d
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changeset
|
855 |
\end{rail} |
f5bd306f5e9d
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changeset
|
856 |
|
f5bd306f5e9d
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changeset
|
857 |
\begin{description} |
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haftmann
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diff
changeset
|
858 |
|
f5bd306f5e9d
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diff
changeset
|
859 |
\item @{command (HOL) "value"}~@{text t} evaluates and prints a |
f5bd306f5e9d
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haftmann
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|
860 |
term; optionally @{text modes} can be specified, which are |
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|
861 |
appended to the current print mode (see also \cite{isabelle-ref}). |
f5bd306f5e9d
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changeset
|
862 |
Internally, the evaluation is performed by registered evaluators, |
f5bd306f5e9d
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diff
changeset
|
863 |
which are invoked sequentially until a result is returned. |
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diff
changeset
|
864 |
Alternatively a specific evaluator can be selected using square |
f5bd306f5e9d
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diff
changeset
|
865 |
brackets; available evaluators include @{text nbe} for |
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
866 |
\emph{normalization by evaluation} and \emph{code} for code |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
867 |
generation in SML. |
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
868 |
|
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
869 |
\item @{command (HOL) "quickcheck"} tests the current goal for |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
870 |
counter examples using a series of arbitrary assignments for its |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
871 |
free variables; by default the first subgoal is tested, an other |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
872 |
can be selected explicitly using an optional goal index. |
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
873 |
A number of configuration options are supported for |
f5bd306f5e9d
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parents:
31254
diff
changeset
|
874 |
@{command (HOL) "quickcheck"}, notably: |
f5bd306f5e9d
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diff
changeset
|
875 |
|
f5bd306f5e9d
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diff
changeset
|
876 |
\begin{description} |
f5bd306f5e9d
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haftmann
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31254
diff
changeset
|
877 |
|
f5bd306f5e9d
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parents:
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diff
changeset
|
878 |
\item[size] specifies the maximum size of the search space for |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
879 |
assignment values. |
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
880 |
|
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
881 |
\item[iterations] sets how many sets of assignments are |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
882 |
generated for each particular size. |
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
883 |
|
f5bd306f5e9d
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parents:
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diff
changeset
|
884 |
\end{description} |
f5bd306f5e9d
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haftmann
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diff
changeset
|
885 |
|
f5bd306f5e9d
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parents:
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diff
changeset
|
886 |
These option can be given within square brackets. |
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
887 |
|
f5bd306f5e9d
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parents:
31254
diff
changeset
|
888 |
\item @{command (HOL) "quickcheck_params"} changes quickcheck |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
889 |
configuration options persitently. |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
890 |
|
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
891 |
\end{description} |
f5bd306f5e9d
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haftmann
parents:
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diff
changeset
|
892 |
*} |
f5bd306f5e9d
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haftmann
parents:
31254
diff
changeset
|
893 |
|
f5bd306f5e9d
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parents:
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diff
changeset
|
894 |
|
28603 | 895 |
section {* Invoking automated reasoning tools -- The Sledgehammer *} |
