--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/IsaMakefile Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,33 @@
+
+## targets
+
+default: Thy
+images:
+test: Thy
+
+all: images test
+
+
+## global settings
+
+SRC = $(ISABELLE_HOME)/src
+OUT = $(ISABELLE_OUTPUT)
+LOG = $(OUT)/log
+
+USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
+
+
+## Thy
+
+THY = $(LOG)/HOL-Thy.gz
+
+Thy: $(THY)
+
+$(THY): Thy/ROOT.ML Thy/Setup.thy Thy/Classes.thy ../antiquote_setup.ML ../more_antiquote.ML
+ @$(USEDIR) HOL Thy
+
+
+## clean
+
+clean:
+ @rm -f $(THY)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/Makefile Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,35 @@
+
+## targets
+
+default: dvi
+
+
+## dependencies
+
+include ../Makefile.in
+
+NAME = classes
+
+FILES = $(NAME).tex classes.tex Thy/document/Classes.tex \
+ style.sty ../iman.sty ../extra.sty ../isar.sty \
+ ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty \
+ ../manual.bib ../proof.sty
+
+dvi: $(NAME).dvi
+
+$(NAME).dvi: $(FILES) isabelle_isar.eps
+ $(LATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(LATEX) $(NAME)
+ $(LATEX) $(NAME)
+
+pdf: $(NAME).pdf
+
+$(NAME).pdf: $(FILES) isabelle_isar.pdf
+ $(PDFLATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(FIXBOOKMARKS) $(NAME).out
+ $(PDFLATEX) $(NAME)
+ $(PDFLATEX) $(NAME)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/Thy/Classes.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,635 @@
+theory Classes
+imports Main Setup
+begin
+
+chapter {* Haskell-style classes with Isabelle/Isar *}
+
+section {* Introduction *}
+
+text {*
+ Type classes were introduces by Wadler and Blott \cite{wadler89how}
+ into the Haskell language, to allow for a reasonable implementation
+ of overloading\footnote{throughout this tutorial, we are referring
+ to classical Haskell 1.0 type classes, not considering
+ later additions in expressiveness}.
+ As a canonical example, a polymorphic equality function
+ @{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} which is overloaded on different
+ types for @{text "\<alpha>"}, which is achieved by splitting introduction
+ of the @{text eq} function from its overloaded definitions by means
+ of @{text class} and @{text instance} declarations:
+
+ \begin{quote}
+
+ \noindent@{text "class eq where"}\footnote{syntax here is a kind of isabellized Haskell} \\
+ \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
+
+ \medskip\noindent@{text "instance nat \<Colon> eq where"} \\
+ \hspace*{2ex}@{text "eq 0 0 = True"} \\
+ \hspace*{2ex}@{text "eq 0 _ = False"} \\
+ \hspace*{2ex}@{text "eq _ 0 = False"} \\
+ \hspace*{2ex}@{text "eq (Suc n) (Suc m) = eq n m"}
+
+ \medskip\noindent@{text "instance (\<alpha>\<Colon>eq, \<beta>\<Colon>eq) pair \<Colon> eq where"} \\
+ \hspace*{2ex}@{text "eq (x1, y1) (x2, y2) = eq x1 x2 \<and> eq y1 y2"}
+
+ \medskip\noindent@{text "class ord extends eq where"} \\
+ \hspace*{2ex}@{text "less_eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
+ \hspace*{2ex}@{text "less \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
+
+ \end{quote}
+
+ \noindent Type variables are annotated with (finitely many) classes;
+ these annotations are assertions that a particular polymorphic type
+ provides definitions for overloaded functions.
+
+ Indeed, type classes not only allow for simple overloading
+ but form a generic calculus, an instance of order-sorted
+ algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
+
+ From a software engeneering point of view, type classes
+ roughly correspond to interfaces in object-oriented languages like Java;
+ so, it is naturally desirable that type classes do not only
+ provide functions (class parameters) but also state specifications
+ implementations must obey. For example, the @{text "class eq"}
+ above could be given the following specification, demanding that
+ @{text "class eq"} is an equivalence relation obeying reflexivity,
+ symmetry and transitivity:
+
+ \begin{quote}
+
+ \noindent@{text "class eq where"} \\
+ \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
+ @{text "satisfying"} \\
+ \hspace*{2ex}@{text "refl: eq x x"} \\
+ \hspace*{2ex}@{text "sym: eq x y \<longleftrightarrow> eq x y"} \\
+ \hspace*{2ex}@{text "trans: eq x y \<and> eq y z \<longrightarrow> eq x z"}
+
+ \end{quote}
+
+ \noindent From a theoretic point of view, type classes are lightweight
+ modules; Haskell type classes may be emulated by
+ SML functors \cite{classes_modules}.
+ Isabelle/Isar offers a discipline of type classes which brings
+ all those aspects together:
+
+ \begin{enumerate}
+ \item specifying abstract parameters together with
+ corresponding specifications,
+ \item instantiating those abstract parameters by a particular
+ type
+ \item in connection with a ``less ad-hoc'' approach to overloading,
+ \item with a direct link to the Isabelle module system
+ (aka locales \cite{kammueller-locales}).
+ \end{enumerate}
+
+ \noindent Isar type classes also directly support code generation
+ in a Haskell like fashion.
+
+ This tutorial demonstrates common elements of structured specifications
+ and abstract reasoning with type classes by the algebraic hierarchy of
+ semigroups, monoids and groups. Our background theory is that of
+ Isabelle/HOL \cite{isa-tutorial}, for which some
+ familiarity is assumed.
+
+ Here we merely present the look-and-feel for end users.
+ Internally, those are mapped to more primitive Isabelle concepts.
+ See \cite{Haftmann-Wenzel:2006:classes} for more detail.
+*}
+
+section {* A simple algebra example \label{sec:example} *}
+
+subsection {* Class definition *}
+
+text {*
+ Depending on an arbitrary type @{text "\<alpha>"}, class @{text
+ "semigroup"} introduces a binary operator @{text "(\<otimes>)"} that is
+ assumed to be associative:
+*}
+
+class %quote semigroup =
+ fixes mult :: "\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" (infixl "\<otimes>" 70)
+ assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
+
+text {*
+ \noindent This @{command class} specification consists of two
+ parts: the \qn{operational} part names the class parameter
+ (@{element "fixes"}), the \qn{logical} part specifies properties on them
+ (@{element "assumes"}). The local @{element "fixes"} and
+ @{element "assumes"} are lifted to the theory toplevel,
+ yielding the global
+ parameter @{term [source] "mult \<Colon> \<alpha>\<Colon>semigroup \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"} and the
+ global theorem @{fact "semigroup.assoc:"}~@{prop [source] "\<And>x y
+ z \<Colon> \<alpha>\<Colon>semigroup. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"}.
+*}
+
+
+subsection {* Class instantiation \label{sec:class_inst} *}
+
+text {*
+ The concrete type @{typ int} is made a @{class semigroup}
+ instance by providing a suitable definition for the class parameter
+ @{text "(\<otimes>)"} and a proof for the specification of @{fact assoc}.
+ This is accomplished by the @{command instantiation} target:
+*}
+
+instantiation %quote int :: semigroup
+begin
+
+definition %quote
+ mult_int_def: "i \<otimes> j = i + (j\<Colon>int)"
+
+instance %quote proof
+ fix i j k :: int have "(i + j) + k = i + (j + k)" by simp
+ then show "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
+ unfolding mult_int_def .
+qed
+
+end %quote
+
+text {*
+ \noindent @{command instantiation} allows to define class parameters
+ at a particular instance using common specification tools (here,
+ @{command definition}). The concluding @{command instance}
+ opens a proof that the given parameters actually conform
+ to the class specification. Note that the first proof step
+ is the @{method default} method,
+ which for such instance proofs maps to the @{method intro_classes} method.
+ This boils down an instance judgement to the relevant primitive
+ proof goals and should conveniently always be the first method applied
+ in an instantiation proof.
+
+ From now on, the type-checker will consider @{typ int}
+ as a @{class semigroup} automatically, i.e.\ any general results
+ are immediately available on concrete instances.
+
+ \medskip Another instance of @{class semigroup} are the natural numbers:
+*}
+
+instantiation %quote nat :: semigroup
+begin
+
+primrec %quote mult_nat where
+ "(0\<Colon>nat) \<otimes> n = n"
+ | "Suc m \<otimes> n = Suc (m \<otimes> n)"
+
+instance %quote proof
+ fix m n q :: nat
+ show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
+ by (induct m) auto
+qed
+
+end %quote
+
+text {*
+ \noindent Note the occurence of the name @{text mult_nat}
+ in the primrec declaration; by default, the local name of
+ a class operation @{text f} to instantiate on type constructor
+ @{text \<kappa>} are mangled as @{text f_\<kappa>}. In case of uncertainty,
+ these names may be inspected using the @{command "print_context"} command
+ or the corresponding ProofGeneral button.
+*}
+
+subsection {* Lifting and parametric types *}
+
+text {*
+ Overloaded definitions giving on class instantiation
+ may include recursion over the syntactic structure of types.
+ As a canonical example, we model product semigroups
+ using our simple algebra:
+*}
+
+instantiation %quote * :: (semigroup, semigroup) semigroup
+begin
+
+definition %quote
+ mult_prod_def: "p\<^isub>1 \<otimes> p\<^isub>2 = (fst p\<^isub>1 \<otimes> fst p\<^isub>2, snd p\<^isub>1 \<otimes> snd p\<^isub>2)"
+
+instance %quote proof
+ fix p\<^isub>1 p\<^isub>2 p\<^isub>3 :: "\<alpha>\<Colon>semigroup \<times> \<beta>\<Colon>semigroup"
+ show "p\<^isub>1 \<otimes> p\<^isub>2 \<otimes> p\<^isub>3 = p\<^isub>1 \<otimes> (p\<^isub>2 \<otimes> p\<^isub>3)"
+ unfolding mult_prod_def by (simp add: assoc)
+qed
+
+end %quote
+
+text {*
+ \noindent Associativity from product semigroups is
+ established using
+ the definition of @{text "(\<otimes>)"} on products and the hypothetical
+ associativity of the type components; these hypotheses
+ are facts due to the @{class semigroup} constraints imposed
+ on the type components by the @{command instance} proposition.
+ Indeed, this pattern often occurs with parametric types
+ and type classes.
+*}
+
+
+subsection {* Subclassing *}
+
+text {*
+ We define a subclass @{text monoidl} (a semigroup with a left-hand neutral)
+ by extending @{class semigroup}
+ with one additional parameter @{text neutral} together
+ with its property:
+*}
+
+class %quote monoidl = semigroup +
+ fixes neutral :: "\<alpha>" ("\<one>")
+ assumes neutl: "\<one> \<otimes> x = x"
+
+text {*
+ \noindent Again, we prove some instances, by
+ providing suitable parameter definitions and proofs for the
+ additional specifications. Observe that instantiations
+ for types with the same arity may be simultaneous:
+*}
+
+instantiation %quote nat and int :: monoidl
+begin
+
+definition %quote
+ neutral_nat_def: "\<one> = (0\<Colon>nat)"
+
+definition %quote
+ neutral_int_def: "\<one> = (0\<Colon>int)"
+
+instance %quote proof
+ fix n :: nat
+ show "\<one> \<otimes> n = n"
+ unfolding neutral_nat_def by simp
+next
+ fix k :: int
+ show "\<one> \<otimes> k = k"
+ unfolding neutral_int_def mult_int_def by simp
+qed
+
+end %quote
+
+instantiation %quote * :: (monoidl, monoidl) monoidl
+begin
+
+definition %quote
+ neutral_prod_def: "\<one> = (\<one>, \<one>)"
+
+instance %quote proof
+ fix p :: "\<alpha>\<Colon>monoidl \<times> \<beta>\<Colon>monoidl"
+ show "\<one> \<otimes> p = p"
+ unfolding neutral_prod_def mult_prod_def by (simp add: neutl)
+qed
+
+end %quote
+
+text {*
+ \noindent Fully-fledged monoids are modelled by another subclass
+ which does not add new parameters but tightens the specification:
+*}
+
+class %quote monoid = monoidl +
+ assumes neutr: "x \<otimes> \<one> = x"
+
+instantiation %quote nat and int :: monoid
+begin
+
+instance %quote proof
+ fix n :: nat
+ show "n \<otimes> \<one> = n"
+ unfolding neutral_nat_def by (induct n) simp_all
+next
+ fix k :: int
+ show "k \<otimes> \<one> = k"
+ unfolding neutral_int_def mult_int_def by simp
+qed
+
+end %quote
+
+instantiation %quote * :: (monoid, monoid) monoid
+begin
+
+instance %quote proof
+ fix p :: "\<alpha>\<Colon>monoid \<times> \<beta>\<Colon>monoid"
+ show "p \<otimes> \<one> = p"
+ unfolding neutral_prod_def mult_prod_def by (simp add: neutr)
+qed
+
+end %quote
+
+text {*
+ \noindent To finish our small algebra example, we add a @{text group} class
+ with a corresponding instance:
+*}
+
+class %quote group = monoidl +
+ fixes inverse :: "\<alpha> \<Rightarrow> \<alpha>" ("(_\<div>)" [1000] 999)
+ assumes invl: "x\<div> \<otimes> x = \<one>"
+
+instantiation %quote int :: group
+begin
+
+definition %quote
+ inverse_int_def: "i\<div> = - (i\<Colon>int)"
+
+instance %quote proof
+ fix i :: int
+ have "-i + i = 0" by simp
+ then show "i\<div> \<otimes> i = \<one>"
+ unfolding mult_int_def neutral_int_def inverse_int_def .
+qed
+
+end %quote
+
+
+section {* Type classes as locales *}
+
+subsection {* A look behind the scene *}
+
+text {*
+ The example above gives an impression how Isar type classes work
+ in practice. As stated in the introduction, classes also provide
+ a link to Isar's locale system. Indeed, the logical core of a class
+ is nothing else than a locale:
+*}
+
+class %quote idem =
+ fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
+ assumes idem: "f (f x) = f x"
+
+text {*
+ \noindent essentially introduces the locale
+*} setup %invisible {* Sign.add_path "foo" *}
+
+locale %quote idem =
+ fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
+ assumes idem: "f (f x) = f x"
+
+text {* \noindent together with corresponding constant(s): *}
+
+consts %quote f :: "\<alpha> \<Rightarrow> \<alpha>"
+
+text {*
+ \noindent The connection to the type system is done by means
+ of a primitive axclass
+*} setup %invisible {* Sign.add_path "foo" *}
+
+axclass %quote idem < type
+ idem: "f (f x) = f x" setup %invisible {* Sign.parent_path *}
+
+text {* \noindent together with a corresponding interpretation: *}
+
+interpretation %quote idem_class:
+ idem "f \<Colon> (\<alpha>\<Colon>idem) \<Rightarrow> \<alpha>"
+proof qed (rule idem)
+
+text {*
+ \noindent This gives you at hand the full power of the Isabelle module system;
+ conclusions in locale @{text idem} are implicitly propagated
+ to class @{text idem}.
+*} setup %invisible {* Sign.parent_path *}
+
+subsection {* Abstract reasoning *}
+
+text {*
+ Isabelle locales enable reasoning at a general level, while results
+ are implicitly transferred to all instances. For example, we can
+ now establish the @{text "left_cancel"} lemma for groups, which
+ states that the function @{text "(x \<otimes>)"} is injective:
+*}
+
+lemma %quote (in group) left_cancel: "x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"
+proof
+ assume "x \<otimes> y = x \<otimes> z"
+ then have "x\<div> \<otimes> (x \<otimes> y) = x\<div> \<otimes> (x \<otimes> z)" by simp
+ then have "(x\<div> \<otimes> x) \<otimes> y = (x\<div> \<otimes> x) \<otimes> z" using assoc by simp
+ then show "y = z" using neutl and invl by simp
+next
+ assume "y = z"
+ then show "x \<otimes> y = x \<otimes> z" by simp
+qed
+
+text {*
+ \noindent Here the \qt{@{keyword "in"} @{class group}} target specification
+ indicates that the result is recorded within that context for later
+ use. This local theorem is also lifted to the global one @{fact
+ "group.left_cancel:"} @{prop [source] "\<And>x y z \<Colon> \<alpha>\<Colon>group. x \<otimes> y = x \<otimes>
+ z \<longleftrightarrow> y = z"}. Since type @{text "int"} has been made an instance of
+ @{text "group"} before, we may refer to that fact as well: @{prop
+ [source] "\<And>x y z \<Colon> int. x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"}.
+*}
+
+
+subsection {* Derived definitions *}
+
+text {*
+ Isabelle locales support a concept of local definitions
+ in locales:
+*}
+
+primrec %quote (in monoid) pow_nat :: "nat \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
+ "pow_nat 0 x = \<one>"
+ | "pow_nat (Suc n) x = x \<otimes> pow_nat n x"
+
+text {*
+ \noindent If the locale @{text group} is also a class, this local
+ definition is propagated onto a global definition of
+ @{term [source] "pow_nat \<Colon> nat \<Rightarrow> \<alpha>\<Colon>monoid \<Rightarrow> \<alpha>\<Colon>monoid"}
+ with corresponding theorems
+
+ @{thm pow_nat.simps [no_vars]}.
+
+ \noindent As you can see from this example, for local
+ definitions you may use any specification tool
+ which works together with locales (e.g. \cite{krauss2006}).
+*}
+
+
+subsection {* A functor analogy *}
+
+text {*
+ We introduced Isar classes by analogy to type classes
+ functional programming; if we reconsider this in the
+ context of what has been said about type classes and locales,
+ we can drive this analogy further by stating that type
+ classes essentially correspond to functors which have
+ a canonical interpretation as type classes.
+ Anyway, there is also the possibility of other interpretations.
+ For example, also @{text list}s form a monoid with
+ @{text append} and @{term "[]"} as operations, but it
+ seems inappropriate to apply to lists
+ the same operations as for genuinely algebraic types.
+ In such a case, we simply can do a particular interpretation
+ of monoids for lists:
+*}
+
+interpretation %quote list_monoid!: monoid append "[]"
+ proof qed auto
+
+text {*
+ \noindent This enables us to apply facts on monoids
+ to lists, e.g. @{thm list_monoid.neutl [no_vars]}.
+
+ When using this interpretation pattern, it may also
+ be appropriate to map derived definitions accordingly:
+*}
+
+primrec %quote replicate :: "nat \<Rightarrow> \<alpha> list \<Rightarrow> \<alpha> list" where
+ "replicate 0 _ = []"
+ | "replicate (Suc n) xs = xs @ replicate n xs"
+
+interpretation %quote list_monoid!: monoid append "[]" where
+ "monoid.pow_nat append [] = replicate"
+proof -
+ interpret monoid append "[]" ..
+ show "monoid.pow_nat append [] = replicate"
+ proof
+ fix n
+ show "monoid.pow_nat append [] n = replicate n"
+ by (induct n) auto
+ qed
+qed intro_locales
+
+
+subsection {* Additional subclass relations *}
+
+text {*
+ Any @{text "group"} is also a @{text "monoid"}; this
+ can be made explicit by claiming an additional
+ subclass relation,
+ together with a proof of the logical difference:
+*}
+
+subclass %quote (in group) monoid
+proof
+ fix x
+ from invl have "x\<div> \<otimes> x = \<one>" by simp
+ with assoc [symmetric] neutl invl have "x\<div> \<otimes> (x \<otimes> \<one>) = x\<div> \<otimes> x" by simp
+ with left_cancel show "x \<otimes> \<one> = x" by simp
+qed
+
+text {*
+ \noindent The logical proof is carried out on the locale level.
+ Afterwards it is propagated
+ to the type system, making @{text group} an instance of
+ @{text monoid} by adding an additional edge
+ to the graph of subclass relations
+ (cf.\ \figref{fig:subclass}).
+
+ \begin{figure}[htbp]
+ \begin{center}
+ \small
+ \unitlength 0.6mm
+ \begin{picture}(40,60)(0,0)
+ \put(20,60){\makebox(0,0){@{text semigroup}}}
+ \put(20,40){\makebox(0,0){@{text monoidl}}}
+ \put(00,20){\makebox(0,0){@{text monoid}}}
+ \put(40,00){\makebox(0,0){@{text group}}}
+ \put(20,55){\vector(0,-1){10}}
+ \put(15,35){\vector(-1,-1){10}}
+ \put(25,35){\vector(1,-3){10}}
+ \end{picture}
+ \hspace{8em}
+ \begin{picture}(40,60)(0,0)
+ \put(20,60){\makebox(0,0){@{text semigroup}}}
+ \put(20,40){\makebox(0,0){@{text monoidl}}}
+ \put(00,20){\makebox(0,0){@{text monoid}}}
+ \put(40,00){\makebox(0,0){@{text group}}}
+ \put(20,55){\vector(0,-1){10}}
+ \put(15,35){\vector(-1,-1){10}}
+ \put(05,15){\vector(3,-1){30}}
+ \end{picture}
+ \caption{Subclass relationship of monoids and groups:
+ before and after establishing the relationship
+ @{text "group \<subseteq> monoid"}; transitive edges are left out.}
+ \label{fig:subclass}
+ \end{center}
+ \end{figure}
+7
+ For illustration, a derived definition
+ in @{text group} which uses @{text pow_nat}:
+*}
+
+definition %quote (in group) pow_int :: "int \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
+ "pow_int k x = (if k >= 0
+ then pow_nat (nat k) x
+ else (pow_nat (nat (- k)) x)\<div>)"
+
+text {*
+ \noindent yields the global definition of
+ @{term [source] "pow_int \<Colon> int \<Rightarrow> \<alpha>\<Colon>group \<Rightarrow> \<alpha>\<Colon>group"}
+ with the corresponding theorem @{thm pow_int_def [no_vars]}.
+*}
+
+subsection {* A note on syntax *}
+
+text {*
+ As a commodity, class context syntax allows to refer
+ to local class operations and their global counterparts
+ uniformly; type inference resolves ambiguities. For example:
+*}
+
+context %quote semigroup
+begin
+
+term %quote "x \<otimes> y" -- {* example 1 *}
+term %quote "(x\<Colon>nat) \<otimes> y" -- {* example 2 *}
+
+end %quote
+
+term %quote "x \<otimes> y" -- {* example 3 *}
+
+text {*
+ \noindent Here in example 1, the term refers to the local class operation
+ @{text "mult [\<alpha>]"}, whereas in example 2 the type constraint
+ enforces the global class operation @{text "mult [nat]"}.
+ In the global context in example 3, the reference is
+ to the polymorphic global class operation @{text "mult [?\<alpha> \<Colon> semigroup]"}.
+*}
+
+section {* Further issues *}
+
+subsection {* Type classes and code generation *}
+
+text {*
+ Turning back to the first motivation for type classes,
+ namely overloading, it is obvious that overloading
+ stemming from @{command class} statements and
+ @{command instantiation}
+ targets naturally maps to Haskell type classes.
+ The code generator framework \cite{isabelle-codegen}
+ takes this into account. Concerning target languages
+ lacking type classes (e.g.~SML), type classes
+ are implemented by explicit dictionary construction.
+ As example, let's go back to the power function:
+*}
+
+definition %quote example :: int where
+ "example = pow_int 10 (-2)"
+
+text {*
+ \noindent This maps to Haskell as:
+*}
+
+text %quote {*@{code_stmts example (Haskell)}*}
+
+text {*
+ \noindent The whole code in SML with explicit dictionary passing:
+*}
+
+text %quote {*@{code_stmts example (SML)}*}
+
+subsection {* Inspecting the type class universe *}
+
+text {*
+ To facilitate orientation in complex subclass structures,
+ two diagnostics commands are provided:
+
+ \begin{description}
+
+ \item[@{command "print_classes"}] print a list of all classes
+ together with associated operations etc.
+
+ \item[@{command "class_deps"}] visualizes the subclass relation
+ between all classes as a Hasse diagram.
+
+ \end{description}
+*}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/Thy/ROOT.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,6 @@
+
+(* $Id$ *)
+
+no_document use_thy "Setup";
+
+use_thy "Classes";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/Thy/Setup.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,34 @@
+theory Setup
+imports Main Code_Integer
+uses
+ "../../../antiquote_setup"
+ "../../../more_antiquote"
+begin
+
+ML {* Code_Target.code_width := 74 *}
+
+syntax
+ "_alpha" :: "type" ("\<alpha>")
+ "_alpha_ofsort" :: "sort \<Rightarrow> type" ("\<alpha>()\<Colon>_" [0] 1000)
+ "_beta" :: "type" ("\<beta>")
+ "_beta_ofsort" :: "sort \<Rightarrow> type" ("\<beta>()\<Colon>_" [0] 1000)
+
+parse_ast_translation {*
+ let
+ fun alpha_ast_tr [] = Syntax.Variable "'a"
+ | alpha_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
+ fun alpha_ofsort_ast_tr [ast] =
+ Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'a", ast]
+ | alpha_ofsort_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
+ fun beta_ast_tr [] = Syntax.Variable "'b"
+ | beta_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
+ fun beta_ofsort_ast_tr [ast] =
+ Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'b", ast]
+ | beta_ofsort_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
+ in [
+ ("_alpha", alpha_ast_tr), ("_alpha_ofsort", alpha_ofsort_ast_tr),
+ ("_beta", beta_ast_tr), ("_beta_ofsort", beta_ofsort_ast_tr)
+ ] end
+*}
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/Thy/document/Classes.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,1346 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Classes}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{theory}\isamarkupfalse%
+\ Classes\isanewline
+\isakeyword{imports}\ Main\ Setup\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupchapter{Haskell-style classes with Isabelle/Isar%
+}
+\isamarkuptrue%
+%
+\isamarkupsection{Introduction%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Type classes were introduces by Wadler and Blott \cite{wadler89how}
+ into the Haskell language, to allow for a reasonable implementation
+ of overloading\footnote{throughout this tutorial, we are referring
+ to classical Haskell 1.0 type classes, not considering
+ later additions in expressiveness}.
+ As a canonical example, a polymorphic equality function
+ \isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} which is overloaded on different
+ types for \isa{{\isasymalpha}}, which is achieved by splitting introduction
+ of the \isa{eq} function from its overloaded definitions by means
+ of \isa{class} and \isa{instance} declarations:
+
+ \begin{quote}
+
+ \noindent\isa{class\ eq\ where}\footnote{syntax here is a kind of isabellized Haskell} \\
+ \hspace*{2ex}\isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool}
+
+ \medskip\noindent\isa{instance\ nat\ {\isasymColon}\ eq\ where} \\
+ \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isadigit{0}}\ {\isacharequal}\ True} \\
+ \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isacharunderscore}\ {\isacharequal}\ False} \\
+ \hspace*{2ex}\isa{eq\ {\isacharunderscore}\ {\isadigit{0}}\ {\isacharequal}\ False} \\
+ \hspace*{2ex}\isa{eq\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharparenleft}Suc\ m{\isacharparenright}\ {\isacharequal}\ eq\ n\ m}
+
+ \medskip\noindent\isa{instance\ {\isacharparenleft}{\isasymalpha}{\isasymColon}eq{\isacharcomma}\ {\isasymbeta}{\isasymColon}eq{\isacharparenright}\ pair\ {\isasymColon}\ eq\ where} \\
+ \hspace*{2ex}\isa{eq\ {\isacharparenleft}x{\isadigit{1}}{\isacharcomma}\ y{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ eq\ x{\isadigit{1}}\ x{\isadigit{2}}\ {\isasymand}\ eq\ y{\isadigit{1}}\ y{\isadigit{2}}}
+
+ \medskip\noindent\isa{class\ ord\ extends\ eq\ where} \\
+ \hspace*{2ex}\isa{less{\isacharunderscore}eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} \\
+ \hspace*{2ex}\isa{less\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool}
+
+ \end{quote}
+
+ \noindent Type variables are annotated with (finitely many) classes;
+ these annotations are assertions that a particular polymorphic type
+ provides definitions for overloaded functions.
+
+ Indeed, type classes not only allow for simple overloading
+ but form a generic calculus, an instance of order-sorted
+ algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
+
+ From a software engeneering point of view, type classes
+ roughly correspond to interfaces in object-oriented languages like Java;
+ so, it is naturally desirable that type classes do not only
+ provide functions (class parameters) but also state specifications
+ implementations must obey. For example, the \isa{class\ eq}
+ above could be given the following specification, demanding that
+ \isa{class\ eq} is an equivalence relation obeying reflexivity,
+ symmetry and transitivity:
+
+ \begin{quote}
+
+ \noindent\isa{class\ eq\ where} \\
+ \hspace*{2ex}\isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} \\
+ \isa{satisfying} \\
+ \hspace*{2ex}\isa{refl{\isacharcolon}\ eq\ x\ x} \\
+ \hspace*{2ex}\isa{sym{\isacharcolon}\ eq\ x\ y\ {\isasymlongleftrightarrow}\ eq\ x\ y} \\
+ \hspace*{2ex}\isa{trans{\isacharcolon}\ eq\ x\ y\ {\isasymand}\ eq\ y\ z\ {\isasymlongrightarrow}\ eq\ x\ z}
+
+ \end{quote}
+
+ \noindent From a theoretic point of view, type classes are lightweight
+ modules; Haskell type classes may be emulated by
+ SML functors \cite{classes_modules}.
+ Isabelle/Isar offers a discipline of type classes which brings
+ all those aspects together:
+
+ \begin{enumerate}
+ \item specifying abstract parameters together with
+ corresponding specifications,
+ \item instantiating those abstract parameters by a particular
+ type
+ \item in connection with a ``less ad-hoc'' approach to overloading,
+ \item with a direct link to the Isabelle module system
+ (aka locales \cite{kammueller-locales}).
+ \end{enumerate}
+
+ \noindent Isar type classes also directly support code generation
+ in a Haskell like fashion.
+
+ This tutorial demonstrates common elements of structured specifications
+ and abstract reasoning with type classes by the algebraic hierarchy of
+ semigroups, monoids and groups. Our background theory is that of
+ Isabelle/HOL \cite{isa-tutorial}, for which some
+ familiarity is assumed.
+
+ Here we merely present the look-and-feel for end users.
+ Internally, those are mapped to more primitive Isabelle concepts.
+ See \cite{Haftmann-Wenzel:2006:classes} for more detail.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{A simple algebra example \label{sec:example}%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Class definition%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Depending on an arbitrary type \isa{{\isasymalpha}}, class \isa{semigroup} introduces a binary operator \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} that is
+ assumed to be associative:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{class}\isamarkupfalse%
+\ semigroup\ {\isacharequal}\isanewline
+\ \ \isakeyword{fixes}\ mult\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymotimes}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} specification consists of two
+ parts: the \qn{operational} part names the class parameter
+ (\hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}}), the \qn{logical} part specifies properties on them
+ (\hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}}). The local \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}} and
+ \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}} are lifted to the theory toplevel,
+ yielding the global
+ parameter \isa{{\isachardoublequote}mult\ {\isasymColon}\ {\isasymalpha}{\isasymColon}semigroup\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequote}} and the
+ global theorem \hyperlink{fact.semigroup.assoc:}{\mbox{\isa{semigroup{\isachardot}assoc{\isacharcolon}}}}~\isa{{\isachardoublequote}{\isasymAnd}x\ y\ z\ {\isasymColon}\ {\isasymalpha}{\isasymColon}semigroup{\isachardot}\ {\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequote}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Class instantiation \label{sec:class_inst}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The concrete type \isa{int} is made a \isa{semigroup}
+ instance by providing a suitable definition for the class parameter
+ \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} and a proof for the specification of \hyperlink{fact.assoc}{\mbox{\isa{assoc}}}.
+ This is accomplished by the \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} target:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{instantiation}\isamarkupfalse%
+\ int\ {\isacharcolon}{\isacharcolon}\ semigroup\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ mult{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymotimes}\ j\ {\isacharequal}\ i\ {\isacharplus}\ {\isacharparenleft}j{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ i\ j\ k\ {\isacharcolon}{\isacharcolon}\ int\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isacharplus}\ j{\isacharparenright}\ {\isacharplus}\ k\ {\isacharequal}\ i\ {\isacharplus}\ {\isacharparenleft}j\ {\isacharplus}\ k{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isasymotimes}\ j{\isacharparenright}\ {\isasymotimes}\ k\ {\isacharequal}\ i\ {\isasymotimes}\ {\isacharparenleft}j\ {\isasymotimes}\ k{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} allows to define class parameters
+ at a particular instance using common specification tools (here,
+ \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}). The concluding \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}
+ opens a proof that the given parameters actually conform
+ to the class specification. Note that the first proof step
+ is the \hyperlink{method.default}{\mbox{\isa{default}}} method,
+ which for such instance proofs maps to the \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}} method.
+ This boils down an instance judgement to the relevant primitive
+ proof goals and should conveniently always be the first method applied
+ in an instantiation proof.
+
+ From now on, the type-checker will consider \isa{int}
+ as a \isa{semigroup} automatically, i.e.\ any general results
+ are immediately available on concrete instances.
+
+ \medskip Another instance of \isa{semigroup} are the natural numbers:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{instantiation}\isamarkupfalse%
+\ nat\ {\isacharcolon}{\isacharcolon}\ semigroup\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{primrec}\isamarkupfalse%
+\ mult{\isacharunderscore}nat\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}Suc\ m\ {\isasymotimes}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}m\ {\isasymotimes}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ m\ n\ q\ {\isacharcolon}{\isacharcolon}\ nat\ \isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}m\ {\isasymotimes}\ n\ {\isasymotimes}\ q\ {\isacharequal}\ m\ {\isasymotimes}\ {\isacharparenleft}n\ {\isasymotimes}\ q{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ m{\isacharparenright}\ auto\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Note the occurence of the name \isa{mult{\isacharunderscore}nat}
+ in the primrec declaration; by default, the local name of
+ a class operation \isa{f} to instantiate on type constructor
+ \isa{{\isasymkappa}} are mangled as \isa{f{\isacharunderscore}{\isasymkappa}}. In case of uncertainty,
+ these names may be inspected using the \hyperlink{command.print-context}{\mbox{\isa{\isacommand{print{\isacharunderscore}context}}}} command
+ or the corresponding ProofGeneral button.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Lifting and parametric types%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Overloaded definitions giving on class instantiation
+ may include recursion over the syntactic structure of types.
+ As a canonical example, we model product semigroups
+ using our simple algebra:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{instantiation}\isamarkupfalse%
+\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ mult{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}fst\ p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ fst\ p\isactrlisub {\isadigit{2}}{\isacharcomma}\ snd\ p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ snd\ p\isactrlisub {\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ p\isactrlisub {\isadigit{1}}\ p\isactrlisub {\isadigit{2}}\ p\isactrlisub {\isadigit{3}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}semigroup\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}semigroup{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{2}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{3}}\ {\isacharequal}\ p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ {\isacharparenleft}p\isactrlisub {\isadigit{2}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{3}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
+\isacommand{qed}\isamarkupfalse%
+\ \ \ \ \ \ \isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Associativity from product semigroups is
+ established using
+ the definition of \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} on products and the hypothetical
+ associativity of the type components; these hypotheses
+ are facts due to the \isa{semigroup} constraints imposed
+ on the type components by the \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} proposition.
