src/Doc/Codegen/Evaluation.thy
author haftmann
Thu Jan 15 13:39:41 2015 +0100 (2015-01-15)
changeset 59377 056945909f60
parent 59335 e743ce816cf6
child 59378 065f349852e6
permissions -rw-r--r--
modernized cartouches
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theory Evaluation
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imports Setup
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begin
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section \<open>Evaluation \label{sec:evaluation}\<close>
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text \<open>
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  Recalling \secref{sec:principle}, code generation turns a system of
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  equations into a program with the \emph{same} equational semantics.
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  As a consequence, this program can be used as a \emph{rewrite
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  engine} for terms: rewriting a term @{term "t"} using a program to a
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  term @{term "t'"} yields the theorems @{prop "t \<equiv> t'"}.  This
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  application of code generation in the following is referred to as
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  \emph{evaluation}.
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\<close>
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subsection \<open>Evaluation techniques\<close>
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text \<open>
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  The existing infrastructure provides a rich palette of evaluation
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  techniques, each comprising different aspects:
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  \begin{description}
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    \item[Expressiveness.]  Depending on how good symbolic computation
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      is supported, the class of terms which can be evaluated may be
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      bigger or smaller.
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    \item[Efficiency.]  The more machine-near the technique, the
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      faster it is.
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    \item[Trustability.]  Techniques which a huge (and also probably
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      more configurable infrastructure) are more fragile and less
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      trustable.
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  \end{description}
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\<close>
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subsubsection \<open>The simplifier (@{text simp})\<close>
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text \<open>
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  The simplest way for evaluation is just using the simplifier with
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  the original code equations of the underlying program.  This gives
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  fully symbolic evaluation and highest trustablity, with the usual
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  performance of the simplifier.  Note that for operations on abstract
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  datatypes (cf.~\secref{sec:invariant}), the original theorems as
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  given by the users are used, not the modified ones.
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\<close>
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subsubsection \<open>Normalization by evaluation (@{text nbe})\<close>
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text \<open>
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  Normalization by evaluation @{cite "Aehlig-Haftmann-Nipkow:2008:nbe"}
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  provides a comparably fast partially symbolic evaluation which
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  permits also normalization of functions and uninterpreted symbols;
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  the stack of code to be trusted is considerable.
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\<close>
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subsubsection \<open>Evaluation in ML (@{text code})\<close>
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text \<open>
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  Highest performance can be achieved by evaluation in ML, at the cost
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  of being restricted to ground results and a layered stack of code to
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  be trusted, including code generator configurations by the user.
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  Evaluation is carried out in a target language \emph{Eval} which
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  inherits from \emph{SML} but for convenience uses parts of the
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  Isabelle runtime environment.  The soundness of computation carried
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  out there depends crucially on the correctness of the code
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  generator setup; this is one of the reasons why you should not use
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  adaptation (see \secref{sec:adaptation}) frivolously.
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\<close>
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subsection \<open>Aspects of evaluation\<close>
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text \<open>
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  Each of the techniques can be combined with different aspects.  The
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  most important distinction is between dynamic and static evaluation.
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  Dynamic evaluation takes the code generator configuration \qt{as it
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  is} at the point where evaluation is issued.  Best example is the
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  @{command_def value} command which allows ad-hoc evaluation of
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  terms:
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\<close>
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value %quote "42 / (12 :: rat)"
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text \<open>
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  \noindent @{command value} tries first to evaluate using ML, falling
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  back to normalization by evaluation if this fails.
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  A particular technique may be specified in square brackets, e.g.
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\<close>
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value %quote [nbe] "42 / (12 :: rat)"
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text \<open>
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  To employ dynamic evaluation in the document generation, there is also
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  a @{text value} antiquotation with the same evaluation techniques 
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  as @{command value}.
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  Static evaluation freezes the code generator configuration at a
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  certain point and uses this context whenever evaluation is issued
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  later on.  This is particularly appropriate for proof procedures
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  which use evaluation, since then the behaviour of evaluation is not
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  changed or even compromised later on by actions of the user.
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  As a technical complication, terms after evaluation in ML must be
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  turned into Isabelle's internal term representation again.  Since
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  this is also configurable, it is never fully trusted.  For this
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  reason, evaluation in ML comes with further aspects:
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  \begin{description}
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    \item[Plain evaluation.]  A term is normalized using the provided
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      term reconstruction from ML to Isabelle; for applications which
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      do not need to be fully trusted.
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    \item[Property conversion.]  Evaluates propositions; since these
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      are monomorphic, the term reconstruction is fixed once and for all
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      and therefore trustable.