896 |
||
897 |
text {* |
|
898 |
Isabelle/HOL includes a generic \emph{ATP manager} that allows |
|
899 |
external automated reasoning tools to crunch a pending goal. |
|
900 |
Supported provers include E\footnote{\url{http://www.eprover.org}}, |
|
901 |
SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire. |
|
902 |
There is also a wrapper to invoke provers remotely via the |
|
903 |
SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}} |
|
904 |
web service. |
|
905 |
||
906 |
The problem passed to external provers consists of the goal together |
|
907 |
with a smart selection of lemmas from the current theory context. |
|
908 |
The result of a successful proof search is some source text that |
|
909 |
usually reconstructs the proof within Isabelle, without requiring |
|
910 |
external provers again. The Metis |
|
911 |
prover\footnote{\url{http://www.gilith.com/software/metis/}} that is |
|
912 |
integrated into Isabelle/HOL is being used here. |
|
913 |
||
914 |
In this mode of operation, heavy means of automated reasoning are |
|
915 |
used as a strong relevance filter, while the main proof checking |
|
916 |
works via explicit inferences going through the Isabelle kernel. |
|
917 |
Moreover, rechecking Isabelle proof texts with already specified |
|
918 |
auxiliary facts is much faster than performing fully automated |
|
919 |
search over and over again. |
|
920 |
||
921 |
\begin{matharray}{rcl} |
|
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|
922 |
@{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
9ec4482c9201
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changeset
|
923 |
@{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
9ec4482c9201
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changeset
|
924 |
@{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\ |
9ec4482c9201
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changeset
|
925 |
@{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\ |
29112
f2b45eea6dac
added 'atp_messages' command, which displays recent messages synchronously;
wenzelm
parents:
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diff
changeset
|
926 |
@{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\ |
28761
9ec4482c9201
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|
927 |
@{method_def (HOL) metis} & : & @{text method} \\ |
28603 | 928 |
\end{matharray} |
929 |
||
930 |
\begin{rail} |
|
31912
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|
931 |
'sledgehammer' ( nameref * ) |
28603 | 932 |
; |
29112
f2b45eea6dac
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changeset
|
933 |
'atp\_messages' ('(' nat ')')? |
29114 | 934 |
; |
28603 | 935 |
|
936 |
'metis' thmrefs |
|
937 |
; |
|
938 |
\end{rail} |
|
939 |
||
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|
940 |
\begin{description} |
28603 | 941 |
|
28760
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changeset
|
942 |
\item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"} |
cbc435f7b16b
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|
943 |
invokes the specified automated theorem provers on the first |
cbc435f7b16b
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changeset
|
944 |
subgoal. Provers are run in parallel, the first successful result |
cbc435f7b16b
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changeset
|
945 |
is displayed, and the other attempts are terminated. |
28603 | 946 |
|
947 |
Provers are defined in the theory context, see also @{command (HOL) |
|
948 |
print_atps}. If no provers are given as arguments to @{command |
|
949 |
(HOL) sledgehammer}, the system refers to the default defined as |
|
950 |
``ATP provers'' preference by the user interface. |
|
951 |
||
952 |
There are additional preferences for timeout (default: 60 seconds), |
|
953 |
and the maximum number of independent prover processes (default: 5); |
|
954 |
excessive provers are automatically terminated. |
|
955 |
||
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|
956 |
\item @{command (HOL) print_atps} prints the list of automated |
28603 | 957 |
theorem provers available to the @{command (HOL) sledgehammer} |
958 |
command. |
|
959 |
||
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changeset
|
960 |
\item @{command (HOL) atp_info} prints information about presently |
28603 | 961 |
running provers, including elapsed runtime, and the remaining time |
962 |
until timeout. |
|
963 |
||
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changeset
|
964 |
\item @{command (HOL) atp_kill} terminates all presently running |
28603 | 965 |
provers. |
966 |
||
29112
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diff
changeset
|
967 |
\item @{command (HOL) atp_messages} displays recent messages issued |
f2b45eea6dac
added 'atp_messages' command, which displays recent messages synchronously;
wenzelm
parents:
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diff
changeset
|
968 |