+ Indeed, this pattern often occurs with parametric types
+ and type classes.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Subclassing%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+We define a subclass \isa{monoidl} (a semigroup with a left-hand neutral)
+ by extending \isa{semigroup}
+ with one additional parameter \isa{neutral} together
+ with its property:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{class}\isamarkupfalse%
+\ monoidl\ {\isacharequal}\ semigroup\ {\isacharplus}\isanewline
+\ \ \isakeyword{fixes}\ neutral\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ neutl{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Again, we prove some instances, by
+ providing suitable parameter definitions and proofs for the
+ additional specifications. Observe that instantiations
+ for types with the same arity may be simultaneous:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{instantiation}\isamarkupfalse%
+\ nat\ \isakeyword{and}\ int\ {\isacharcolon}{\isacharcolon}\ monoidl\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ neutral{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ n\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ neutral{\isacharunderscore}nat{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ k\ {\isacharcolon}{\isacharcolon}\ int\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ k\ {\isacharequal}\ k{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ neutral{\isacharunderscore}int{\isacharunderscore}def\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{instantiation}\isamarkupfalse%
+\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}monoidl{\isacharcomma}\ monoidl{\isacharparenright}\ monoidl\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ neutral{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isasymone}{\isacharcomma}\ {\isasymone}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ p\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}monoidl\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}monoidl{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ neutral{\isacharunderscore}prod{\isacharunderscore}def\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutl{\isacharparenright}\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Fully-fledged monoids are modelled by another subclass
+ which does not add new parameters but tightens the specification:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{class}\isamarkupfalse%
+\ monoid\ {\isacharequal}\ monoidl\ {\isacharplus}\isanewline
+\ \ \isakeyword{assumes}\ neutr{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instantiation}\isamarkupfalse%
+\ nat\ \isakeyword{and}\ int\ {\isacharcolon}{\isacharcolon}\ monoid\ \isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ n\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}n\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ neutral{\isacharunderscore}nat{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ n{\isacharparenright}\ simp{\isacharunderscore}all\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ k\ {\isacharcolon}{\isacharcolon}\ int\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}k\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ k{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ neutral{\isacharunderscore}int{\isacharunderscore}def\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{instantiation}\isamarkupfalse%
+\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}monoid{\isacharcomma}\ monoid{\isacharparenright}\ monoid\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\ \isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ p\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}monoid\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}monoid{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}p\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ neutral{\isacharunderscore}prod{\isacharunderscore}def\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutr{\isacharparenright}\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent To finish our small algebra example, we add a \isa{group} class
+ with a corresponding instance:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{class}\isamarkupfalse%
+\ group\ {\isacharequal}\ monoidl\ {\isacharplus}\isanewline
+\ \ \isakeyword{fixes}\ inverse\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}{\isachardoublequoteopen}{\isacharparenleft}{\isacharunderscore}{\isasymdiv}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ invl{\isacharcolon}\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instantiation}\isamarkupfalse%
+\ int\ {\isacharcolon}{\isacharcolon}\ group\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ inverse{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}i{\isasymdiv}\ {\isacharequal}\ {\isacharminus}\ {\isacharparenleft}i{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ i\ {\isacharcolon}{\isacharcolon}\ int\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharminus}i\ {\isacharplus}\ i\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}i{\isasymdiv}\ {\isasymotimes}\ i\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
+\ mult{\isacharunderscore}int{\isacharunderscore}def\ neutral{\isacharunderscore}int{\isacharunderscore}def\ inverse{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isamarkupsection{Type classes as locales%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{A look behind the scene%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The example above gives an impression how Isar type classes work
+ in practice. As stated in the introduction, classes also provide
+ a link to Isar's locale system. Indeed, the logical core of a class
+ is nothing else than a locale:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{class}\isamarkupfalse%
+\ idem\ {\isacharequal}\isanewline
+\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{assumes}\ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent essentially introduces the locale%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadeliminvisible
+\ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{setup}\isamarkupfalse%
+\ {\isacharverbatimopen}\ Sign{\isachardot}add{\isacharunderscore}path\ {\isachardoublequote}foo{\isachardoublequote}\ {\isacharverbatimclose}%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+\isanewline
+%
+\isadelimquote
+\isanewline
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{locale}\isamarkupfalse%
+\ idem\ {\isacharequal}\isanewline
+\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{assumes}\ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent together with corresponding constant(s):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{consts}\isamarkupfalse%
+\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The connection to the type system is done by means
+ of a primitive axclass%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadeliminvisible
+\ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{setup}\isamarkupfalse%
+\ {\isacharverbatimopen}\ Sign{\isachardot}add{\isacharunderscore}path\ {\isachardoublequote}foo{\isachardoublequote}\ {\isacharverbatimclose}%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+\isanewline
+%
+\isadelimquote
+\isanewline
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{axclass}\isamarkupfalse%
+\ idem\ {\isacharless}\ type\isanewline
+\ \ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isadeliminvisible
+\ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{setup}\isamarkupfalse%
+\ {\isacharverbatimopen}\ Sign{\isachardot}parent{\isacharunderscore}path\ {\isacharverbatimclose}%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\begin{isamarkuptext}%
+\noindent together with a corresponding interpretation:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{interpretation}\isamarkupfalse%
+\ idem{\isacharunderscore}class{\isacharcolon}\isanewline
+\ \ idem\ {\isachardoublequoteopen}f\ {\isasymColon}\ {\isacharparenleft}{\isasymalpha}{\isasymColon}idem{\isacharparenright}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ \isacommand{qed}\isamarkupfalse%
+\ {\isacharparenleft}rule\ idem{\isacharparenright}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This gives you at hand the full power of the Isabelle module system;
+ conclusions in locale \isa{idem} are implicitly propagated
+ to class \isa{idem}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadeliminvisible
+\ %
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{setup}\isamarkupfalse%
+\ {\isacharverbatimopen}\ Sign{\isachardot}parent{\isacharunderscore}path\ {\isacharverbatimclose}%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isamarkupsubsection{Abstract reasoning%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle locales enable reasoning at a general level, while results
+ are implicitly transferred to all instances. For example, we can
+ now establish the \isa{left{\isacharunderscore}cancel} lemma for groups, which
+ states that the function \isa{{\isacharparenleft}x\ {\isasymotimes}{\isacharparenright}} is injective:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ left{\isacharunderscore}cancel{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z\ {\isasymlongleftrightarrow}\ y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isacharequal}\ x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasymdiv}\ {\isasymotimes}\ x{\isacharparenright}\ {\isasymotimes}\ y\ {\isacharequal}\ {\isacharparenleft}x{\isasymdiv}\ {\isasymotimes}\ x{\isacharparenright}\ {\isasymotimes}\ z{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
+\ assoc\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
+\ neutl\ \isakeyword{and}\ invl\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Here the \qt{\hyperlink{keyword.in}{\mbox{\isa{\isakeyword{in}}}} \isa{group}} target specification
+ indicates that the result is recorded within that context for later
+ use. This local theorem is also lifted to the global one \hyperlink{fact.group.left-cancel:}{\mbox{\isa{group{\isachardot}left{\isacharunderscore}cancel{\isacharcolon}}}} \isa{{\isachardoublequote}{\isasymAnd}x\ y\ z\ {\isasymColon}\ {\isasymalpha}{\isasymColon}group{\isachardot}\ x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z\ {\isasymlongleftrightarrow}\ y\ {\isacharequal}\ z{\isachardoublequote}}. Since type \isa{int} has been made an instance of
+ \isa{group} before, we may refer to that fact as well: \isa{{\isachardoublequote}{\isasymAnd}x\ y\ z\ {\isasymColon}\ int{\isachardot}\ x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z\ {\isasymlongleftrightarrow}\ y\ {\isacharequal}\ z{\isachardoublequote}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Derived definitions%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle locales support a concept of local definitions
+ in locales:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{primrec}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ monoid{\isacharparenright}\ pow{\isacharunderscore}nat\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}pow{\isacharunderscore}nat\ {\isadigit{0}}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}pow{\isacharunderscore}nat\ {\isacharparenleft}Suc\ n{\isacharparenright}\ x\ {\isacharequal}\ x\ {\isasymotimes}\ pow{\isacharunderscore}nat\ n\ x{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent If the locale \isa{group} is also a class, this local
+ definition is propagated onto a global definition of
+ \isa{{\isachardoublequote}pow{\isacharunderscore}nat\ {\isasymColon}\ nat\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}monoid\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}monoid{\isachardoublequote}}
+ with corresponding theorems
+
+ \isa{pow{\isacharunderscore}nat\ {\isadigit{0}}\ x\ {\isacharequal}\ {\isasymone}\isasep\isanewline%
+pow{\isacharunderscore}nat\ {\isacharparenleft}Suc\ n{\isacharparenright}\ x\ {\isacharequal}\ x\ {\isasymotimes}\ pow{\isacharunderscore}nat\ n\ x}.
+
+ \noindent As you can see from this example, for local
+ definitions you may use any specification tool
+ which works together with locales (e.g. \cite{krauss2006}).%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{A functor analogy%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+We introduced Isar classes by analogy to type classes
+ functional programming; if we reconsider this in the
+ context of what has been said about type classes and locales,
+ we can drive this analogy further by stating that type
+ classes essentially correspond to functors which have
+ a canonical interpretation as type classes.
+ Anyway, there is also the possibility of other interpretations.
+ For example, also \isa{list}s form a monoid with
+ \isa{append} and \isa{{\isacharbrackleft}{\isacharbrackright}} as operations, but it
+ seems inappropriate to apply to lists
+ the same operations as for genuinely algebraic types.
+ In such a case, we simply can do a particular interpretation
+ of monoids for lists:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{interpretation}\isamarkupfalse%
+\ list{\isacharunderscore}monoid{\isacharbang}{\isacharcolon}\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ \isacommand{qed}\isamarkupfalse%
+\ auto%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This enables us to apply facts on monoids
+ to lists, e.g. \isa{{\isacharbrackleft}{\isacharbrackright}\ {\isacharat}\ x\ {\isacharequal}\ x}.
+
+ When using this interpretation pattern, it may also
+ be appropriate to map derived definitions accordingly:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{primrec}\isamarkupfalse%
+\ replicate\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isasymalpha}\ list\ {\isasymRightarrow}\ {\isasymalpha}\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}replicate\ {\isadigit{0}}\ {\isacharunderscore}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}replicate\ {\isacharparenleft}Suc\ n{\isacharparenright}\ xs\ {\isacharequal}\ xs\ {\isacharat}\ replicate\ n\ xs{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{interpretation}\isamarkupfalse%
+\ list{\isacharunderscore}monoid{\isacharbang}{\isacharcolon}\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ replicate{\isachardoublequoteclose}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \isacommand{interpret}\isamarkupfalse%
+\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ replicate{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ n\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ n\ {\isacharequal}\ replicate\ n{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ n{\isacharparenright}\ auto\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+\ intro{\isacharunderscore}locales%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isamarkupsubsection{Additional subclass relations%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Any \isa{group} is also a \isa{monoid}; this
+ can be made explicit by claiming an additional
+ subclass relation,
+ together with a proof of the logical difference:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{subclass}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ monoid\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ x\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ invl\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{with}\isamarkupfalse%
+\ assoc\ {\isacharbrackleft}symmetric{\isacharbrackright}\ neutl\ invl\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ {\isasymone}{\isacharparenright}\ {\isacharequal}\ x{\isasymdiv}\ {\isasymotimes}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{with}\isamarkupfalse%
+\ left{\isacharunderscore}cancel\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The logical proof is carried out on the locale level.
+ Afterwards it is propagated
+ to the type system, making \isa{group} an instance of
+ \isa{monoid} by adding an additional edge
+ to the graph of subclass relations
+ (cf.\ \figref{fig:subclass}).
+
+ \begin{figure}[htbp]
+ \begin{center}
+ \small
+ \unitlength 0.6mm
+ \begin{picture}(40,60)(0,0)
+ \put(20,60){\makebox(0,0){\isa{semigroup}}}
+ \put(20,40){\makebox(0,0){\isa{monoidl}}}
+ \put(00,20){\makebox(0,0){\isa{monoid}}}
+ \put(40,00){\makebox(0,0){\isa{group}}}
+ \put(20,55){\vector(0,-1){10}}
+ \put(15,35){\vector(-1,-1){10}}
+ \put(25,35){\vector(1,-3){10}}
+ \end{picture}
+ \hspace{8em}
+ \begin{picture}(40,60)(0,0)
+ \put(20,60){\makebox(0,0){\isa{semigroup}}}
+ \put(20,40){\makebox(0,0){\isa{monoidl}}}
+ \put(00,20){\makebox(0,0){\isa{monoid}}}
+ \put(40,00){\makebox(0,0){\isa{group}}}
+ \put(20,55){\vector(0,-1){10}}
+ \put(15,35){\vector(-1,-1){10}}
+ \put(05,15){\vector(3,-1){30}}
+ \end{picture}
+ \caption{Subclass relationship of monoids and groups:
+ before and after establishing the relationship
+ \isa{group\ {\isasymsubseteq}\ monoid}; transitive edges left out.}
+ \label{fig:subclass}
+ \end{center}
+ \end{figure}
+7
+ For illustration, a derived definition
+ in \isa{group} which uses \isa{pow{\isacharunderscore}nat}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{definition}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ pow{\isacharunderscore}int\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}int\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}pow{\isacharunderscore}int\ k\ x\ {\isacharequal}\ {\isacharparenleft}if\ k\ {\isachargreater}{\isacharequal}\ {\isadigit{0}}\isanewline
+\ \ \ \ then\ pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ k{\isacharparenright}\ x\isanewline
+\ \ \ \ else\ {\isacharparenleft}pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ {\isacharparenleft}{\isacharminus}\ k{\isacharparenright}{\isacharparenright}\ x{\isacharparenright}{\isasymdiv}{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent yields the global definition of
+ \isa{{\isachardoublequote}pow{\isacharunderscore}int\ {\isasymColon}\ int\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}group\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}group{\isachardoublequote}}
+ with the corresponding theorem \isa{pow{\isacharunderscore}int\ k\ x\ {\isacharequal}\ {\isacharparenleft}if\ {\isadigit{0}}\ {\isasymle}\ k\ then\ pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ k{\isacharparenright}\ x\ else\ {\isacharparenleft}pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ {\isacharparenleft}{\isacharminus}\ k{\isacharparenright}{\isacharparenright}\ x{\isacharparenright}{\isasymdiv}{\isacharparenright}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{A note on syntax%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+As a commodity, class context syntax allows to refer
+ to local class operations and their global counterparts
+ uniformly; type inference resolves ambiguities. For example:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{context}\isamarkupfalse%
+\ semigroup\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{term}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
+\isamarkupcmt{example 1%
+}
+\isanewline
+\isacommand{term}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasymColon}nat{\isacharparenright}\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
+\isamarkupcmt{example 2%
+}
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{term}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
+\isamarkupcmt{example 3%
+}
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Here in example 1, the term refers to the local class operation
+ \isa{mult\ {\isacharbrackleft}{\isasymalpha}{\isacharbrackright}}, whereas in example 2 the type constraint
+ enforces the global class operation \isa{mult\ {\isacharbrackleft}nat{\isacharbrackright}}.
+ In the global context in example 3, the reference is
+ to the polymorphic global class operation \isa{mult\ {\isacharbrackleft}{\isacharquery}{\isasymalpha}\ {\isasymColon}\ semigroup{\isacharbrackright}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Further issues%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Type classes and code generation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Turning back to the first motivation for type classes,
+ namely overloading, it is obvious that overloading
+ stemming from \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} statements and
+ \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}}
+ targets naturally maps to Haskell type classes.
+ The code generator framework \cite{isabelle-codegen}
+ takes this into account. Concerning target languages
+ lacking type classes (e.g.~SML), type classes
+ are implemented by explicit dictionary construction.
+ As example, let's go back to the power function:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{definition}\isamarkupfalse%
+\ example\ {\isacharcolon}{\isacharcolon}\ int\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}example\ {\isacharequal}\ pow{\isacharunderscore}int\ {\isadigit{1}}{\isadigit{0}}\ {\isacharparenleft}{\isacharminus}{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This maps to Haskell as:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}module Example where {\char123}\\
+\hspace*{0pt}\\
+\hspace*{0pt}\\
+\hspace*{0pt}data Nat = Zero{\char95}nat | Suc Nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}nat{\char95}aux ::~Integer -> Nat -> Nat;\\
+\hspace*{0pt}nat{\char95}aux i n = (if i <= 0 then n else nat{\char95}aux (i - 1) (Suc n));\\
+\hspace*{0pt}\\
+\hspace*{0pt}nat ::~Integer -> Nat;\\
+\hspace*{0pt}nat i = nat{\char95}aux i Zero{\char95}nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}class Semigroup a where {\char123}\\
+\hspace*{0pt} ~mult ::~a -> a -> a;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}class (Semigroup a) => Monoidl a where {\char123}\\
+\hspace*{0pt} ~neutral ::~a;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}class (Monoidl a) => Monoid a where {\char123}\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}class (Monoid a) => Group a where {\char123}\\
+\hspace*{0pt} ~inverse ::~a -> a;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}inverse{\char95}int ::~Integer -> Integer;\\
+\hspace*{0pt}inverse{\char95}int i = negate i;\\
+\hspace*{0pt}\\
+\hspace*{0pt}neutral{\char95}int ::~Integer;\\
+\hspace*{0pt}neutral{\char95}int = 0;\\
+\hspace*{0pt}\\
+\hspace*{0pt}mult{\char95}int ::~Integer -> Integer -> Integer;\\
+\hspace*{0pt}mult{\char95}int i j = i + j;\\
+\hspace*{0pt}\\
+\hspace*{0pt}instance Semigroup Integer where {\char123}\\
+\hspace*{0pt} ~mult = mult{\char95}int;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}instance Monoidl Integer where {\char123}\\
+\hspace*{0pt} ~neutral = neutral{\char95}int;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}instance Monoid Integer where {\char123}\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}instance Group Integer where {\char123}\\
+\hspace*{0pt} ~inverse = inverse{\char95}int;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}pow{\char95}nat ::~forall a.~(Monoid a) => Nat -> a -> a;\\
+\hspace*{0pt}pow{\char95}nat Zero{\char95}nat x = neutral;\\
+\hspace*{0pt}pow{\char95}nat (Suc n) x = mult x (pow{\char95}nat n x);\\
+\hspace*{0pt}\\
+\hspace*{0pt}pow{\char95}int ::~forall a.~(Group a) => Integer -> a -> a;\\
+\hspace*{0pt}pow{\char95}int k x =\\
+\hspace*{0pt} ~(if 0 <= k then pow{\char95}nat (nat k) x\\
+\hspace*{0pt} ~~~else inverse (pow{\char95}nat (nat (negate k)) x));\\
+\hspace*{0pt}\\
+\hspace*{0pt}example ::~Integer;\\
+\hspace*{0pt}example = pow{\char95}int 10 (-2);\\
+\hspace*{0pt}\\
+\hspace*{0pt}{\char125}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The whole code in SML with explicit dictionary passing:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun nat{\char95}aux i n =\\
+\hspace*{0pt} ~(if IntInf.<= (i,~(0 :~IntInf.int)) then n\\
+\hspace*{0pt} ~~~else nat{\char95}aux (IntInf.- (i,~(1 :~IntInf.int))) (Suc n));\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun nat i = nat{\char95}aux i Zero{\char95}nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a semigroup = {\char123}mult :~'a -> 'a -> 'a{\char125};\\
+\hspace*{0pt}fun mult (A{\char95}:'a semigroup) = {\char35}mult A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a monoidl =\\
+\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}semigroup{\char95}monoidl :~'a semigroup,~neutral :~'a{\char125};\\
+\hspace*{0pt}fun semigroup{\char95}monoidl (A{\char95}:'a monoidl) = {\char35}Classes{\char95}{\char95}semigroup{\char95}monoidl A{\char95};\\
+\hspace*{0pt}fun neutral (A{\char95}:'a monoidl) = {\char35}neutral A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a monoid = {\char123}Classes{\char95}{\char95}monoidl{\char95}monoid :~'a monoidl{\char125};\\
+\hspace*{0pt}fun monoidl{\char95}monoid (A{\char95}:'a monoid) = {\char35}Classes{\char95}{\char95}monoidl{\char95}monoid A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a group = {\char123}Classes{\char95}{\char95}monoid{\char95}group :~'a monoid,~inverse :~'a -> 'a{\char125};\\
+\hspace*{0pt}fun monoid{\char95}group (A{\char95}:'a group) = {\char35}Classes{\char95}{\char95}monoid{\char95}group A{\char95};\\
+\hspace*{0pt}fun inverse (A{\char95}:'a group) = {\char35}inverse A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun inverse{\char95}int i = IntInf.{\char126}~i;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val neutral{\char95}int :~IntInf.int = (0 :~IntInf.int)\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun mult{\char95}int i j = IntInf.+ (i,~j);\\
+\hspace*{0pt}\\
+\hspace*{0pt}val semigroup{\char95}int = {\char123}mult = mult{\char95}int{\char125}~:~IntInf.int semigroup;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val monoidl{\char95}int =\\
+\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}semigroup{\char95}monoidl = semigroup{\char95}int,~neutral = neutral{\char95}int{\char125}~:\\
+\hspace*{0pt} ~IntInf.int monoidl;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val monoid{\char95}int = {\char123}Classes{\char95}{\char95}monoidl{\char95}monoid = monoidl{\char95}int{\char125}~:\\
+\hspace*{0pt} ~IntInf.int monoid;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val group{\char95}int =\\
+\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}monoid{\char95}group = monoid{\char95}int,~inverse = inverse{\char95}int{\char125}~:\\
+\hspace*{0pt} ~IntInf.int group;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun pow{\char95}nat A{\char95}~Zero{\char95}nat x = neutral (monoidl{\char95}monoid A{\char95})\\
+\hspace*{0pt} ~| pow{\char95}nat A{\char95}~(Suc n) x =\\
+\hspace*{0pt} ~~~mult ((semigroup{\char95}monoidl o monoidl{\char95}monoid) A{\char95}) x (pow{\char95}nat A{\char95}~n x);\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun pow{\char95}int A{\char95}~k x =\\
+\hspace*{0pt} ~(if IntInf.<= ((0 :~IntInf.int),~k)\\
+\hspace*{0pt} ~~~then pow{\char95}nat (monoid{\char95}group A{\char95}) (nat k) x\\
+\hspace*{0pt} ~~~else inverse A{\char95}~(pow{\char95}nat (monoid{\char95}group A{\char95}) (nat (IntInf.{\char126}~k)) x));\\
+\hspace*{0pt}\\
+\hspace*{0pt}val example :~IntInf.int =\\
+\hspace*{0pt} ~pow{\char95}int group{\char95}int (10 :~IntInf.int) ({\char126}2 :~IntInf.int)\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isamarkupsubsection{Inspecting the type class universe%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+To facilitate orientation in complex subclass structures,
+ two diagnostics commands are provided:
+
+ \begin{description}
+
+ \item[\hyperlink{command.print-classes}{\mbox{\isa{\isacommand{print{\isacharunderscore}classes}}}}] print a list of all classes
+ together with associated operations etc.
+
+ \item[\hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isacharunderscore}deps}}}}] visualizes the subclass relation
+ between all classes as a Hasse diagram.
+
+ \end{description}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/classes.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,50 @@
+
+\documentclass[12pt,a4paper,fleqn]{report}
+\usepackage{latexsym,graphicx}
+\usepackage[refpage]{nomencl}
+\usepackage{../iman,../extra,../isar,../proof}
+\usepackage{../isabelle,../isabellesym}
+\usepackage{style}
+\usepackage{../pdfsetup}
+
+
+\hyphenation{Isabelle}
+\hyphenation{Isar}
+\isadroptag{theory}
+
+\title{\includegraphics[scale=0.5]{isabelle_isar}
+ \\[4ex] Haskell-style type classes with Isabelle/Isar}
+\author{\emph{Florian Haftmann}}
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+ This tutorial introduces the look-and-feel of Isar type classes
+ to the end-user; Isar type classes are a convenient mechanism
+ for organizing specifications, overcoming some drawbacks
+ of raw axiomatic type classes. Essentially, they combine
+ an operational aspect (in the manner of Haskell) with
+ a logical aspect, both managed uniformly.
+\end{abstract}
+
+\thispagestyle{empty}\clearpage
+
+\pagenumbering{roman}
+\clearfirst
+
+\input{Thy/document/Classes.tex}
+
+\begingroup
+\bibliographystyle{plain} \small\raggedright\frenchspacing
+\bibliography{../manual}
+\endgroup
+
+\end{document}
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Classes/style.sty Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,43 @@
+
+%% toc
+\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
+\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
+
+%% references
+\newcommand{\secref}[1]{\S\ref{#1}}
+\newcommand{\figref}[1]{figure~\ref{#1}}
+
+%% logical markup
+\newcommand{\strong}[1]{{\bfseries {#1}}}
+\newcommand{\qn}[1]{\emph{#1}}
+
+%% typographic conventions
+\newcommand{\qt}[1]{``{#1}''}
+
+%% verbatim text
+\newcommand{\isatypewriter}{\fontsize{9pt}{0pt}\tt\renewcommand{\baselinestretch}{1}\setlength{\baselineskip}{9pt}}
+
+%% quoted segments
+\makeatletter
+\isakeeptag{quote}
+\newenvironment{quotesegment}{\begin{quote}\isa@parindent\parindent\parindent0pt\isa@parskip\parskip\parskip0pt}{\end{quote}}
+\renewcommand{\isatagquote}{\begin{quotesegment}}
+\renewcommand{\endisatagquote}{\end{quotesegment}}
+\makeatother
+
+%% presentation
+\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
+
+\pagestyle{headings}
+\binperiod
+\underscoreoff
+
+\renewcommand{\isadigit}[1]{\isamath{#1}}
+
+\isabellestyle{it}
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "implementation"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/IsaMakefile Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,33 @@
+
+## targets
+
+default: Thy
+images:
+test: Thy
+
+all: images test
+
+
+## global settings
+
+SRC = $(ISABELLE_HOME)/src
+OUT = $(ISABELLE_OUTPUT)
+LOG = $(OUT)/log
+
+USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
+
+
+## Thy
+
+THY = $(LOG)/HOL-Thy.gz
+
+Thy: $(THY)
+
+$(THY): Thy/ROOT.ML Thy/*.thy ../antiquote_setup.ML ../more_antiquote.ML
+ @$(USEDIR) HOL Thy
+
+
+## clean
+
+clean:
+ @rm -f $(THY)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Makefile Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,35 @@
+
+## targets
+
+default: dvi
+
+
+## dependencies
+
+include ../Makefile.in
+
+NAME = codegen
+
+FILES = $(NAME).tex Thy/document/*.tex \
+ style.sty ../iman.sty ../extra.sty ../isar.sty \
+ ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty \
+ ../manual.bib ../proof.sty
+
+dvi: $(NAME).dvi
+
+$(NAME).dvi: $(FILES) isabelle_isar.eps codegen_process.ps
+ $(LATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(LATEX) $(NAME)
+ $(LATEX) $(NAME)
+
+pdf: $(NAME).pdf
+
+$(NAME).pdf: $(FILES) isabelle_isar.pdf codegen_process.pdf
+ $(PDFLATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(FIXBOOKMARKS) $(NAME).out
+ $(PDFLATEX) $(NAME)
+ $(PDFLATEX) $(NAME)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/Adaption.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,361 @@
+theory Adaption
+imports Setup
+begin
+
+setup %invisible {* Code_Target.extend_target ("\<SML>", ("SML", K I)) *}
+
+section {* Adaption to target languages \label{sec:adaption} *}
+
+subsection {* Adapting code generation *}
+
+text {*
+ The aspects of code generation introduced so far have two aspects
+ in common:
+
+ \begin{itemize}
+ \item They act uniformly, without reference to a specific
+ target language.
+ \item They are \emph{safe} in the sense that as long as you trust
+ the code generator meta theory and implementation, you cannot
+ produce programs that yield results which are not derivable
+ in the logic.
+ \end{itemize}
+
+ \noindent In this section we will introduce means to \emph{adapt} the serialiser
+ to a specific target language, i.e.~to print program fragments
+ in a way which accommodates \qt{already existing} ingredients of
+ a target language environment, for three reasons:
+
+ \begin{itemize}
+ \item improving readability and aesthetics of generated code
+ \item gaining efficiency
+ \item interface with language parts which have no direct counterpart
+ in @{text "HOL"} (say, imperative data structures)
+ \end{itemize}
+
+ \noindent Generally, you should avoid using those features yourself
+ \emph{at any cost}:
+
+ \begin{itemize}
+ \item The safe configuration methods act uniformly on every target language,
+ whereas for adaption you have to treat each target language separate.
+ \item Application is extremely tedious since there is no abstraction
+ which would allow for a static check, making it easy to produce garbage.
+ \item More or less subtle errors can be introduced unconsciously.
+ \end{itemize}
+
+ \noindent However, even if you ought refrain from setting up adaption
+ yourself, already the @{text "HOL"} comes with some reasonable default
+ adaptions (say, using target language list syntax). There also some
+ common adaption cases which you can setup by importing particular
+ library theories. In order to understand these, we provide some clues here;
+ these however are not supposed to replace a careful study of the sources.
+*}
+
+subsection {* The adaption principle *}
+
+text {*
+ The following figure illustrates what \qt{adaption} is conceptually
+ supposed to be:
+
+ \begin{figure}[here]
+ \begin{tikzpicture}[scale = 0.5]
+ \tikzstyle water=[color = blue, thick]
+ \tikzstyle ice=[color = black, very thick, cap = round, join = round, fill = white]
+ \tikzstyle process=[color = green, semithick, ->]
+ \tikzstyle adaption=[color = red, semithick, ->]
+ \tikzstyle target=[color = black]
+ \foreach \x in {0, ..., 24}
+ \draw[style=water] (\x, 0.25) sin + (0.25, 0.25) cos + (0.25, -0.25) sin
+ + (0.25, -0.25) cos + (0.25, 0.25);
+ \draw[style=ice] (1, 0) --
+ (3, 6) node[above, fill=white] {logic} -- (5, 0) -- cycle;
+ \draw[style=ice] (9, 0) --
+ (11, 6) node[above, fill=white] {intermediate language} -- (13, 0) -- cycle;
+ \draw[style=ice] (15, -6) --
+ (19, 6) node[above, fill=white] {target language} -- (23, -6) -- cycle;
+ \draw[style=process]
+ (3.5, 3) .. controls (7, 5) .. node[fill=white] {translation} (10.5, 3);
+ \draw[style=process]
+ (11.5, 3) .. controls (15, 5) .. node[fill=white] (serialisation) {serialisation} (18.5, 3);
+ \node (adaption) at (11, -2) [style=adaption] {adaption};
+ \node at (19, 3) [rotate=90] {generated};
+ \node at (19.5, -5) {language};
+ \node at (19.5, -3) {library};
+ \node (includes) at (19.5, -1) {includes};
+ \node (reserved) at (16.5, -3) [rotate=72] {reserved}; % proper 71.57
+ \draw[style=process]
+ (includes) -- (serialisation);
+ \draw[style=process]
+ (reserved) -- (serialisation);
+ \draw[style=adaption]
+ (adaption) -- (serialisation);
+ \draw[style=adaption]
+ (adaption) -- (includes);
+ \draw[style=adaption]
+ (adaption) -- (reserved);
+ \end{tikzpicture}
+ \caption{The adaption principle}
+ \label{fig:adaption}
+ \end{figure}
+
+ \noindent In the tame view, code generation acts as broker between
+ @{text logic}, @{text "intermediate language"} and
+ @{text "target language"} by means of @{text translation} and
+ @{text serialisation}; for the latter, the serialiser has to observe
+ the structure of the @{text language} itself plus some @{text reserved}
+ keywords which have to be avoided for generated code.
+ However, if you consider @{text adaption} mechanisms, the code generated
+ by the serializer is just the tip of the iceberg:
+
+ \begin{itemize}
+ \item @{text serialisation} can be \emph{parametrised} such that
+ logical entities are mapped to target-specific ones
+ (e.g. target-specific list syntax,
+ see also \secref{sec:adaption_mechanisms})
+ \item Such parametrisations can involve references to a
+ target-specific standard @{text library} (e.g. using
+ the @{text Haskell} @{verbatim Maybe} type instead
+ of the @{text HOL} @{type "option"} type);
+ if such are used, the corresponding identifiers
+ (in our example, @{verbatim Maybe}, @{verbatim Nothing}
+ and @{verbatim Just}) also have to be considered @{text reserved}.
+ \item Even more, the user can enrich the library of the
+ target-language by providing code snippets
+ (\qt{@{text "includes"}}) which are prepended to
+ any generated code (see \secref{sec:include}); this typically
+ also involves further @{text reserved} identifiers.
+ \end{itemize}
+
+ \noindent As figure \ref{fig:adaption} illustrates, all these adaption mechanisms
+ have to act consistently; it is at the discretion of the user
+ to take care for this.
+*}
+
+subsection {* Common adaption patterns *}
+
+text {*
+ The @{theory HOL} @{theory Main} theory already provides a code
+ generator setup
+ which should be suitable for most applications. Common extensions
+ and modifications are available by certain theories of the @{text HOL}
+ library; beside being useful in applications, they may serve
+ as a tutorial for customising the code generator setup (see below
+ \secref{sec:adaption_mechanisms}).
+
+ \begin{description}
+
+ \item[@{theory "Code_Integer"}] represents @{text HOL} integers by big
+ integer literals in target languages.
+ \item[@{theory "Code_Char"}] represents @{text HOL} characters by
+ character literals in target languages.
+ \item[@{theory "Code_Char_chr"}] like @{text "Code_Char"},
+ but also offers treatment of character codes; includes
+ @{theory "Code_Char"}.
+ \item[@{theory "Efficient_Nat"}] \label{eff_nat} implements natural numbers by integers,
+ which in general will result in higher efficiency; pattern
+ matching with @{term "0\<Colon>nat"} / @{const "Suc"}
+ is eliminated; includes @{theory "Code_Integer"}
+ and @{theory "Code_Index"}.
+ \item[@{theory "Code_Index"}] provides an additional datatype
+ @{typ index} which is mapped to target-language built-in integers.
+ Useful for code setups which involve e.g. indexing of
+ target-language arrays.
+ \item[@{theory "Code_Message"}] provides an additional datatype
+ @{typ message_string} which is isomorphic to strings;
+ @{typ message_string}s are mapped to target-language strings.
+ Useful for code setups which involve e.g. printing (error) messages.
+
+ \end{description}
+
+ \begin{warn}
+ When importing any of these theories, they should form the last
+ items in an import list. Since these theories adapt the
+ code generator setup in a non-conservative fashion,
+ strange effects may occur otherwise.
+ \end{warn}
+*}
+
+
+subsection {* Parametrising serialisation \label{sec:adaption_mechanisms} *}
+
+text {*
+ Consider the following function and its corresponding
+ SML code:
+*}
+
+primrec %quote in_interval :: "nat \<times> nat \<Rightarrow> nat \<Rightarrow> bool" where
+ "in_interval (k, l) n \<longleftrightarrow> k \<le> n \<and> n \<le> l"
+(*<*)
+code_type %invisible bool
+ (SML)
+code_const %invisible True and False and "op \<and>" and Not
+ (SML and and and)
+(*>*)
+text %quote {*@{code_stmts in_interval (SML)}*}
+
+text {*
+ \noindent Though this is correct code, it is a little bit unsatisfactory:
+ boolean values and operators are materialised as distinguished
+ entities with have nothing to do with the SML-built-in notion
+ of \qt{bool}. This results in less readable code;
+ additionally, eager evaluation may cause programs to
+ loop or break which would perfectly terminate when
+ the existing SML @{verbatim "bool"} would be used. To map
+ the HOL @{typ bool} on SML @{verbatim "bool"}, we may use
+ \qn{custom serialisations}:
+*}
+
+code_type %quotett bool
+ (SML "bool")
+code_const %quotett True and False and "op \<and>"
+ (SML "true" and "false" and "_ andalso _")
+
+text {*
+ \noindent The @{command code_type} command takes a type constructor
+ as arguments together with a list of custom serialisations.
+ Each custom serialisation starts with a target language
+ identifier followed by an expression, which during
+ code serialisation is inserted whenever the type constructor
+ would occur. For constants, @{command code_const} implements
+ the corresponding mechanism. Each ``@{verbatim "_"}'' in
+ a serialisation expression is treated as a placeholder
+ for the type constructor's (the constant's) arguments.
+*}
+
+text %quote {*@{code_stmts in_interval (SML)}*}
+
+text {*
+ \noindent This still is not perfect: the parentheses
+ around the \qt{andalso} expression are superfluous.
+ Though the serialiser
+ by no means attempts to imitate the rich Isabelle syntax
+ framework, it provides some common idioms, notably
+ associative infixes with precedences which may be used here:
+*}
+
+code_const %quotett "op \<and>"
+ (SML infixl 1 "andalso")
+
+text %quote {*@{code_stmts in_interval (SML)}*}
+
+text {*
+ \noindent The attentive reader may ask how we assert that no generated
+ code will accidentally overwrite. For this reason the serialiser has
+ an internal table of identifiers which have to be avoided to be used
+ for new declarations. Initially, this table typically contains the
+ keywords of the target language. It can be extended manually, thus avoiding
+ accidental overwrites, using the @{command "code_reserved"} command:
+*}
+
+code_reserved %quote "\<SML>" bool true false andalso
+
+text {*
+ \noindent Next, we try to map HOL pairs to SML pairs, using the
+ infix ``@{verbatim "*"}'' type constructor and parentheses:
+*}
+(*<*)
+code_type %invisible *
+ (SML)
+code_const %invisible Pair
+ (SML)
+(*>*)
+code_type %quotett *
+ (SML infix 2 "*")
+code_const %quotett Pair
+ (SML "!((_),/ (_))")
+
+text {*
+ \noindent The initial bang ``@{verbatim "!"}'' tells the serialiser
+ never to put
+ parentheses around the whole expression (they are already present),
+ while the parentheses around argument place holders
+ tell not to put parentheses around the arguments.
+ The slash ``@{verbatim "/"}'' (followed by arbitrary white space)
+ inserts a space which may be used as a break if necessary
+ during pretty printing.
+
+ These examples give a glimpse what mechanisms
+ custom serialisations provide; however their usage
+ requires careful thinking in order not to introduce
+ inconsistencies -- or, in other words:
+ custom serialisations are completely axiomatic.
+
+ A further noteworthy details is that any special
+ character in a custom serialisation may be quoted
+ using ``@{verbatim "'"}''; thus, in
+ ``@{verbatim "fn '_ => _"}'' the first
+ ``@{verbatim "_"}'' is a proper underscore while the
+ second ``@{verbatim "_"}'' is a placeholder.
+*}
+
+
+subsection {* @{text Haskell} serialisation *}
+
+text {*
+ For convenience, the default
+ @{text HOL} setup for @{text Haskell} maps the @{class eq} class to
+ its counterpart in @{text Haskell}, giving custom serialisations
+ for the class @{class eq} (by command @{command code_class}) and its operation
+ @{const HOL.eq}
+*}
+
+code_class %quotett eq
+ (Haskell "Eq")
+
+code_const %quotett "op ="
+ (Haskell infixl 4 "==")
+
+text {*
+ \noindent A problem now occurs whenever a type which
+ is an instance of @{class eq} in @{text HOL} is mapped
+ on a @{text Haskell}-built-in type which is also an instance
+ of @{text Haskell} @{text Eq}:
+*}
+
+typedecl %quote bar
+
+instantiation %quote bar :: eq
+begin
+
+definition %quote "eq_class.eq (x\<Colon>bar) y \<longleftrightarrow> x = y"
+
+instance %quote by default (simp add: eq_bar_def)
+
+end %quote
+
+code_type %quotett bar
+ (Haskell "Integer")
+
+text {*
+ \noindent The code generator would produce
+ an additional instance, which of course is rejected by the @{text Haskell}
+ compiler.
+ To suppress this additional instance, use
+ @{text "code_instance"}:
+*}
+
+code_instance %quotett bar :: eq
+ (Haskell -)
+
+
+subsection {* Enhancing the target language context \label{sec:include} *}
+
+text {*
+ In rare cases it is necessary to \emph{enrich} the context of a
+ target language; this is accomplished using the @{command "code_include"}
+ command:
+*}
+
+code_include %quotett Haskell "Errno"
+{*errno i = error ("Error number: " ++ show i)*}
+
+code_reserved %quotett Haskell Errno
+
+text {*
+ \noindent Such named @{text include}s are then prepended to every generated code.
+ Inspect such code in order to find out how @{command "code_include"} behaves
+ with respect to a particular target language.
+*}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/Codegen.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,2 @@
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/Further.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,113 @@
+theory Further
+imports Setup
+begin
+
+section {* Further issues \label{sec:further} *}
+
+subsection {* Further reading *}
+
+text {*
+ Do dive deeper into the issue of code generation, you should visit
+ the Isabelle/Isar Reference Manual \cite{isabelle-isar-ref} which
+ contains exhaustive syntax diagrams.
+*}
+
+subsection {* Modules *}
+
+text {*
+ When invoking the @{command export_code} command it is possible to leave
+ out the @{keyword "module_name"} part; then code is distributed over
+ different modules, where the module name space roughly is induced
+ by the @{text Isabelle} theory name space.
+
+ Then sometimes the awkward situation occurs that dependencies between
+ definitions introduce cyclic dependencies between modules, which in the
+ @{text Haskell} world leaves you to the mercy of the @{text Haskell} implementation
+ you are using, while for @{text SML}/@{text OCaml} code generation is not possible.
+
+ A solution is to declare module names explicitly.
+ Let use assume the three cyclically dependent
+ modules are named \emph{A}, \emph{B} and \emph{C}.
+ Then, by stating
+*}
+
+code_modulename %quote SML
+ A ABC
+ B ABC
+ C ABC
+
+text {*
+ we explicitly map all those modules on \emph{ABC},
+ resulting in an ad-hoc merge of this three modules
+ at serialisation time.