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    \item[Conversion.]  Evaluates an arbitrary term @{term "t"} first
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      by plain evaluation and certifies the result @{term "t'"} by
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      checking the equation @{term "t \<equiv> t'"} using property
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      conversion.
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  \end{description}
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  \noindent The picture is further complicated by the roles of
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  exceptions.  Here three cases have to be distinguished:
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  \begin{itemize}
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    \item Evaluation of @{term t} terminates with a result @{term
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      "t'"}.
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    \item Evaluation of @{term t} terminates which en exception
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      indicating a pattern match failure or a non-implemented
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      function.  As sketched in \secref{sec:partiality}, this can be
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      interpreted as partiality.
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    \item Evaluation raises any other kind of exception.
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  \end{itemize}
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  \noindent For conversions, the first case yields the equation @{term
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  "t = t'"}, the second defaults to reflexivity @{term "t = t"}.
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  Exceptions of the third kind are propagated to the user.
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  By default return values of plain evaluation are optional, yielding
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  @{text "SOME t'"} in the first case, @{text "NONE"} in the
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  second, and propagating the exception in the third case.  A strict
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  variant of plain evaluation either yields @{text "t'"} or propagates
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  any exception, a liberal variant captures any exception in a result
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  of type @{text "Exn.result"}.
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  For property conversion (which coincides with conversion except for
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  evaluation in ML), methods are provided which solve a given goal by
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  evaluation.
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\<close>
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subsection \<open>Schematic overview\<close>
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text \<open>
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  \newcommand{\ttsize}{\fontsize{5.8pt}{8pt}\selectfont}
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  \fontsize{9pt}{12pt}\selectfont
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  \begin{tabular}{ll||c|c|c}
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    & & @{text simp} & @{text nbe} & @{text code} \tabularnewline \hline \hline
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    \multirow{5}{1ex}{\rotatebox{90}{dynamic}}
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      & interactive evaluation 
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      & @{command value} @{text "[simp]"} & @{command value} @{text "[nbe]"} & @{command value} @{text "[code]"}
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      \tabularnewline
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    & plain evaluation & & & \ttsize@{ML "Code_Evaluation.dynamic_value"} \tabularnewline \cline{2-5}
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    & evaluation method & @{method code_simp} & @{method normalization} & @{method eval} \tabularnewline
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    & property conversion & & & \ttsize@{ML "Code_Runtime.dynamic_holds_conv"} \tabularnewline \cline{2-5}
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    & conversion & \ttsize@{ML "Code_Simp.dynamic_conv"} & \ttsize@{ML "Nbe.dynamic_conv"}
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      & \ttsize@{ML "Code_Evaluation.dynamic_conv"} \tabularnewline \hline \hline
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    \multirow{3}{1ex}{\rotatebox{90}{static}}
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    & plain evaluation & & & \ttsize@{ML "Code_Evaluation.static_value"} \tabularnewline \cline{2-5}
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    & property conversion & &
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      & \ttsize@{ML "Code_Runtime.static_holds_conv"} \tabularnewline \cline{2-5}
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    & conversion & \ttsize@{ML "Code_Simp.static_conv"}
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      & \ttsize@{ML "Nbe.static_conv"}
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      & \ttsize@{ML "Code_Evaluation.static_conv"}
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  \end{tabular}
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\<close>
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subsection \<open>Preprocessing HOL terms into evaluable shape\<close>
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text \<open>
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  When integration decision procedures developed inside HOL into HOL itself,
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  it is necessary to somehow get from the Isabelle/ML representation to
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  the representation used by the decision procedure itself (``reification'').
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  One option is to hardcode it using code antiquotations (see \secref{sec:code_antiq}).
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  Another option is to use pre-existing infrastructure in HOL:
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  @{ML "Reification.conv"} and @{ML "Reification.tac"}
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  An simplistic example:
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\<close>
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datatype %quote form_ord = T | F | Less nat nat
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  | And form_ord form_ord | Or form_ord form_ord | Neg form_ord
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primrec %quote interp :: "form_ord \<Rightarrow> 'a::order list \<Rightarrow> bool"
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where
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  "interp T vs \<longleftrightarrow> True"
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| "interp F vs \<longleftrightarrow> False"
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| "interp (Less i j) vs \<longleftrightarrow> vs ! i < vs ! j"
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| "interp (And f1 f2) vs \<longleftrightarrow> interp f1 vs \<and> interp f2 vs"
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| "interp (Or f1 f2) vs \<longleftrightarrow> interp f1 vs \<or> interp f2 vs"
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| "interp (Neg f) vs \<longleftrightarrow> \<not> interp f vs"
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text \<open>
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  The datatype @{type form_ord} represents formulae whose semantics is given by
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  @{const interp}.  Note that values are represented by variable indices (@{typ nat})
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  whose concrete values are given in list @{term vs}.