by automated theorem provers. This allows to examine results that |
f2b45eea6dac
added 'atp_messages' command, which displays recent messages synchronously;
wenzelm
parents:
28761
diff
changeset
|
969 |
might have got lost due to the asynchronous nature of default |
f2b45eea6dac
added 'atp_messages' command, which displays recent messages synchronously;
wenzelm
parents:
28761
diff
changeset
|
970 |
@{command (HOL) sledgehammer} output. An optional message limit may |
f2b45eea6dac
added 'atp_messages' command, which displays recent messages synchronously;
wenzelm
parents:
28761
diff
changeset
|
971 |
be specified (default 5). |
f2b45eea6dac
added 'atp_messages' command, which displays recent messages synchronously;
wenzelm
parents:
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diff
changeset
|
972 |
|
28760
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unified use of declaration environment with IsarImplementation;
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diff
changeset
|
973 |
\item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
974 |
with the given facts. Metis is an automated proof tool of medium |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
975 |
strength, but is fully integrated into Isabelle/HOL, with explicit |
cbc435f7b16b
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wenzelm
parents:
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diff
changeset
|
976 |
inferences going through the kernel. Thus its results are |
28603 | 977 |
guaranteed to be ``correct by construction''. |
978 |
||
979 |
Note that all facts used with Metis need to be specified as explicit |
|
980 |
arguments. There are no rule declarations as for other Isabelle |
|
981 |
provers, like @{method blast} or @{method fast}. |
|
982 |
||
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|
983 |
\end{description} |
28603 | 984 |
*} |
985 |
||
986 |
||
28752 | 987 |
section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *} |
26849 | 988 |
|
989 |
text {* |
|
27123
11fcdd5897dd
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diff
changeset
|
990 |
The following tools of Isabelle/HOL support cases analysis and |
11fcdd5897dd
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wenzelm
parents:
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diff
changeset
|
991 |
induction in unstructured tactic scripts; see also |
11fcdd5897dd
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wenzelm
parents:
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diff
changeset
|
992 |
\secref{sec:cases-induct} for proper Isar versions of similar ideas. |
26849 | 993 |
|
994 |
\begin{matharray}{rcl} |
|
28761
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changeset
|
995 |
@{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
9ec4482c9201
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wenzelm
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diff
changeset
|
996 |
@{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
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updated/refined types of Isar language elements, removed special LaTeX macros;
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parents:
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diff
changeset
|
997 |
@{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\ |
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diff
changeset
|
998 |
@{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
26849 | 999 |
\end{matharray} |
1000 |
||
1001 |
\begin{rail} |
|
1002 |
'case\_tac' goalspec? term rule? |
|
1003 |
; |
|
1004 |
'induct\_tac' goalspec? (insts * 'and') rule? |
|
1005 |
; |
|
1006 |
'ind\_cases' (prop +) ('for' (name +)) ? |
|
1007 |
; |
|
1008 |
'inductive\_cases' (thmdecl? (prop +) + 'and') |
|
1009 |
; |
|
1010 |
||
1011 |
rule: ('rule' ':' thmref) |
|
1012 |
; |
|
1013 |
\end{rail} |
|
1014 |
||
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diff
changeset
|
1015 |
\begin{description} |
26849 | 1016 |
|
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1017 |
\item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1018 |
to reason about inductive types. Rules are selected according to |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1019 |
the declarations by the @{attribute cases} and @{attribute induct} |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1020 |
attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL) |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1021 |
datatype} package already takes care of this. |
27123
11fcdd5897dd
case_tac/induct_tac: use same declarations as cases/induct;
wenzelm
parents:
27103
diff
changeset
|
1022 |
|
11fcdd5897dd
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parents:
27103
diff
changeset
|
1023 |
These unstructured tactics feature both goal addressing and dynamic |
26849 | 1024 |
instantiation. Note that named rule cases are \emph{not} provided |
27123
11fcdd5897dd
case_tac/induct_tac: use same declarations as cases/induct;
wenzelm