+*}
+
+subsection {* Evaluation oracle *}
+
+text {*
+ Code generation may also be used to \emph{evaluate} expressions
+ (using @{text SML} as target language of course).
+ For instance, the @{command value} allows to reduce an expression to a
+ normal form with respect to the underlying code equations:
+*}
+
+value %quote "42 / (12 :: rat)"
+
+text {*
+ \noindent will display @{term "7 / (2 :: rat)"}.
+
+ The @{method eval} method tries to reduce a goal by code generation to @{term True}
+ and solves it in that case, but fails otherwise:
+*}
+
+lemma %quote "42 / (12 :: rat) = 7 / 2"
+ by %quote eval
+
+text {*
+ \noindent The soundness of the @{method eval} method depends crucially
+ on the correctness of the code generator; this is one of the reasons
+ why you should not use adaption (see \secref{sec:adaption}) frivolously.
+*}
+
+subsection {* Code antiquotation *}
+
+text {*
+ In scenarios involving techniques like reflection it is quite common
+ that code generated from a theory forms the basis for implementing
+ a proof procedure in @{text SML}. To facilitate interfacing of generated code
+ with system code, the code generator provides a @{text code} antiquotation:
+*}
+
+datatype %quote form = T | F | And form form | Or form form
+
+ML %quote {*
+ fun eval_form @{code T} = true
+ | eval_form @{code F} = false
+ | eval_form (@{code And} (p, q)) =
+ eval_form p andalso eval_form q
+ | eval_form (@{code Or} (p, q)) =
+ eval_form p orelse eval_form q;
+*}
+
+text {*
+ \noindent @{text code} takes as argument the name of a constant; after the
+ whole @{text SML} is read, the necessary code is generated transparently
+ and the corresponding constant names are inserted. This technique also
+ allows to use pattern matching on constructors stemming from compiled
+ @{text datatypes}.
+
+ For a less simplistic example, theory @{theory Ferrack} is
+ a good reference.
+*}
+
+subsection {* Imperative data structures *}
+
+text {*
+ If you consider imperative data structures as inevitable for a specific
+ application, you should consider
+ \emph{Imperative Functional Programming with Isabelle/HOL}
+ (\cite{bulwahn-et-al:2008:imperative});
+ the framework described there is available in theory @{theory Imperative_HOL}.
+*}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/Introduction.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,206 @@
+theory Introduction
+imports Setup
+begin
+
+chapter {* Code generation from @{text "Isabelle/HOL"} theories *}
+
+section {* Introduction and Overview *}
+
+text {*
+ This tutorial introduces a generic code generator for the
+ @{text Isabelle} system.
+ Generic in the sense that the
+ \qn{target language} for which code shall ultimately be
+ generated is not fixed but may be an arbitrary state-of-the-art
+ functional programming language (currently, the implementation
+ supports @{text SML} \cite{SML}, @{text OCaml} \cite{OCaml} and @{text Haskell}
+ \cite{haskell-revised-report}).
+
+ Conceptually the code generator framework is part
+ of Isabelle's @{theory Pure} meta logic framework; the logic
+ @{theory HOL} which is an extension of @{theory Pure}
+ already comes with a reasonable framework setup and thus provides
+ a good working horse for raising code-generation-driven
+ applications. So, we assume some familiarity and experience
+ with the ingredients of the @{theory HOL} distribution theories.
+ (see also \cite{isa-tutorial}).
+
+ The code generator aims to be usable with no further ado
+ in most cases while allowing for detailed customisation.
+ This manifests in the structure of this tutorial: after a short
+ conceptual introduction with an example (\secref{sec:intro}),
+ we discuss the generic customisation facilities (\secref{sec:program}).
+ A further section (\secref{sec:adaption}) is dedicated to the matter of
+ \qn{adaption} to specific target language environments. After some
+ further issues (\secref{sec:further}) we conclude with an overview
+ of some ML programming interfaces (\secref{sec:ml}).
+
+ \begin{warn}
+ Ultimately, the code generator which this tutorial deals with
+ is supposed to replace the existing code generator
+ by Stefan Berghofer \cite{Berghofer-Nipkow:2002}.
+ So, for the moment, there are two distinct code generators
+ in Isabelle. In case of ambiguity, we will refer to the framework
+ described here as @{text "generic code generator"}, to the
+ other as @{text "SML code generator"}.
+ Also note that while the framework itself is
+ object-logic independent, only @{theory HOL} provides a reasonable
+ framework setup.
+ \end{warn}
+
+*}
+
+subsection {* Code generation via shallow embedding \label{sec:intro} *}
+
+text {*
+ The key concept for understanding @{text Isabelle}'s code generation is
+ \emph{shallow embedding}, i.e.~logical entities like constants, types and
+ classes are identified with corresponding concepts in the target language.
+
+ Inside @{theory HOL}, the @{command datatype} and
+ @{command definition}/@{command primrec}/@{command fun} declarations form
+ the core of a functional programming language. The default code generator setup
+ allows to turn those into functional programs immediately.
+ This means that \qt{naive} code generation can proceed without further ado.
+ For example, here a simple \qt{implementation} of amortised queues:
+*}
+
+datatype %quote 'a queue = AQueue "'a list" "'a list"
+
+definition %quote empty :: "'a queue" where
+ "empty = AQueue [] []"
+
+primrec %quote enqueue :: "'a \<Rightarrow> 'a queue \<Rightarrow> 'a queue" where
+ "enqueue x (AQueue xs ys) = AQueue (x # xs) ys"
+
+fun %quote dequeue :: "'a queue \<Rightarrow> 'a option \<times> 'a queue" where
+ "dequeue (AQueue [] []) = (None, AQueue [] [])"
+ | "dequeue (AQueue xs (y # ys)) = (Some y, AQueue xs ys)"
+ | "dequeue (AQueue xs []) =
+ (case rev xs of y # ys \<Rightarrow> (Some y, AQueue [] ys))"
+
+text {* \noindent Then we can generate code e.g.~for @{text SML} as follows: *}
+
+export_code %quote empty dequeue enqueue in SML
+ module_name Example file "examples/example.ML"
+
+text {* \noindent resulting in the following code: *}
+
+text %quote {*@{code_stmts empty enqueue dequeue (SML)}*}
+
+text {*
+ \noindent The @{command export_code} command takes a space-separated list of
+ constants for which code shall be generated; anything else needed for those
+ is added implicitly. Then follows a target language identifier
+ (@{text SML}, @{text OCaml} or @{text Haskell}) and a freely chosen module name.
+ A file name denotes the destination to store the generated code. Note that
+ the semantics of the destination depends on the target language: for
+ @{text SML} and @{text OCaml} it denotes a \emph{file}, for @{text Haskell}
+ it denotes a \emph{directory} where a file named as the module name
+ (with extension @{text ".hs"}) is written:
+*}
+
+export_code %quote empty dequeue enqueue in Haskell
+ module_name Example file "examples/"
+
+text {*
+ \noindent This is how the corresponding code in @{text Haskell} looks like:
+*}
+
+text %quote {*@{code_stmts empty enqueue dequeue (Haskell)}*}
+
+text {*
+ \noindent This demonstrates the basic usage of the @{command export_code} command;
+ for more details see \secref{sec:further}.
+*}
+
+subsection {* Code generator architecture \label{sec:concept} *}
+
+text {*
+ What you have seen so far should be already enough in a lot of cases. If you
+ are content with this, you can quit reading here. Anyway, in order to customise
+ and adapt the code generator, it is inevitable to gain some understanding
+ how it works.
+
+ \begin{figure}[h]
+ \begin{tikzpicture}[x = 4.2cm, y = 1cm]
+ \tikzstyle entity=[rounded corners, draw, thick, color = black, fill = white];
+ \tikzstyle process=[ellipse, draw, thick, color = green, fill = white];
+ \tikzstyle process_arrow=[->, semithick, color = green];
+ \node (HOL) at (0, 4) [style=entity] {@{text "Isabelle/HOL"} theory};
+ \node (eqn) at (2, 2) [style=entity] {code equations};
+ \node (iml) at (2, 0) [style=entity] {intermediate language};
+ \node (seri) at (1, 0) [style=process] {serialisation};
+ \node (SML) at (0, 3) [style=entity] {@{text SML}};
+ \node (OCaml) at (0, 2) [style=entity] {@{text OCaml}};
+ \node (further) at (0, 1) [style=entity] {@{text "\<dots>"}};
+ \node (Haskell) at (0, 0) [style=entity] {@{text Haskell}};
+ \draw [style=process_arrow] (HOL) .. controls (2, 4) ..
+ node [style=process, near start] {selection}
+ node [style=process, near end] {preprocessing}
+ (eqn);
+ \draw [style=process_arrow] (eqn) -- node (transl) [style=process] {translation} (iml);
+ \draw [style=process_arrow] (iml) -- (seri);
+ \draw [style=process_arrow] (seri) -- (SML);
+ \draw [style=process_arrow] (seri) -- (OCaml);
+ \draw [style=process_arrow, dashed] (seri) -- (further);
+ \draw [style=process_arrow] (seri) -- (Haskell);
+ \end{tikzpicture}
+ \caption{Code generator architecture}
+ \label{fig:arch}
+ \end{figure}
+
+ The code generator employs a notion of executability
+ for three foundational executable ingredients known
+ from functional programming:
+ \emph{code equations}, \emph{datatypes}, and
+ \emph{type classes}. A code equation as a first approximation
+ is a theorem of the form @{text "f t\<^isub>1 t\<^isub>2 \<dots> t\<^isub>n \<equiv> t"}
+ (an equation headed by a constant @{text f} with arguments
+ @{text "t\<^isub>1 t\<^isub>2 \<dots> t\<^isub>n"} and right hand side @{text t}).
+ Code generation aims to turn code equations
+ into a functional program. This is achieved by three major
+ components which operate sequentially, i.e. the result of one is
+ the input
+ of the next in the chain, see diagram \ref{fig:arch}:
+
+ \begin{itemize}
+
+ \item Out of the vast collection of theorems proven in a
+ \qn{theory}, a reasonable subset modelling
+ code equations is \qn{selected}.
+
+ \item On those selected theorems, certain
+ transformations are carried out
+ (\qn{preprocessing}). Their purpose is to turn theorems
+ representing non- or badly executable
+ specifications into equivalent but executable counterparts.
+ The result is a structured collection of \qn{code theorems}.
+
+ \item Before the selected code equations are continued with,
+ they can be \qn{preprocessed}, i.e. subjected to theorem
+ transformations. This \qn{preprocessor} is an interface which
+ allows to apply
+ the full expressiveness of ML-based theorem transformations
+ to code generation; motivating examples are shown below, see
+ \secref{sec:preproc}.
+ The result of the preprocessing step is a structured collection
+ of code equations.
+
+ \item These code equations are \qn{translated} to a program
+ in an abstract intermediate language. Think of it as a kind
+ of \qt{Mini-Haskell} with four \qn{statements}: @{text data}
+ (for datatypes), @{text fun} (stemming from code equations),
+ also @{text class} and @{text inst} (for type classes).
+
+ \item Finally, the abstract program is \qn{serialised} into concrete
+ source code of a target language.
+
+ \end{itemize}
+
+ \noindent From these steps, only the two last are carried out outside the logic; by
+ keeping this layer as thin as possible, the amount of code to trust is
+ kept to a minimum.
+*}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/ML.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,177 @@
+theory "ML"
+imports Setup
+begin
+
+section {* ML system interfaces \label{sec:ml} *}
+
+text {*
+ Since the code generator framework not only aims to provide
+ a nice Isar interface but also to form a base for
+ code-generation-based applications, here a short
+ description of the most important ML interfaces.
+*}
+
+subsection {* Executable theory content: @{text Code} *}
+
+text {*
+ This Pure module implements the core notions of
+ executable content of a theory.
+*}
+
+subsubsection {* Managing executable content *}
+
+text %mlref {*
+ \begin{mldecls}
+ @{index_ML Code.add_eqn: "thm -> theory -> theory"} \\
+ @{index_ML Code.del_eqn: "thm -> theory -> theory"} \\
+ @{index_ML Code.add_eqnl: "string * (thm * bool) list lazy -> theory -> theory"} \\
+ @{index_ML Code.map_pre: "(simpset -> simpset) -> theory -> theory"} \\
+ @{index_ML Code.map_post: "(simpset -> simpset) -> theory -> theory"} \\
+ @{index_ML Code.add_functrans: "string * (theory -> (thm * bool) list -> (thm * bool) list option)
+ -> theory -> theory"} \\
+ @{index_ML Code.del_functrans: "string -> theory -> theory"} \\
+ @{index_ML Code.add_datatype: "(string * typ) list -> theory -> theory"} \\
+ @{index_ML Code.get_datatype: "theory -> string
+ -> (string * sort) list * (string * typ list) list"} \\
+ @{index_ML Code.get_datatype_of_constr: "theory -> string -> string option"}
+ \end{mldecls}
+
+ \begin{description}
+
+ \item @{ML Code.add_eqn}~@{text "thm"}~@{text "thy"} adds function
+ theorem @{text "thm"} to executable content.
+
+ \item @{ML Code.del_eqn}~@{text "thm"}~@{text "thy"} removes function
+ theorem @{text "thm"} from executable content, if present.
+
+ \item @{ML Code.add_eqnl}~@{text "(const, lthms)"}~@{text "thy"} adds
+ suspended code equations @{text lthms} for constant
+ @{text const} to executable content.
+
+ \item @{ML Code.map_pre}~@{text "f"}~@{text "thy"} changes
+ the preprocessor simpset.
+
+ \item @{ML Code.add_functrans}~@{text "(name, f)"}~@{text "thy"} adds
+ function transformer @{text f} (named @{text name}) to executable content;
+ @{text f} is a transformer of the code equations belonging
+ to a certain function definition, depending on the
+ current theory context. Returning @{text NONE} indicates that no
+ transformation took place; otherwise, the whole process will be iterated
+ with the new code equations.
+
+ \item @{ML Code.del_functrans}~@{text "name"}~@{text "thy"} removes
+ function transformer named @{text name} from executable content.
+
+ \item @{ML Code.add_datatype}~@{text cs}~@{text thy} adds
+ a datatype to executable content, with generation
+ set @{text cs}.
+
+ \item @{ML Code.get_datatype_of_constr}~@{text "thy"}~@{text "const"}
+ returns type constructor corresponding to
+ constructor @{text const}; returns @{text NONE}
+ if @{text const} is no constructor.
+
+ \end{description}
+*}
+
+subsection {* Auxiliary *}
+
+text %mlref {*
+ \begin{mldecls}
+ @{index_ML Code_Unit.read_const: "theory -> string -> string"} \\
+ @{index_ML Code_Unit.head_eqn: "theory -> thm -> string * ((string * sort) list * typ)"} \\
+ @{index_ML Code_Unit.rewrite_eqn: "simpset -> thm -> thm"} \\
+ \end{mldecls}
+
+ \begin{description}
+
+ \item @{ML Code_Unit.read_const}~@{text thy}~@{text s}
+ reads a constant as a concrete term expression @{text s}.
+
+ \item @{ML Code_Unit.head_eqn}~@{text thy}~@{text thm}
+ extracts the constant and its type from a code equation @{text thm}.
+
+ \item @{ML Code_Unit.rewrite_eqn}~@{text ss}~@{text thm}
+ rewrites a code equation @{text thm} with a simpset @{text ss};
+ only arguments and right hand side are rewritten,
+ not the head of the code equation.
+
+ \end{description}
+
+*}
+
+subsection {* Implementing code generator applications *}
+
+text {*
+ Implementing code generator applications on top
+ of the framework set out so far usually not only
+ involves using those primitive interfaces
+ but also storing code-dependent data and various
+ other things.
+*}
+
+subsubsection {* Data depending on the theory's executable content *}
+
+text {*
+ Due to incrementality of code generation, changes in the
+ theory's executable content have to be propagated in a
+ certain fashion. Additionally, such changes may occur
+ not only during theory extension but also during theory
+ merge, which is a little bit nasty from an implementation
+ point of view. The framework provides a solution
+ to this technical challenge by providing a functorial
+ data slot @{ML_functor CodeDataFun}; on instantiation
+ of this functor, the following types and operations
+ are required:
+
+ \medskip
+ \begin{tabular}{l}
+ @{text "type T"} \\
+ @{text "val empty: T"} \\
+ @{text "val purge: theory \<rightarrow> string list option \<rightarrow> T \<rightarrow> T"}
+ \end{tabular}
+
+ \begin{description}
+
+ \item @{text T} the type of data to store.
+
+ \item @{text empty} initial (empty) data.
+
+ \item @{text purge}~@{text thy}~@{text consts} propagates changes in executable content;
+ @{text consts} indicates the kind
+ of change: @{ML NONE} stands for a fundamental change
+ which invalidates any existing code, @{text "SOME consts"}
+ hints that executable content for constants @{text consts}
+ has changed.
+
+ \end{description}
+
+ \noindent An instance of @{ML_functor CodeDataFun} provides the following
+ interface:
+
+ \medskip
+ \begin{tabular}{l}
+ @{text "get: theory \<rightarrow> T"} \\
+ @{text "change: theory \<rightarrow> (T \<rightarrow> T) \<rightarrow> T"} \\
+ @{text "change_yield: theory \<rightarrow> (T \<rightarrow> 'a * T) \<rightarrow> 'a * T"}
+ \end{tabular}
+
+ \begin{description}
+
+ \item @{text get} retrieval of the current data.
+
+ \item @{text change} update of current data (cached!)
+ by giving a continuation.
+
+ \item @{text change_yield} update with side result.
+
+ \end{description}
+*}
+
+text {*
+ \bigskip
+
+ \emph{Happy proving, happy hacking!}
+*}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/Program.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,534 @@
+theory Program
+imports Introduction
+begin
+
+section {* Turning Theories into Programs \label{sec:program} *}
+
+subsection {* The @{text "Isabelle/HOL"} default setup *}
+
+text {*
+ We have already seen how by default equations stemming from
+ @{command definition}/@{command primrec}/@{command fun}
+ statements are used for code generation. This default behaviour
+ can be changed, e.g. by providing different code equations.
+ All kinds of customisation shown in this section is \emph{safe}
+ in the sense that the user does not have to worry about
+ correctness -- all programs generatable that way are partially
+ correct.
+*}
+
+subsection {* Selecting code equations *}
+
+text {*
+ Coming back to our introductory example, we
+ could provide an alternative code equations for @{const dequeue}
+ explicitly:
+*}
+
+lemma %quote [code]:
+ "dequeue (AQueue xs []) =
+ (if xs = [] then (None, AQueue [] [])
+ else dequeue (AQueue [] (rev xs)))"
+ "dequeue (AQueue xs (y # ys)) =
+ (Some y, AQueue xs ys)"
+ by (cases xs, simp_all) (cases "rev xs", simp_all)
+
+text {*
+ \noindent The annotation @{text "[code]"} is an @{text Isar}
+ @{text attribute} which states that the given theorems should be
+ considered as code equations for a @{text fun} statement --
+ the corresponding constant is determined syntactically. The resulting code:
+*}
+
+text %quote {*@{code_stmts dequeue (consts) dequeue (Haskell)}*}
+
+text {*
+ \noindent You may note that the equality test @{term "xs = []"} has been
+ replaced by the predicate @{term "null xs"}. This is due to the default
+ setup in the \qn{preprocessor} to be discussed further below (\secref{sec:preproc}).
+
+ Changing the default constructor set of datatypes is also
+ possible. See \secref{sec:datatypes} for an example.
+
+ As told in \secref{sec:concept}, code generation is based
+ on a structured collection of code theorems.
+ For explorative purpose, this collection
+ may be inspected using the @{command code_thms} command:
+*}
+
+code_thms %quote dequeue
+
+text {*
+ \noindent prints a table with \emph{all} code equations
+ for @{const dequeue}, including
+ \emph{all} code equations those equations depend
+ on recursively.
+
+ Similarly, the @{command code_deps} command shows a graph
+ visualising dependencies between code equations.
+*}
+
+subsection {* @{text class} and @{text instantiation} *}
+
+text {*
+ Concerning type classes and code generation, let us examine an example
+ from abstract algebra:
+*}
+
+class %quote semigroup =
+ fixes mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<otimes>" 70)
+ assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
+
+class %quote monoid = semigroup +
+ fixes neutral :: 'a ("\<one>")
+ assumes neutl: "\<one> \<otimes> x = x"
+ and neutr: "x \<otimes> \<one> = x"
+
+instantiation %quote nat :: monoid
+begin
+
+primrec %quote mult_nat where
+ "0 \<otimes> n = (0\<Colon>nat)"
+ | "Suc m \<otimes> n = n + m \<otimes> n"
+
+definition %quote neutral_nat where
+ "\<one> = Suc 0"
+
+lemma %quote add_mult_distrib:
+ fixes n m q :: nat
+ shows "(n + m) \<otimes> q = n \<otimes> q + m \<otimes> q"
+ by (induct n) simp_all
+
+instance %quote proof
+ fix m n q :: nat
+ show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
+ by (induct m) (simp_all add: add_mult_distrib)
+ show "\<one> \<otimes> n = n"
+ by (simp add: neutral_nat_def)
+ show "m \<otimes> \<one> = m"
+ by (induct m) (simp_all add: neutral_nat_def)
+qed
+
+end %quote
+
+text {*
+ \noindent We define the natural operation of the natural numbers
+ on monoids:
+*}
+
+primrec %quote (in monoid) pow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" where
+ "pow 0 a = \<one>"
+ | "pow (Suc n) a = a \<otimes> pow n a"
+
+text {*
+ \noindent This we use to define the discrete exponentiation function:
+*}
+
+definition %quote bexp :: "nat \<Rightarrow> nat" where
+ "bexp n = pow n (Suc (Suc 0))"
+
+text {*
+ \noindent The corresponding code:
+*}
+
+text %quote {*@{code_stmts bexp (Haskell)}*}
+
+text {*
+ \noindent This is a convenient place to show how explicit dictionary construction
+ manifests in generated code (here, the same example in @{text SML}):
+*}
+
+text %quote {*@{code_stmts bexp (SML)}*}
+
+text {*
+ \noindent Note the parameters with trailing underscore (@{verbatim "A_"})
+ which are the dictionary parameters.
+*}
+
+subsection {* The preprocessor \label{sec:preproc} *}
+
+text {*
+ Before selected function theorems are turned into abstract
+ code, a chain of definitional transformation steps is carried
+ out: \emph{preprocessing}. In essence, the preprocessor
+ consists of two components: a \emph{simpset} and \emph{function transformers}.
+
+ The \emph{simpset} allows to employ the full generality of the Isabelle
+ simplifier. Due to the interpretation of theorems
+ as code equations, rewrites are applied to the right
+ hand side and the arguments of the left hand side of an
+ equation, but never to the constant heading the left hand side.
+ An important special case are \emph{inline theorems} which may be
+ declared and undeclared using the
+ \emph{code inline} or \emph{code inline del} attribute respectively.
+
+ Some common applications:
+*}
+
+text_raw {*
+ \begin{itemize}
+*}
+
+text {*
+ \item replacing non-executable constructs by executable ones:
+*}
+
+lemma %quote [code inline]:
+ "x \<in> set xs \<longleftrightarrow> x mem xs" by (induct xs) simp_all
+
+text {*
+ \item eliminating superfluous constants:
+*}
+
+lemma %quote [code inline]:
+ "1 = Suc 0" by simp
+
+text {*
+ \item replacing executable but inconvenient constructs:
+*}
+
+lemma %quote [code inline]:
+ "xs = [] \<longleftrightarrow> List.null xs" by (induct xs) simp_all
+
+text_raw {*
+ \end{itemize}
+*}
+
+text {*
+ \noindent \emph{Function transformers} provide a very general interface,
+ transforming a list of function theorems to another
+ list of function theorems, provided that neither the heading
+ constant nor its type change. The @{term "0\<Colon>nat"} / @{const Suc}
+ pattern elimination implemented in
+ theory @{text Efficient_Nat} (see \secref{eff_nat}) uses this
+ interface.
+
+ \noindent The current setup of the preprocessor may be inspected using
+ the @{command print_codesetup} command.
+ @{command code_thms} provides a convenient
+ mechanism to inspect the impact of a preprocessor setup
+ on code equations.
+
+ \begin{warn}
+ The attribute \emph{code unfold}
+ associated with the @{text "SML code generator"} also applies to
+ the @{text "generic code generator"}:
+ \emph{code unfold} implies \emph{code inline}.
+ \end{warn}
+*}
+
+subsection {* Datatypes \label{sec:datatypes} *}
+
+text {*
+ Conceptually, any datatype is spanned by a set of
+ \emph{constructors} of type @{text "\<tau> = \<dots> \<Rightarrow> \<kappa> \<alpha>\<^isub>1 \<dots> \<alpha>\<^isub>n"} where @{text
+ "{\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>n}"} is exactly the set of \emph{all} type variables in
+ @{text "\<tau>"}. The HOL datatype package by default registers any new
+ datatype in the table of datatypes, which may be inspected using the
+ @{command print_codesetup} command.
+
+ In some cases, it is appropriate to alter or extend this table. As
+ an example, we will develop an alternative representation of the
+ queue example given in \secref{sec:intro}. The amortised
+ representation is convenient for generating code but exposes its
+ \qt{implementation} details, which may be cumbersome when proving
+ theorems about it. Therefore, here a simple, straightforward
+ representation of queues:
+*}
+
+datatype %quote 'a queue = Queue "'a list"
+
+definition %quote empty :: "'a queue" where
+ "empty = Queue []"
+
+primrec %quote enqueue :: "'a \<Rightarrow> 'a queue \<Rightarrow> 'a queue" where
+ "enqueue x (Queue xs) = Queue (xs @ [x])"
+
+fun %quote dequeue :: "'a queue \<Rightarrow> 'a option \<times> 'a queue" where
+ "dequeue (Queue []) = (None, Queue [])"
+ | "dequeue (Queue (x # xs)) = (Some x, Queue xs)"
+
+text {*
+ \noindent This we can use directly for proving; for executing,
+ we provide an alternative characterisation:
+*}
+
+definition %quote AQueue :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a queue" where
+ "AQueue xs ys = Queue (ys @ rev xs)"
+
+code_datatype %quote AQueue
+
+text {*
+ \noindent Here we define a \qt{constructor} @{const "AQueue"} which
+ is defined in terms of @{text "Queue"} and interprets its arguments
+ according to what the \emph{content} of an amortised queue is supposed
+ to be. Equipped with this, we are able to prove the following equations
+ for our primitive queue operations which \qt{implement} the simple
+ queues in an amortised fashion:
+*}
+
+lemma %quote empty_AQueue [code]:
+ "empty = AQueue [] []"
+ unfolding AQueue_def empty_def by simp
+
+lemma %quote enqueue_AQueue [code]:
+ "enqueue x (AQueue xs ys) = AQueue (x # xs) ys"
+ unfolding AQueue_def by simp
+
+lemma %quote dequeue_AQueue [code]:
+ "dequeue (AQueue xs []) =
+ (if xs = [] then (None, AQueue [] [])
+ else dequeue (AQueue [] (rev xs)))"
+ "dequeue (AQueue xs (y # ys)) = (Some y, AQueue xs ys)"
+ unfolding AQueue_def by simp_all
+
+text {*
+ \noindent For completeness, we provide a substitute for the
+ @{text case} combinator on queues:
+*}
+
+definition %quote
+ aqueue_case_def: "aqueue_case = queue_case"
+
+lemma %quote aqueue_case [code, code inline]:
+ "queue_case = aqueue_case"
+ unfolding aqueue_case_def ..
+
+lemma %quote case_AQueue [code]:
+ "aqueue_case f (AQueue xs ys) = f (ys @ rev xs)"
+ unfolding aqueue_case_def AQueue_def by simp
+
+text {*
+ \noindent The resulting code looks as expected:
+*}
+
+text %quote {*@{code_stmts empty enqueue dequeue (SML)}*}
+
+text {*
+ \noindent From this example, it can be glimpsed that using own
+ constructor sets is a little delicate since it changes the set of
+ valid patterns for values of that type. Without going into much
+ detail, here some practical hints:
+
+ \begin{itemize}
+
+ \item When changing the constructor set for datatypes, take care
+ to provide an alternative for the @{text case} combinator
+ (e.g.~by replacing it using the preprocessor).
+
+ \item Values in the target language need not to be normalised --
+ different values in the target language may represent the same
+ value in the logic.
+
+ \item Usually, a good methodology to deal with the subtleties of
+ pattern matching is to see the type as an abstract type: provide
+ a set of operations which operate on the concrete representation
+ of the type, and derive further operations by combinations of
+ these primitive ones, without relying on a particular
+ representation.
+
+ \end{itemize}
+*}
+
+
+subsection {* Equality and wellsortedness *}
+
+text {*
+ Surely you have already noticed how equality is treated
+ by the code generator:
+*}
+
+primrec %quote collect_duplicates :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+ "collect_duplicates xs ys [] = xs"
+ | "collect_duplicates xs ys (z#zs) = (if z \<in> set xs
+ then if z \<in> set ys
+ then collect_duplicates xs ys zs
+ else collect_duplicates xs (z#ys) zs
+ else collect_duplicates (z#xs) (z#ys) zs)"
+
+text {*
+ \noindent The membership test during preprocessing is rewritten,
+ resulting in @{const List.member}, which itself
+ performs an explicit equality check.
+*}
+
+text %quote {*@{code_stmts collect_duplicates (SML)}*}
+
+text {*
+ \noindent Obviously, polymorphic equality is implemented the Haskell
+ way using a type class. How is this achieved? HOL introduces
+ an explicit class @{class eq} with a corresponding operation
+ @{const eq_class.eq} such that @{thm eq [no_vars]}.
+ The preprocessing framework does the rest by propagating the
+ @{class eq} constraints through all dependent code equations.
+ For datatypes, instances of @{class eq} are implicitly derived
+ when possible. For other types, you may instantiate @{text eq}
+ manually like any other type class.
+
+ Though this @{text eq} class is designed to get rarely in
+ the way, a subtlety
+ enters the stage when definitions of overloaded constants
+ are dependent on operational equality. For example, let
+ us define a lexicographic ordering on tuples
+ (also see theory @{theory Product_ord}):
+*}
+
+instantiation %quote "*" :: (order, order) order
+begin
+
+definition %quote [code del]:
+ "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
+
+definition %quote [code del]:
+ "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
+
+instance %quote proof
+qed (auto simp: less_eq_prod_def less_prod_def intro: order_less_trans)
+
+end %quote
+
+lemma %quote order_prod [code]:
+ "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
+ x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
+ "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
+ x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
+ by (simp_all add: less_prod_def less_eq_prod_def)
+
+text {*
+ \noindent Then code generation will fail. Why? The definition
+ of @{term "op \<le>"} depends on equality on both arguments,
+ which are polymorphic and impose an additional @{class eq}
+ class constraint, which the preprocessor does not propagate
+ (for technical reasons).
+
+ The solution is to add @{class eq} explicitly to the first sort arguments in the
+ code theorems:
+*}
+
+lemma %quote order_prod_code [code]:
+ "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
+ x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
+ "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
+ x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
+ by (simp_all add: less_prod_def less_eq_prod_def)
+
+text {*
+ \noindent Then code generation succeeds:
+*}
+
+text %quote {*@{code_stmts "op \<le> \<Colon> _ \<times> _ \<Rightarrow> _ \<times> _ \<Rightarrow> bool" (SML)}*}
+
+text {*
+ In some cases, the automatically derived code equations
+ for equality on a particular type may not be appropriate.
+ As example, watch the following datatype representing
+ monomorphic parametric types (where type constructors
+ are referred to by natural numbers):
+*}
+
+datatype %quote monotype = Mono nat "monotype list"
+(*<*)
+lemma monotype_eq:
+ "eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<equiv>
+ eq_class.eq tyco1 tyco2 \<and> eq_class.eq typargs1 typargs2" by (simp add: eq)
+(*>*)
+
+text {*
+ \noindent Then code generation for SML would fail with a message
+ that the generated code contains illegal mutual dependencies:
+ the theorem @{thm monotype_eq [no_vars]} already requires the
+ instance @{text "monotype \<Colon> eq"}, which itself requires
+ @{thm monotype_eq [no_vars]}; Haskell has no problem with mutually
+ recursive @{text instance} and @{text function} definitions,
+ but the SML serialiser does not support this.
+
+ In such cases, you have to provide your own equality equations
+ involving auxiliary constants. In our case,
+ @{const [show_types] list_all2} can do the job:
+*}
+
+lemma %quote monotype_eq_list_all2 [code]:
+ "eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<longleftrightarrow>
+ eq_class.eq tyco1 tyco2 \<and> list_all2 eq_class.eq typargs1 typargs2"
+ by (simp add: eq list_all2_eq [symmetric])
+
+text {*
+ \noindent does not depend on instance @{text "monotype \<Colon> eq"}:
+*}
+
+text %quote {*@{code_stmts "eq_class.eq :: monotype \<Rightarrow> monotype \<Rightarrow> bool" (SML)}*}
+
+
+subsection {* Explicit partiality *}
+
+text {*
+ Partiality usually enters the game by partial patterns, as
+ in the following example, again for amortised queues:
+*}
+
+definition %quote strict_dequeue :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
+ "strict_dequeue q = (case dequeue q
+ of (Some x, q') \<Rightarrow> (x, q'))"
+
+lemma %quote strict_dequeue_AQueue [code]:
+ "strict_dequeue (AQueue xs (y # ys)) = (y, AQueue xs ys)"
+ "strict_dequeue (AQueue xs []) =
+ (case rev xs of y # ys \<Rightarrow> (y, AQueue [] ys))"
+ by (simp_all add: strict_dequeue_def dequeue_AQueue split: list.splits)
+
+text {*
+ \noindent In the corresponding code, there is no equation
+ for the pattern @{term "AQueue [] []"}:
+*}
+
+text %quote {*@{code_stmts strict_dequeue (consts) strict_dequeue (Haskell)}*}
+
+text {*
+ \noindent In some cases it is desirable to have this
+ pseudo-\qt{partiality} more explicitly, e.g.~as follows:
+*}
+
+axiomatization %quote empty_queue :: 'a
+
+definition %quote strict_dequeue' :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
+ "strict_dequeue' q = (case dequeue q of (Some x, q') \<Rightarrow> (x, q') | _ \<Rightarrow> empty_queue)"
+
+lemma %quote strict_dequeue'_AQueue [code]:
+ "strict_dequeue' (AQueue xs []) = (if xs = [] then empty_queue
+ else strict_dequeue' (AQueue [] (rev xs)))"
+ "strict_dequeue' (AQueue xs (y # ys)) =
+ (y, AQueue xs ys)"
+ by (simp_all add: strict_dequeue'_def dequeue_AQueue split: list.splits)
+
+text {*
+ Observe that on the right hand side of the definition of @{const
+ "strict_dequeue'"} the constant @{const empty_queue} occurs
+ which is unspecified.
+
+ Normally, if constants without any code equations occur in a
+ program, the code generator complains (since in most cases this is
+ not what the user expects). But such constants can also be thought
+ of as function definitions with no equations which always fail,
+ since there is never a successful pattern match on the left hand
+ side. In order to categorise a constant into that category
+ explicitly, use @{command "code_abort"}:
+*}
+
+code_abort %quote empty_queue
+
+text {*
+ \noindent Then the code generator will just insert an error or
+ exception at the appropriate position:
+*}
+
+text %quote {*@{code_stmts strict_dequeue' (consts) empty_queue strict_dequeue' (Haskell)}*}
+
+text {*
+ \noindent This feature however is rarely needed in practice.
+ Note also that the @{text HOL} default setup already declares
+ @{const undefined} as @{command "code_abort"}, which is most
+ likely to be used in such situations.
+*}
+
+end
+
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/ROOT.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,11 @@
+
+(* $Id$ *)
+
+no_document use_thy "Setup";
+no_document use_thys ["Efficient_Nat"];
+
+use_thy "Introduction";
+use_thy "Program";
+use_thy "Adaption";
+use_thy "Further";
+use_thy "ML";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/Setup.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,12 @@
+theory Setup
+imports Complex_Main
+uses "../../../antiquote_setup.ML" "../../../more_antiquote.ML"
+begin
+
+ML {* no_document use_thys
+ ["Efficient_Nat", "Code_Char_chr", "Product_ord", "~~/src/HOL/Imperative_HOL/Imperative_HOL",
+ "~~/src/HOL/Decision_Procs/Ferrack"] *}
+
+ML_val {* Code_Target.code_width := 74 *}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/document/Adaption.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,679 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Adaption}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{theory}\isamarkupfalse%
+\ Adaption\isanewline
+\isakeyword{imports}\ Setup\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+\isanewline
+%
+\endisadelimtheory
+%
+\isadeliminvisible
+\isanewline
+%
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{setup}\isamarkupfalse%
+\ {\isacharverbatimopen}\ Code{\isacharunderscore}Target{\isachardot}extend{\isacharunderscore}target\ {\isacharparenleft}{\isachardoublequote}{\isasymSML}{\isachardoublequote}{\isacharcomma}\ {\isacharparenleft}{\isachardoublequote}SML{\isachardoublequote}{\isacharcomma}\ K\ I{\isacharparenright}{\isacharparenright}\ {\isacharverbatimclose}%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isamarkupsection{Adaption to target languages \label{sec:adaption}%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Adapting code generation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The aspects of code generation introduced so far have two aspects
+ in common:
+
+ \begin{itemize}
+ \item They act uniformly, without reference to a specific
+ target language.
+ \item They are \emph{safe} in the sense that as long as you trust
+ the code generator meta theory and implementation, you cannot
+ produce programs that yield results which are not derivable
+ in the logic.
+ \end{itemize}
+
+ \noindent In this section we will introduce means to \emph{adapt} the serialiser
+ to a specific target language, i.e.~to print program fragments
+ in a way which accommodates \qt{already existing} ingredients of
+ a target language environment, for three reasons:
+
+ \begin{itemize}
+ \item improving readability and aesthetics of generated code
+ \item gaining efficiency
+ \item interface with language parts which have no direct counterpart
+ in \isa{HOL} (say, imperative data structures)
+ \end{itemize}
+
+ \noindent Generally, you should avoid using those features yourself
+ \emph{at any cost}:
+
+ \begin{itemize}
+ \item The safe configuration methods act uniformly on every target language,
+ whereas for adaption you have to treat each target language separate.
+ \item Application is extremely tedious since there is no abstraction
+ which would allow for a static check, making it easy to produce garbage.