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\<close>
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ML (*<*) \<open>\<close>
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schematic_lemma "thm": fixes x y z :: "'a::order" shows "x < y \<and> x < z \<equiv> ?P"
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ML_prf 
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(*>*) \<open>val thm =
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  Reification.conv @{context} @{thms interp.simps} @{cterm "x < y \<and> x < z"}\<close> (*<*)
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by (tactic \<open>ALLGOALS (rtac thm)\<close>)
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(*>*) 
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text \<open>
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  By virtue of @{fact interp.simps}, @{ML "Reification.conv"} provides a conversion
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  which, for this concrete example, yields @{thm thm [no_vars]}.  Note that the argument
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  to @{const interp} does not contain any free variables and can thus be evaluated
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  using evaluation.
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  A less meager example can be found in the AFP, session @{text "Regular-Sets"},
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  theory @{text Regexp_Method}.
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\<close>
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subsection \<open>Intimate connection between logic and system runtime\<close>
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text \<open>
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  The toolbox of static evaluation conversions forms a reasonable base
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  to interweave generated code and system tools.  However in some
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  situations more direct interaction is desirable.
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\<close>
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subsubsection \<open>Static embedding of generated code into system runtime -- the @{text code} antiquotation \label{sec:code_antiq}\<close>
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text \<open>
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  The @{text code} antiquotation allows to include constants from
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  generated code directly into ML system code, as in the following toy
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  example:
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\<close>
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datatype %quote form = T | F | And form form | Or form form (*<*)
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(*>*) ML %quotett \<open>
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  fun eval_form @{code T} = true
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    | eval_form @{code F} = false
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    | eval_form (@{code And} (p, q)) =
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        eval_form p andalso eval_form q
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    | eval_form (@{code Or} (p, q)) =
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        eval_form p orelse eval_form q;
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\<close>
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text \<open>
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  \noindent @{text code} takes as argument the name of a constant;
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  after the whole ML is read, the necessary code is generated
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  transparently and the corresponding constant names are inserted.
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  This technique also allows to use pattern matching on constructors
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  stemming from compiled datatypes.  Note that the @{text code}
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  antiquotation may not refer to constants which carry adaptations;
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  here you have to refer to the corresponding adapted code directly.
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  For a less simplistic example, theory @{text Approximation} in
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  the @{text Decision_Procs} session is a good reference.
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\<close>
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subsubsection \<open>Static embedding of generated code into system runtime -- @{text code_reflect}\<close>
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text \<open>
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  The @{text code} antiquoation is lightweight, but the generated code
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  is only accessible while the ML section is processed.  Sometimes this
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  is not appropriate, especially if the generated code contains datatype
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  declarations which are shared with other parts of the system.  In these
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  cases, @{command_def code_reflect} can be used:
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\<close>
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code_reflect %quote Sum_Type
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  datatypes sum = Inl | Inr
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  functions "Sum_Type.sum.projl" "Sum_Type.sum.projr"
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text \<open>
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  \noindent @{command_def code_reflect} takes a structure name and
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  references to datatypes and functions; for these code is compiled
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  into the named ML structure and the \emph{Eval} target is modified
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  in a way that future code generation will reference these
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  precompiled versions of the given datatypes and functions.  This
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  also allows to refer to the referenced datatypes and functions from
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  arbitrary ML code as well.
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  A typical example for @{command code_reflect} can be found in the
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  @{theory Predicate} theory.
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\<close>
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subsubsection \<open>Separate compilation -- @{text code_reflect}\<close>
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text \<open>
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  For technical reasons it is sometimes necessary to separate
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  generation and compilation of code which is supposed to be used in
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  the system runtime.  For this @{command code_reflect} with an
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  optional @{text "file"} argument can be used:
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\<close>
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code_reflect %quote Rat
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  datatypes rat = Frct
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  functions Fract
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    "(plus :: rat \<Rightarrow> rat \<Rightarrow> rat)" "(minus :: rat \<Rightarrow> rat \<Rightarrow> rat)"
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    "(times :: rat \<Rightarrow> rat \<Rightarrow> rat)" "(divide :: rat \<Rightarrow> rat \<Rightarrow> rat)"
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  file "$ISABELLE_TMP/examples/rat.ML"
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text \<open>
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  \noindent This merely generates the referenced code to the given
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  file which can be included into the system runtime later on.
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\<close>
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end
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