parents:
27103
diff
changeset
|
1025 |
as would be by the proper @{method cases} and @{method induct} proof |
11fcdd5897dd
case_tac/induct_tac: use same declarations as cases/induct;
wenzelm
parents:
27103
diff
changeset
|
1026 |
methods (see \secref{sec:cases-induct}). Unlike the @{method |
11fcdd5897dd
case_tac/induct_tac: use same declarations as cases/induct;
wenzelm
parents:
27103
diff
changeset
|
1027 |
induct} method, @{method induct_tac} does not handle structured rule |
11fcdd5897dd
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wenzelm
parents:
27103
diff
changeset
|
1028 |
statements, only the compact object-logic conclusion of the subgoal |
11fcdd5897dd
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wenzelm
parents:
27103
diff
changeset
|
1029 |
being addressed. |
26849 | 1030 |
|
28760
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diff
changeset
|
1031 |
\item @{method (HOL) ind_cases} and @{command (HOL) |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1032 |
"inductive_cases"} provide an interface to the internal @{ML_text |
26860 | 1033 |
mk_cases} operation. Rules are simplified in an unrestricted |
1034 |
forward manner. |
|
26849 | 1035 |
|
1036 |
While @{method (HOL) ind_cases} is a proof method to apply the |
|
1037 |
result immediately as elimination rules, @{command (HOL) |
|
1038 |
"inductive_cases"} provides case split theorems at the theory level |
|
1039 |
for later use. The @{keyword "for"} argument of the @{method (HOL) |
|
1040 |
ind_cases} method allows to specify a list of variables that should |
|
1041 |
be generalized before applying the resulting rule. |
|
1042 |
||
28760
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diff
changeset
|
1043 |
\end{description} |
26849 | 1044 |
*} |
1045 |
||
1046 |
||
1047 |
section {* Executable code *} |
|
1048 |
||
1049 |
text {* |
|
1050 |
Isabelle/Pure provides two generic frameworks to support code |
|
1051 |
generation from executable specifications. Isabelle/HOL |
|
1052 |
instantiates these mechanisms in a way that is amenable to end-user |
|
1053 |
applications. |
|
1054 |
||
1055 |
One framework generates code from both functional and relational |
|
1056 |
programs to SML. See \cite{isabelle-HOL} for further information |
|
1057 |
(this actually covers the new-style theory format as well). |
|
1058 |
||
1059 |
\begin{matharray}{rcl} |
|
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updated/refined types of Isar language elements, removed special LaTeX macros;
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diff
changeset
|
1060 |
@{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
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diff
changeset
|
1061 |
@{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
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diff
changeset
|
1062 |
@{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1063 |
@{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
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diff
changeset
|
1064 |
@{attribute_def (HOL) code} & : & @{text attribute} \\ |
26849 | 1065 |
\end{matharray} |
1066 |
||
1067 |
\begin{rail} |
|
1068 |
( 'code\_module' | 'code\_library' ) modespec ? name ? \\ |
|
1069 |
( 'file' name ) ? ( 'imports' ( name + ) ) ? \\ |
|
1070 |
'contains' ( ( name '=' term ) + | term + ) |
|
1071 |
; |
|
1072 |
||
1073 |
modespec: '(' ( name * ) ')' |
|
1074 |
; |
|
1075 |
||
1076 |
'consts\_code' (codespec +) |
|
1077 |
; |
|
1078 |
||
1079 |
codespec: const template attachment ? |
|
1080 |
; |
|
1081 |
||
1082 |
'types\_code' (tycodespec +) |
|
1083 |
; |
|
1084 |
||
1085 |
tycodespec: name template attachment ? |
|
1086 |
; |
|
1087 |
||
1088 |
const: term |
|
1089 |
; |
|
1090 |
||
1091 |
template: '(' string ')' |
|
1092 |
; |
|
1093 |
||
1094 |
attachment: 'attach' modespec ? verblbrace text verbrbrace |
|
1095 |
; |
|
1096 |
||
1097 |
'code' (name)? |
|
1098 |
; |
|
1099 |
\end{rail} |
|
1100 |
||
1101 |
\medskip The other framework generates code from functional programs |
|
1102 |
(including overloading using type classes) to SML \cite{SML}, OCaml |
|
1103 |
\cite{OCaml} and Haskell \cite{haskell-revised-report}. |
|
1104 |
Conceptually, code generation is split up in three steps: |
|
1105 |
\emph{selection} of code theorems, \emph{translation} into an |
|
1106 |
abstract executable view and \emph{serialization} to a specific |
|
1107 |
\emph{target language}. See \cite{isabelle-codegen} for an |
|
1108 |
introduction on how to use it. |
|
1109 |
||
1110 |
\begin{matharray}{rcl} |
|
28761
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updated/refined types of Isar language elements, removed special LaTeX macros;
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28760
diff
changeset
|
1111 |
@{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
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diff
changeset
|
1112 |
@{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