+ \item More or less subtle errors can be introduced unconsciously.
+ \end{itemize}
+
+ \noindent However, even if you ought refrain from setting up adaption
+ yourself, already the \isa{HOL} comes with some reasonable default
+ adaptions (say, using target language list syntax). There also some
+ common adaption cases which you can setup by importing particular
+ library theories. In order to understand these, we provide some clues here;
+ these however are not supposed to replace a careful study of the sources.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{The adaption principle%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The following figure illustrates what \qt{adaption} is conceptually
+ supposed to be:
+
+ \begin{figure}[here]
+ \begin{tikzpicture}[scale = 0.5]
+ \tikzstyle water=[color = blue, thick]
+ \tikzstyle ice=[color = black, very thick, cap = round, join = round, fill = white]
+ \tikzstyle process=[color = green, semithick, ->]
+ \tikzstyle adaption=[color = red, semithick, ->]
+ \tikzstyle target=[color = black]
+ \foreach \x in {0, ..., 24}
+ \draw[style=water] (\x, 0.25) sin + (0.25, 0.25) cos + (0.25, -0.25) sin
+ + (0.25, -0.25) cos + (0.25, 0.25);
+ \draw[style=ice] (1, 0) --
+ (3, 6) node[above, fill=white] {logic} -- (5, 0) -- cycle;
+ \draw[style=ice] (9, 0) --
+ (11, 6) node[above, fill=white] {intermediate language} -- (13, 0) -- cycle;
+ \draw[style=ice] (15, -6) --
+ (19, 6) node[above, fill=white] {target language} -- (23, -6) -- cycle;
+ \draw[style=process]
+ (3.5, 3) .. controls (7, 5) .. node[fill=white] {translation} (10.5, 3);
+ \draw[style=process]
+ (11.5, 3) .. controls (15, 5) .. node[fill=white] (serialisation) {serialisation} (18.5, 3);
+ \node (adaption) at (11, -2) [style=adaption] {adaption};
+ \node at (19, 3) [rotate=90] {generated};
+ \node at (19.5, -5) {language};
+ \node at (19.5, -3) {library};
+ \node (includes) at (19.5, -1) {includes};
+ \node (reserved) at (16.5, -3) [rotate=72] {reserved}; % proper 71.57
+ \draw[style=process]
+ (includes) -- (serialisation);
+ \draw[style=process]
+ (reserved) -- (serialisation);
+ \draw[style=adaption]
+ (adaption) -- (serialisation);
+ \draw[style=adaption]
+ (adaption) -- (includes);
+ \draw[style=adaption]
+ (adaption) -- (reserved);
+ \end{tikzpicture}
+ \caption{The adaption principle}
+ \label{fig:adaption}
+ \end{figure}
+
+ \noindent In the tame view, code generation acts as broker between
+ \isa{logic}, \isa{intermediate\ language} and
+ \isa{target\ language} by means of \isa{translation} and
+ \isa{serialisation}; for the latter, the serialiser has to observe
+ the structure of the \isa{language} itself plus some \isa{reserved}
+ keywords which have to be avoided for generated code.
+ However, if you consider \isa{adaption} mechanisms, the code generated
+ by the serializer is just the tip of the iceberg:
+
+ \begin{itemize}
+ \item \isa{serialisation} can be \emph{parametrised} such that
+ logical entities are mapped to target-specific ones
+ (e.g. target-specific list syntax,
+ see also \secref{sec:adaption_mechanisms})
+ \item Such parametrisations can involve references to a
+ target-specific standard \isa{library} (e.g. using
+ the \isa{Haskell} \verb|Maybe| type instead
+ of the \isa{HOL} \isa{option} type);
+ if such are used, the corresponding identifiers
+ (in our example, \verb|Maybe|, \verb|Nothing|
+ and \verb|Just|) also have to be considered \isa{reserved}.
+ \item Even more, the user can enrich the library of the
+ target-language by providing code snippets
+ (\qt{\isa{includes}}) which are prepended to
+ any generated code (see \secref{sec:include}); this typically
+ also involves further \isa{reserved} identifiers.
+ \end{itemize}
+
+ \noindent As figure \ref{fig:adaption} illustrates, all these adaption mechanisms
+ have to act consistently; it is at the discretion of the user
+ to take care for this.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Common adaption patterns%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} \hyperlink{theory.Main}{\mbox{\isa{Main}}} theory already provides a code
+ generator setup
+ which should be suitable for most applications. Common extensions
+ and modifications are available by certain theories of the \isa{HOL}
+ library; beside being useful in applications, they may serve
+ as a tutorial for customising the code generator setup (see below
+ \secref{sec:adaption_mechanisms}).
+
+ \begin{description}
+
+ \item[\hyperlink{theory.Code-Integer}{\mbox{\isa{Code{\isacharunderscore}Integer}}}] represents \isa{HOL} integers by big
+ integer literals in target languages.
+ \item[\hyperlink{theory.Code-Char}{\mbox{\isa{Code{\isacharunderscore}Char}}}] represents \isa{HOL} characters by
+ character literals in target languages.
+ \item[\hyperlink{theory.Code-Char-chr}{\mbox{\isa{Code{\isacharunderscore}Char{\isacharunderscore}chr}}}] like \isa{Code{\isacharunderscore}Char},
+ but also offers treatment of character codes; includes
+ \hyperlink{theory.Code-Char}{\mbox{\isa{Code{\isacharunderscore}Char}}}.
+ \item[\hyperlink{theory.Efficient-Nat}{\mbox{\isa{Efficient{\isacharunderscore}Nat}}}] \label{eff_nat} implements natural numbers by integers,
+ which in general will result in higher efficiency; pattern
+ matching with \isa{{\isadigit{0}}} / \isa{Suc}
+ is eliminated; includes \hyperlink{theory.Code-Integer}{\mbox{\isa{Code{\isacharunderscore}Integer}}}
+ and \hyperlink{theory.Code-Index}{\mbox{\isa{Code{\isacharunderscore}Index}}}.
+ \item[\hyperlink{theory.Code-Index}{\mbox{\isa{Code{\isacharunderscore}Index}}}] provides an additional datatype
+ \isa{index} which is mapped to target-language built-in integers.
+ Useful for code setups which involve e.g. indexing of
+ target-language arrays.
+ \item[\hyperlink{theory.Code-Message}{\mbox{\isa{Code{\isacharunderscore}Message}}}] provides an additional datatype
+ \isa{message{\isacharunderscore}string} which is isomorphic to strings;
+ \isa{message{\isacharunderscore}string}s are mapped to target-language strings.
+ Useful for code setups which involve e.g. printing (error) messages.
+
+ \end{description}
+
+ \begin{warn}
+ When importing any of these theories, they should form the last
+ items in an import list. Since these theories adapt the
+ code generator setup in a non-conservative fashion,
+ strange effects may occur otherwise.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Parametrising serialisation \label{sec:adaption_mechanisms}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Consider the following function and its corresponding
+ SML code:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{primrec}\isamarkupfalse%
+\ in{\isacharunderscore}interval\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymtimes}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}in{\isacharunderscore}interval\ {\isacharparenleft}k{\isacharcomma}\ l{\isacharparenright}\ n\ {\isasymlongleftrightarrow}\ k\ {\isasymle}\ n\ {\isasymand}\ n\ {\isasymle}\ l{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isataginvisible
+%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype boola = True | False;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun anda x True = x\\
+\hspace*{0pt} ~| anda x False = False\\
+\hspace*{0pt} ~| anda True x = x\\
+\hspace*{0pt} ~| anda False x = False;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun less{\char95}nat m (Suc n) = less{\char95}eq{\char95}nat m n\\
+\hspace*{0pt} ~| less{\char95}nat n Zero{\char95}nat = False\\
+\hspace*{0pt}and less{\char95}eq{\char95}nat (Suc m) n = less{\char95}nat m n\\
+\hspace*{0pt} ~| less{\char95}eq{\char95}nat Zero{\char95}nat n = True;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun in{\char95}interval (k,~l) n = anda (less{\char95}eq{\char95}nat k n) (less{\char95}eq{\char95}nat n l);\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Though this is correct code, it is a little bit unsatisfactory:
+ boolean values and operators are materialised as distinguished
+ entities with have nothing to do with the SML-built-in notion
+ of \qt{bool}. This results in less readable code;
+ additionally, eager evaluation may cause programs to
+ loop or break which would perfectly terminate when
+ the existing SML \verb|bool| would be used. To map
+ the HOL \isa{bool} on SML \verb|bool|, we may use
+ \qn{custom serialisations}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isatagquotett
+\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
+\ bool\isanewline
+\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}bool{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ True\ \isakeyword{and}\ False\ \isakeyword{and}\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}true{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}false{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharunderscore}\ andalso\ {\isacharunderscore}{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagquotett
+{\isafoldquotett}%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\begin{isamarkuptext}%
+\noindent The \hyperlink{command.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} command takes a type constructor
+ as arguments together with a list of custom serialisations.
+ Each custom serialisation starts with a target language
+ identifier followed by an expression, which during
+ code serialisation is inserted whenever the type constructor
+ would occur. For constants, \hyperlink{command.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} implements
+ the corresponding mechanism. Each ``\verb|_|'' in
+ a serialisation expression is treated as a placeholder
+ for the type constructor's (the constant's) arguments.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun less{\char95}nat m (Suc n) = less{\char95}eq{\char95}nat m n\\
+\hspace*{0pt} ~| less{\char95}nat n Zero{\char95}nat = false\\
+\hspace*{0pt}and less{\char95}eq{\char95}nat (Suc m) n = less{\char95}nat m n\\
+\hspace*{0pt} ~| less{\char95}eq{\char95}nat Zero{\char95}nat n = true;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun in{\char95}interval (k,~l) n = (less{\char95}eq{\char95}nat k n) andalso (less{\char95}eq{\char95}nat n l);\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This still is not perfect: the parentheses
+ around the \qt{andalso} expression are superfluous.
+ Though the serialiser
+ by no means attempts to imitate the rich Isabelle syntax
+ framework, it provides some common idioms, notably
+ associative infixes with precedences which may be used here:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isatagquotett
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharparenleft}SML\ \isakeyword{infixl}\ {\isadigit{1}}\ {\isachardoublequoteopen}andalso{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagquotett
+{\isafoldquotett}%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun less{\char95}nat m (Suc n) = less{\char95}eq{\char95}nat m n\\
+\hspace*{0pt} ~| less{\char95}nat n Zero{\char95}nat = false\\
+\hspace*{0pt}and less{\char95}eq{\char95}nat (Suc m) n = less{\char95}nat m n\\
+\hspace*{0pt} ~| less{\char95}eq{\char95}nat Zero{\char95}nat n = true;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun in{\char95}interval (k,~l) n = less{\char95}eq{\char95}nat k n andalso less{\char95}eq{\char95}nat n l;\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The attentive reader may ask how we assert that no generated
+ code will accidentally overwrite. For this reason the serialiser has
+ an internal table of identifiers which have to be avoided to be used
+ for new declarations. Initially, this table typically contains the
+ keywords of the target language. It can be extended manually, thus avoiding
+ accidental overwrites, using the \hyperlink{command.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} command:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{code{\isacharunderscore}reserved}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymSML}{\isachardoublequoteclose}\ bool\ true\ false\ andalso%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Next, we try to map HOL pairs to SML pairs, using the
+ infix ``\verb|*|'' type constructor and parentheses:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isataginvisible
+%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isatagquotett
+\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
+\ {\isacharasterisk}\isanewline
+\ \ {\isacharparenleft}SML\ \isakeyword{infix}\ {\isadigit{2}}\ {\isachardoublequoteopen}{\isacharasterisk}{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ Pair\isanewline
+\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}{\isacharbang}{\isacharparenleft}{\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharcomma}{\isacharslash}\ {\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagquotett
+{\isafoldquotett}%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\begin{isamarkuptext}%
+\noindent The initial bang ``\verb|!|'' tells the serialiser
+ never to put
+ parentheses around the whole expression (they are already present),
+ while the parentheses around argument place holders
+ tell not to put parentheses around the arguments.
+ The slash ``\verb|/|'' (followed by arbitrary white space)
+ inserts a space which may be used as a break if necessary
+ during pretty printing.
+
+ These examples give a glimpse what mechanisms
+ custom serialisations provide; however their usage
+ requires careful thinking in order not to introduce
+ inconsistencies -- or, in other words:
+ custom serialisations are completely axiomatic.
+
+ A further noteworthy details is that any special
+ character in a custom serialisation may be quoted
+ using ``\verb|'|''; thus, in
+ ``\verb|fn '_ => _|'' the first
+ ``\verb|_|'' is a proper underscore while the
+ second ``\verb|_|'' is a placeholder.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{\isa{Haskell} serialisation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+For convenience, the default
+ \isa{HOL} setup for \isa{Haskell} maps the \isa{eq} class to
+ its counterpart in \isa{Haskell}, giving custom serialisations
+ for the class \isa{eq} (by command \hyperlink{command.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}) and its operation
+ \isa{eq{\isacharunderscore}class{\isachardot}eq}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isatagquotett
+\isacommand{code{\isacharunderscore}class}\isamarkupfalse%
+\ eq\isanewline
+\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Eq{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\isanewline
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ {\isacharequal}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharparenleft}Haskell\ \isakeyword{infixl}\ {\isadigit{4}}\ {\isachardoublequoteopen}{\isacharequal}{\isacharequal}{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagquotett
+{\isafoldquotett}%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\begin{isamarkuptext}%
+\noindent A problem now occurs whenever a type which
+ is an instance of \isa{eq} in \isa{HOL} is mapped
+ on a \isa{Haskell}-built-in type which is also an instance
+ of \isa{Haskell} \isa{Eq}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{typedecl}\isamarkupfalse%
+\ bar\isanewline
+\isanewline
+\isacommand{instantiation}\isamarkupfalse%
+\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ {\isachardoublequoteopen}eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}x{\isasymColon}bar{\isacharparenright}\ y\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{by}\isamarkupfalse%
+\ default\ {\isacharparenleft}simp\ add{\isacharcolon}\ eq{\isacharunderscore}bar{\isacharunderscore}def{\isacharparenright}\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+\isanewline
+%
+\isadelimquotett
+\isanewline
+%
+\endisadelimquotett
+%
+\isatagquotett
+\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
+\ bar\isanewline
+\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Integer{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagquotett
+{\isafoldquotett}%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\begin{isamarkuptext}%
+\noindent The code generator would produce
+ an additional instance, which of course is rejected by the \isa{Haskell}
+ compiler.
+ To suppress this additional instance, use
+ \isa{code{\isacharunderscore}instance}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isatagquotett
+\isacommand{code{\isacharunderscore}instance}\isamarkupfalse%
+\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
+\ \ {\isacharparenleft}Haskell\ {\isacharminus}{\isacharparenright}%
+\endisatagquotett
+{\isafoldquotett}%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isamarkupsubsection{Enhancing the target language context \label{sec:include}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In rare cases it is necessary to \emph{enrich} the context of a
+ target language; this is accomplished using the \hyperlink{command.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}
+ command:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\isatagquotett
+\isacommand{code{\isacharunderscore}include}\isamarkupfalse%
+\ Haskell\ {\isachardoublequoteopen}Errno{\isachardoublequoteclose}\isanewline
+{\isacharverbatimopen}errno\ i\ {\isacharequal}\ error\ {\isacharparenleft}{\isachardoublequote}Error\ number{\isacharcolon}\ {\isachardoublequote}\ {\isacharplus}{\isacharplus}\ show\ i{\isacharparenright}{\isacharverbatimclose}\isanewline
+\isanewline
+\isacommand{code{\isacharunderscore}reserved}\isamarkupfalse%
+\ Haskell\ Errno%
+\endisatagquotett
+{\isafoldquotett}%
+%
+\isadelimquotett
+%
+\endisadelimquotett
+%
+\begin{isamarkuptext}%
+\noindent Such named \isa{include}s are then prepended to every generated code.
+ Inspect such code in order to find out how \hyperlink{command.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} behaves
+ with respect to a particular target language.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/document/Codegen.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,1690 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Codegen}%
+%
+\isadelimtheory
+\isanewline
+\isanewline
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isamarkupchapter{Code generation from Isabelle theories%
+}
+\isamarkuptrue%
+%
+\isamarkupsection{Introduction%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Motivation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Executing formal specifications as programs is a well-established
+ topic in the theorem proving community. With increasing
+ application of theorem proving systems in the area of
+ software development and verification, its relevance manifests
+ for running test cases and rapid prototyping. In logical
+ calculi like constructive type theory,
+ a notion of executability is implicit due to the nature
+ of the calculus. In contrast, specifications in Isabelle
+ can be highly non-executable. In order to bridge
+ the gap between logic and executable specifications,
+ an explicit non-trivial transformation has to be applied:
+ code generation.
+
+ This tutorial introduces a generic code generator for the
+ Isabelle system \cite{isa-tutorial}.
+ Generic in the sense that the
+ \qn{target language} for which code shall ultimately be
+ generated is not fixed but may be an arbitrary state-of-the-art
+ functional programming language (currently, the implementation
+ supports SML \cite{SML}, OCaml \cite{OCaml} and Haskell
+ \cite{haskell-revised-report}).
+ We aim to provide a
+ versatile environment
+ suitable for software development and verification,
+ structuring the process
+ of code generation into a small set of orthogonal principles
+ while achieving a big coverage of application areas
+ with maximum flexibility.
+
+ Conceptually the code generator framework is part
+ of Isabelle's \isa{Pure} meta logic; the object logic
+ \isa{HOL} which is an extension of \isa{Pure}
+ already comes with a reasonable framework setup and thus provides
+ a good working horse for raising code-generation-driven
+ applications. So, we assume some familiarity and experience
+ with the ingredients of the \isa{HOL} \emph{Main} theory
+ (see also \cite{isa-tutorial}).%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Overview%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The code generator aims to be usable with no further ado
+ in most cases while allowing for detailed customization.
+ This manifests in the structure of this tutorial:
+ we start with a generic example \secref{sec:example}
+ and introduce code generation concepts \secref{sec:concept}.
+ Section
+ \secref{sec:basics} explains how to use the framework naively,
+ presuming a reasonable default setup. Then, section
+ \secref{sec:advanced} deals with advanced topics,
+ introducing further aspects of the code generator framework
+ in a motivation-driven manner. Last, section \secref{sec:ml}
+ introduces the framework's internal programming interfaces.
+
+ \begin{warn}
+ Ultimately, the code generator which this tutorial deals with
+ is supposed to replace the already established code generator
+ by Stefan Berghofer \cite{Berghofer-Nipkow:2002}.
+ So, for the moment, there are two distinct code generators
+ in Isabelle.
+ Also note that while the framework itself is
+ object-logic independent, only \isa{HOL} provides a reasonable
+ framework setup.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{An example: a simple theory of search trees \label{sec:example}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+When writing executable specifications using \isa{HOL},
+ it is convenient to use
+ three existing packages: the datatype package for defining
+ datatypes, the function package for (recursive) functions,
+ and the class package for overloaded definitions.
+
+ We develope a small theory of search trees; trees are represented
+ as a datatype with key type \isa{{\isacharprime}a} and value type \isa{{\isacharprime}b}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree\ {\isacharequal}\ Leaf\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}linorder{\isachardoublequoteclose}\ {\isacharprime}b\isanewline
+\ \ {\isacharbar}\ Branch\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharprime}a{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent Note that we have constrained the type of keys
+ to the class of total orders, \isa{linorder}.
+
+ We define \isa{find} and \isa{update} functions:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{primrec}\isamarkupfalse%
+\isanewline
+\ \ find\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a{\isasymColon}linorder{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ option{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}find\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\ it\ {\isacharequal}\ {\isacharparenleft}if\ it\ {\isacharequal}\ key\ then\ Some\ val\ else\ None{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}find\ {\isacharparenleft}Branch\ t{\isadigit{1}}\ key\ t{\isadigit{2}}{\isacharparenright}\ it\ {\isacharequal}\ {\isacharparenleft}if\ it\ {\isasymle}\ key\ then\ find\ t{\isadigit{1}}\ it\ else\ find\ t{\isadigit{2}}\ it{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{fun}\isamarkupfalse%
+\isanewline
+\ \ update\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}linorder\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}\isanewline
+\ \ \ \ if\ it\ {\isacharequal}\ key\ then\ Leaf\ key\ entry\isanewline
+\ \ \ \ \ \ else\ if\ it\ {\isasymle}\ key\isanewline
+\ \ \ \ \ \ then\ Branch\ {\isacharparenleft}Leaf\ it\ entry{\isacharparenright}\ it\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\isanewline
+\ \ \ \ \ \ else\ Branch\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\ it\ {\isacharparenleft}Leaf\ it\ entry{\isacharparenright}\isanewline
+\ \ \ {\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ {\isacharparenleft}Branch\ t{\isadigit{1}}\ key\ t{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}\isanewline
+\ \ \ \ if\ it\ {\isasymle}\ key\isanewline
+\ \ \ \ \ \ then\ {\isacharparenleft}Branch\ {\isacharparenleft}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ t{\isadigit{1}}{\isacharparenright}\ key\ t{\isadigit{2}}{\isacharparenright}\isanewline
+\ \ \ \ \ \ else\ {\isacharparenleft}Branch\ t{\isadigit{1}}\ key\ {\isacharparenleft}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ t{\isadigit{2}}{\isacharparenright}{\isacharparenright}\isanewline
+\ \ \ {\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent For testing purpose, we define a small example
+ using natural numbers \isa{nat} (which are a \isa{linorder})
+ as keys and list of nats as values:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ example\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}nat{\isacharcomma}\ nat\ list{\isacharparenright}\ searchtree{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}example\ {\isacharequal}\ update\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isacharcomma}\ {\isacharbrackleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharbrackright}{\isacharparenright}\ {\isacharparenleft}update\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isacharcomma}\ {\isacharbrackleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isacharbrackright}{\isacharparenright}\isanewline
+\ \ \ \ {\isacharparenleft}update\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ {\isacharbrackleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharbrackright}{\isacharparenright}\ {\isacharparenleft}Leaf\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent Then we generate code%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ example\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}tree{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent which looks like:
+ \lstsml{Thy/examples/tree.ML}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Code generation concepts and process \label{sec:concept}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=0.7\textwidth]{codegen_process}
+ \caption{code generator -- processing overview}
+ \label{fig:process}
+ \end{figure}
+
+ The code generator employs a notion of executability
+ for three foundational executable ingredients known
+ from functional programming:
+ \emph{defining equations}, \emph{datatypes}, and
+ \emph{type classes}. A defining equation as a first approximation
+ is a theorem of the form \isa{f\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n\ {\isasymequiv}\ t}
+ (an equation headed by a constant \isa{f} with arguments
+ \isa{t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n} and right hand side \isa{t}).
+ Code generation aims to turn defining equations
+ into a functional program by running through
+ a process (see figure \ref{fig:process}):
+
+ \begin{itemize}
+
+ \item Out of the vast collection of theorems proven in a
+ \qn{theory}, a reasonable subset modeling
+ defining equations is \qn{selected}.
+
+ \item On those selected theorems, certain
+ transformations are carried out
+ (\qn{preprocessing}). Their purpose is to turn theorems
+ representing non- or badly executable
+ specifications into equivalent but executable counterparts.
+ The result is a structured collection of \qn{code theorems}.
+
+ \item These \qn{code theorems} then are \qn{translated}
+ into an Haskell-like intermediate
+ language.
+
+ \item Finally, out of the intermediate language the final
+ code in the desired \qn{target language} is \qn{serialized}.
+
+ \end{itemize}
+
+ From these steps, only the two last are carried out
+ outside the logic; by keeping this layer as
+ thin as possible, the amount of code to trust is
+ kept to a minimum.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Basics \label{sec:basics}%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Invoking the code generator%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Thanks to a reasonable setup of the \isa{HOL} theories, in
+ most cases code generation proceeds without further ado:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{primrec}\isamarkupfalse%
+\isanewline
+\ \ fac\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ \ \ {\isachardoublequoteopen}fac\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}fac\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ Suc\ n\ {\isacharasterisk}\ fac\ n{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent This executable specification is now turned to SML code:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ fac\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}fac{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent The \isa{{\isasymEXPORTCODE}} command takes a space-separated list of
+ constants together with \qn{serialization directives}
+ These start with a \qn{target language}
+ identifier, followed by a file specification
+ where to write the generated code to.
+
+ Internally, the defining equations for all selected
+ constants are taken, including any transitively required
+ constants, datatypes and classes, resulting in the following
+ code:
+
+ \lstsml{Thy/examples/fac.ML}
+
+ The code generator will complain when a required
+ ingredient does not provide a executable counterpart,
+ e.g.~generating code
+ for constants not yielding
+ a defining equation (e.g.~the Hilbert choice
+ operation \isa{SOME}):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ pick{\isacharunderscore}some\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}pick{\isacharunderscore}some\ xs\ {\isacharequal}\ {\isacharparenleft}SOME\ x{\isachardot}\ x\ {\isasymin}\ set\ xs{\isacharparenright}{\isachardoublequoteclose}%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+\isanewline
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ pick{\isacharunderscore}some\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}fail{\isacharunderscore}const{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent will fail.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Theorem selection%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The list of all defining equations in a theory may be inspected
+ using the \isa{{\isasymPRINTCODESETUP}} command:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{print{\isacharunderscore}codesetup}\isamarkupfalse%
+%
+\begin{isamarkuptext}%
+\noindent which displays a table of constant with corresponding
+ defining equations (the additional stuff displayed
+ shall not bother us for the moment).
+
+ The typical \isa{HOL} tools are already set up in a way that
+ function definitions introduced by \isa{{\isasymDEFINITION}},
+ \isa{{\isasymPRIMREC}}, \isa{{\isasymFUN}},
+ \isa{{\isasymFUNCTION}}, \isa{{\isasymCONSTDEFS}},
+ \isa{{\isasymRECDEF}} are implicitly propagated
+ to this defining equation table. Specific theorems may be
+ selected using an attribute: \emph{code func}. As example,
+ a weight selector function:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{primrec}\isamarkupfalse%
+\isanewline
+\ \ pick\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}nat\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ list\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}pick\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ n\ {\isacharequal}\ {\isacharparenleft}let\ {\isacharparenleft}k{\isacharcomma}\ v{\isacharparenright}\ {\isacharequal}\ x\ in\isanewline
+\ \ \ \ if\ n\ {\isacharless}\ k\ then\ v\ else\ pick\ xs\ {\isacharparenleft}n\ {\isacharminus}\ k{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent We want to eliminate the explicit destruction
+ of \isa{x} to \isa{{\isacharparenleft}k{\isacharcomma}\ v{\isacharparenright}}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}pick\ {\isacharparenleft}{\isacharparenleft}k{\isacharcomma}\ v{\isacharparenright}{\isacharhash}xs{\isacharparenright}\ n\ {\isacharequal}\ {\isacharparenleft}if\ n\ {\isacharless}\ k\ then\ v\ else\ pick\ xs\ {\isacharparenleft}n\ {\isacharminus}\ k{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ pick\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}pick{\isadigit{1}}{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent This theorem now is used for generating code:
+
+ \lstsml{Thy/examples/pick1.ML}
+
+ \noindent The policy is that \emph{default equations} stemming from
+ \isa{{\isasymDEFINITION}},
+ \isa{{\isasymPRIMREC}}, \isa{{\isasymFUN}},
+ \isa{{\isasymFUNCTION}}, \isa{{\isasymCONSTDEFS}},
+ \isa{{\isasymRECDEF}} statements are discarded as soon as an
+ equation is explicitly selected by means of \emph{code func}.
+ Further applications of \emph{code func} add theorems incrementally,
+ but syntactic redundancies are implicitly dropped. For example,
+ using a modified version of the \isa{fac} function
+ as defining equation, the then redundant (since
+ syntactically subsumed) original defining equations
+ are dropped.
+
+ \begin{warn}
+ The attributes \emph{code} and \emph{code del}
+ associated with the existing code generator also apply to
+ the new one: \emph{code} implies \emph{code func},
+ and \emph{code del} implies \emph{code func del}.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Type classes%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Type classes enter the game via the Isar class package.
+ For a short introduction how to use it, see \cite{isabelle-classes};
+ here we just illustrate its impact on code generation.
+
+ In a target language, type classes may be represented
+ natively (as in the case of Haskell). For languages
+ like SML, they are implemented using \emph{dictionaries}.
+ Our following example specifies a class \qt{null},
+ assigning to each of its inhabitants a \qt{null} value:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{class}\isamarkupfalse%
+\ null\ {\isacharequal}\ type\ {\isacharplus}\isanewline
+\ \ \isakeyword{fixes}\ null\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\isanewline
+\isanewline
+\isacommand{primrec}\isamarkupfalse%
+\isanewline
+\ \ head\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}null\ list\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}head\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ null{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}head\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent We provide some instances for our \isa{null}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{instantiation}\isamarkupfalse%
+\ option\ \isakeyword{and}\ list\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharparenright}\ null\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ {\isachardoublequoteopen}null\ {\isacharequal}\ None{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ {\isachardoublequoteopen}null\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\begin{isamarkuptext}%
+\noindent Constructing a dummy example:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ {\isachardoublequoteopen}dummy\ {\isacharequal}\ head\ {\isacharbrackleft}Some\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ None{\isacharbrackright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+Type classes offer a suitable occasion to introduce
+ the Haskell serializer. Its usage is almost the same
+ as SML, but, in accordance with conventions
+ some Haskell systems enforce, each module ends
+ up in a single file. The module hierarchy is reflected in
+ the file system, with root directory given as file specification.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ dummy\ \isakeyword{in}\ Haskell\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lsthaskell{Thy/examples/Codegen.hs}
+ \noindent (we have left out all other modules).
+
+ \medskip
+
+ The whole code in SML with explicit dictionary passing:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ dummy\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}class{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/class.ML}
+
+ \medskip
+
+ \noindent or in OCaml:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ dummy\ \isakeyword{in}\ OCaml\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}class{\isachardot}ocaml{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/class.ocaml}
+
+ \medskip The explicit association of constants
+ to classes can be inspected using the \isa{{\isasymPRINTCLASSES}}
+ command.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Recipes and advanced topics \label{sec:advanced}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In this tutorial, we do not attempt to give an exhaustive
+ description of the code generator framework; instead,
+ we cast a light on advanced topics by introducing
+ them together with practically motivated examples. Concerning
+ further reading, see
+
+ \begin{itemize}
+
+ \item the Isabelle/Isar Reference Manual \cite{isabelle-isar-ref}
+ for exhaustive syntax diagrams.
+ \item or \cite{Haftmann-Nipkow:2007:codegen} which deals with foundational issues
+ of the code generator framework.
+
+ \end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Library theories \label{sec:library}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The \isa{HOL} \isa{Main} theory already provides a code
+ generator setup
+ which should be suitable for most applications. Common extensions
+ and modifications are available by certain theories of the \isa{HOL}
+ library; beside being useful in applications, they may serve
+ as a tutorial for customizing the code generator setup.
+
+ \begin{description}
+
+ \item[\isa{Code{\isacharunderscore}Integer}] represents \isa{HOL} integers by big
+ integer literals in target languages.
+ \item[\isa{Code{\isacharunderscore}Char}] represents \isa{HOL} characters by
+ character literals in target languages.
+ \item[\isa{Code{\isacharunderscore}Char{\isacharunderscore}chr}] like \isa{Code{\isacharunderscore}Char},
+ but also offers treatment of character codes; includes
+ \isa{Code{\isacharunderscore}Integer}.
+ \item[\isa{Efficient{\isacharunderscore}Nat}] \label{eff_nat} implements natural numbers by integers,
+ which in general will result in higher efficency; pattern
+ matching with \isa{{\isadigit{0}}} / \isa{Suc}
+ is eliminated; includes \isa{Code{\isacharunderscore}Integer}.
+ \item[\isa{Code{\isacharunderscore}Index}] provides an additional datatype
+ \isa{index} which is mapped to target-language built-in integers.
+ Useful for code setups which involve e.g. indexing of
+ target-language arrays.
+ \item[\isa{Code{\isacharunderscore}Message}] provides an additional datatype
+ \isa{message{\isacharunderscore}string} which is isomorphic to strings;
+ \isa{message{\isacharunderscore}string}s are mapped to target-language strings.
+ Useful for code setups which involve e.g. printing (error) messages.
+
+ \end{description}
+
+ \begin{warn}
+ When importing any of these theories, they should form the last
+ items in an import list. Since these theories adapt the
+ code generator setup in a non-conservative fashion,
+ strange effects may occur otherwise.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Preprocessing%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Before selected function theorems are turned into abstract
+ code, a chain of definitional transformation steps is carried
+ out: \emph{preprocessing}. In essence, the preprocessor
+ consists of two components: a \emph{simpset} and \emph{function transformers}.
+
+ The \emph{simpset} allows to employ the full generality of the Isabelle
+ simplifier. Due to the interpretation of theorems
+ as defining equations, rewrites are applied to the right
+ hand side and the arguments of the left hand side of an
+ equation, but never to the constant heading the left hand side.
+ An important special case are \emph{inline theorems} which may be
+ declared an undeclared using the
+ \emph{code inline} or \emph{code inline del} attribute respectively.
+ Some common applications:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{itemize}
+%
+\begin{isamarkuptext}%
+\item replacing non-executable constructs by executable ones:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ set\ xs\ {\isasymlongleftrightarrow}\ x\ mem\ xs{\isachardoublequoteclose}%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\item eliminating superfluous constants:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\item replacing executable but inconvenient constructs:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\ \ \isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ \ \ {\isachardoublequoteopen}xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymlongleftrightarrow}\ List{\isachardot}null\ xs{\isachardoublequoteclose}%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\end{itemize}
+%
+\begin{isamarkuptext}%
+\emph{Function transformers} provide a very general interface,
+ transforming a list of function theorems to another
+ list of function theorems, provided that neither the heading
+ constant nor its type change. The \isa{{\isadigit{0}}} / \isa{Suc}
+ pattern elimination implemented in
+ theory \isa{Efficient{\isacharunderscore}Nat} (see \secref{eff_nat}) uses this
+ interface.
+
+ \noindent The current setup of the preprocessor may be inspected using
+ the \isa{{\isasymPRINTCODESETUP}} command.
+
+ \begin{warn}
+ The attribute \emph{code unfold}
+ associated with the existing code generator also applies to
+ the new one: \emph{code unfold} implies \emph{code inline}.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Concerning operational equality%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Surely you have already noticed how equality is treated
+ by the code generator:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{primrec}\isamarkupfalse%
+\isanewline
+\ \ collect{\isacharunderscore}duplicates\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ \ \ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ xs{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharparenleft}z{\isacharhash}zs{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ z\ {\isasymin}\ set\ xs\isanewline
+\ \ \ \ \ \ then\ if\ z\ {\isasymin}\ set\ ys\isanewline
+\ \ \ \ \ \ \ \ then\ collect{\isacharunderscore}duplicates\ xs\ ys\ zs\isanewline
+\ \ \ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ xs\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs\isanewline
+\ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ {\isacharparenleft}z{\isacharhash}xs{\isacharparenright}\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+The membership test during preprocessing is rewritten,
+ resulting in \isa{op\ mem}, which itself
+ performs an explicit equality check.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ collect{\isacharunderscore}duplicates\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}collect{\isacharunderscore}duplicates{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/collect_duplicates.ML}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Obviously, polymorphic equality is implemented the Haskell
+ way using a type class. How is this achieved? HOL introduces
+ an explicit class \isa{eq} with a corresponding operation
+ \isa{eq{\isacharunderscore}class{\isachardot}eq} such that \isa{eq{\isacharunderscore}class{\isachardot}eq\ x\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharequal}\ y{\isacharparenright}}.
+ The preprocessing framework does the rest.
+ For datatypes, instances of \isa{eq} are implicitly derived
+ when possible. For other types, you may instantiate \isa{eq}
+ manually like any other type class.
+
+ Though this \isa{eq} class is designed to get rarely in
+ the way, a subtlety
+ enters the stage when definitions of overloaded constants
+ are dependent on operational equality. For example, let
+ us define a lexicographic ordering on tuples:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{instantiation}\isamarkupfalse%
+\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}ord{\isacharcomma}\ ord{\isacharparenright}\ ord\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ {\isacharbrackleft}code\ func\ del{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}p{\isadigit{1}}\ {\isacharless}\ p{\isadigit{2}}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}let\ {\isacharparenleft}x{\isadigit{1}}{\isacharcomma}\ y{\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{1}}{\isacharsemicolon}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{2}}\ in\isanewline
+\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ {\isacharbrackleft}code\ func\ del{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}p{\isadigit{1}}\ {\isasymle}\ p{\isadigit{2}}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}let\ {\isacharparenleft}x{\isadigit{1}}{\isacharcomma}\ y{\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{1}}{\isacharsemicolon}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{2}}\ in\isanewline
+\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ ord{\isacharunderscore}prod\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}ord{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}ord{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{unfolding}\isamarkupfalse%
+\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp{\isacharunderscore}all%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Then code generation will fail. Why? The definition
+ of \isa{op\ {\isasymle}} depends on equality on both arguments,
+ which are polymorphic and impose an additional \isa{eq}
+ class constraint, thus violating the type discipline
+ for class operations.
+
+ The solution is to add \isa{eq} explicitly to the first sort arguments in the
+ code theorems:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ ord{\isacharunderscore}prod{\isacharunderscore}code\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}ord{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}ord{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{unfolding}\isamarkupfalse%
+\ ord{\isacharunderscore}prod\ \isacommand{by}\isamarkupfalse%
+\ rule{\isacharplus}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent Then code generation succeeds:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}eq{\isacharcomma}\ ord{\isacharbraceright}\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}ord\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}lexicographic{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/lexicographic.ML}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In general, code theorems for overloaded constants may have more
+ restrictive sort constraints than the underlying instance relation
+ between class and type constructor as long as the whole system of
+ constraints is coregular; code theorems violating coregularity
+ are rejected immediately. Consequently, it might be necessary
+ to delete disturbing theorems in the code theorem table,
+ as we have done here with the original definitions \isa{less{\isacharunderscore}prod{\isacharunderscore}def}
+ and \isa{less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def}.
+
+ In some cases, the automatically derived defining equations
+ for equality on a particular type may not be appropriate.
+ As example, watch the following datatype representing
+ monomorphic parametric types (where type constructors
+ are referred to by natural numbers):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ monotype\ {\isacharequal}\ Mono\ nat\ {\isachardoublequoteopen}monotype\ list{\isachardoublequoteclose}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Then code generation for SML would fail with a message
+ that the generated code conains illegal mutual dependencies:
+ the theorem \isa{Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}\ {\isacharequal}\ Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}\ {\isasymequiv}\ tyco{\isadigit{1}}\ {\isacharequal}\ tyco{\isadigit{2}}\ {\isasymand}\ typargs{\isadigit{1}}\ {\isacharequal}\ typargs{\isadigit{2}}} already requires the
+ instance \isa{monotype\ {\isasymColon}\ eq}, which itself requires
+ \isa{Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}\ {\isacharequal}\ Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}\ {\isasymequiv}\ tyco{\isadigit{1}}\ {\isacharequal}\ tyco{\isadigit{2}}\ {\isasymand}\ typargs{\isadigit{1}}\ {\isacharequal}\ typargs{\isadigit{2}}}; Haskell has no problem with mutually
+ recursive \isa{instance} and \isa{function} definitions,
+ but the SML serializer does not support this.
+
+ In such cases, you have to provide you own equality equations
+ involving auxiliary constants. In our case,
+ \isa{list{\isacharunderscore}all{\isadigit{2}}} can do the job:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ monotype{\isacharunderscore}eq{\isacharunderscore}list{\isacharunderscore}all{\isadigit{2}}\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}\ {\isacharequal}\ Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ \ tyco{\isadigit{1}}\ {\isacharequal}\ tyco{\isadigit{2}}\ {\isasymand}\ list{\isacharunderscore}all{\isadigit{2}}\ {\isacharparenleft}op\ {\isacharequal}{\isacharparenright}\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ list{\isacharunderscore}all{\isadigit{2}}{\isacharunderscore}eq\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+does not depend on instance \isa{monotype\ {\isasymColon}\ eq}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ {\isacharequal}\ {\isacharcolon}{\isacharcolon}\ monotype\ {\isasymRightarrow}\ monotype\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}monotype{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/monotype.ML}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Programs as sets of theorems%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+As told in \secref{sec:concept}, code generation is based
+ on a structured collection of code theorems.
+ For explorative purpose, this collection
+ may be inspected using the \isa{{\isasymCODETHMS}} command:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{code{\isacharunderscore}thms}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ mod\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent prints a table with \emph{all} defining equations
+ for \isa{op\ mod}, including
+ \emph{all} defining equations those equations depend
+ on recursivly. \isa{{\isasymCODETHMS}} provides a convenient
+ mechanism to inspect the impact of a preprocessor setup
+ on defining equations.
+
+ Similarly, the \isa{{\isasymCODEDEPS}} command shows a graph
+ visualizing dependencies between defining equations.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Constructor sets for datatypes%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Conceptually, any datatype is spanned by a set of
+ \emph{constructors} of type \isa{{\isasymtau}\ {\isacharequal}\ {\isasymdots}\ {\isasymRightarrow}\ {\isasymkappa}\ {\isasymalpha}\isactrlisub {\isadigit{1}}\ {\isasymdots}\ {\isasymalpha}\isactrlisub n}
+ where \isa{{\isacharbraceleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub n{\isacharbraceright}} is excactly the set of \emph{all}
+ type variables in \isa{{\isasymtau}}. The HOL datatype package
+ by default registers any new datatype in the table
+ of datatypes, which may be inspected using
+ the \isa{{\isasymPRINTCODESETUP}} command.
+
+ In some cases, it may be convenient to alter or
+ extend this table; as an example, we will develope an alternative
+ representation of natural numbers as binary digits, whose
+ size does increase logarithmically with its value, not linear
+ \footnote{Indeed, the \isa{Efficient{\isacharunderscore}Nat} theory (see \ref{eff_nat})
+ does something similar}. First, the digit representation:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ Dig{\isadigit{0}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ n\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ n{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ Dig{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent We will use these two ">digits"< to represent natural numbers
+ in binary digits, e.g.:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isadigit{4}}{\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{2}}\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}Dig{\isadigit{1}}\ {\isacharparenleft}Dig{\isadigit{0}}\ {\isacharparenleft}Dig{\isadigit{1}}\ {\isacharparenleft}Dig{\isadigit{0}}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ Dig{\isadigit{0}}{\isacharunderscore}def\ Dig{\isadigit{1}}{\isacharunderscore}def{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent Of course we also have to provide proper code equations for
+ the operations, e.g. \isa{op\ {\isacharplus}}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ plus{\isacharunderscore}Dig\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharplus}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}m\ {\isacharplus}\ {\isadigit{0}}\ {\isacharequal}\ m{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharplus}\ Dig{\isadigit{0}}\ n\ {\isacharequal}\ Dig{\isadigit{1}}\ n{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ m\ {\isacharplus}\ {\isadigit{1}}\ {\isacharequal}\ Dig{\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharplus}\ Dig{\isadigit{1}}\ n\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ m\ {\isacharplus}\ {\isadigit{1}}\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}m\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ m\ {\isacharplus}\ Dig{\isadigit{0}}\ n\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ m\ {\isacharplus}\ Dig{\isadigit{1}}\ n\ {\isacharequal}\ Dig{\isadigit{1}}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ m\ {\isacharplus}\ Dig{\isadigit{0}}\ n\ {\isacharequal}\ Dig{\isadigit{1}}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ m\ {\isacharplus}\ Dig{\isadigit{1}}\ n\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}m\ {\isacharplus}\ n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ Dig{\isadigit{0}}{\isacharunderscore}def\ Dig{\isadigit{1}}{\isacharunderscore}def{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent We then instruct the code generator to view \isa{{\isadigit{0}}},
+ \isa{{\isadigit{1}}}, \isa{Dig{\isadigit{0}}} and \isa{Dig{\isadigit{1}}} as
+ datatype constructors:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isadigit{0}}{\isasymColon}nat{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isasymColon}nat{\isachardoublequoteclose}\ Dig{\isadigit{0}}\ Dig{\isadigit{1}}%
+\begin{isamarkuptext}%
+\noindent For the former constructor \isa{Suc}, we provide a code
+ equation and remove some parts of the default code generator setup
+ which are an obstacle here:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ Suc{\isacharunderscore}Dig\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}Suc\ n\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{declare}\isamarkupfalse%
+\ One{\isacharunderscore}nat{\isacharunderscore}def\ {\isacharbrackleft}code\ inline\ del{\isacharbrackright}\isanewline
+\isacommand{declare}\isamarkupfalse%
+\ add{\isacharunderscore}Suc{\isacharunderscore}shift\ {\isacharbrackleft}code\ func\ del{\isacharbrackright}%
+\begin{isamarkuptext}%
+\noindent This yields the following code:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ {\isacharplus}\ {\isasymColon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}nat{\isacharunderscore}binary{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/nat_binary.ML}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\medskip
+
+ From this example, it can be easily glimpsed that using own constructor sets
+ is a little delicate since it changes the set of valid patterns for values
+ of that type. Without going into much detail, here some practical hints:
+
+ \begin{itemize}
+ \item When changing the constuctor set for datatypes, take care to
+ provide an alternative for the \isa{case} combinator (e.g. by replacing
+ it using the preprocessor).
+ \item Values in the target language need not to be normalized -- different
+ values in the target language may represent the same value in the
+ logic (e.g. \isa{Dig{\isadigit{1}}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}}).
+ \item Usually, a good methodology to deal with the subleties of pattern
+ matching is to see the type as an abstract type: provide a set
+ of operations which operate on the concrete representation of the type,
+ and derive further operations by combinations of these primitive ones,
+ without relying on a particular representation.
+ \end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isataginvisible
+\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isadigit{0}}{\isacharcolon}{\isacharcolon}nat{\isachardoublequoteclose}\ Suc\isanewline
+\isacommand{declare}\isamarkupfalse%
+\ plus{\isacharunderscore}Dig\ {\isacharbrackleft}code\ func\ del{\isacharbrackright}\isanewline
+\isacommand{declare}\isamarkupfalse%
+\ One{\isacharunderscore}nat{\isacharunderscore}def\ {\isacharbrackleft}code\ inline{\isacharbrackright}\isanewline
+\isacommand{declare}\isamarkupfalse%
+\ add{\isacharunderscore}Suc{\isacharunderscore}shift\ {\isacharbrackleft}code\ func{\isacharbrackright}\ \isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharplus}\ n\ {\isacharequal}\ {\isacharparenleft}n\ {\isasymColon}\ nat{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp%
+\endisataginvisible
+{\isafoldinvisible}%
+%
+\isadeliminvisible
+%
+\endisadeliminvisible
+%
+\isamarkupsubsection{Customizing serialization%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Basics%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Consider the following function and its corresponding
+ SML code:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{primrec}\isamarkupfalse%
+\isanewline
+\ \ in{\isacharunderscore}interval\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymtimes}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}in{\isacharunderscore}interval\ {\isacharparenleft}k{\isacharcomma}\ l{\isacharparenright}\ n\ {\isasymlongleftrightarrow}\ k\ {\isasymle}\ n\ {\isasymand}\ n\ {\isasymle}\ l{\isachardoublequoteclose}%
+\isadelimtt
+%
+\endisadelimtt
+%
+\isatagtt
+%
+\endisatagtt
+{\isafoldtt}%
+%
+\isadelimtt
+%
+\endisadelimtt
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ in{\isacharunderscore}interval\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}bool{\isacharunderscore}literal{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/bool_literal.ML}
+
+ \noindent Though this is correct code, it is a little bit unsatisfactory:
+ boolean values and operators are materialized as distinguished
+ entities with have nothing to do with the SML-builtin notion
+ of \qt{bool}. This results in less readable code;
+ additionally, eager evaluation may cause programs to
+ loop or break which would perfectly terminate when
+ the existing SML \qt{bool} would be used. To map
+ the HOL \qt{bool} on SML \qt{bool}, we may use
+ \qn{custom serializations}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\isatagtt
+\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
+\ bool\isanewline
+\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}bool{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ True\ \isakeyword{and}\ False\ \isakeyword{and}\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}true{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}false{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharunderscore}\ andalso\ {\isacharunderscore}{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagtt
+{\isafoldtt}%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\begin{isamarkuptext}%
+The \isa{{\isasymCODETYPE}} commad takes a type constructor
+ as arguments together with a list of custom serializations.
+ Each custom serialization starts with a target language
+ identifier followed by an expression, which during
+ code serialization is inserted whenever the type constructor
+ would occur. For constants, \isa{{\isasymCODECONST}} implements
+ the corresponding mechanism. Each ``\verb|_|'' in
+ a serialization expression is treated as a placeholder
+ for the type constructor's (the constant's) arguments.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ in{\isacharunderscore}interval\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}bool{\isacharunderscore}mlbool{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/bool_mlbool.ML}
+
+ \noindent This still is not perfect: the parentheses
+ around the \qt{andalso} expression are superfluous.
+ Though the serializer
+ by no means attempts to imitate the rich Isabelle syntax
+ framework, it provides some common idioms, notably
+ associative infixes with precedences which may be used here:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\isatagtt
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharparenleft}SML\ \isakeyword{infixl}\ {\isadigit{1}}\ {\isachardoublequoteopen}andalso{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagtt
+{\isafoldtt}%
+%
+\isadelimtt
+%
+\endisadelimtt
+\isanewline
+\isanewline
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ in{\isacharunderscore}interval\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}bool{\isacharunderscore}infix{\isachardot}ML{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\lstsml{Thy/examples/bool_infix.ML}
+
+ \medskip
+
+ Next, we try to map HOL pairs to SML pairs, using the
+ infix ``\verb|*|'' type constructor and parentheses:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\isatagtt
+\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
+\ {\isacharasterisk}\isanewline
+\ \ {\isacharparenleft}SML\ \isakeyword{infix}\ {\isadigit{2}}\ {\isachardoublequoteopen}{\isacharasterisk}{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ Pair\isanewline
+\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}{\isacharbang}{\isacharparenleft}{\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharcomma}{\isacharslash}\ {\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagtt
+{\isafoldtt}%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\begin{isamarkuptext}%
+The initial bang ``\verb|!|'' tells the serializer to never put
+ parentheses around the whole expression (they are already present),
+ while the parentheses around argument place holders
+ tell not to put parentheses around the arguments.
+ The slash ``\verb|/|'' (followed by arbitrary white space)
+ inserts a space which may be used as a break if necessary
+ during pretty printing.
+
+ These examples give a glimpse what mechanisms
+ custom serializations provide; however their usage
+ requires careful thinking in order not to introduce
+ inconsistencies -- or, in other words:
+ custom serializations are completely axiomatic.
+
+ A further noteworthy details is that any special
+ character in a custom serialization may be quoted
+ using ``\verb|'|''; thus, in
+ ``\verb|fn '_ => _|'' the first
+ ``\verb|_|'' is a proper underscore while the
+ second ``\verb|_|'' is a placeholder.
+
+ The HOL theories provide further
+ examples for custom serializations.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Haskell serialization%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+For convenience, the default
+ HOL setup for Haskell maps the \isa{eq} class to
+ its counterpart in Haskell, giving custom serializations
+ for the class (\isa{{\isasymCODECLASS}}) and its operation:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\isatagtt
+\isacommand{code{\isacharunderscore}class}\isamarkupfalse%
+\ eq\isanewline
+\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Eq{\isachardoublequoteclose}\ \isakeyword{where}\ {\isachardoublequoteopen}op\ {\isacharequal}{\isachardoublequoteclose}\ {\isasymequiv}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharequal}{\isacharequal}{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\isanewline
+\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
+\ {\isachardoublequoteopen}op\ {\isacharequal}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharparenleft}Haskell\ \isakeyword{infixl}\ {\isadigit{4}}\ {\isachardoublequoteopen}{\isacharequal}{\isacharequal}{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagtt
+{\isafoldtt}%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\begin{isamarkuptext}%
+A problem now occurs whenever a type which
+ is an instance of \isa{eq} in HOL is mapped
+ on a Haskell-builtin type which is also an instance
+ of Haskell \isa{Eq}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{typedecl}\isamarkupfalse%
+\ bar\isanewline
+\isanewline
+\isacommand{instantiation}\isamarkupfalse%
+\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ {\isachardoublequoteopen}eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}x{\isasymColon}bar{\isacharparenright}\ y\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ default\ {\isacharparenleft}simp\ add{\isacharcolon}\ eq{\isacharunderscore}bar{\isacharunderscore}def{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+\isanewline
+%
+\isadelimtt
+\isanewline
+%
+\endisadelimtt
+%
+\isatagtt
+\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
+\ bar\isanewline
+\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Integer{\isachardoublequoteclose}{\isacharparenright}%
+\endisatagtt
+{\isafoldtt}%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\begin{isamarkuptext}%
+The code generator would produce
+ an additional instance, which of course is rejected.
+ To suppress this additional instance, use
+ \isa{{\isasymCODEINSTANCE}}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\isatagtt
+\isacommand{code{\isacharunderscore}instance}\isamarkupfalse%
+\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
+\ \ {\isacharparenleft}Haskell\ {\isacharminus}{\isacharparenright}%
+\endisatagtt
+{\isafoldtt}%
+%
+\isadelimtt
+%
+\endisadelimtt
+%
+\isamarkupsubsubsection{Pretty printing%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The serializer provides ML interfaces to set up
+ pretty serializations for expressions like lists, numerals
+ and characters; these are
+ monolithic stubs and should only be used with the
+ theories introduced in \secref{sec:library}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Cyclic module dependencies%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Sometimes the awkward situation occurs that dependencies
+ between definitions introduce cyclic dependencies
+ between modules, which in the Haskell world leaves
+ you to the mercy of the Haskell implementation you are using,
+ while for SML code generation is not possible.
+
+ A solution is to declare module names explicitly.
+ Let use assume the three cyclically dependent
+ modules are named \emph{A}, \emph{B} and \emph{C}.
+ Then, by stating%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{code{\isacharunderscore}modulename}\isamarkupfalse%
+\ SML\isanewline
+\ \ A\ ABC\isanewline
+\ \ B\ ABC\isanewline
+\ \ C\ ABC%
+\begin{isamarkuptext}%
+we explicitly map all those modules on \emph{ABC},
+ resulting in an ad-hoc merge of this three modules
+ at serialization time.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Incremental code generation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Code generation is \emph{incremental}: theorems
+ and abstract intermediate code are cached and extended on demand.
+ The cache may be partially or fully dropped if the underlying
+ executable content of the theory changes.
+ Implementation of caching is supposed to transparently
+ hid away the details from the user. Anyway, caching
+ reaches the surface by using a slightly more general form
+ of the \isa{{\isasymCODETHMS}}, \isa{{\isasymCODEDEPS}}
+ and \isa{{\isasymEXPORTCODE}} commands: the list of constants
+ may be omitted. Then, all constants with code theorems
+ in the current cache are referred to.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{ML interfaces \label{sec:ml}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Since the code generator framework not only aims to provide
+ a nice Isar interface but also to form a base for
+ code-generation-based applications, here a short
+ description of the most important ML interfaces.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Executable theory content: \isa{Code}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This Pure module implements the core notions of
+ executable content of a theory.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Managing executable content%
+}
+\isamarkuptrue%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isatagmlref
+%
+\begin{isamarkuptext}%
+\begin{mldecls}
+ \indexml{Code.add\_func}\verb|Code.add_func: thm -> theory -> theory| \\
+ \indexml{Code.del\_func}\verb|Code.del_func: thm -> theory -> theory| \\
+ \indexml{Code.add\_funcl}\verb|Code.add_funcl: string * thm list Susp.T -> theory -> theory| \\
+ \indexml{Code.map\_pre}\verb|Code.map_pre: (MetaSimplifier.simpset -> MetaSimplifier.simpset) -> theory -> theory| \\
+ \indexml{Code.map\_post}\verb|Code.map_post: (MetaSimplifier.simpset -> MetaSimplifier.simpset) -> theory -> theory| \\
+ \indexml{Code.add\_functrans}\verb|Code.add_functrans: string * (theory -> thm list -> thm list option)|\isasep\isanewline%
+\verb| -> theory -> theory| \\
+ \indexml{Code.del\_functrans}\verb|Code.del_functrans: string -> theory -> theory| \\
+ \indexml{Code.add\_datatype}\verb|Code.add_datatype: (string * typ) list -> theory -> theory| \\
+ \indexml{Code.get\_datatype}\verb|Code.get_datatype: theory -> string|\isasep\isanewline%
+\verb| -> (string * sort) list * (string * typ list) list| \\
+ \indexml{Code.get\_datatype\_of\_constr}\verb|Code.get_datatype_of_constr: theory -> string -> string option|
+ \end{mldecls}
+
+ \begin{description}
+
+ \item \verb|Code.add_func|~\isa{thm}~\isa{thy} adds function
+ theorem \isa{thm} to executable content.
+
+ \item \verb|Code.del_func|~\isa{thm}~\isa{thy} removes function
+ theorem \isa{thm} from executable content, if present.
+
+ \item \verb|Code.add_funcl|~\isa{{\isacharparenleft}const{\isacharcomma}\ lthms{\isacharparenright}}~\isa{thy} adds
+ suspended defining equations \isa{lthms} for constant
+ \isa{const} to executable content.
+
+ \item \verb|Code.map_pre|~\isa{f}~\isa{thy} changes
+ the preprocessor simpset.
+
+ \item \verb|Code.add_functrans|~\isa{{\isacharparenleft}name{\isacharcomma}\ f{\isacharparenright}}~\isa{thy} adds
+ function transformer \isa{f} (named \isa{name}) to executable content;
+ \isa{f} is a transformer of the defining equations belonging
+ to a certain function definition, depending on the
+ current theory context. Returning \isa{NONE} indicates that no
+ transformation took place; otherwise, the whole process will be iterated
+ with the new defining equations.
+
+ \item \verb|Code.del_functrans|~\isa{name}~\isa{thy} removes
+ function transformer named \isa{name} from executable content.
+
+ \item \verb|Code.add_datatype|~\isa{cs}~\isa{thy} adds
+ a datatype to executable content, with generation
+ set \isa{cs}.
+
+ \item \verb|Code.get_datatype_of_constr|~\isa{thy}~\isa{const}
+ returns type constructor corresponding to
+ constructor \isa{const}; returns \isa{NONE}
+ if \isa{const} is no constructor.
+
+ \end{description}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagmlref
+{\isafoldmlref}%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isamarkupsubsection{Auxiliary%
+}
+\isamarkuptrue%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isatagmlref
+%
+\begin{isamarkuptext}%
+\begin{mldecls}
+ \indexml{Code\_Unit.read\_const}\verb|Code_Unit.read_const: theory -> string -> string| \\
+ \indexml{Code\_Unit.head\_func}\verb|Code_Unit.head_func: thm -> string * ((string * sort) list * typ)| \\
+ \indexml{Code\_Unit.rewrite\_func}\verb|Code_Unit.rewrite_func: MetaSimplifier.simpset -> thm -> thm| \\
+ \end{mldecls}
+
+ \begin{description}
+
+ \item \verb|Code_Unit.read_const|~\isa{thy}~\isa{s}
+ reads a constant as a concrete term expression \isa{s}.
+
+ \item \verb|Code_Unit.head_func|~\isa{thm}
+ extracts the constant and its type from a defining equation \isa{thm}.
+
+ \item \verb|Code_Unit.rewrite_func|~\isa{ss}~\isa{thm}
+ rewrites a defining equation \isa{thm} with a simpset \isa{ss};
+ only arguments and right hand side are rewritten,
+ not the head of the defining equation.
+
+ \end{description}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagmlref
+{\isafoldmlref}%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isamarkupsubsection{Implementing code generator applications%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Implementing code generator applications on top
+ of the framework set out so far usually not only
+ involves using those primitive interfaces
+ but also storing code-dependent data and various
+ other things.
+
+ \begin{warn}
+ Some interfaces discussed here have not reached
+ a final state yet.
+ Changes likely to occur in future.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Data depending on the theory's executable content%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Due to incrementality of code generation, changes in the
+ theory's executable content have to be propagated in a
+ certain fashion. Additionally, such changes may occur
+ not only during theory extension but also during theory
+ merge, which is a little bit nasty from an implementation
+ point of view. The framework provides a solution
+ to this technical challenge by providing a functorial
+ data slot \verb|CodeDataFun|; on instantiation
+ of this functor, the following types and operations
+ are required:
+
+ \medskip
+ \begin{tabular}{l}
+ \isa{type\ T} \\
+ \isa{val\ empty{\isacharcolon}\ T} \\
+ \isa{val\ merge{\isacharcolon}\ Pretty{\isachardot}pp\ {\isasymrightarrow}\ T\ {\isacharasterisk}\ T\ {\isasymrightarrow}\ T} \\
+ \isa{val\ purge{\isacharcolon}\ theory\ option\ {\isasymrightarrow}\ CodeUnit{\isachardot}const\ list\ option\ {\isasymrightarrow}\ T\ {\isasymrightarrow}\ T}
+ \end{tabular}
+
+ \begin{description}
+
+ \item \isa{T} the type of data to store.
+
+ \item \isa{empty} initial (empty) data.
+
+ \item \isa{merge} merging two data slots.
+
+ \item \isa{purge}~\isa{thy}~\isa{consts} propagates changes in executable content;
+ if possible, the current theory context is handed over
+ as argument \isa{thy} (if there is no current theory context (e.g.~during
+ theory merge, \verb|NONE|); \isa{consts} indicates the kind
+ of change: \verb|NONE| stands for a fundamental change
+ which invalidates any existing code, \isa{SOME\ consts}
+ hints that executable content for constants \isa{consts}
+ has changed.
+
+ \end{description}
+
+ An instance of \verb|CodeDataFun| provides the following
+ interface:
+
+ \medskip
+ \begin{tabular}{l}
+ \isa{get{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} \\
+ \isa{change{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ T{\isacharparenright}\ {\isasymrightarrow}\ T} \\
+ \isa{change{\isacharunderscore}yield{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T{\isacharparenright}\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T}
+ \end{tabular}
+
+ \begin{description}
+
+ \item \isa{get} retrieval of the current data.
+
+ \item \isa{change} update of current data (cached!)
+ by giving a continuation.
+
+ \item \isa{change{\isacharunderscore}yield} update with side result.
+
+ \end{description}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\emph{Happy proving, happy hacking!}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/document/Further.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,234 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Further}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{theory}\isamarkupfalse%
+\ Further\isanewline
+\isakeyword{imports}\ Setup\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsection{Further issues \label{sec:further}%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Further reading%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Do dive deeper into the issue of code generation, you should visit
+ the Isabelle/Isar Reference Manual \cite{isabelle-isar-ref} which
+ contains exhaustive syntax diagrams.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Modules%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+When invoking the \hyperlink{command.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} command it is possible to leave
+ out the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} part; then code is distributed over
+ different modules, where the module name space roughly is induced
+ by the \isa{Isabelle} theory name space.
+
+ Then sometimes the awkward situation occurs that dependencies between
+ definitions introduce cyclic dependencies between modules, which in the
+ \isa{Haskell} world leaves you to the mercy of the \isa{Haskell} implementation
+ you are using, while for \isa{SML}/\isa{OCaml} code generation is not possible.
+
+ A solution is to declare module names explicitly.
+ Let use assume the three cyclically dependent
+ modules are named \emph{A}, \emph{B} and \emph{C}.
+ Then, by stating%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{code{\isacharunderscore}modulename}\isamarkupfalse%
+\ SML\isanewline
+\ \ A\ ABC\isanewline
+\ \ B\ ABC\isanewline
+\ \ C\ ABC%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+we explicitly map all those modules on \emph{ABC},
+ resulting in an ad-hoc merge of this three modules
+ at serialisation time.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Evaluation oracle%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Code generation may also be used to \emph{evaluate} expressions
+ (using \isa{SML} as target language of course).
+ For instance, the \hyperlink{command.value}{\mbox{\isa{\isacommand{value}}}} allows to reduce an expression to a
+ normal form with respect to the underlying code equations:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{value}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{2}}\ {\isacharslash}\ {\isacharparenleft}{\isadigit{1}}{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ rat{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent will display \isa{{\isadigit{7}}\ {\isacharslash}\ {\isadigit{2}}}.
+
+ The \hyperlink{method.eval}{\mbox{\isa{eval}}} method tries to reduce a goal by code generation to \isa{True}
+ and solves it in that case, but fails otherwise:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{2}}\ {\isacharslash}\ {\isacharparenleft}{\isadigit{1}}{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ rat{\isacharparenright}\ {\isacharequal}\ {\isadigit{7}}\ {\isacharslash}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ eval%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The soundness of the \hyperlink{method.eval}{\mbox{\isa{eval}}} method depends crucially
+ on the correctness of the code generator; this is one of the reasons
+ why you should not use adaption (see \secref{sec:adaption}) frivolously.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Code antiquotation%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In scenarios involving techniques like reflection it is quite common
+ that code generated from a theory forms the basis for implementing
+ a proof procedure in \isa{SML}. To facilitate interfacing of generated code
+ with system code, the code generator provides a \isa{code} antiquotation:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{datatype}\isamarkupfalse%
+\ form\ {\isacharequal}\ T\ {\isacharbar}\ F\ {\isacharbar}\ And\ form\ form\ {\isacharbar}\ Or\ form\ form\isanewline
+\isanewline
+\isacommand{ML}\isamarkupfalse%
+\ {\isacharverbatimopen}\isanewline
+\ \ fun\ eval{\isacharunderscore}form\ %
+\isaantiq
+code\ T%
+\endisaantiq
+\ {\isacharequal}\ true\isanewline
+\ \ \ \ {\isacharbar}\ eval{\isacharunderscore}form\ %
+\isaantiq
+code\ F%
+\endisaantiq
+\ {\isacharequal}\ false\isanewline
+\ \ \ \ {\isacharbar}\ eval{\isacharunderscore}form\ {\isacharparenleft}%
+\isaantiq
+code\ And%
+\endisaantiq
+\ {\isacharparenleft}p{\isacharcomma}\ q{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ \ \ \ eval{\isacharunderscore}form\ p\ andalso\ eval{\isacharunderscore}form\ q\isanewline
+\ \ \ \ {\isacharbar}\ eval{\isacharunderscore}form\ {\isacharparenleft}%
+\isaantiq
+code\ Or%
+\endisaantiq
+\ {\isacharparenleft}p{\isacharcomma}\ q{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ \ \ \ eval{\isacharunderscore}form\ p\ orelse\ eval{\isacharunderscore}form\ q{\isacharsemicolon}\isanewline
+{\isacharverbatimclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent \isa{code} takes as argument the name of a constant; after the
+ whole \isa{SML} is read, the necessary code is generated transparently
+ and the corresponding constant names are inserted. This technique also
+ allows to use pattern matching on constructors stemming from compiled
+ \isa{datatypes}.
+
+ For a less simplistic example, theory \hyperlink{theory.Ferrack}{\mbox{\isa{Ferrack}}} is
+ a good reference.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Imperative data structures%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+If you consider imperative data structures as inevitable for a specific
+ application, you should consider
+ \emph{Imperative Functional Programming with Isabelle/HOL}
+ (\cite{bulwahn-et-al:2008:imperative});
+ the framework described there is available in theory \hyperlink{theory.Imperative-HOL}{\mbox{\isa{Imperative{\isacharunderscore}HOL}}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/document/Introduction.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,390 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Introduction}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{theory}\isamarkupfalse%
+\ Introduction\isanewline
+\isakeyword{imports}\ Setup\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupchapter{Code generation from \isa{Isabelle{\isacharslash}HOL} theories%
+}
+\isamarkuptrue%
+%
+\isamarkupsection{Introduction and Overview%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This tutorial introduces a generic code generator for the
+ \isa{Isabelle} system.
+ Generic in the sense that the
+ \qn{target language} for which code shall ultimately be
+ generated is not fixed but may be an arbitrary state-of-the-art
+ functional programming language (currently, the implementation
+ supports \isa{SML} \cite{SML}, \isa{OCaml} \cite{OCaml} and \isa{Haskell}
+ \cite{haskell-revised-report}).
+
+ Conceptually the code generator framework is part
+ of Isabelle's \hyperlink{theory.Pure}{\mbox{\isa{Pure}}} meta logic framework; the logic
+ \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} which is an extension of \hyperlink{theory.Pure}{\mbox{\isa{Pure}}}
+ already comes with a reasonable framework setup and thus provides
+ a good working horse for raising code-generation-driven
+ applications. So, we assume some familiarity and experience
+ with the ingredients of the \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} distribution theories.
+ (see also \cite{isa-tutorial}).
+
+ The code generator aims to be usable with no further ado
+ in most cases while allowing for detailed customisation.
+ This manifests in the structure of this tutorial: after a short
+ conceptual introduction with an example (\secref{sec:intro}),
+ we discuss the generic customisation facilities (\secref{sec:program}).
+ A further section (\secref{sec:adaption}) is dedicated to the matter of
+ \qn{adaption} to specific target language environments. After some
+ further issues (\secref{sec:further}) we conclude with an overview
+ of some ML programming interfaces (\secref{sec:ml}).
+
+ \begin{warn}
+ Ultimately, the code generator which this tutorial deals with
+ is supposed to replace the existing code generator
+ by Stefan Berghofer \cite{Berghofer-Nipkow:2002}.
+ So, for the moment, there are two distinct code generators
+ in Isabelle. In case of ambiguity, we will refer to the framework
+ described here as \isa{generic\ code\ generator}, to the
+ other as \isa{SML\ code\ generator}.
+ Also note that while the framework itself is
+ object-logic independent, only \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} provides a reasonable
+ framework setup.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Code generation via shallow embedding \label{sec:intro}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The key concept for understanding \isa{Isabelle}'s code generation is
+ \emph{shallow embedding}, i.e.~logical entities like constants, types and
+ classes are identified with corresponding concepts in the target language.
+
+ Inside \hyperlink{theory.HOL}{\mbox{\isa{HOL}}}, the \hyperlink{command.datatype}{\mbox{\isa{\isacommand{datatype}}}} and
+ \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}/\hyperlink{command.primrec}{\mbox{\isa{\isacommand{primrec}}}}/\hyperlink{command.fun}{\mbox{\isa{\isacommand{fun}}}} declarations form
+ the core of a functional programming language. The default code generator setup
+ allows to turn those into functional programs immediately.
+ This means that \qt{naive} code generation can proceed without further ado.
+ For example, here a simple \qt{implementation} of amortised queues:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{datatype}\isamarkupfalse%
+\ {\isacharprime}a\ queue\ {\isacharequal}\ AQueue\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ empty\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{primrec}\isamarkupfalse%
+\ enqueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ AQueue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}\ ys{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{fun}\isamarkupfalse%
+\ dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ option\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ \ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ \ {\isacharparenleft}case\ rev\ xs\ of\ y\ {\isacharhash}\ ys\ {\isasymRightarrow}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ ys{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Then we can generate code e.g.~for \isa{SML} as follows:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ empty\ dequeue\ enqueue\ \isakeyword{in}\ SML\isanewline
+\ \ \isakeyword{module{\isacharunderscore}name}\ Example\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}example{\isachardot}ML{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent resulting in the following code:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun foldl f a [] = a\\
+\hspace*{0pt} ~| foldl f a (x ::~xs) = foldl f (f a x) xs;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun rev xs = foldl (fn xsa => fn x => x ::~xsa) [] xs;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun list{\char95}case f1 f2 (a ::~lista) = f2 a lista\\
+\hspace*{0pt} ~| list{\char95}case f1 f2 [] = f1;\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype 'a queue = AQueue of 'a list * 'a list;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val empty :~'a queue = AQueue ([],~[])\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun dequeue (AQueue ([],~[])) = (NONE,~AQueue ([],~[]))\\
+\hspace*{0pt} ~| dequeue (AQueue (xs,~y ::~ys)) = (SOME y,~AQueue (xs,~ys))\\
+\hspace*{0pt} ~| dequeue (AQueue (v ::~va,~[])) =\\
+\hspace*{0pt} ~~~let\\
+\hspace*{0pt} ~~~~~val y ::~ys = rev (v ::~va);\\
+\hspace*{0pt} ~~~in\\
+\hspace*{0pt} ~~~~~(SOME y,~AQueue ([],~ys))\\
+\hspace*{0pt} ~~~end;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun enqueue x (AQueue (xs,~ys)) = AQueue (x ::~xs,~ys);\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The \hyperlink{command.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} command takes a space-separated list of
+ constants for which code shall be generated; anything else needed for those
+ is added implicitly. Then follows a target language identifier
+ (\isa{SML}, \isa{OCaml} or \isa{Haskell}) and a freely chosen module name.
+ A file name denotes the destination to store the generated code. Note that
+ the semantics of the destination depends on the target language: for
+ \isa{SML} and \isa{OCaml} it denotes a \emph{file}, for \isa{Haskell}
+ it denotes a \emph{directory} where a file named as the module name
+ (with extension \isa{{\isachardot}hs}) is written:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
+\ empty\ dequeue\ enqueue\ \isakeyword{in}\ Haskell\isanewline
+\ \ \isakeyword{module{\isacharunderscore}name}\ Example\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This is how the corresponding code in \isa{Haskell} looks like:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}module Example where {\char123}\\
+\hspace*{0pt}\\
+\hspace*{0pt}\\
+\hspace*{0pt}foldla ::~forall a b.~(a -> b -> a) -> a -> [b] -> a;\\
+\hspace*{0pt}foldla f a [] = a;\\
+\hspace*{0pt}foldla f a (x :~xs) = foldla f (f a x) xs;\\
+\hspace*{0pt}\\
+\hspace*{0pt}rev ::~forall a.~[a] -> [a];\\
+\hspace*{0pt}rev xs = foldla ({\char92}~xsa x -> x :~xsa) [] xs;\\
+\hspace*{0pt}\\
+\hspace*{0pt}list{\char95}case ::~forall t a.~t -> (a -> [a] -> t) -> [a] -> t;\\
+\hspace*{0pt}list{\char95}case f1 f2 (a :~list) = f2 a list;\\
+\hspace*{0pt}list{\char95}case f1 f2 [] = f1;\\
+\hspace*{0pt}\\
+\hspace*{0pt}data Queue a = AQueue [a] [a];\\
+\hspace*{0pt}\\
+\hspace*{0pt}empty ::~forall a.~Queue a;\\
+\hspace*{0pt}empty = AQueue [] [];\\
+\hspace*{0pt}\\
+\hspace*{0pt}dequeue ::~forall a.~Queue a -> (Maybe a,~Queue a);\\
+\hspace*{0pt}dequeue (AQueue [] []) = (Nothing,~AQueue [] []);\\
+\hspace*{0pt}dequeue (AQueue xs (y :~ys)) = (Just y,~AQueue xs ys);\\
+\hspace*{0pt}dequeue (AQueue (v :~va) []) =\\
+\hspace*{0pt} ~let {\char123}\\
+\hspace*{0pt} ~~~(y :~ys) = rev (v :~va);\\
+\hspace*{0pt} ~{\char125}~in (Just y,~AQueue [] ys);\\
+\hspace*{0pt}\\
+\hspace*{0pt}enqueue ::~forall a.~a -> Queue a -> Queue a;\\
+\hspace*{0pt}enqueue x (AQueue xs ys) = AQueue (x :~xs) ys;\\
+\hspace*{0pt}\\
+\hspace*{0pt}{\char125}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This demonstrates the basic usage of the \hyperlink{command.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} command;
+ for more details see \secref{sec:further}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Code generator architecture \label{sec:concept}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+What you have seen so far should be already enough in a lot of cases. If you
+ are content with this, you can quit reading here. Anyway, in order to customise
+ and adapt the code generator, it is inevitable to gain some understanding
+ how it works.
+
+ \begin{figure}[h]
+ \begin{tikzpicture}[x = 4.2cm, y = 1cm]
+ \tikzstyle entity=[rounded corners, draw, thick, color = black, fill = white];
+ \tikzstyle process=[ellipse, draw, thick, color = green, fill = white];
+ \tikzstyle process_arrow=[->, semithick, color = green];
+ \node (HOL) at (0, 4) [style=entity] {\isa{Isabelle{\isacharslash}HOL} theory};
+ \node (eqn) at (2, 2) [style=entity] {code equations};
+ \node (iml) at (2, 0) [style=entity] {intermediate language};
+ \node (seri) at (1, 0) [style=process] {serialisation};
+ \node (SML) at (0, 3) [style=entity] {\isa{SML}};
+ \node (OCaml) at (0, 2) [style=entity] {\isa{OCaml}};
+ \node (further) at (0, 1) [style=entity] {\isa{{\isasymdots}}};
+ \node (Haskell) at (0, 0) [style=entity] {\isa{Haskell}};
+ \draw [style=process_arrow] (HOL) .. controls (2, 4) ..