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diff
changeset
|
1113 |
@{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
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diff
changeset
|
1114 |
@{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1115 |
@{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1116 |
@{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1117 |
@{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1118 |
@{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1119 |
@{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
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diff
changeset
|
1120 |
@{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1121 |
@{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1122 |
@{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1123 |
@{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
parents:
28760
diff
changeset
|
1124 |
@{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
31254 | 1125 |
@{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
28761
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
wenzelm
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diff
changeset
|
1126 |
@{attribute_def (HOL) code} & : & @{text attribute} \\ |
26849 | 1127 |
\end{matharray} |
1128 |
||
1129 |
\begin{rail} |
|
1130 |
'export\_code' ( constexpr + ) ? \\ |
|
1131 |
( ( 'in' target ( 'module\_name' string ) ? \\ |
|
1132 |
( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ? |
|
1133 |
; |
|
1134 |
||
1135 |
'code\_thms' ( constexpr + ) ? |
|
1136 |
; |
|
1137 |
||
1138 |
'code\_deps' ( constexpr + ) ? |
|
1139 |
; |
|
1140 |
||
1141 |
const: term |
|
1142 |
; |
|
1143 |
||
1144 |
constexpr: ( const | 'name.*' | '*' ) |
|
1145 |
; |
|
1146 |
||
1147 |
typeconstructor: nameref |
|
1148 |
; |
|
1149 |
||
1150 |
class: nameref |
|
1151 |
; |
|
1152 |
||
1153 |
target: 'OCaml' | 'SML' | 'Haskell' |
|
1154 |
; |
|
1155 |
||
1156 |
'code\_datatype' const + |
|
1157 |
; |
|
1158 |
||
1159 |
'code\_const' (const + 'and') \\ |
|
1160 |
( ( '(' target ( syntax ? + 'and' ) ')' ) + ) |
|
1161 |
; |
|
1162 |
||
1163 |
'code\_type' (typeconstructor + 'and') \\ |
|
1164 |
( ( '(' target ( syntax ? + 'and' ) ')' ) + ) |
|
1165 |
; |
|
1166 |
||
1167 |
'code\_class' (class + 'and') \\ |
|
28687 | 1168 |
( ( '(' target \\ ( string ? + 'and' ) ')' ) + ) |
26849 | 1169 |
; |
1170 |
||
1171 |
'code\_instance' (( typeconstructor '::' class ) + 'and') \\ |
|
1172 |
( ( '(' target ( '-' ? + 'and' ) ')' ) + ) |
|
1173 |
; |
|
1174 |
||
1175 |
'code\_monad' const const target |
|
1176 |
; |
|
1177 |
||
1178 |
'code\_reserved' target ( string + ) |
|
1179 |
; |
|
1180 |
||
1181 |
'code\_include' target ( string ( string | '-') ) |
|
1182 |
; |
|
1183 |
||
1184 |
'code\_modulename' target ( ( string string ) + ) |
|
1185 |
; |
|
1186 |
||
27452 | 1187 |
'code\_abort' ( const + ) |
26849 | 1188 |
; |
1189 |
||
1190 |
syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string |
|
1191 |
; |
|
1192 |
||
31998
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1193 |
'code' ( 'del' ) ? |
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1194 |
; |
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1195 |
|
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1196 |
'code_unfold' ( 'del' ) ? |
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1197 |
; |
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1198 |
|
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1199 |
'code_post' ( 'del' ) ? |
26849 | 1200 |
; |
1201 |
\end{rail} |
|
1202 |
||
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changeset
|
1203 |
\begin{description} |
26849 | 1204 |
|
28760
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changeset
|
1205 |
\item @{command (HOL) "export_code"} is the canonical interface for |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1206 |
generating and serializing code: for a given list of constants, code |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1207 |
is generated for the specified target languages. Abstract code is |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1208 |
cached incrementally. If no constant is given, the currently cached |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
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diff
changeset
|
1209 |
code is serialized. If no serialization instruction is given, only |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
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diff
changeset
|
1210 |
abstract code is cached. |
26849 | 1211 |
|
1212 |
Constants may be specified by giving them literally, referring to |
|
1213 |
all executable contants within a certain theory by giving @{text |
|
1214 |
"name.*"}, or referring to \emph{all} executable constants currently |
|
1215 |
available by giving @{text "*"}. |
|
1216 |
||
1217 |
By default, for each involved theory one corresponding name space |
|
1218 |