+ node [style=process, near start] {selection}
+ node [style=process, near end] {preprocessing}
+ (eqn);
+ \draw [style=process_arrow] (eqn) -- node (transl) [style=process] {translation} (iml);
+ \draw [style=process_arrow] (iml) -- (seri);
+ \draw [style=process_arrow] (seri) -- (SML);
+ \draw [style=process_arrow] (seri) -- (OCaml);
+ \draw [style=process_arrow, dashed] (seri) -- (further);
+ \draw [style=process_arrow] (seri) -- (Haskell);
+ \end{tikzpicture}
+ \caption{Code generator architecture}
+ \label{fig:arch}
+ \end{figure}
+
+ The code generator employs a notion of executability
+ for three foundational executable ingredients known
+ from functional programming:
+ \emph{code equations}, \emph{datatypes}, and
+ \emph{type classes}. A code equation as a first approximation
+ is a theorem of the form \isa{f\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n\ {\isasymequiv}\ t}
+ (an equation headed by a constant \isa{f} with arguments
+ \isa{t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n} and right hand side \isa{t}).
+ Code generation aims to turn code equations
+ into a functional program. This is achieved by three major
+ components which operate sequentially, i.e. the result of one is
+ the input
+ of the next in the chain, see diagram \ref{fig:arch}:
+
+ \begin{itemize}
+
+ \item Out of the vast collection of theorems proven in a
+ \qn{theory}, a reasonable subset modelling
+ code equations is \qn{selected}.
+
+ \item On those selected theorems, certain
+ transformations are carried out
+ (\qn{preprocessing}). Their purpose is to turn theorems
+ representing non- or badly executable
+ specifications into equivalent but executable counterparts.
+ The result is a structured collection of \qn{code theorems}.
+
+ \item Before the selected code equations are continued with,
+ they can be \qn{preprocessed}, i.e. subjected to theorem
+ transformations. This \qn{preprocessor} is an interface which
+ allows to apply
+ the full expressiveness of ML-based theorem transformations
+ to code generation; motivating examples are shown below, see
+ \secref{sec:preproc}.
+ The result of the preprocessing step is a structured collection
+ of code equations.
+
+ \item These code equations are \qn{translated} to a program
+ in an abstract intermediate language. Think of it as a kind
+ of \qt{Mini-Haskell} with four \qn{statements}: \isa{data}
+ (for datatypes), \isa{fun} (stemming from code equations),
+ also \isa{class} and \isa{inst} (for type classes).
+
+ \item Finally, the abstract program is \qn{serialised} into concrete
+ source code of a target language.
+
+ \end{itemize}
+
+ \noindent From these steps, only the two last are carried out outside the logic; by
+ keeping this layer as thin as possible, the amount of code to trust is
+ kept to a minimum.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/document/ML.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,255 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{ML}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{theory}\isamarkupfalse%
+\ {\isachardoublequoteopen}ML{\isachardoublequoteclose}\isanewline
+\isakeyword{imports}\ Setup\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsection{ML system interfaces \label{sec:ml}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Since the code generator framework not only aims to provide
+ a nice Isar interface but also to form a base for
+ code-generation-based applications, here a short
+ description of the most important ML interfaces.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Executable theory content: \isa{Code}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This Pure module implements the core notions of
+ executable content of a theory.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Managing executable content%
+}
+\isamarkuptrue%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isatagmlref
+%
+\begin{isamarkuptext}%
+\begin{mldecls}
+ \indexdef{}{ML}{Code.add\_eqn}\verb|Code.add_eqn: thm -> theory -> theory| \\
+ \indexdef{}{ML}{Code.del\_eqn}\verb|Code.del_eqn: thm -> theory -> theory| \\
+ \indexdef{}{ML}{Code.add\_eqnl}\verb|Code.add_eqnl: string * (thm * bool) list lazy -> theory -> theory| \\
+ \indexdef{}{ML}{Code.map\_pre}\verb|Code.map_pre: (simpset -> simpset) -> theory -> theory| \\
+ \indexdef{}{ML}{Code.map\_post}\verb|Code.map_post: (simpset -> simpset) -> theory -> theory| \\
+ \indexdef{}{ML}{Code.add\_functrans}\verb|Code.add_functrans: string * (theory -> (thm * bool) list -> (thm * bool) list option)|\isasep\isanewline%
+\verb| -> theory -> theory| \\
+ \indexdef{}{ML}{Code.del\_functrans}\verb|Code.del_functrans: string -> theory -> theory| \\
+ \indexdef{}{ML}{Code.add\_datatype}\verb|Code.add_datatype: (string * typ) list -> theory -> theory| \\
+ \indexdef{}{ML}{Code.get\_datatype}\verb|Code.get_datatype: theory -> string|\isasep\isanewline%
+\verb| -> (string * sort) list * (string * typ list) list| \\
+ \indexdef{}{ML}{Code.get\_datatype\_of\_constr}\verb|Code.get_datatype_of_constr: theory -> string -> string option|
+ \end{mldecls}
+
+ \begin{description}
+
+ \item \verb|Code.add_eqn|~\isa{thm}~\isa{thy} adds function
+ theorem \isa{thm} to executable content.
+
+ \item \verb|Code.del_eqn|~\isa{thm}~\isa{thy} removes function
+ theorem \isa{thm} from executable content, if present.
+
+ \item \verb|Code.add_eqnl|~\isa{{\isacharparenleft}const{\isacharcomma}\ lthms{\isacharparenright}}~\isa{thy} adds
+ suspended code equations \isa{lthms} for constant
+ \isa{const} to executable content.
+
+ \item \verb|Code.map_pre|~\isa{f}~\isa{thy} changes
+ the preprocessor simpset.
+
+ \item \verb|Code.add_functrans|~\isa{{\isacharparenleft}name{\isacharcomma}\ f{\isacharparenright}}~\isa{thy} adds
+ function transformer \isa{f} (named \isa{name}) to executable content;
+ \isa{f} is a transformer of the code equations belonging
+ to a certain function definition, depending on the
+ current theory context. Returning \isa{NONE} indicates that no
+ transformation took place; otherwise, the whole process will be iterated
+ with the new code equations.
+
+ \item \verb|Code.del_functrans|~\isa{name}~\isa{thy} removes
+ function transformer named \isa{name} from executable content.
+
+ \item \verb|Code.add_datatype|~\isa{cs}~\isa{thy} adds
+ a datatype to executable content, with generation
+ set \isa{cs}.
+
+ \item \verb|Code.get_datatype_of_constr|~\isa{thy}~\isa{const}
+ returns type constructor corresponding to
+ constructor \isa{const}; returns \isa{NONE}
+ if \isa{const} is no constructor.
+
+ \end{description}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagmlref
+{\isafoldmlref}%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isamarkupsubsection{Auxiliary%
+}
+\isamarkuptrue%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isatagmlref
+%
+\begin{isamarkuptext}%
+\begin{mldecls}
+ \indexdef{}{ML}{Code\_Unit.read\_const}\verb|Code_Unit.read_const: theory -> string -> string| \\
+ \indexdef{}{ML}{Code\_Unit.head\_eqn}\verb|Code_Unit.head_eqn: theory -> thm -> string * ((string * sort) list * typ)| \\
+ \indexdef{}{ML}{Code\_Unit.rewrite\_eqn}\verb|Code_Unit.rewrite_eqn: simpset -> thm -> thm| \\
+ \end{mldecls}
+
+ \begin{description}
+
+ \item \verb|Code_Unit.read_const|~\isa{thy}~\isa{s}
+ reads a constant as a concrete term expression \isa{s}.
+
+ \item \verb|Code_Unit.head_eqn|~\isa{thy}~\isa{thm}
+ extracts the constant and its type from a code equation \isa{thm}.
+
+ \item \verb|Code_Unit.rewrite_eqn|~\isa{ss}~\isa{thm}
+ rewrites a code equation \isa{thm} with a simpset \isa{ss};
+ only arguments and right hand side are rewritten,
+ not the head of the code equation.
+
+ \end{description}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagmlref
+{\isafoldmlref}%
+%
+\isadelimmlref
+%
+\endisadelimmlref
+%
+\isamarkupsubsection{Implementing code generator applications%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Implementing code generator applications on top
+ of the framework set out so far usually not only
+ involves using those primitive interfaces
+ but also storing code-dependent data and various
+ other things.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Data depending on the theory's executable content%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Due to incrementality of code generation, changes in the
+ theory's executable content have to be propagated in a
+ certain fashion. Additionally, such changes may occur
+ not only during theory extension but also during theory
+ merge, which is a little bit nasty from an implementation
+ point of view. The framework provides a solution
+ to this technical challenge by providing a functorial
+ data slot \verb|CodeDataFun|; on instantiation
+ of this functor, the following types and operations
+ are required:
+
+ \medskip
+ \begin{tabular}{l}
+ \isa{type\ T} \\
+ \isa{val\ empty{\isacharcolon}\ T} \\
+ \isa{val\ purge{\isacharcolon}\ theory\ {\isasymrightarrow}\ string\ list\ option\ {\isasymrightarrow}\ T\ {\isasymrightarrow}\ T}
+ \end{tabular}
+
+ \begin{description}
+
+ \item \isa{T} the type of data to store.
+
+ \item \isa{empty} initial (empty) data.
+
+ \item \isa{purge}~\isa{thy}~\isa{consts} propagates changes in executable content;
+ \isa{consts} indicates the kind
+ of change: \verb|NONE| stands for a fundamental change
+ which invalidates any existing code, \isa{SOME\ consts}
+ hints that executable content for constants \isa{consts}
+ has changed.
+
+ \end{description}
+
+ \noindent An instance of \verb|CodeDataFun| provides the following
+ interface:
+
+ \medskip
+ \begin{tabular}{l}
+ \isa{get{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} \\
+ \isa{change{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ T{\isacharparenright}\ {\isasymrightarrow}\ T} \\
+ \isa{change{\isacharunderscore}yield{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T{\isacharparenright}\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T}
+ \end{tabular}
+
+ \begin{description}
+
+ \item \isa{get} retrieval of the current data.
+
+ \item \isa{change} update of current data (cached!)
+ by giving a continuation.
+
+ \item \isa{change{\isacharunderscore}yield} update with side result.
+
+ \end{description}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\bigskip
+
+ \emph{Happy proving, happy hacking!}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/document/Program.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,1250 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Program}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{theory}\isamarkupfalse%
+\ Program\isanewline
+\isakeyword{imports}\ Introduction\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsection{Turning Theories into Programs \label{sec:program}%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{The \isa{Isabelle{\isacharslash}HOL} default setup%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+We have already seen how by default equations stemming from
+ \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}/\hyperlink{command.primrec}{\mbox{\isa{\isacommand{primrec}}}}/\hyperlink{command.fun}{\mbox{\isa{\isacommand{fun}}}}
+ statements are used for code generation. This default behaviour
+ can be changed, e.g. by providing different code equations.
+ All kinds of customisation shown in this section is \emph{safe}
+ in the sense that the user does not have to worry about
+ correctness -- all programs generatable that way are partially
+ correct.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Selecting code equations%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Coming back to our introductory example, we
+ could provide an alternative code equations for \isa{dequeue}
+ explicitly:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ else\ dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}cases\ xs{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}\ {\isacharparenleft}cases\ {\isachardoublequoteopen}rev\ xs{\isachardoublequoteclose}{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The annotation \isa{{\isacharbrackleft}code{\isacharbrackright}} is an \isa{Isar}
+ \isa{attribute} which states that the given theorems should be
+ considered as code equations for a \isa{fun} statement --
+ the corresponding constant is determined syntactically. The resulting code:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}dequeue ::~forall a.~Queue a -> (Maybe a,~Queue a);\\
+\hspace*{0pt}dequeue (AQueue xs (y :~ys)) = (Just y,~AQueue xs ys);\\
+\hspace*{0pt}dequeue (AQueue xs []) =\\
+\hspace*{0pt} ~(if nulla xs then (Nothing,~AQueue [] [])\\
+\hspace*{0pt} ~~~else dequeue (AQueue [] (rev xs)));%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent You may note that the equality test \isa{xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}} has been
+ replaced by the predicate \isa{null\ xs}. This is due to the default
+ setup in the \qn{preprocessor} to be discussed further below (\secref{sec:preproc}).
+
+ Changing the default constructor set of datatypes is also
+ possible. See \secref{sec:datatypes} for an example.
+
+ As told in \secref{sec:concept}, code generation is based
+ on a structured collection of code theorems.
+ For explorative purpose, this collection
+ may be inspected using the \hyperlink{command.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} command:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{code{\isacharunderscore}thms}\isamarkupfalse%
+\ dequeue%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent prints a table with \emph{all} code equations
+ for \isa{dequeue}, including
+ \emph{all} code equations those equations depend
+ on recursively.
+
+ Similarly, the \hyperlink{command.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} command shows a graph
+ visualising dependencies between code equations.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{\isa{class} and \isa{instantiation}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Concerning type classes and code generation, let us examine an example
+ from abstract algebra:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{class}\isamarkupfalse%
+\ semigroup\ {\isacharequal}\isanewline
+\ \ \isakeyword{fixes}\ mult\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymotimes}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{class}\isamarkupfalse%
+\ monoid\ {\isacharequal}\ semigroup\ {\isacharplus}\isanewline
+\ \ \isakeyword{fixes}\ neutral\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ neutl{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ neutr{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instantiation}\isamarkupfalse%
+\ nat\ {\isacharcolon}{\isacharcolon}\ monoid\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{primrec}\isamarkupfalse%
+\ mult{\isacharunderscore}nat\ \isakeyword{where}\isanewline
+\ \ \ \ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymotimes}\ n\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}Suc\ m\ {\isasymotimes}\ n\ {\isacharequal}\ n\ {\isacharplus}\ m\ {\isasymotimes}\ n{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ neutral{\isacharunderscore}nat\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ add{\isacharunderscore}mult{\isacharunderscore}distrib{\isacharcolon}\isanewline
+\ \ \isakeyword{fixes}\ n\ m\ q\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
+\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharplus}\ m{\isacharparenright}\ {\isasymotimes}\ q\ {\isacharequal}\ n\ {\isasymotimes}\ q\ {\isacharplus}\ m\ {\isasymotimes}\ q{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ n{\isacharparenright}\ simp{\isacharunderscore}all\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ m\ n\ q\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}m\ {\isasymotimes}\ n\ {\isasymotimes}\ q\ {\isacharequal}\ m\ {\isasymotimes}\ {\isacharparenleft}n\ {\isasymotimes}\ q{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ m{\isacharparenright}\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ add{\isacharunderscore}mult{\isacharunderscore}distrib{\isacharparenright}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharparenright}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}m\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ m{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ m{\isacharparenright}\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharparenright}\isanewline
+\isacommand{qed}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent We define the natural operation of the natural numbers
+ on monoids:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{primrec}\isamarkupfalse%
+\ {\isacharparenleft}\isakeyword{in}\ monoid{\isacharparenright}\ pow\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ \ \ {\isachardoublequoteopen}pow\ {\isadigit{0}}\ a\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}pow\ {\isacharparenleft}Suc\ n{\isacharparenright}\ a\ {\isacharequal}\ a\ {\isasymotimes}\ pow\ n\ a{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This we use to define the discrete exponentiation function:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{definition}\isamarkupfalse%
+\ bexp\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}bexp\ n\ {\isacharequal}\ pow\ n\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The corresponding code:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}module Example where {\char123}\\
+\hspace*{0pt}\\
+\hspace*{0pt}\\
+\hspace*{0pt}data Nat = Zero{\char95}nat | Suc Nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}class Semigroup a where {\char123}\\
+\hspace*{0pt} ~mult ::~a -> a -> a;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}class (Semigroup a) => Monoid a where {\char123}\\
+\hspace*{0pt} ~neutral ::~a;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}pow ::~forall a.~(Monoid a) => Nat -> a -> a;\\
+\hspace*{0pt}pow Zero{\char95}nat a = neutral;\\
+\hspace*{0pt}pow (Suc n) a = mult a (pow n a);\\
+\hspace*{0pt}\\
+\hspace*{0pt}plus{\char95}nat ::~Nat -> Nat -> Nat;\\
+\hspace*{0pt}plus{\char95}nat (Suc m) n = plus{\char95}nat m (Suc n);\\
+\hspace*{0pt}plus{\char95}nat Zero{\char95}nat n = n;\\
+\hspace*{0pt}\\
+\hspace*{0pt}neutral{\char95}nat ::~Nat;\\
+\hspace*{0pt}neutral{\char95}nat = Suc Zero{\char95}nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}mult{\char95}nat ::~Nat -> Nat -> Nat;\\
+\hspace*{0pt}mult{\char95}nat Zero{\char95}nat n = Zero{\char95}nat;\\
+\hspace*{0pt}mult{\char95}nat (Suc m) n = plus{\char95}nat n (mult{\char95}nat m n);\\
+\hspace*{0pt}\\
+\hspace*{0pt}instance Semigroup Nat where {\char123}\\
+\hspace*{0pt} ~mult = mult{\char95}nat;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}instance Monoid Nat where {\char123}\\
+\hspace*{0pt} ~neutral = neutral{\char95}nat;\\
+\hspace*{0pt}{\char125};\\
+\hspace*{0pt}\\
+\hspace*{0pt}bexp ::~Nat -> Nat;\\
+\hspace*{0pt}bexp n = pow n (Suc (Suc Zero{\char95}nat));\\
+\hspace*{0pt}\\
+\hspace*{0pt}{\char125}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This is a convenient place to show how explicit dictionary construction
+ manifests in generated code (here, the same example in \isa{SML}):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a semigroup = {\char123}mult :~'a -> 'a -> 'a{\char125};\\
+\hspace*{0pt}fun mult (A{\char95}:'a semigroup) = {\char35}mult A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a monoid = {\char123}Program{\char95}{\char95}semigroup{\char95}monoid :~'a semigroup,~neutral :~'a{\char125};\\
+\hspace*{0pt}fun semigroup{\char95}monoid (A{\char95}:'a monoid) = {\char35}Program{\char95}{\char95}semigroup{\char95}monoid A{\char95};\\
+\hspace*{0pt}fun neutral (A{\char95}:'a monoid) = {\char35}neutral A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun pow A{\char95}~Zero{\char95}nat a = neutral A{\char95}\\
+\hspace*{0pt} ~| pow A{\char95}~(Suc n) a = mult (semigroup{\char95}monoid A{\char95}) a (pow A{\char95}~n a);\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun plus{\char95}nat (Suc m) n = plus{\char95}nat m (Suc n)\\
+\hspace*{0pt} ~| plus{\char95}nat Zero{\char95}nat n = n;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val neutral{\char95}nat :~nat = Suc Zero{\char95}nat\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun mult{\char95}nat Zero{\char95}nat n = Zero{\char95}nat\\
+\hspace*{0pt} ~| mult{\char95}nat (Suc m) n = plus{\char95}nat n (mult{\char95}nat m n);\\
+\hspace*{0pt}\\
+\hspace*{0pt}val semigroup{\char95}nat = {\char123}mult = mult{\char95}nat{\char125}~:~nat semigroup;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val monoid{\char95}nat =\\
+\hspace*{0pt} ~{\char123}Program{\char95}{\char95}semigroup{\char95}monoid = semigroup{\char95}nat,~neutral = neutral{\char95}nat{\char125}~:\\
+\hspace*{0pt} ~nat monoid;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun bexp n = pow monoid{\char95}nat n (Suc (Suc Zero{\char95}nat));\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Note the parameters with trailing underscore (\verb|A_|)
+ which are the dictionary parameters.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{The preprocessor \label{sec:preproc}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Before selected function theorems are turned into abstract
+ code, a chain of definitional transformation steps is carried
+ out: \emph{preprocessing}. In essence, the preprocessor
+ consists of two components: a \emph{simpset} and \emph{function transformers}.
+
+ The \emph{simpset} allows to employ the full generality of the Isabelle
+ simplifier. Due to the interpretation of theorems
+ as code equations, rewrites are applied to the right
+ hand side and the arguments of the left hand side of an
+ equation, but never to the constant heading the left hand side.
+ An important special case are \emph{inline theorems} which may be
+ declared and undeclared using the
+ \emph{code inline} or \emph{code inline del} attribute respectively.
+
+ Some common applications:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{itemize}
+%
+\begin{isamarkuptext}%
+\item replacing non-executable constructs by executable ones:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}x\ {\isasymin}\ set\ xs\ {\isasymlongleftrightarrow}\ x\ mem\ xs{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\item eliminating superfluous constants:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\item replacing executable but inconvenient constructs:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymlongleftrightarrow}\ List{\isachardot}null\ xs{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\end{itemize}
+%
+\begin{isamarkuptext}%
+\noindent \emph{Function transformers} provide a very general interface,
+ transforming a list of function theorems to another
+ list of function theorems, provided that neither the heading
+ constant nor its type change. The \isa{{\isadigit{0}}} / \isa{Suc}
+ pattern elimination implemented in
+ theory \isa{Efficient{\isacharunderscore}Nat} (see \secref{eff_nat}) uses this
+ interface.
+
+ \noindent The current setup of the preprocessor may be inspected using
+ the \hyperlink{command.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} command.
+ \hyperlink{command.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} provides a convenient
+ mechanism to inspect the impact of a preprocessor setup
+ on code equations.
+
+ \begin{warn}
+ The attribute \emph{code unfold}
+ associated with the \isa{SML\ code\ generator} also applies to
+ the \isa{generic\ code\ generator}:
+ \emph{code unfold} implies \emph{code inline}.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Datatypes \label{sec:datatypes}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Conceptually, any datatype is spanned by a set of
+ \emph{constructors} of type \isa{{\isasymtau}\ {\isacharequal}\ {\isasymdots}\ {\isasymRightarrow}\ {\isasymkappa}\ {\isasymalpha}\isactrlisub {\isadigit{1}}\ {\isasymdots}\ {\isasymalpha}\isactrlisub n} where \isa{{\isacharbraceleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub n{\isacharbraceright}} is exactly the set of \emph{all} type variables in
+ \isa{{\isasymtau}}. The HOL datatype package by default registers any new
+ datatype in the table of datatypes, which may be inspected using the
+ \hyperlink{command.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} command.
+
+ In some cases, it is appropriate to alter or extend this table. As
+ an example, we will develop an alternative representation of the
+ queue example given in \secref{sec:intro}. The amortised
+ representation is convenient for generating code but exposes its
+ \qt{implementation} details, which may be cumbersome when proving
+ theorems about it. Therefore, here a simple, straightforward
+ representation of queues:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{datatype}\isamarkupfalse%
+\ {\isacharprime}a\ queue\ {\isacharequal}\ Queue\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ empty\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{primrec}\isamarkupfalse%
+\ enqueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}Queue\ xs{\isacharparenright}\ {\isacharequal}\ Queue\ {\isacharparenleft}xs\ {\isacharat}\ {\isacharbrackleft}x{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{fun}\isamarkupfalse%
+\ dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ option\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ \ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}None{\isacharcomma}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ x{\isacharcomma}\ Queue\ xs{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This we can use directly for proving; for executing,
+ we provide an alternative characterisation:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{definition}\isamarkupfalse%
+\ AQueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}AQueue\ xs\ ys\ {\isacharequal}\ Queue\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
+\ AQueue%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Here we define a \qt{constructor} \isa{Program{\isachardot}AQueue} which
+ is defined in terms of \isa{Queue} and interprets its arguments
+ according to what the \emph{content} of an amortised queue is supposed
+ to be. Equipped with this, we are able to prove the following equations
+ for our primitive queue operations which \qt{implement} the simple
+ queues in an amortised fashion:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ empty{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{unfolding}\isamarkupfalse%
+\ AQueue{\isacharunderscore}def\ empty{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ enqueue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ AQueue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}\ ys{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{unfolding}\isamarkupfalse%
+\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ dequeue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
+\ \ \ \ else\ dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{unfolding}\isamarkupfalse%
+\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp{\isacharunderscore}all%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent For completeness, we provide a substitute for the
+ \isa{case} combinator on queues:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{definition}\isamarkupfalse%
+\isanewline
+\ \ aqueue{\isacharunderscore}case{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}aqueue{\isacharunderscore}case\ {\isacharequal}\ queue{\isacharunderscore}case{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ aqueue{\isacharunderscore}case\ {\isacharbrackleft}code{\isacharcomma}\ code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}queue{\isacharunderscore}case\ {\isacharequal}\ aqueue{\isacharunderscore}case{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{unfolding}\isamarkupfalse%
+\ aqueue{\isacharunderscore}case{\isacharunderscore}def\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ case{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}aqueue{\isacharunderscore}case\ f\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{unfolding}\isamarkupfalse%
+\ aqueue{\isacharunderscore}case{\isacharunderscore}def\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The resulting code looks as expected:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun foldl f a [] = a\\
+\hspace*{0pt} ~| foldl f a (x ::~xs) = foldl f (f a x) xs;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun rev xs = foldl (fn xsa => fn x => x ::~xsa) [] xs;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun null [] = true\\
+\hspace*{0pt} ~| null (x ::~xs) = false;\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype 'a queue = AQueue of 'a list * 'a list;\\
+\hspace*{0pt}\\
+\hspace*{0pt}val empty :~'a queue = AQueue ([],~[])\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun dequeue (AQueue (xs,~y ::~ys)) = (SOME y,~AQueue (xs,~ys))\\
+\hspace*{0pt} ~| dequeue (AQueue (xs,~[])) =\\
+\hspace*{0pt} ~~~(if null xs then (NONE,~AQueue ([],~[]))\\
+\hspace*{0pt} ~~~~~else dequeue (AQueue ([],~rev xs)));\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun enqueue x (AQueue (xs,~ys)) = AQueue (x ::~xs,~ys);\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent From this example, it can be glimpsed that using own
+ constructor sets is a little delicate since it changes the set of
+ valid patterns for values of that type. Without going into much
+ detail, here some practical hints:
+
+ \begin{itemize}
+
+ \item When changing the constructor set for datatypes, take care
+ to provide an alternative for the \isa{case} combinator
+ (e.g.~by replacing it using the preprocessor).
+
+ \item Values in the target language need not to be normalised --
+ different values in the target language may represent the same
+ value in the logic.
+
+ \item Usually, a good methodology to deal with the subtleties of
+ pattern matching is to see the type as an abstract type: provide
+ a set of operations which operate on the concrete representation
+ of the type, and derive further operations by combinations of
+ these primitive ones, without relying on a particular
+ representation.
+
+ \end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Equality and wellsortedness%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Surely you have already noticed how equality is treated
+ by the code generator:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{primrec}\isamarkupfalse%
+\ collect{\isacharunderscore}duplicates\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ xs{\isachardoublequoteclose}\isanewline
+\ \ {\isacharbar}\ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharparenleft}z{\isacharhash}zs{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ z\ {\isasymin}\ set\ xs\isanewline
+\ \ \ \ \ \ then\ if\ z\ {\isasymin}\ set\ ys\isanewline
+\ \ \ \ \ \ \ \ then\ collect{\isacharunderscore}duplicates\ xs\ ys\ zs\isanewline
+\ \ \ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ xs\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs\isanewline
+\ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ {\isacharparenleft}z{\isacharhash}xs{\isacharparenright}\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs{\isacharparenright}{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent The membership test during preprocessing is rewritten,
+ resulting in \isa{op\ mem}, which itself
+ performs an explicit equality check.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a eq = {\char123}eq :~'a -> 'a -> bool{\char125};\\
+\hspace*{0pt}fun eq (A{\char95}:'a eq) = {\char35}eq A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun eqop A{\char95}~a b = eq A{\char95}~a b;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun member A{\char95}~x [] = false\\
+\hspace*{0pt} ~| member A{\char95}~x (y ::~ys) = eqop A{\char95}~x y orelse member A{\char95}~x ys;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun collect{\char95}duplicates A{\char95}~xs ys [] = xs\\
+\hspace*{0pt} ~| collect{\char95}duplicates A{\char95}~xs ys (z ::~zs) =\\
+\hspace*{0pt} ~~~(if member A{\char95}~z xs\\
+\hspace*{0pt} ~~~~~then (if member A{\char95}~z ys then collect{\char95}duplicates A{\char95}~xs ys zs\\
+\hspace*{0pt} ~~~~~~~~~~~~else collect{\char95}duplicates A{\char95}~xs (z ::~ys) zs)\\
+\hspace*{0pt} ~~~~~else collect{\char95}duplicates A{\char95}~(z ::~xs) (z ::~ys) zs);\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Obviously, polymorphic equality is implemented the Haskell
+ way using a type class. How is this achieved? HOL introduces
+ an explicit class \isa{eq} with a corresponding operation
+ \isa{eq{\isacharunderscore}class{\isachardot}eq} such that \isa{eq{\isacharunderscore}class{\isachardot}eq\ {\isacharequal}\ op\ {\isacharequal}}.
+ The preprocessing framework does the rest by propagating the
+ \isa{eq} constraints through all dependent code equations.
+ For datatypes, instances of \isa{eq} are implicitly derived
+ when possible. For other types, you may instantiate \isa{eq}
+ manually like any other type class.
+
+ Though this \isa{eq} class is designed to get rarely in
+ the way, a subtlety
+ enters the stage when definitions of overloaded constants
+ are dependent on operational equality. For example, let
+ us define a lexicographic ordering on tuples
+ (also see theory \hyperlink{theory.Product-ord}{\mbox{\isa{Product{\isacharunderscore}ord}}}):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{instantiation}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharasterisk}{\isachardoublequoteclose}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}order{\isacharcomma}\ order{\isacharparenright}\ order\isanewline
+\isakeyword{begin}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ {\isacharbrackleft}code\ del{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}x\ {\isasymle}\ y\ {\isasymlongleftrightarrow}\ fst\ x\ {\isacharless}\ fst\ y\ {\isasymor}\ fst\ x\ {\isacharequal}\ fst\ y\ {\isasymand}\ snd\ x\ {\isasymle}\ snd\ y{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ {\isacharbrackleft}code\ del{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}x\ {\isacharless}\ y\ {\isasymlongleftrightarrow}\ fst\ x\ {\isacharless}\ fst\ y\ {\isasymor}\ fst\ x\ {\isacharequal}\ fst\ y\ {\isasymand}\ snd\ x\ {\isacharless}\ snd\ y{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{instance}\isamarkupfalse%
+\ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+\ {\isacharparenleft}auto\ simp{\isacharcolon}\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}prod{\isacharunderscore}def\ intro{\isacharcolon}\ order{\isacharunderscore}less{\isacharunderscore}trans{\isacharparenright}\isanewline
+\isanewline
+\isacommand{end}\isamarkupfalse%
+\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ order{\isacharunderscore}prod\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}order{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}order{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Then code generation will fail. Why? The definition
+ of \isa{op\ {\isasymle}} depends on equality on both arguments,
+ which are polymorphic and impose an additional \isa{eq}
+ class constraint, which the preprocessor does not propagate
+ (for technical reasons).
+
+ The solution is to add \isa{eq} explicitly to the first sort arguments in the
+ code theorems:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ order{\isacharunderscore}prod{\isacharunderscore}code\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}order{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}order{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Then code generation succeeds:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a eq = {\char123}eq :~'a -> 'a -> bool{\char125};\\
+\hspace*{0pt}fun eq (A{\char95}:'a eq) = {\char35}eq A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a ord = {\char123}less{\char95}eq :~'a -> 'a -> bool,~less :~'a -> 'a -> bool{\char125};\\
+\hspace*{0pt}fun less{\char95}eq (A{\char95}:'a ord) = {\char35}less{\char95}eq A{\char95};\\
+\hspace*{0pt}fun less (A{\char95}:'a ord) = {\char35}less A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun eqop A{\char95}~a b = eq A{\char95}~a b;\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a preorder = {\char123}Orderings{\char95}{\char95}ord{\char95}preorder :~'a ord{\char125};\\
+\hspace*{0pt}fun ord{\char95}preorder (A{\char95}:'a preorder) = {\char35}Orderings{\char95}{\char95}ord{\char95}preorder A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}type 'a order = {\char123}Orderings{\char95}{\char95}preorder{\char95}order :~'a preorder{\char125};\\
+\hspace*{0pt}fun preorder{\char95}order (A{\char95}:'a order) = {\char35}Orderings{\char95}{\char95}preorder{\char95}order A{\char95};\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun less{\char95}eqa (A1{\char95},~A2{\char95}) B{\char95}~(x1,~y1) (x2,~y2) =\\
+\hspace*{0pt} ~less ((ord{\char95}preorder o preorder{\char95}order) A2{\char95}) x1 x2 orelse\\
+\hspace*{0pt} ~~~eqop A1{\char95}~x1 x2 andalso\\
+\hspace*{0pt} ~~~~~less{\char95}eq ((ord{\char95}preorder o preorder{\char95}order) B{\char95}) y1 y2\\
+\hspace*{0pt} ~| less{\char95}eqa (A1{\char95},~A2{\char95}) B{\char95}~(x1,~y1) (x2,~y2) =\\
+\hspace*{0pt} ~~~less ((ord{\char95}preorder o preorder{\char95}order) A2{\char95}) x1 x2 orelse\\
+\hspace*{0pt} ~~~~~eqop A1{\char95}~x1 x2 andalso\\
+\hspace*{0pt} ~~~~~~~less{\char95}eq ((ord{\char95}preorder o preorder{\char95}order) B{\char95}) y1 y2;\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+In some cases, the automatically derived code equations
+ for equality on a particular type may not be appropriate.
+ As example, watch the following datatype representing
+ monomorphic parametric types (where type constructors
+ are referred to by natural numbers):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{datatype}\isamarkupfalse%
+\ monotype\ {\isacharequal}\ Mono\ nat\ {\isachardoublequoteopen}monotype\ list{\isachardoublequoteclose}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent Then code generation for SML would fail with a message
+ that the generated code contains illegal mutual dependencies:
+ the theorem \isa{eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}{\isacharparenright}\ {\isasymequiv}\ eq{\isacharunderscore}class{\isachardot}eq\ tyco{\isadigit{1}}\ tyco{\isadigit{2}}\ {\isasymand}\ eq{\isacharunderscore}class{\isachardot}eq\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}} already requires the
+ instance \isa{monotype\ {\isasymColon}\ eq}, which itself requires
+ \isa{eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}{\isacharparenright}\ {\isasymequiv}\ eq{\isacharunderscore}class{\isachardot}eq\ tyco{\isadigit{1}}\ tyco{\isadigit{2}}\ {\isasymand}\ eq{\isacharunderscore}class{\isachardot}eq\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}}; Haskell has no problem with mutually
+ recursive \isa{instance} and \isa{function} definitions,
+ but the SML serialiser does not support this.
+
+ In such cases, you have to provide your own equality equations
+ involving auxiliary constants. In our case,
+ \isa{list{\isacharunderscore}all{\isadigit{2}}} can do the job:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{lemma}\isamarkupfalse%
+\ monotype{\isacharunderscore}eq{\isacharunderscore}list{\isacharunderscore}all{\isadigit{2}}\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
+\ \ \ \ \ eq{\isacharunderscore}class{\isachardot}eq\ tyco{\isadigit{1}}\ tyco{\isadigit{2}}\ {\isasymand}\ list{\isacharunderscore}all{\isadigit{2}}\ eq{\isacharunderscore}class{\isachardot}eq\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp\ add{\isacharcolon}\ eq\ list{\isacharunderscore}all{\isadigit{2}}{\isacharunderscore}eq\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent does not depend on instance \isa{monotype\ {\isasymColon}\ eq}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}structure Example = \\
+\hspace*{0pt}struct\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun null [] = true\\
+\hspace*{0pt} ~| null (x ::~xs) = false;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun eq{\char95}nat (Suc a) Zero{\char95}nat = false\\
+\hspace*{0pt} ~| eq{\char95}nat Zero{\char95}nat (Suc a) = false\\
+\hspace*{0pt} ~| eq{\char95}nat (Suc nat) (Suc nat') = eq{\char95}nat nat nat'\\
+\hspace*{0pt} ~| eq{\char95}nat Zero{\char95}nat Zero{\char95}nat = true;\\
+\hspace*{0pt}\\
+\hspace*{0pt}datatype monotype = Mono of nat * monotype list;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun list{\char95}all2 p (x ::~xs) (y ::~ys) = p x y andalso list{\char95}all2 p xs ys\\
+\hspace*{0pt} ~| list{\char95}all2 p xs [] = null xs\\
+\hspace*{0pt} ~| list{\char95}all2 p [] ys = null ys;\\
+\hspace*{0pt}\\
+\hspace*{0pt}fun eq{\char95}monotype (Mono (tyco1,~typargs1)) (Mono (tyco2,~typargs2)) =\\
+\hspace*{0pt} ~eq{\char95}nat tyco1 tyco2 andalso list{\char95}all2 eq{\char95}monotype typargs1 typargs2;\\
+\hspace*{0pt}\\
+\hspace*{0pt}end;~(*struct Example*)%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isamarkupsubsection{Explicit partiality%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Partiality usually enters the game by partial patterns, as
+ in the following example, again for amortised queues:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{definition}\isamarkupfalse%
+\ strict{\isacharunderscore}dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ q\ {\isacharequal}\ {\isacharparenleft}case\ dequeue\ q\isanewline
+\ \ \ \ of\ {\isacharparenleft}Some\ x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ strict{\isacharunderscore}dequeue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ {\isacharparenleft}case\ rev\ xs\ of\ y\ {\isacharhash}\ ys\ {\isasymRightarrow}\ {\isacharparenleft}y{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ ys{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ strict{\isacharunderscore}dequeue{\isacharunderscore}def\ dequeue{\isacharunderscore}AQueue\ split{\isacharcolon}\ list{\isachardot}splits{\isacharparenright}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent In the corresponding code, there is no equation
+ for the pattern \isa{Program{\isachardot}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}strict{\char95}dequeue ::~forall a.~Queue a -> (a,~Queue a);\\
+\hspace*{0pt}strict{\char95}dequeue (AQueue xs []) =\\
+\hspace*{0pt} ~let {\char123}\\
+\hspace*{0pt} ~~~(y :~ys) = rev xs;\\
+\hspace*{0pt} ~{\char125}~in (y,~AQueue [] ys);\\
+\hspace*{0pt}strict{\char95}dequeue (AQueue xs (y :~ys)) = (y,~AQueue xs ys);%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent In some cases it is desirable to have this
+ pseudo-\qt{partiality} more explicitly, e.g.~as follows:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{axiomatization}\isamarkupfalse%
+\ empty{\isacharunderscore}queue\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\isanewline
+\isanewline
+\isacommand{definition}\isamarkupfalse%
+\ strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ q\ {\isacharequal}\ {\isacharparenleft}case\ dequeue\ q\ of\ {\isacharparenleft}Some\ x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isacharbar}\ {\isacharunderscore}\ {\isasymRightarrow}\ empty{\isacharunderscore}queue{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ strict{\isacharunderscore}dequeue{\isacharprime}{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ empty{\isacharunderscore}queue\isanewline
+\ \ \ \ \ else\ strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
+\ \ \ \ \ {\isacharparenleft}y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ strict{\isacharunderscore}dequeue{\isacharprime}{\isacharunderscore}def\ dequeue{\isacharunderscore}AQueue\ split{\isacharcolon}\ list{\isachardot}splits{\isacharparenright}%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+Observe that on the right hand side of the definition of \isa{strict{\isacharunderscore}dequeue{\isacharprime}} the constant \isa{empty{\isacharunderscore}queue} occurs
+ which is unspecified.
+
+ Normally, if constants without any code equations occur in a
+ program, the code generator complains (since in most cases this is
+ not what the user expects). But such constants can also be thought
+ of as function definitions with no equations which always fail,
+ since there is never a successful pattern match on the left hand
+ side. In order to categorise a constant into that category
+ explicitly, use \hyperlink{command.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+\isacommand{code{\isacharunderscore}abort}\isamarkupfalse%
+\ empty{\isacharunderscore}queue%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent Then the code generator will just insert an error or
+ exception at the appropriate position:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\isatagquote
+%
+\begin{isamarkuptext}%
+\isatypewriter%
+\noindent%
+\hspace*{0pt}empty{\char95}queue ::~forall a.~a;\\
+\hspace*{0pt}empty{\char95}queue = error {\char34}empty{\char95}queue{\char34};\\
+\hspace*{0pt}\\
+\hspace*{0pt}strict{\char95}dequeue' ::~forall a.~Queue a -> (a,~Queue a);\\
+\hspace*{0pt}strict{\char95}dequeue' (AQueue xs (y :~ys)) = (y,~AQueue xs ys);\\
+\hspace*{0pt}strict{\char95}dequeue' (AQueue xs []) =\\
+\hspace*{0pt} ~(if nulla xs then empty{\char95}queue\\
+\hspace*{0pt} ~~~else strict{\char95}dequeue' (AQueue [] (rev xs)));%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\endisatagquote
+{\isafoldquote}%
+%
+\isadelimquote
+%
+\endisadelimquote
+%
+\begin{isamarkuptext}%
+\noindent This feature however is rarely needed in practice.
+ Note also that the \isa{HOL} default setup already declares
+ \isa{undefined} as \hyperlink{command.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}, which is most
+ likely to be used in such situations.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\ \end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/Codegen.hs Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,23 @@
+module Codegen where {
+
+import qualified Nat;
+
+class Null a where {
+ nulla :: a;
+};
+
+heada :: forall a. (Codegen.Null a) => [a] -> a;
+heada (x : xs) = x;
+heada [] = Codegen.nulla;
+
+null_option :: forall a. Maybe a;
+null_option = Nothing;
+
+instance Codegen.Null (Maybe a) where {
+ nulla = Codegen.null_option;
+};
+
+dummy :: Maybe Nat.Nat;
+dummy = Codegen.heada [Just (Nat.Suc Nat.Zero_nat), Nothing];
+
+}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/Example.hs Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,33 @@
+{-# OPTIONS_GHC -fglasgow-exts #-}
+
+module Example where {
+
+
+foldla :: forall a b. (a -> b -> a) -> a -> [b] -> a;
+foldla f a [] = a;
+foldla f a (x : xs) = foldla f (f a x) xs;
+
+rev :: forall a. [a] -> [a];
+rev xs = foldla (\ xsa x -> x : xsa) [] xs;
+
+list_case :: forall t a. t -> (a -> [a] -> t) -> [a] -> t;
+list_case f1 f2 (a : list) = f2 a list;
+list_case f1 f2 [] = f1;
+
+data Queue a = AQueue [a] [a];
+
+empty :: forall a. Queue a;
+empty = AQueue [] [];
+
+dequeue :: forall a. Queue a -> (Maybe a, Queue a);
+dequeue (AQueue [] []) = (Nothing, AQueue [] []);
+dequeue (AQueue xs (y : ys)) = (Just y, AQueue xs ys);
+dequeue (AQueue (v : va) []) =
+ let {
+ (y : ys) = rev (v : va);
+ } in (Just y, AQueue [] ys);
+
+enqueue :: forall a. a -> Queue a -> Queue a;
+enqueue x (AQueue xs ys) = AQueue (x : xs) ys;
+
+}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/arbitrary.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,9 @@
+structure Codegen =
+struct
+
+val arbitrary_option : 'a option = NONE;
+
+fun dummy_option [] = arbitrary_option
+ | dummy_option (x :: xs) = SOME x;
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/bool_infix.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,19 @@
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+fun less_nat m (Suc n) = less_eq_nat m n
+ | less_nat n Zero_nat = false
+and less_eq_nat (Suc m) n = less_nat m n
+ | less_eq_nat Zero_nat n = true;
+
+end; (*struct Nat*)
+
+structure Codegen =
+struct
+
+fun in_interval (k, l) n =
+ Nat.less_eq_nat k n andalso Nat.less_eq_nat n l;
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/bool_literal.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,31 @@
+structure HOL =
+struct
+
+datatype boola = False | True;
+
+fun anda x True = x
+ | anda x False = False
+ | anda True x = x
+ | anda False x = False;
+
+end; (*struct HOL*)
+
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+fun less_nat m (Suc n) = less_eq_nat m n
+ | less_nat n Zero_nat = HOL.False
+and less_eq_nat (Suc m) n = less_nat m n
+ | less_eq_nat Zero_nat n = HOL.True;
+
+end; (*struct Nat*)
+
+structure Codegen =
+struct
+
+fun in_interval (k, l) n =
+ HOL.anda (Nat.less_eq_nat k n) (Nat.less_eq_nat n l);
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/bool_mlbool.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,19 @@
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+fun less_nat m (Suc n) = less_eq_nat m n
+ | less_nat n Zero_nat = false
+and less_eq_nat (Suc m) n = less_nat m n
+ | less_eq_nat Zero_nat n = true;
+
+end; (*struct Nat*)
+
+structure Codegen =
+struct
+
+fun in_interval (k, l) n =
+ (Nat.less_eq_nat k n) andalso (Nat.less_eq_nat n l);
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/class.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,24 @@
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+end; (*struct Nat*)
+
+structure Codegen =
+struct
+
+type 'a null = {null : 'a};
+fun null (A_:'a null) = #null A_;
+
+fun head A_ (x :: xs) = x
+ | head A_ [] = null A_;
+
+val null_option : 'a option = NONE;
+
+fun null_optiona () = {null = null_option} : ('a option) null;
+
+val dummy : Nat.nat option =
+ head (null_optiona ()) [SOME (Nat.Suc Nat.Zero_nat), NONE];
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/class.ocaml Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,24 @@
+module Nat =
+struct
+
+type nat = Suc of nat | Zero_nat;;
+
+end;; (*struct Nat*)
+
+module Codegen =
+struct
+
+type 'a null = {null : 'a};;
+let null _A = _A.null;;
+
+let rec head _A = function x :: xs -> x
+ | [] -> null _A;;
+
+let rec null_option = None;;
+
+let null_optiona () = ({null = null_option} : ('a option) null);;
+
+let rec dummy
+ = head (null_optiona ()) [Some (Nat.Suc Nat.Zero_nat); None];;
+
+end;; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/collect_duplicates.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,30 @@
+structure HOL =
+struct
+
+type 'a eq = {eq : 'a -> 'a -> bool};
+fun eq (A_:'a eq) = #eq A_;
+
+fun eqop A_ a = eq A_ a;
+
+end; (*struct HOL*)
+
+structure List =
+struct
+
+fun member A_ x (y :: ys) =
+ (if HOL.eqop A_ y x then true else member A_ x ys)
+ | member A_ x [] = false;
+
+end; (*struct List*)
+
+structure Codegen =
+struct
+
+fun collect_duplicates A_ xs ys (z :: zs) =
+ (if List.member A_ z xs
+ then (if List.member A_ z ys then collect_duplicates A_ xs ys zs
+ else collect_duplicates A_ xs (z :: ys) zs)
+ else collect_duplicates A_ (z :: xs) (z :: ys) zs)
+ | collect_duplicates A_ xs ys [] = xs;
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/dirty_set.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,102 @@
+structure ROOT =
+struct
+
+structure Nat =
+struct
+
+datatype nat = Zero_nat | Suc of nat;
+
+end; (*struct Nat*)
+
+structure Integer =
+struct
+
+datatype bit = B0 | B1;
+
+datatype int = Pls | Min | Bit of int * bit | Number_of_int of int;
+
+fun pred (Bit (k, B0)) = Bit (pred k, B1)
+ | pred (Bit (k, B1)) = Bit (k, B0)
+ | pred Min = Bit (Min, B0)
+ | pred Pls = Min;
+
+fun uminus_int (Number_of_int w) = Number_of_int (uminus_int w)
+ | uminus_int (Bit (k, B0)) = Bit (uminus_int k, B0)
+ | uminus_int (Bit (k, B1)) = Bit (pred (uminus_int k), B1)
+ | uminus_int Min = Bit (Pls, B1)
+ | uminus_int Pls = Pls;
+
+fun succ (Bit (k, B0)) = Bit (k, B1)
+ | succ (Bit (k, B1)) = Bit (succ k, B0)
+ | succ Min = Pls
+ | succ Pls = Bit (Pls, B1);
+
+fun plus_int (Number_of_int v) (Number_of_int w) =
+ Number_of_int (plus_int v w)
+ | plus_int k Min = pred k
+ | plus_int k Pls = k
+ | plus_int (Bit (k, B1)) (Bit (l, B1)) = Bit (plus_int k (succ l), B0)
+ | plus_int (Bit (k, B1)) (Bit (l, B0)) = Bit (plus_int k l, B1)
+ | plus_int (Bit (k, B0)) (Bit (l, b)) = Bit (plus_int k l, b)
+ | plus_int Min k = pred k
+ | plus_int Pls k = k;
+
+fun minus_int (Number_of_int v) (Number_of_int w) =
+ Number_of_int (plus_int v (uminus_int w))
+ | minus_int z w = plus_int z (uminus_int w);
+
+fun less_eq_int (Number_of_int k) (Number_of_int l) = less_eq_int k l
+ | less_eq_int (Bit (k1, B1)) (Bit (k2, B0)) = less_int k1 k2
+ | less_eq_int (Bit (k1, v)) (Bit (k2, B1)) = less_eq_int k1 k2
+ | less_eq_int (Bit (k1, B0)) (Bit (k2, v)) = less_eq_int k1 k2
+ | less_eq_int (Bit (k, v)) Min = less_eq_int k Min
+ | less_eq_int (Bit (k, B1)) Pls = less_int k Pls
+ | less_eq_int (Bit (k, B0)) Pls = less_eq_int k Pls
+ | less_eq_int Min (Bit (k, B1)) = less_eq_int Min k
+ | less_eq_int Min (Bit (k, B0)) = less_int Min k
+ | less_eq_int Min Min = true
+ | less_eq_int Min Pls = true
+ | less_eq_int Pls (Bit (k, v)) = less_eq_int Pls k
+ | less_eq_int Pls Min = false
+ | less_eq_int Pls Pls = true
+and less_int (Number_of_int k) (Number_of_int l) = less_int k l
+ | less_int (Bit (k1, B0)) (Bit (k2, B1)) = less_eq_int k1 k2
+ | less_int (Bit (k1, B1)) (Bit (k2, v)) = less_int k1 k2
+ | less_int (Bit (k1, v)) (Bit (k2, B0)) = less_int k1 k2
+ | less_int (Bit (k, B1)) Min = less_int k Min
+ | less_int (Bit (k, B0)) Min = less_eq_int k Min
+ | less_int (Bit (k, v)) Pls = less_int k Pls
+ | less_int Min (Bit (k, v)) = less_int Min k
+ | less_int Min Min = false
+ | less_int Min Pls = true
+ | less_int Pls (Bit (k, B1)) = less_eq_int Pls k
+ | less_int Pls (Bit (k, B0)) = less_int Pls k
+ | less_int Pls Min = false
+ | less_int Pls Pls = false;
+
+fun nat_aux n i =
+ (if less_eq_int i (Number_of_int Pls) then n
+ else nat_aux (Nat.Suc n)
+ (minus_int i (Number_of_int (Bit (Pls, B1)))));
+
+fun nat i = nat_aux Nat.Zero_nat i;
+
+end; (*struct Integer*)
+
+structure Codegen =
+struct
+
+val dummy_set : (Nat.nat -> Nat.nat) list = Nat.Suc :: [];
+
+val foobar_set : Nat.nat list =
+ Nat.Zero_nat ::
+ (Nat.Suc Nat.Zero_nat ::
+ (Integer.nat
+ (Integer.Number_of_int
+ (Integer.Bit
+ (Integer.Bit (Integer.Pls, Integer.B1), Integer.B0)))
+ :: []));
+
+end; (*struct Codegen*)
+
+end; (*struct ROOT*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/example.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,27 @@
+structure Example =
+struct
+
+fun foldl f a [] = a
+ | foldl f a (x :: xs) = foldl f (f a x) xs;
+
+fun rev xs = foldl (fn xsa => fn x => x :: xsa) [] xs;
+
+fun list_case f1 f2 (a :: lista) = f2 a lista
+ | list_case f1 f2 [] = f1;
+
+datatype 'a queue = AQueue of 'a list * 'a list;
+
+val empty : 'a queue = AQueue ([], [])
+
+fun dequeue (AQueue ([], [])) = (NONE, AQueue ([], []))
+ | dequeue (AQueue (xs, y :: ys)) = (SOME y, AQueue (xs, ys))
+ | dequeue (AQueue (v :: va, [])) =
+ let
+ val y :: ys = rev (v :: va);
+ in
+ (SOME y, AQueue ([], ys))
+ end;
+
+fun enqueue x (AQueue (xs, ys)) = AQueue (x :: xs, ys);
+
+end; (*struct Example*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/fac.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,22 @@
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+val one_nat : nat = Suc Zero_nat;
+
+fun plus_nat (Suc m) n = plus_nat m (Suc n)
+ | plus_nat Zero_nat n = n;
+
+fun times_nat (Suc m) n = plus_nat n (times_nat m n)
+ | times_nat Zero_nat n = Zero_nat;
+
+end; (*struct Nat*)
+
+structure Codegen =
+struct
+
+fun fac (Nat.Suc n) = Nat.times_nat (Nat.Suc n) (fac n)
+ | fac Nat.Zero_nat = Nat.one_nat;
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/integers.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,59 @@
+structure ROOT =
+struct
+
+structure Integer =
+struct
+
+datatype bit = B0 | B1;
+
+datatype int = Pls | Min | Bit of int * bit | Number_of_int of int;
+
+fun pred (Bit (k, B0)) = Bit (pred k, B1)
+ | pred (Bit (k, B1)) = Bit (k, B0)
+ | pred Min = Bit (Min, B0)
+ | pred Pls = Min;
+
+fun succ (Bit (k, B0)) = Bit (k, B1)
+ | succ (Bit (k, B1)) = Bit (succ k, B0)
+ | succ Min = Pls
+ | succ Pls = Bit (Pls, B1);
+
+fun plus_int (Number_of_int v) (Number_of_int w) =
+ Number_of_int (plus_int v w)
+ | plus_int k Min = pred k
+ | plus_int k Pls = k
+ | plus_int (Bit (k, B1)) (Bit (l, B1)) = Bit (plus_int k (succ l), B0)
+ | plus_int (Bit (k, B1)) (Bit (l, B0)) = Bit (plus_int k l, B1)
+ | plus_int (Bit (k, B0)) (Bit (l, b)) = Bit (plus_int k l, b)
+ | plus_int Min k = pred k
+ | plus_int Pls k = k;
+
+fun uminus_int (Number_of_int w) = Number_of_int (uminus_int w)
+ | uminus_int (Bit (k, B0)) = Bit (uminus_int k, B0)
+ | uminus_int (Bit (k, B1)) = Bit (pred (uminus_int k), B1)
+ | uminus_int Min = Bit (Pls, B1)
+ | uminus_int Pls = Pls;
+
+fun times_int (Number_of_int v) (Number_of_int w) =
+ Number_of_int (times_int v w)
+ | times_int (Bit (k, B0)) l = Bit (times_int k l, B0)
+ | times_int (Bit (k, B1)) l = plus_int (Bit (times_int k l, B0)) l
+ | times_int Min k = uminus_int k
+ | times_int Pls w = Pls;
+
+end; (*struct Integer*)
+
+structure Codegen =
+struct
+
+fun double_inc k =
+ Integer.plus_int
+ (Integer.times_int
+ (Integer.Number_of_int
+ (Integer.Bit (Integer.Bit (Integer.Pls, Integer.B1), Integer.B0)))
+ k)
+ (Integer.Number_of_int (Integer.Bit (Integer.Pls, Integer.B1)));
+
+end; (*struct Codegen*)
+
+end; (*struct ROOT*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/lexicographic.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,19 @@
+structure HOL =
+struct
+
+type 'a eq = {eq : 'a -> 'a -> bool};
+fun eq (A_:'a eq) = #eq A_;
+
+type 'a ord = {less_eq : 'a -> 'a -> bool, less : 'a -> 'a -> bool};
+fun less_eq (A_:'a ord) = #less_eq A_;
+fun less (A_:'a ord) = #less A_;
+
+end; (*struct HOL*)
+
+structure Codegen =
+struct
+
+fun less_eq (A1_, A2_) B_ (x1, y1) (x2, y2) =
+ HOL.less A2_ x1 x2 orelse HOL.eq A1_ x1 x2 andalso HOL.less_eq B_ y1 y2;
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/lookup.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,13 @@
+structure ROOT =
+struct
+
+structure Codegen =
+struct
+
+fun lookup ((k, v) :: xs) l =
+ (if ((k : string) = l) then SOME v else lookup xs l)
+ | lookup [] l = NONE;
+
+end; (*struct Codegen*)
+
+end; (*struct ROOT*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/monotype.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,34 @@
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+fun eq_nat (Suc a) Zero_nat = false
+ | eq_nat Zero_nat (Suc a) = false
+ | eq_nat (Suc nat) (Suc nat') = eq_nat nat nat'
+ | eq_nat Zero_nat Zero_nat = true;
+
+end; (*struct Nat*)
+
+structure List =
+struct
+
+fun null (x :: xs) = false
+ | null [] = true;
+
+fun list_all2 p (x :: xs) (y :: ys) = p x y andalso list_all2 p xs ys
+ | list_all2 p xs [] = null xs
+ | list_all2 p [] ys = null ys;
+
+end; (*struct List*)
+
+structure Codegen =
+struct
+
+datatype monotype = Mono of Nat.nat * monotype list;
+
+fun eq_monotype (Mono (tyco1, typargs1)) (Mono (tyco2, typargs2)) =
+ Nat.eq_nat tyco1 tyco2 andalso
+ List.list_all2 eq_monotype typargs1 typargs2;
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/nat_binary.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,17 @@
+structure Nat =
+struct
+
+datatype nat = Dig1 of nat | Dig0 of nat | One_nat | Zero_nat;
+
+fun plus_nat (Dig1 m) (Dig1 n) = Dig0 (plus_nat (plus_nat m n) One_nat)
+ | plus_nat (Dig1 m) (Dig0 n) = Dig1 (plus_nat m n)
+ | plus_nat (Dig0 m) (Dig1 n) = Dig1 (plus_nat m n)
+ | plus_nat (Dig0 m) (Dig0 n) = Dig0 (plus_nat m n)
+ | plus_nat (Dig1 m) One_nat = Dig0 (plus_nat m One_nat)
+ | plus_nat One_nat (Dig1 n) = Dig0 (plus_nat n One_nat)
+ | plus_nat (Dig0 m) One_nat = Dig1 m
+ | plus_nat One_nat (Dig0 n) = Dig1 n
+ | plus_nat m Zero_nat = m
+ | plus_nat Zero_nat n = n;
+
+end; (*struct Nat*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/pick1.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,44 @@
+structure HOL =
+struct
+
+fun leta s f = f s;
+
+end; (*struct HOL*)
+
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+fun less_nat m (Suc n) = less_eq_nat m n
+ | less_nat n Zero_nat = false
+and less_eq_nat (Suc m) n = less_nat m n
+ | less_eq_nat Zero_nat n = true;
+
+fun minus_nat (Suc m) (Suc n) = minus_nat m n
+ | minus_nat Zero_nat n = Zero_nat
+ | minus_nat m Zero_nat = m;
+
+end; (*struct Nat*)
+
+structure Product_Type =
+struct
+
+fun split f (a, b) = f a b;
+
+end; (*struct Product_Type*)
+
+structure Codegen =
+struct
+
+fun pick ((k, v) :: xs) n =
+ (if Nat.less_nat n k then v else pick xs (Nat.minus_nat n k))
+ | pick (x :: xs) n =
+ let
+ val a = x;
+ val (k, v) = a;
+ in
+ (if Nat.less_nat n k then v else pick xs (Nat.minus_nat n k))
+ end;
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/Thy/examples/tree.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,95 @@
+structure HOL =
+struct
+
+type 'a eq = {eq : 'a -> 'a -> bool};
+fun eq (A_:'a eq) = #eq A_;
+
+type 'a ord = {less_eq : 'a -> 'a -> bool, less : 'a -> 'a -> bool};
+fun less_eq (A_:'a ord) = #less_eq A_;
+fun less (A_:'a ord) = #less A_;
+
+fun eqop A_ a = eq A_ a;
+
+end; (*struct HOL*)
+
+structure Orderings =
+struct
+
+type 'a preorder = {Orderings__ord_preorder : 'a HOL.ord};
+fun ord_preorder (A_:'a preorder) = #Orderings__ord_preorder A_;
+
+type 'a order = {Orderings__preorder_order : 'a preorder};
+fun preorder_order (A_:'a order) = #Orderings__preorder_order A_;
+
+type 'a linorder = {Orderings__order_linorder : 'a order};
+fun order_linorder (A_:'a linorder) = #Orderings__order_linorder A_;
+
+end; (*struct Orderings*)
+
+structure Nat =
+struct
+
+datatype nat = Suc of nat | Zero_nat;
+
+fun eq_nat (Suc a) Zero_nat = false
+ | eq_nat Zero_nat (Suc a) = false
+ | eq_nat (Suc nat) (Suc nat') = eq_nat nat nat'
+ | eq_nat Zero_nat Zero_nat = true;
+
+val eq_nata = {eq = eq_nat} : nat HOL.eq;
+
+fun less_nat m (Suc n) = less_eq_nat m n
+ | less_nat n Zero_nat = false
+and less_eq_nat (Suc m) n = less_nat m n
+ | less_eq_nat Zero_nat n = true;
+
+val ord_nat = {less_eq = less_eq_nat, less = less_nat} : nat HOL.ord;
+
+val preorder_nat = {Orderings__ord_preorder = ord_nat} :
+ nat Orderings.preorder;
+
+val order_nat = {Orderings__preorder_order = preorder_nat} :
+ nat Orderings.order;
+
+val linorder_nat = {Orderings__order_linorder = order_nat} :
+ nat Orderings.linorder;
+
+end; (*struct Nat*)
+
+structure Codegen =
+struct
+
+datatype ('a, 'b) searchtree =
+ Branch of ('a, 'b) searchtree * 'a * ('a, 'b) searchtree |
+ Leaf of 'a * 'b;
+
+fun update (A1_, A2_) (it, entry) (Leaf (key, vala)) =
+ (if HOL.eqop A1_ it key then Leaf (key, entry)
+ else (if HOL.less_eq
+ ((Orderings.ord_preorder o Orderings.preorder_order o
+ Orderings.order_linorder)
+ A2_)
+ it key
+ then Branch (Leaf (it, entry), it, Leaf (key, vala))
+ else Branch (Leaf (key, vala), it, Leaf (it, entry))))
+ | update (A1_, A2_) (it, entry) (Branch (t1, key, t2)) =
+ (if HOL.less_eq
+ ((Orderings.ord_preorder o Orderings.preorder_order o
+ Orderings.order_linorder)
+ A2_)
+ it key
+ then Branch (update (A1_, A2_) (it, entry) t1, key, t2)
+ else Branch (t1, key, update (A1_, A2_) (it, entry) t2));
+
+val example : (Nat.nat, (Nat.nat list)) searchtree =
+ update (Nat.eq_nata, Nat.linorder_nat)
+ (Nat.Suc (Nat.Suc (Nat.Suc (Nat.Suc Nat.Zero_nat))),
+ [Nat.Suc (Nat.Suc Nat.Zero_nat), Nat.Suc (Nat.Suc Nat.Zero_nat)])
+ (update (Nat.eq_nata, Nat.linorder_nat)
+ (Nat.Suc (Nat.Suc (Nat.Suc Nat.Zero_nat)),
+ [Nat.Suc (Nat.Suc (Nat.Suc Nat.Zero_nat))])
+ (update (Nat.eq_nata, Nat.linorder_nat)
+ (Nat.Suc (Nat.Suc Nat.Zero_nat), [Nat.Suc (Nat.Suc Nat.Zero_nat)])
+ (Leaf (Nat.Suc Nat.Zero_nat, []))));
+
+end; (*struct Codegen*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/codegen.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,53 @@
+
+\documentclass[12pt,a4paper,fleqn]{report}
+\usepackage{latexsym,graphicx}
+\usepackage[refpage]{nomencl}
+\usepackage{../iman,../extra,../isar,../proof}
+\usepackage{../isabelle,../isabellesym}
+\usepackage{style}
+\usepackage{pgf}
+\usepackage{pgflibraryshapes}
+\usepackage{tikz}
+\usepackage{../pdfsetup}
+
+\hyphenation{Isabelle}
+\hyphenation{Isar}
+\isadroptag{theory}
+
+\title{\includegraphics[scale=0.5]{isabelle_isar}
+ \\[4ex] Code generation from Isabelle/HOL theories}
+\author{\emph{Florian Haftmann}}
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+ This tutorial gives an introduction to a generic code generator framework in Isabelle
+ for generating executable code in functional programming languages from logical
+ specifications in Isabelle/HOL.
+\end{abstract}
+
+\thispagestyle{empty}\clearpage
+
+\pagenumbering{roman}
+\clearfirst
+
+\input{Thy/document/Introduction.tex}
+\input{Thy/document/Program.tex}
+\input{Thy/document/Adaption.tex}
+\input{Thy/document/Further.tex}
+\input{Thy/document/ML.tex}
+
+\begingroup
+\bibliographystyle{plain} \small\raggedright\frenchspacing
+\bibliography{../manual}
+\endgroup
+
+\end{document}
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
Binary file doc-src/Codegen/codegen_process.pdf has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/codegen_process.ps Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,586 @@
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Codegen/style.sty Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,56 @@
+
+%% toc
+\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
+\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
+
+%% references
+\newcommand{\secref}[1]{\S\ref{#1}}
+
+%% logical markup
+\newcommand{\strong}[1]{{\bfseries {#1}}}
+\newcommand{\qn}[1]{\emph{#1}}
+
+%% typographic conventions
+\newcommand{\qt}[1]{``{#1}''}
+
+%% verbatim text
+\newcommand{\isatypewriter}{\fontsize{9pt}{0pt}\tt\renewcommand{\baselinestretch}{1}\setlength{\baselineskip}{9pt}}
+
+%% quoted segments
+\makeatletter
+\isakeeptag{quote}
+\newenvironment{quotesegment}{\begin{quote}\isa@parindent\parindent\parindent0pt\isa@parskip\parskip\parskip0pt}{\end{quote}}
+\renewcommand{\isatagquote}{\begin{quotesegment}}
+\renewcommand{\endisatagquote}{\end{quotesegment}}
+\makeatother
+
+\isakeeptag{quotett}
+\renewcommand{\isatagquotett}{\begin{quotesegment}\isabellestyle{tt}\isastyle}
+\renewcommand{\endisatagquotett}{\end{quotesegment}\isabellestyle{it}\isastyle}
+
+%% a trick
+\newcommand{\isasymSML}{SML}
+
+%% presentation
+\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
+
+\pagestyle{headings}
+\binperiod
+\underscoreoff
+
+\renewcommand{\isadigit}[1]{\isamath{#1}}
+
+%% ml reference
+\newenvironment{mldecls}{\par\noindent\begingroup\footnotesize\def\isanewline{\\}\begin{tabular}{l}}{\end{tabular}\smallskip\endgroup}
+
+\isakeeptag{mlref}
+\renewcommand{\isatagmlref}{\subsection*{\makebox[0pt][r]{\fbox{\ML}~~}Reference}\begingroup\def\isastyletext{\rm}\small}
+\renewcommand{\endisatagmlref}{\endgroup}
+
+\isabellestyle{it}
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "implementation"
+%%% End:
--- a/doc-src/Dirs Mon Mar 02 16:58:39 2009 +0100
+++ b/doc-src/Dirs Tue Mar 03 11:00:51 2009 +0100
@@ -1,1 +1,1 @@
-Intro Ref System Logics HOL ZF Inductive TutorialI IsarOverview IsarRef IsarImplementation Locales LaTeXsugar IsarAdvanced/Classes IsarAdvanced/Codegen IsarAdvanced/Functions
+Intro Ref System Logics HOL ZF Inductive TutorialI IsarOverview IsarRef IsarImplementation Locales LaTeXsugar Classes Codegen Functions
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/IsaMakefile Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,33 @@
+
+## targets
+
+default: Thy
+images:
+test: Thy
+
+all: images test
+
+
+## global settings
+
+SRC = $(ISABELLE_HOME)/src
+OUT = $(ISABELLE_OUTPUT)
+LOG = $(OUT)/log
+
+USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
+
+
+## Thy
+
+THY = $(LOG)/HOL-Thy.gz
+
+Thy: $(THY)
+
+$(THY): Thy/ROOT.ML Thy/Functions.thy
+ @$(USEDIR) HOL Thy
+
+
+## clean
+
+clean:
+ @rm -f $(THY)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/Makefile Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,38 @@
+#
+# $Id$
+#
+
+## targets
+
+default: dvi
+
+
+## dependencies
+
+include ../Makefile.in
+
+NAME = functions
+
+FILES = $(NAME).tex Thy/document/Functions.tex intro.tex conclusion.tex \
+ style.sty ../iman.sty ../extra.sty ../isar.sty \
+ ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty \
+ ../manual.bib ../proof.sty
+
+dvi: $(NAME).dvi
+
+$(NAME).dvi: $(FILES) isabelle_isar.eps
+ $(LATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(LATEX) $(NAME)
+ $(LATEX) $(NAME)
+
+pdf: $(NAME).pdf
+
+$(NAME).pdf: $(FILES) isabelle_isar.pdf
+ $(PDFLATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(FIXBOOKMARKS) $(NAME).out
+ $(PDFLATEX) $(NAME)
+ $(PDFLATEX) $(NAME)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/Thy/Functions.thy Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,1264 @@
+(* Title: doc-src/IsarAdvanced/Functions/Thy/Fundefs.thy
+ Author: Alexander Krauss, TU Muenchen
+
+Tutorial for function definitions with the new "function" package.
+*)
+
+theory Functions
+imports Main
+begin
+
+section {* Function Definitions for Dummies *}
+
+text {*
+ In most cases, defining a recursive function is just as simple as other definitions:
+*}
+
+fun fib :: "nat \<Rightarrow> nat"
+where
+ "fib 0 = 1"
+| "fib (Suc 0) = 1"
+| "fib (Suc (Suc n)) = fib n + fib (Suc n)"
+
+text {*
+ The syntax is rather self-explanatory: We introduce a function by
+ giving its name, its type,
+ and a set of defining recursive equations.
+ If we leave out the type, the most general type will be
+ inferred, which can sometimes lead to surprises: Since both @{term
+ "1::nat"} and @{text "+"} are overloaded, we would end up
+ with @{text "fib :: nat \<Rightarrow> 'a::{one,plus}"}.
+*}
+
+text {*
+ The function always terminates, since its argument gets smaller in
+ every recursive call.
+ Since HOL is a logic of total functions, termination is a
+ fundamental requirement to prevent inconsistencies\footnote{From the
+ \qt{definition} @{text "f(n) = f(n) + 1"} we could prove
+ @{text "0 = 1"} by subtracting @{text "f(n)"} on both sides.}.
+ Isabelle tries to prove termination automatically when a definition
+ is made. In \S\ref{termination}, we will look at cases where this
+ fails and see what to do then.
+*}
+
+subsection {* Pattern matching *}
+
+text {* \label{patmatch}
+ Like in functional programming, we can use pattern matching to
+ define functions. At the moment we will only consider \emph{constructor
+ patterns}, which only consist of datatype constructors and
+ variables. Furthermore, patterns must be linear, i.e.\ all variables
+ on the left hand side of an equation must be distinct. In
+ \S\ref{genpats} we discuss more general pattern matching.
+
+ If patterns overlap, the order of the equations is taken into
+ account. The following function inserts a fixed element between any
+ two elements of a list:
+*}
+
+fun sep :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ "sep a (x#y#xs) = x # a # sep a (y # xs)"
+| "sep a xs = xs"
+
+text {*
+ Overlapping patterns are interpreted as \qt{increments} to what is
+ already there: The second equation is only meant for the cases where
+ the first one does not match. Consequently, Isabelle replaces it
+ internally by the remaining cases, making the patterns disjoint:
+*}
+
+thm sep.simps
+
+text {* @{thm [display] sep.simps[no_vars]} *}
+
+text {*
+ \noindent The equations from function definitions are automatically used in
+ simplification:
+*}
+
+lemma "sep 0 [1, 2, 3] = [1, 0, 2, 0, 3]"
+by simp
+
+subsection {* Induction *}
+
+text {*
+
+ Isabelle provides customized induction rules for recursive
+ functions. These rules follow the recursive structure of the
+ definition. Here is the rule @{text sep.induct} arising from the
+ above definition of @{const sep}:
+
+ @{thm [display] sep.induct}
+
+ We have a step case for list with at least two elements, and two
+ base cases for the zero- and the one-element list. Here is a simple
+ proof about @{const sep} and @{const map}
+*}
+
+lemma "map f (sep x ys) = sep (f x) (map f ys)"
+apply (induct x ys rule: sep.induct)
+
+txt {*
+ We get three cases, like in the definition.
+
+ @{subgoals [display]}
+*}
+
+apply auto
+done
+text {*
+
+ With the \cmd{fun} command, you can define about 80\% of the
+ functions that occur in practice. The rest of this tutorial explains
+ the remaining 20\%.
+*}
+
+
+section {* fun vs.\ function *}
+
+text {*
+ The \cmd{fun} command provides a
+ convenient shorthand notation for simple function definitions. In
+ this mode, Isabelle tries to solve all the necessary proof obligations
+ automatically. If any proof fails, the definition is
+ rejected. This can either mean that the definition is indeed faulty,
+ or that the default proof procedures are just not smart enough (or
+ rather: not designed) to handle the definition.
+
+ By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
+ solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
+
+\end{isamarkuptext}
+
+
+\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
+\cmd{fun} @{text "f :: \<tau>"}\\%
+\cmd{where}\\%
+\hspace*{2ex}{\it equations}\\%
+\hspace*{2ex}\vdots\vspace*{6pt}
+\end{minipage}\right]
+\quad\equiv\quad
+\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
+\cmd{function} @{text "("}\cmd{sequential}@{text ") f :: \<tau>"}\\%
+\cmd{where}\\%
+\hspace*{2ex}{\it equations}\\%
+\hspace*{2ex}\vdots\\%
+\cmd{by} @{text "pat_completeness auto"}\\%
+\cmd{termination by} @{text "lexicographic_order"}\vspace{6pt}
+\end{minipage}
+\right]\]
+
+\begin{isamarkuptext}
+ \vspace*{1em}
+ \noindent Some details have now become explicit:
+
+ \begin{enumerate}
+ \item The \cmd{sequential} option enables the preprocessing of
+ pattern overlaps which we already saw. Without this option, the equations
+ must already be disjoint and complete. The automatic completion only
+ works with constructor patterns.
+
+ \item A function definition produces a proof obligation which
+ expresses completeness and compatibility of patterns (we talk about
+ this later). The combination of the methods @{text "pat_completeness"} and
+ @{text "auto"} is used to solve this proof obligation.
+
+ \item A termination proof follows the definition, started by the
+ \cmd{termination} command. This will be explained in \S\ref{termination}.
+ \end{enumerate}
+ Whenever a \cmd{fun} command fails, it is usually a good idea to
+ expand the syntax to the more verbose \cmd{function} form, to see
+ what is actually going on.
+ *}
+
+
+section {* Termination *}
+
+text {*\label{termination}
+ The method @{text "lexicographic_order"} is the default method for
+ termination proofs. It can prove termination of a
+ certain class of functions by searching for a suitable lexicographic
+ combination of size measures. Of course, not all functions have such
+ a simple termination argument. For them, we can specify the termination
+ relation manually.
+*}
+
+subsection {* The {\tt relation} method *}
+text{*
+ Consider the following function, which sums up natural numbers up to
+ @{text "N"}, using a counter @{text "i"}:
+*}
+
+function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+ "sum i N = (if i > N then 0 else i + sum (Suc i) N)"
+by pat_completeness auto
+
+text {*
+ \noindent The @{text "lexicographic_order"} method fails on this example, because none of the
+ arguments decreases in the recursive call, with respect to the standard size ordering.
+ To prove termination manually, we must provide a custom wellfounded relation.
+
+ The termination argument for @{text "sum"} is based on the fact that
+ the \emph{difference} between @{text "i"} and @{text "N"} gets
+ smaller in every step, and that the recursion stops when @{text "i"}
+ is greater than @{text "N"}. Phrased differently, the expression
+ @{text "N + 1 - i"} always decreases.
+
+ We can use this expression as a measure function suitable to prove termination.
+*}
+
+termination sum
+apply (relation "measure (\<lambda>(i,N). N + 1 - i)")
+
+txt {*
+ The \cmd{termination} command sets up the termination goal for the
+ specified function @{text "sum"}. If the function name is omitted, it
+ implicitly refers to the last function definition.
+
+ The @{text relation} method takes a relation of
+ type @{typ "('a \<times> 'a) set"}, where @{typ "'a"} is the argument type of
+ the function. If the function has multiple curried arguments, then
+ these are packed together into a tuple, as it happened in the above
+ example.