module is generated. Alternativly, a module name may be specified |
|
1219 |
after the @{keyword "module_name"} keyword; then \emph{all} code is |
|
1220 |
placed in this module. |
|
1221 |
||
1222 |
For \emph{SML} and \emph{OCaml}, the file specification refers to a |
|
1223 |
single file; for \emph{Haskell}, it refers to a whole directory, |
|
1224 |
where code is generated in multiple files reflecting the module |
|
1225 |
hierarchy. The file specification ``@{text "-"}'' denotes standard |
|
1226 |
output. For \emph{SML}, omitting the file specification compiles |
|
1227 |
code internally in the context of the current ML session. |
|
1228 |
||
1229 |
Serializers take an optional list of arguments in parentheses. For |
|
1230 |
\emph{Haskell} a module name prefix may be given using the ``@{text |
|
1231 |
"root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim |
|
1232 |
"deriving (Read, Show)"}'' clause to each appropriate datatype |
|
1233 |
declaration. |
|
1234 |
||
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diff
changeset
|
1235 |
\item @{command (HOL) "code_thms"} prints a list of theorems |
26849 | 1236 |
representing the corresponding program containing all given |
1237 |
constants; if no constants are given, the currently cached code |
|
1238 |
theorems are printed. |
|
1239 |
||
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
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diff
changeset
|
1240 |
\item @{command (HOL) "code_deps"} visualizes dependencies of |
26849 | 1241 |
theorems representing the corresponding program containing all given |
1242 |
constants; if no constants are given, the currently cached code |
|
1243 |
theorems are visualized. |
|
1244 |
||
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
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diff
changeset
|
1245 |
\item @{command (HOL) "code_datatype"} specifies a constructor set |
26849 | 1246 |
for a logical type. |
1247 |
||
28760
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unified use of declaration environment with IsarImplementation;
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diff
changeset
|
1248 |
\item @{command (HOL) "code_const"} associates a list of constants |
26849 | 1249 |
with target-specific serializations; omitting a serialization |
1250 |
deletes an existing serialization. |
|
1251 |
||
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1252 |
\item @{command (HOL) "code_type"} associates a list of type |
26849 | 1253 |
constructors with target-specific serializations; omitting a |
1254 |
serialization deletes an existing serialization. |
|
1255 |
||
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1256 |
\item @{command (HOL) "code_class"} associates a list of classes |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1257 |
with target-specific class names; omitting a serialization deletes |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1258 |
an existing serialization. This applies only to \emph{Haskell}. |
26849 | 1259 |
|
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1260 |
\item @{command (HOL) "code_instance"} declares a list of type |
26849 | 1261 |
constructor / class instance relations as ``already present'' for a |
1262 |
given target. Omitting a ``@{text "-"}'' deletes an existing |
|
1263 |
``already present'' declaration. This applies only to |
|
1264 |
\emph{Haskell}. |
|
1265 |
||
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1266 |
\item @{command (HOL) "code_monad"} provides an auxiliary mechanism |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1267 |
to generate monadic code for Haskell. |
26849 | 1268 |
|
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1269 |
\item @{command (HOL) "code_reserved"} declares a list of names as |
26849 | 1270 |
reserved for a given target, preventing it to be shadowed by any |
1271 |
generated code. |
|
1272 |
||
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1273 |
\item @{command (HOL) "code_include"} adds arbitrary named content |
27706 | 1274 |
(``include'') to generated code. A ``@{text "-"}'' as last argument |
26849 | 1275 |
will remove an already added ``include''. |
1276 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1277 |
\item @{command (HOL) "code_modulename"} declares aliasings from one |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1278 |
module name onto another. |
26849 | 1279 |
|
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1280 |
\item @{command (HOL) "code_abort"} declares constants which are not |
29560 | 1281 |
required to have a definition by means of code equations; if |
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1282 |
needed these are implemented by program abort instead. |
26849 | 1283 |
|
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1284 |
\item @{attribute (HOL) code} explicitly selects (or with option |
29560 | 1285 |
``@{text "del"}'' deselects) a code equation for code |
1286 |
generation. Usually packages introducing code equations provide |
|
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1287 |
a reasonable default setup for selection. |
26849 | 1288 |
|
31998
2c7a24f74db9
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haftmann
parents:
31912
diff
changeset
|
1289 |
\item @{attribute (HOL) code_inline} declares (or with |
28562 | 1290 |
option ``@{text "del"}'' removes) inlining theorems which are |
29560 | 1291 |
applied as rewrite rules to any code equation during |
26849 | 1292 |
preprocessing. |
1293 |
||
31998
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1294 |
\item @{attribute (HOL) code_post} declares (or with |
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1295 |
option ``@{text "del"}'' removes) theorems which are |
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1296 |
applied as rewrite rules to any result of an evaluation. |
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31912
diff
changeset
|
1297 |
|
28760
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unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1298 |
\item @{command (HOL) "print_codesetup"} gives an overview on |
31254 | 1299 |
selected code equations and code generator datatypes. |
1300 |
||
1301 |
\item @{command (HOL) "print_codeproc"} prints the setup |
|
1302 |
of the code generator preprocessor. |
|
26849 | 1303 |
|
28760
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wenzelm
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28752
diff
changeset
|
1304 |
\end{description} |
26849 | 1305 |
*} |
1306 |
||
27045 | 1307 |
|
1308 |
section {* Definition by specification \label{sec:hol-specification} *} |
|
1309 |
||
1310 |
text {* |
|
1311 |
\begin{matharray}{rcl} |
|
28761
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updated/refined types of Isar language elements, removed special LaTeX macros;
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28760
diff
changeset
|
1312 |
@{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
9ec4482c9201
updated/refined types of Isar language elements, removed special LaTeX macros;
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parents:
28760
diff
changeset
|
1313 |
@{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
27045 | 1314 |
\end{matharray} |
1315 |
||
1316 |
\begin{rail} |
|
1317 |
('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +) |
|
1318 |
; |
|
1319 |
decl: ((name ':')? term '(' 'overloaded' ')'?) |
|
1320 |
\end{rail} |
|
1321 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
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diff
changeset
|
1322 |
\begin{description} |
27045 | 1323 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1324 |
\item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a |
27045 | 1325 |
goal stating the existence of terms with the properties specified to |
1326 |
hold for the constants given in @{text decls}. After finishing the |
|
1327 |
proof, the theory will be augmented with definitions for the given |
|
1328 |
constants, as well as with theorems stating the properties for these |
|
1329 |
constants. |
|
1330 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1331 |
\item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1332 |
a goal stating the existence of terms with the properties specified |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1333 |
to hold for the constants given in @{text decls}. After finishing |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1334 |
the proof, the theory will be augmented with axioms expressing the |
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1335 |
properties given in the first place. |
27045 | 1336 |
|
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1337 |
\item @{text decl} declares a constant to be defined by the |
27045 | 1338 |
specification given. The definition for the constant @{text c} is |
1339 |
bound to the name @{text c_def} unless a theorem name is given in |
|
1340 |
the declaration. Overloaded constants should be declared as such. |
|
1341 |
||
28760
cbc435f7b16b
unified use of declaration environment with IsarImplementation;
wenzelm
parents:
28752
diff
changeset
|
1342 |
\end{description} |
27045 | 1343 |
|
1344 |
Whether to use @{command (HOL) "specification"} or @{command (HOL) |
|
1345 |
"ax_specification"} is to some extent a matter of style. @{command |
|
1346 |
(HOL) "specification"} introduces no new axioms, and so by |
|
1347 |
construction cannot introduce inconsistencies, whereas @{command |
|
1348 |
(HOL) "ax_specification"} does introduce axioms, but only after the |
|
1349 |
user has explicitly proven it to be safe. A practical issue must be |
|
1350 |
considered, though: After introducing two constants with the same |
|
1351 |
properties using @{command (HOL) "specification"}, one can prove |
|
1352 |
that the two constants are, in fact, equal. If this might be a |
|
1353 |
problem, one should use @{command (HOL) "ax_specification"}. |
|
1354 |
*} |
|
1355 |
||
26840 | 1356 |
end |