+
+ The predefined function @{term[source] "measure :: ('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set"} constructs a
+ wellfounded relation from a mapping into the natural numbers (a
+ \emph{measure function}).
+
+ After the invocation of @{text "relation"}, we must prove that (a)
+ the relation we supplied is wellfounded, and (b) that the arguments
+ of recursive calls indeed decrease with respect to the
+ relation:
+
+ @{subgoals[display,indent=0]}
+
+ These goals are all solved by @{text "auto"}:
+*}
+
+apply auto
+done
+
+text {*
+ Let us complicate the function a little, by adding some more
+ recursive calls:
+*}
+
+function foo :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+ "foo i N = (if i > N
+ then (if N = 0 then 0 else foo 0 (N - 1))
+ else i + foo (Suc i) N)"
+by pat_completeness auto
+
+text {*
+ When @{text "i"} has reached @{text "N"}, it starts at zero again
+ and @{text "N"} is decremented.
+ This corresponds to a nested
+ loop where one index counts up and the other down. Termination can
+ be proved using a lexicographic combination of two measures, namely
+ the value of @{text "N"} and the above difference. The @{const
+ "measures"} combinator generalizes @{text "measure"} by taking a
+ list of measure functions.
+*}
+
+termination
+by (relation "measures [\<lambda>(i, N). N, \<lambda>(i,N). N + 1 - i]") auto
+
+subsection {* How @{text "lexicographic_order"} works *}
+
+(*fun fails :: "nat \<Rightarrow> nat list \<Rightarrow> nat"
+where
+ "fails a [] = a"
+| "fails a (x#xs) = fails (x + a) (x # xs)"
+*)
+
+text {*
+ To see how the automatic termination proofs work, let's look at an
+ example where it fails\footnote{For a detailed discussion of the
+ termination prover, see \cite{bulwahnKN07}}:
+
+\end{isamarkuptext}
+\cmd{fun} @{text "fails :: \"nat \<Rightarrow> nat list \<Rightarrow> nat\""}\\%
+\cmd{where}\\%
+\hspace*{2ex}@{text "\"fails a [] = a\""}\\%
+|\hspace*{1.5ex}@{text "\"fails a (x#xs) = fails (x + a) (x#xs)\""}\\
+\begin{isamarkuptext}
+
+\noindent Isabelle responds with the following error:
+
+\begin{isabelle}
+*** Unfinished subgoals:\newline
+*** (a, 1, <):\newline
+*** \ 1.~@{text "\<And>x. x = 0"}\newline
+*** (a, 1, <=):\newline
+*** \ 1.~False\newline
+*** (a, 2, <):\newline
+*** \ 1.~False\newline
+*** Calls:\newline
+*** a) @{text "(a, x # xs) -->> (x + a, x # xs)"}\newline
+*** Measures:\newline
+*** 1) @{text "\<lambda>x. size (fst x)"}\newline
+*** 2) @{text "\<lambda>x. size (snd x)"}\newline
+*** Result matrix:\newline
+*** \ \ \ \ 1\ \ 2 \newline
+*** a: ? <= \newline
+*** Could not find lexicographic termination order.\newline
+*** At command "fun".\newline
+\end{isabelle}
+*}
+
+
+text {*
+ The key to this error message is the matrix at the bottom. The rows
+ of that matrix correspond to the different recursive calls (In our
+ case, there is just one). The columns are the function's arguments
+ (expressed through different measure functions, which map the
+ argument tuple to a natural number).
+
+ The contents of the matrix summarize what is known about argument
+ descents: The second argument has a weak descent (@{text "<="}) at the
+ recursive call, and for the first argument nothing could be proved,
+ which is expressed by @{text "?"}. In general, there are the values
+ @{text "<"}, @{text "<="} and @{text "?"}.
+
+ For the failed proof attempts, the unfinished subgoals are also
+ printed. Looking at these will often point to a missing lemma.
+
+% As a more real example, here is quicksort:
+*}
+(*
+function qs :: "nat list \<Rightarrow> nat list"
+where
+ "qs [] = []"
+| "qs (x#xs) = qs [y\<in>xs. y < x] @ x # qs [y\<in>xs. y \<ge> x]"
+by pat_completeness auto
+
+termination apply lexicographic_order
+
+text {* If we try @{text "lexicographic_order"} method, we get the
+ following error *}
+termination by (lexicographic_order simp:l2)
+
+lemma l: "x \<le> y \<Longrightarrow> x < Suc y" by arith
+
+function
+
+*)
+
+section {* Mutual Recursion *}
+
+text {*
+ If two or more functions call one another mutually, they have to be defined
+ in one step. Here are @{text "even"} and @{text "odd"}:
+*}
+
+function even :: "nat \<Rightarrow> bool"
+ and odd :: "nat \<Rightarrow> bool"
+where
+ "even 0 = True"
+| "odd 0 = False"
+| "even (Suc n) = odd n"
+| "odd (Suc n) = even n"
+by pat_completeness auto
+
+text {*
+ To eliminate the mutual dependencies, Isabelle internally
+ creates a single function operating on the sum
+ type @{typ "nat + nat"}. Then, @{const even} and @{const odd} are
+ defined as projections. Consequently, termination has to be proved
+ simultaneously for both functions, by specifying a measure on the
+ sum type:
+*}
+
+termination
+by (relation "measure (\<lambda>x. case x of Inl n \<Rightarrow> n | Inr n \<Rightarrow> n)") auto
+
+text {*
+ We could also have used @{text lexicographic_order}, which
+ supports mutual recursive termination proofs to a certain extent.
+*}
+
+subsection {* Induction for mutual recursion *}
+
+text {*
+
+ When functions are mutually recursive, proving properties about them
+ generally requires simultaneous induction. The induction rule @{text "even_odd.induct"}
+ generated from the above definition reflects this.
+
+ Let us prove something about @{const even} and @{const odd}:
+*}
+
+lemma even_odd_mod2:
+ "even n = (n mod 2 = 0)"
+ "odd n = (n mod 2 = 1)"
+
+txt {*
+ We apply simultaneous induction, specifying the induction variable
+ for both goals, separated by \cmd{and}: *}
+
+apply (induct n and n rule: even_odd.induct)
+
+txt {*
+ We get four subgoals, which correspond to the clauses in the
+ definition of @{const even} and @{const odd}:
+ @{subgoals[display,indent=0]}
+ Simplification solves the first two goals, leaving us with two
+ statements about the @{text "mod"} operation to prove:
+*}
+
+apply simp_all
+
+txt {*
+ @{subgoals[display,indent=0]}
+
+ \noindent These can be handled by Isabelle's arithmetic decision procedures.
+
+*}
+
+apply arith
+apply arith
+done
+
+text {*
+ In proofs like this, the simultaneous induction is really essential:
+ Even if we are just interested in one of the results, the other
+ one is necessary to strengthen the induction hypothesis. If we leave
+ out the statement about @{const odd} and just write @{term True} instead,
+ the same proof fails:
+*}
+
+lemma failed_attempt:
+ "even n = (n mod 2 = 0)"
+ "True"
+apply (induct n rule: even_odd.induct)
+
+txt {*
+ \noindent Now the third subgoal is a dead end, since we have no
+ useful induction hypothesis available:
+
+ @{subgoals[display,indent=0]}
+*}
+
+oops
+
+section {* General pattern matching *}
+text{*\label{genpats} *}
+
+subsection {* Avoiding automatic pattern splitting *}
+
+text {*
+
+ Up to now, we used pattern matching only on datatypes, and the
+ patterns were always disjoint and complete, and if they weren't,
+ they were made disjoint automatically like in the definition of
+ @{const "sep"} in \S\ref{patmatch}.
+
+ This automatic splitting can significantly increase the number of
+ equations involved, and this is not always desirable. The following
+ example shows the problem:
+
+ Suppose we are modeling incomplete knowledge about the world by a
+ three-valued datatype, which has values @{term "T"}, @{term "F"}
+ and @{term "X"} for true, false and uncertain propositions, respectively.
+*}
+
+datatype P3 = T | F | X
+
+text {* \noindent Then the conjunction of such values can be defined as follows: *}
+
+fun And :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
+where
+ "And T p = p"
+| "And p T = p"
+| "And p F = F"
+| "And F p = F"
+| "And X X = X"
+
+
+text {*
+ This definition is useful, because the equations can directly be used
+ as simplification rules. But the patterns overlap: For example,
+ the expression @{term "And T T"} is matched by both the first and
+ the second equation. By default, Isabelle makes the patterns disjoint by
+ splitting them up, producing instances:
+*}
+
+thm And.simps
+
+text {*
+ @{thm[indent=4] And.simps}
+
+ \vspace*{1em}
+ \noindent There are several problems with this:
+
+ \begin{enumerate}
+ \item If the datatype has many constructors, there can be an
+ explosion of equations. For @{const "And"}, we get seven instead of
+ five equations, which can be tolerated, but this is just a small
+ example.
+
+ \item Since splitting makes the equations \qt{less general}, they
+ do not always match in rewriting. While the term @{term "And x F"}
+ can be simplified to @{term "F"} with the original equations, a
+ (manual) case split on @{term "x"} is now necessary.
+
+ \item The splitting also concerns the induction rule @{text
+ "And.induct"}. Instead of five premises it now has seven, which
+ means that our induction proofs will have more cases.
+
+ \item In general, it increases clarity if we get the same definition
+ back which we put in.
+ \end{enumerate}
+
+ If we do not want the automatic splitting, we can switch it off by
+ leaving out the \cmd{sequential} option. However, we will have to
+ prove that our pattern matching is consistent\footnote{This prevents
+ us from defining something like @{term "f x = True"} and @{term "f x
+ = False"} simultaneously.}:
+*}
+
+function And2 :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
+where
+ "And2 T p = p"
+| "And2 p T = p"
+| "And2 p F = F"
+| "And2 F p = F"
+| "And2 X X = X"
+
+txt {*
+ \noindent Now let's look at the proof obligations generated by a
+ function definition. In this case, they are:
+
+ @{subgoals[display,indent=0]}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
+
+ The first subgoal expresses the completeness of the patterns. It has
+ the form of an elimination rule and states that every @{term x} of
+ the function's input type must match at least one of the patterns\footnote{Completeness could
+ be equivalently stated as a disjunction of existential statements:
+@{term "(\<exists>p. x = (T, p)) \<or> (\<exists>p. x = (p, T)) \<or> (\<exists>p. x = (p, F)) \<or>
+ (\<exists>p. x = (F, p)) \<or> (x = (X, X))"}, and you can use the method @{text atomize_elim} to get that form instead.}. If the patterns just involve
+ datatypes, we can solve it with the @{text "pat_completeness"}
+ method:
+*}
+
+apply pat_completeness
+
+txt {*
+ The remaining subgoals express \emph{pattern compatibility}. We do
+ allow that an input value matches multiple patterns, but in this
+ case, the result (i.e.~the right hand sides of the equations) must
+ also be equal. For each pair of two patterns, there is one such
+ subgoal. Usually this needs injectivity of the constructors, which
+ is used automatically by @{text "auto"}.
+*}
+
+by auto
+
+
+subsection {* Non-constructor patterns *}
+
+text {*
+ Most of Isabelle's basic types take the form of inductive datatypes,
+ and usually pattern matching works on the constructors of such types.
+ However, this need not be always the case, and the \cmd{function}
+ command handles other kind of patterns, too.
+
+ One well-known instance of non-constructor patterns are
+ so-called \emph{$n+k$-patterns}, which are a little controversial in
+ the functional programming world. Here is the initial fibonacci
+ example with $n+k$-patterns:
+*}
+
+function fib2 :: "nat \<Rightarrow> nat"
+where
+ "fib2 0 = 1"
+| "fib2 1 = 1"
+| "fib2 (n + 2) = fib2 n + fib2 (Suc n)"
+
+(*<*)ML_val "goals_limit := 1"(*>*)
+txt {*
+ This kind of matching is again justified by the proof of pattern
+ completeness and compatibility.
+ The proof obligation for pattern completeness states that every natural number is
+ either @{term "0::nat"}, @{term "1::nat"} or @{term "n +
+ (2::nat)"}:
+
+ @{subgoals[display,indent=0]}
+
+ This is an arithmetic triviality, but unfortunately the
+ @{text arith} method cannot handle this specific form of an
+ elimination rule. However, we can use the method @{text
+ "atomize_elim"} to do an ad-hoc conversion to a disjunction of
+ existentials, which can then be solved by the arithmetic decision procedure.
+ Pattern compatibility and termination are automatic as usual.
+*}
+(*<*)ML_val "goals_limit := 10"(*>*)
+apply atomize_elim
+apply arith
+apply auto
+done
+termination by lexicographic_order
+text {*
+ We can stretch the notion of pattern matching even more. The
+ following function is not a sensible functional program, but a
+ perfectly valid mathematical definition:
+*}
+
+function ev :: "nat \<Rightarrow> bool"
+where
+ "ev (2 * n) = True"
+| "ev (2 * n + 1) = False"
+apply atomize_elim
+by arith+
+termination by (relation "{}") simp
+
+text {*
+ This general notion of pattern matching gives you a certain freedom
+ in writing down specifications. However, as always, such freedom should
+ be used with care:
+
+ If we leave the area of constructor
+ patterns, we have effectively departed from the world of functional
+ programming. This means that it is no longer possible to use the
+ code generator, and expect it to generate ML code for our
+ definitions. Also, such a specification might not work very well together with
+ simplification. Your mileage may vary.
+*}
+
+
+subsection {* Conditional equations *}
+
+text {*
+ The function package also supports conditional equations, which are
+ similar to guards in a language like Haskell. Here is Euclid's
+ algorithm written with conditional patterns\footnote{Note that the
+ patterns are also overlapping in the base case}:
+*}
+
+function gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+ "gcd x 0 = x"
+| "gcd 0 y = y"
+| "x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (Suc x) (y - x)"
+| "\<not> x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (x - y) (Suc y)"
+by (atomize_elim, auto, arith)
+termination by lexicographic_order
+
+text {*
+ By now, you can probably guess what the proof obligations for the
+ pattern completeness and compatibility look like.
+
+ Again, functions with conditional patterns are not supported by the
+ code generator.
+*}
+
+
+subsection {* Pattern matching on strings *}
+
+text {*
+ As strings (as lists of characters) are normal datatypes, pattern
+ matching on them is possible, but somewhat problematic. Consider the
+ following definition:
+
+\end{isamarkuptext}
+\noindent\cmd{fun} @{text "check :: \"string \<Rightarrow> bool\""}\\%
+\cmd{where}\\%
+\hspace*{2ex}@{text "\"check (''good'') = True\""}\\%
+@{text "| \"check s = False\""}
+\begin{isamarkuptext}
+
+ \noindent An invocation of the above \cmd{fun} command does not
+ terminate. What is the problem? Strings are lists of characters, and
+ characters are a datatype with a lot of constructors. Splitting the
+ catch-all pattern thus leads to an explosion of cases, which cannot
+ be handled by Isabelle.
+
+ There are two things we can do here. Either we write an explicit
+ @{text "if"} on the right hand side, or we can use conditional patterns:
+*}
+
+function check :: "string \<Rightarrow> bool"
+where
+ "check (''good'') = True"
+| "s \<noteq> ''good'' \<Longrightarrow> check s = False"
+by auto
+
+
+section {* Partiality *}
+
+text {*
+ In HOL, all functions are total. A function @{term "f"} applied to
+ @{term "x"} always has the value @{term "f x"}, and there is no notion
+ of undefinedness.
+ This is why we have to do termination
+ proofs when defining functions: The proof justifies that the
+ function can be defined by wellfounded recursion.
+
+ However, the \cmd{function} package does support partiality to a
+ certain extent. Let's look at the following function which looks
+ for a zero of a given function f.
+*}
+
+function (*<*)(domintros, tailrec)(*>*)findzero :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"
+where
+ "findzero f n = (if f n = 0 then n else findzero f (Suc n))"
+by pat_completeness auto
+(*<*)declare findzero.simps[simp del](*>*)
+
+text {*
+ \noindent Clearly, any attempt of a termination proof must fail. And without
+ that, we do not get the usual rules @{text "findzero.simps"} and
+ @{text "findzero.induct"}. So what was the definition good for at all?
+*}
+
+subsection {* Domain predicates *}
+
+text {*
+ The trick is that Isabelle has not only defined the function @{const findzero}, but also
+ a predicate @{term "findzero_dom"} that characterizes the values where the function
+ terminates: the \emph{domain} of the function. If we treat a
+ partial function just as a total function with an additional domain
+ predicate, we can derive simplification and
+ induction rules as we do for total functions. They are guarded
+ by domain conditions and are called @{text psimps} and @{text
+ pinduct}:
+*}
+
+text {*
+ \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.psimps}\end{minipage}
+ \hfill(@{text "findzero.psimps"})
+ \vspace{1em}
+
+ \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.pinduct}\end{minipage}
+ \hfill(@{text "findzero.pinduct"})
+*}
+
+text {*
+ Remember that all we
+ are doing here is use some tricks to make a total function appear
+ as if it was partial. We can still write the term @{term "findzero
+ (\<lambda>x. 1) 0"} and like any other term of type @{typ nat} it is equal
+ to some natural number, although we might not be able to find out
+ which one. The function is \emph{underdefined}.
+
+ But it is defined enough to prove something interesting about it. We
+ can prove that if @{term "findzero f n"}
+ terminates, it indeed returns a zero of @{term f}:
+*}
+
+lemma findzero_zero: "findzero_dom (f, n) \<Longrightarrow> f (findzero f n) = 0"
+
+txt {* \noindent We apply induction as usual, but using the partial induction
+ rule: *}
+
+apply (induct f n rule: findzero.pinduct)
+
+txt {* \noindent This gives the following subgoals:
+
+ @{subgoals[display,indent=0]}
+
+ \noindent The hypothesis in our lemma was used to satisfy the first premise in
+ the induction rule. However, we also get @{term
+ "findzero_dom (f, n)"} as a local assumption in the induction step. This
+ allows to unfold @{term "findzero f n"} using the @{text psimps}
+ rule, and the rest is trivial. Since the @{text psimps} rules carry the
+ @{text "[simp]"} attribute by default, we just need a single step:
+ *}
+apply simp
+done
+
+text {*
+ Proofs about partial functions are often not harder than for total
+ functions. Fig.~\ref{findzero_isar} shows a slightly more
+ complicated proof written in Isar. It is verbose enough to show how
+ partiality comes into play: From the partial induction, we get an
+ additional domain condition hypothesis. Observe how this condition
+ is applied when calls to @{term findzero} are unfolded.
+*}
+
+text_raw {*
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+*}
+lemma "\<lbrakk>findzero_dom (f, n); x \<in> {n ..< findzero f n}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
+proof (induct rule: findzero.pinduct)
+ fix f n assume dom: "findzero_dom (f, n)"
+ and IH: "\<lbrakk>f n \<noteq> 0; x \<in> {Suc n ..< findzero f (Suc n)}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
+ and x_range: "x \<in> {n ..< findzero f n}"
+ have "f n \<noteq> 0"
+ proof
+ assume "f n = 0"
+ with dom have "findzero f n = n" by simp
+ with x_range show False by auto
+ qed
+
+ from x_range have "x = n \<or> x \<in> {Suc n ..< findzero f n}" by auto
+ thus "f x \<noteq> 0"
+ proof
+ assume "x = n"
+ with `f n \<noteq> 0` show ?thesis by simp
+ next
+ assume "x \<in> {Suc n ..< findzero f n}"
+ with dom and `f n \<noteq> 0` have "x \<in> {Suc n ..< findzero f (Suc n)}" by simp
+ with IH and `f n \<noteq> 0`
+ show ?thesis by simp
+ qed
+qed
+text_raw {*
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}\vspace{6pt}\hrule
+\caption{A proof about a partial function}\label{findzero_isar}
+\end{figure}
+*}
+
+subsection {* Partial termination proofs *}
+
+text {*
+ Now that we have proved some interesting properties about our
+ function, we should turn to the domain predicate and see if it is
+ actually true for some values. Otherwise we would have just proved
+ lemmas with @{term False} as a premise.
+
+ Essentially, we need some introduction rules for @{text
+ findzero_dom}. The function package can prove such domain
+ introduction rules automatically. But since they are not used very
+ often (they are almost never needed if the function is total), this
+ functionality is disabled by default for efficiency reasons. So we have to go
+ back and ask for them explicitly by passing the @{text
+ "(domintros)"} option to the function package:
+
+\vspace{1ex}
+\noindent\cmd{function} @{text "(domintros) findzero :: \"(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat\""}\\%
+\cmd{where}\isanewline%
+\ \ \ldots\\
+
+ \noindent Now the package has proved an introduction rule for @{text findzero_dom}:
+*}
+
+thm findzero.domintros
+
+text {*
+ @{thm[display] findzero.domintros}
+
+ Domain introduction rules allow to show that a given value lies in the
+ domain of a function, if the arguments of all recursive calls
+ are in the domain as well. They allow to do a \qt{single step} in a
+ termination proof. Usually, you want to combine them with a suitable
+ induction principle.
+
+ Since our function increases its argument at recursive calls, we
+ need an induction principle which works \qt{backwards}. We will use
+ @{text inc_induct}, which allows to do induction from a fixed number
+ \qt{downwards}:
+
+ \begin{center}@{thm inc_induct}\hfill(@{text "inc_induct"})\end{center}
+
+ Figure \ref{findzero_term} gives a detailed Isar proof of the fact
+ that @{text findzero} terminates if there is a zero which is greater
+ or equal to @{term n}. First we derive two useful rules which will
+ solve the base case and the step case of the induction. The
+ induction is then straightforward, except for the unusual induction
+ principle.
+
+*}
+
+text_raw {*
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+*}
+lemma findzero_termination:
+ assumes "x \<ge> n" and "f x = 0"
+ shows "findzero_dom (f, n)"
+proof -
+ have base: "findzero_dom (f, x)"
+ by (rule findzero.domintros) (simp add:`f x = 0`)
+
+ have step: "\<And>i. findzero_dom (f, Suc i)
+ \<Longrightarrow> findzero_dom (f, i)"
+ by (rule findzero.domintros) simp
+
+ from `x \<ge> n` show ?thesis
+ proof (induct rule:inc_induct)
+ show "findzero_dom (f, x)" by (rule base)
+ next
+ fix i assume "findzero_dom (f, Suc i)"
+ thus "findzero_dom (f, i)" by (rule step)
+ qed
+qed
+text_raw {*
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}\vspace{6pt}\hrule
+\caption{Termination proof for @{text findzero}}\label{findzero_term}
+\end{figure}
+*}
+
+text {*
+ Again, the proof given in Fig.~\ref{findzero_term} has a lot of
+ detail in order to explain the principles. Using more automation, we
+ can also have a short proof:
+*}
+
+lemma findzero_termination_short:
+ assumes zero: "x >= n"
+ assumes [simp]: "f x = 0"
+ shows "findzero_dom (f, n)"
+using zero
+by (induct rule:inc_induct) (auto intro: findzero.domintros)
+
+text {*
+ \noindent It is simple to combine the partial correctness result with the
+ termination lemma:
+*}
+
+lemma findzero_total_correctness:
+ "f x = 0 \<Longrightarrow> f (findzero f 0) = 0"
+by (blast intro: findzero_zero findzero_termination)
+
+subsection {* Definition of the domain predicate *}
+
+text {*
+ Sometimes it is useful to know what the definition of the domain
+ predicate looks like. Actually, @{text findzero_dom} is just an
+ abbreviation:
+
+ @{abbrev[display] findzero_dom}
+
+ The domain predicate is the \emph{accessible part} of a relation @{const
+ findzero_rel}, which was also created internally by the function
+ package. @{const findzero_rel} is just a normal
+ inductive predicate, so we can inspect its definition by
+ looking at the introduction rules @{text findzero_rel.intros}.
+ In our case there is just a single rule:
+
+ @{thm[display] findzero_rel.intros}
+
+ The predicate @{const findzero_rel}
+ describes the \emph{recursion relation} of the function
+ definition. The recursion relation is a binary relation on
+ the arguments of the function that relates each argument to its
+ recursive calls. In general, there is one introduction rule for each
+ recursive call.
+
+ The predicate @{term "accp findzero_rel"} is the accessible part of
+ that relation. An argument belongs to the accessible part, if it can
+ be reached in a finite number of steps (cf.~its definition in @{text
+ "Wellfounded.thy"}).
+
+ Since the domain predicate is just an abbreviation, you can use
+ lemmas for @{const accp} and @{const findzero_rel} directly. Some
+ lemmas which are occasionally useful are @{text accpI}, @{text
+ accp_downward}, and of course the introduction and elimination rules
+ for the recursion relation @{text "findzero.intros"} and @{text "findzero.cases"}.
+*}
+
+(*lemma findzero_nicer_domintros:
+ "f x = 0 \<Longrightarrow> findzero_dom (f, x)"
+ "findzero_dom (f, Suc x) \<Longrightarrow> findzero_dom (f, x)"
+by (rule accpI, erule findzero_rel.cases, auto)+
+*)
+
+subsection {* A Useful Special Case: Tail recursion *}
+
+text {*
+ The domain predicate is our trick that allows us to model partiality
+ in a world of total functions. The downside of this is that we have
+ to carry it around all the time. The termination proof above allowed
+ us to replace the abstract @{term "findzero_dom (f, n)"} by the more
+ concrete @{term "(x \<ge> n \<and> f x = (0::nat))"}, but the condition is still
+ there and can only be discharged for special cases.
+ In particular, the domain predicate guards the unfolding of our
+ function, since it is there as a condition in the @{text psimp}
+ rules.
+
+ Now there is an important special case: We can actually get rid
+ of the condition in the simplification rules, \emph{if the function
+ is tail-recursive}. The reason is that for all tail-recursive
+ equations there is a total function satisfying them, even if they
+ are non-terminating.
+
+% A function is tail recursive, if each call to the function is either
+% equal
+%
+% So the outer form of the
+%
+%if it can be written in the following
+% form:
+% {term[display] "f x = (if COND x then BASE x else f (LOOP x))"}
+
+
+ The function package internally does the right construction and can
+ derive the unconditional simp rules, if we ask it to do so. Luckily,
+ our @{const "findzero"} function is tail-recursive, so we can just go
+ back and add another option to the \cmd{function} command:
+
+\vspace{1ex}
+\noindent\cmd{function} @{text "(domintros, tailrec) findzero :: \"(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat\""}\\%
+\cmd{where}\isanewline%
+\ \ \ldots\\%
+
+
+ \noindent Now, we actually get unconditional simplification rules, even
+ though the function is partial:
+*}
+
+thm findzero.simps
+
+text {*
+ @{thm[display] findzero.simps}
+
+ \noindent Of course these would make the simplifier loop, so we better remove
+ them from the simpset:
+*}
+
+declare findzero.simps[simp del]
+
+text {*
+ Getting rid of the domain conditions in the simplification rules is
+ not only useful because it simplifies proofs. It is also required in
+ order to use Isabelle's code generator to generate ML code
+ from a function definition.
+ Since the code generator only works with equations, it cannot be
+ used with @{text "psimp"} rules. Thus, in order to generate code for
+ partial functions, they must be defined as a tail recursion.
+ Luckily, many functions have a relatively natural tail recursive
+ definition.
+*}
+
+section {* Nested recursion *}
+
+text {*
+ Recursive calls which are nested in one another frequently cause
+ complications, since their termination proof can depend on a partial
+ correctness property of the function itself.
+
+ As a small example, we define the \qt{nested zero} function:
+*}
+
+function nz :: "nat \<Rightarrow> nat"
+where
+ "nz 0 = 0"
+| "nz (Suc n) = nz (nz n)"
+by pat_completeness auto
+
+text {*
+ If we attempt to prove termination using the identity measure on
+ naturals, this fails:
+*}
+
+termination
+ apply (relation "measure (\<lambda>n. n)")
+ apply auto
+
+txt {*
+ We get stuck with the subgoal
+
+ @{subgoals[display]}
+
+ Of course this statement is true, since we know that @{const nz} is
+ the zero function. And in fact we have no problem proving this
+ property by induction.
+*}
+(*<*)oops(*>*)
+lemma nz_is_zero: "nz_dom n \<Longrightarrow> nz n = 0"
+ by (induct rule:nz.pinduct) auto
+
+text {*
+ We formulate this as a partial correctness lemma with the condition
+ @{term "nz_dom n"}. This allows us to prove it with the @{text
+ pinduct} rule before we have proved termination. With this lemma,
+ the termination proof works as expected:
+*}
+
+termination
+ by (relation "measure (\<lambda>n. n)") (auto simp: nz_is_zero)
+
+text {*
+ As a general strategy, one should prove the statements needed for
+ termination as a partial property first. Then they can be used to do
+ the termination proof. This also works for less trivial
+ examples. Figure \ref{f91} defines the 91-function, a well-known
+ challenge problem due to John McCarthy, and proves its termination.
+*}
+
+text_raw {*
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+*}
+
+function f91 :: "nat \<Rightarrow> nat"
+where
+ "f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))"
+by pat_completeness auto
+
+lemma f91_estimate:
+ assumes trm: "f91_dom n"
+ shows "n < f91 n + 11"
+using trm by induct auto
+
+termination
+proof
+ let ?R = "measure (\<lambda>x. 101 - x)"
+ show "wf ?R" ..
+
+ fix n :: nat assume "\<not> 100 < n" -- "Assumptions for both calls"
+
+ thus "(n + 11, n) \<in> ?R" by simp -- "Inner call"
+
+ assume inner_trm: "f91_dom (n + 11)" -- "Outer call"
+ with f91_estimate have "n + 11 < f91 (n + 11) + 11" .
+ with `\<not> 100 < n` show "(f91 (n + 11), n) \<in> ?R" by simp
+qed
+
+text_raw {*
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}
+\vspace{6pt}\hrule
+\caption{McCarthy's 91-function}\label{f91}
+\end{figure}
+*}
+
+
+section {* Higher-Order Recursion *}
+
+text {*
+ Higher-order recursion occurs when recursive calls
+ are passed as arguments to higher-order combinators such as @{const
+ map}, @{term filter} etc.
+ As an example, imagine a datatype of n-ary trees:
+*}
+
+datatype 'a tree =
+ Leaf 'a
+| Branch "'a tree list"
+
+
+text {* \noindent We can define a function which swaps the left and right subtrees recursively, using the
+ list functions @{const rev} and @{const map}: *}
+
+fun mirror :: "'a tree \<Rightarrow> 'a tree"
+where
+ "mirror (Leaf n) = Leaf n"
+| "mirror (Branch l) = Branch (rev (map mirror l))"
+
+text {*
+ Although the definition is accepted without problems, let us look at the termination proof:
+*}
+
+termination proof
+ txt {*
+
+ As usual, we have to give a wellfounded relation, such that the
+ arguments of the recursive calls get smaller. But what exactly are
+ the arguments of the recursive calls when mirror is given as an
+ argument to @{const map}? Isabelle gives us the
+ subgoals
+
+ @{subgoals[display,indent=0]}
+
+ So the system seems to know that @{const map} only
+ applies the recursive call @{term "mirror"} to elements
+ of @{term "l"}, which is essential for the termination proof.
+
+ This knowledge about @{const map} is encoded in so-called congruence rules,
+ which are special theorems known to the \cmd{function} command. The
+ rule for @{const map} is
+
+ @{thm[display] map_cong}
+
+ You can read this in the following way: Two applications of @{const
+ map} are equal, if the list arguments are equal and the functions
+ coincide on the elements of the list. This means that for the value
+ @{term "map f l"} we only have to know how @{term f} behaves on
+ the elements of @{term l}.
+
+ Usually, one such congruence rule is
+ needed for each higher-order construct that is used when defining
+ new functions. In fact, even basic functions like @{const
+ If} and @{const Let} are handled by this mechanism. The congruence
+ rule for @{const If} states that the @{text then} branch is only
+ relevant if the condition is true, and the @{text else} branch only if it
+ is false:
+
+ @{thm[display] if_cong}
+
+ Congruence rules can be added to the
+ function package by giving them the @{term fundef_cong} attribute.
+
+ The constructs that are predefined in Isabelle, usually
+ come with the respective congruence rules.
+ But if you define your own higher-order functions, you may have to
+ state and prove the required congruence rules yourself, if you want to use your
+ functions in recursive definitions.
+*}
+(*<*)oops(*>*)
+
+subsection {* Congruence Rules and Evaluation Order *}
+
+text {*
+ Higher order logic differs from functional programming languages in
+ that it has no built-in notion of evaluation order. A program is
+ just a set of equations, and it is not specified how they must be
+ evaluated.
+
+ However for the purpose of function definition, we must talk about
+ evaluation order implicitly, when we reason about termination.
+ Congruence rules express that a certain evaluation order is
+ consistent with the logical definition.
+
+ Consider the following function.
+*}
+
+function f :: "nat \<Rightarrow> bool"
+where
+ "f n = (n = 0 \<or> f (n - 1))"
+(*<*)by pat_completeness auto(*>*)
+
+text {*
+ For this definition, the termination proof fails. The default configuration
+ specifies no congruence rule for disjunction. We have to add a
+ congruence rule that specifies left-to-right evaluation order:
+
+ \vspace{1ex}
+ \noindent @{thm disj_cong}\hfill(@{text "disj_cong"})
+ \vspace{1ex}
+
+ Now the definition works without problems. Note how the termination
+ proof depends on the extra condition that we get from the congruence
+ rule.
+
+ However, as evaluation is not a hard-wired concept, we
+ could just turn everything around by declaring a different
+ congruence rule. Then we can make the reverse definition:
+*}
+
+lemma disj_cong2[fundef_cong]:
+ "(\<not> Q' \<Longrightarrow> P = P') \<Longrightarrow> (Q = Q') \<Longrightarrow> (P \<or> Q) = (P' \<or> Q')"
+ by blast
+
+fun f' :: "nat \<Rightarrow> bool"
+where
+ "f' n = (f' (n - 1) \<or> n = 0)"
+
+text {*
+ \noindent These examples show that, in general, there is no \qt{best} set of
+ congruence rules.
+
+ However, such tweaking should rarely be necessary in
+ practice, as most of the time, the default set of congruence rules
+ works well.
+*}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/Thy/ROOT.ML Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,4 @@
+
+(* $Id$ *)
+
+use_thy "Functions";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/Thy/document/Functions.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,1985 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Functions}%
+%
+\isadelimtheory
+\isanewline
+\isanewline
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{theory}\isamarkupfalse%
+\ Functions\isanewline
+\isakeyword{imports}\ Main\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsection{Function Definitions for Dummies%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In most cases, defining a recursive function is just as simple as other definitions:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ fib\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}fib\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ fib\ n\ {\isacharplus}\ fib\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+The syntax is rather self-explanatory: We introduce a function by
+ giving its name, its type,
+ and a set of defining recursive equations.
+ If we leave out the type, the most general type will be
+ inferred, which can sometimes lead to surprises: Since both \isa{{\isadigit{1}}} and \isa{{\isacharplus}} are overloaded, we would end up
+ with \isa{fib\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ {\isacharprime}a{\isacharcolon}{\isacharcolon}{\isacharbraceleft}one{\isacharcomma}plus{\isacharbraceright}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The function always terminates, since its argument gets smaller in
+ every recursive call.
+ Since HOL is a logic of total functions, termination is a
+ fundamental requirement to prevent inconsistencies\footnote{From the
+ \qt{definition} \isa{f{\isacharparenleft}n{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}n{\isacharparenright}\ {\isacharplus}\ {\isadigit{1}}} we could prove
+ \isa{{\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}} by subtracting \isa{f{\isacharparenleft}n{\isacharparenright}} on both sides.}.
+ Isabelle tries to prove termination automatically when a definition
+ is made. In \S\ref{termination}, we will look at cases where this
+ fails and see what to do then.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Pattern matching%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{patmatch}
+ Like in functional programming, we can use pattern matching to
+ define functions. At the moment we will only consider \emph{constructor
+ patterns}, which only consist of datatype constructors and
+ variables. Furthermore, patterns must be linear, i.e.\ all variables
+ on the left hand side of an equation must be distinct. In
+ \S\ref{genpats} we discuss more general pattern matching.
+
+ If patterns overlap, the order of the equations is taken into
+ account. The following function inserts a fixed element between any
+ two elements of a list:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ sep\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}sep\ a\ {\isacharparenleft}x{\isacharhash}y{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ x\ {\isacharhash}\ a\ {\isacharhash}\ sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}sep\ a\ xs\ \ \ \ \ \ \ {\isacharequal}\ xs{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+Overlapping patterns are interpreted as \qt{increments} to what is
+ already there: The second equation is only meant for the cases where
+ the first one does not match. Consequently, Isabelle replaces it
+ internally by the remaining cases, making the patterns disjoint:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{thm}\isamarkupfalse%
+\ sep{\isachardot}simps%
+\begin{isamarkuptext}%
+\begin{isabelle}%
+sep\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}\ {\isacharequal}\ x\ {\isacharhash}\ a\ {\isacharhash}\ sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}\isasep\isanewline%
+sep\ a\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\isasep\isanewline%
+sep\ a\ {\isacharbrackleft}v{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}v{\isacharbrackright}%
+\end{isabelle}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent The equations from function definitions are automatically used in
+ simplification:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isachardoublequoteopen}sep\ {\isadigit{0}}\ {\isacharbrackleft}{\isadigit{1}}{\isacharcomma}\ {\isadigit{2}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}{\isadigit{1}}{\isacharcomma}\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{2}}{\isacharcomma}\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsubsection{Induction%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle provides customized induction rules for recursive
+ functions. These rules follow the recursive structure of the
+ definition. Here is the rule \isa{sep{\isachardot}induct} arising from the
+ above definition of \isa{sep}:
+
+ \begin{isabelle}%
+{\isasymlbrakk}{\isasymAnd}a\ x\ y\ xs{\isachardot}\ {\isacharquery}P\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharquery}P\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharsemicolon}\ {\isasymAnd}a{\isachardot}\ {\isacharquery}P\ a\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\ {\isasymAnd}a\ v{\isachardot}\ {\isacharquery}P\ a\ {\isacharbrackleft}v{\isacharbrackright}{\isasymrbrakk}\isanewline
+{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}%
+\end{isabelle}
+
+ We have a step case for list with at least two elements, and two
+ base cases for the zero- and the one-element list. Here is a simple
+ proof about \isa{sep} and \isa{map}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isachardoublequoteopen}map\ f\ {\isacharparenleft}sep\ x\ ys{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}map\ f\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}induct\ x\ ys\ rule{\isacharcolon}\ sep{\isachardot}induct{\isacharparenright}%
+\begin{isamarkuptxt}%
+We get three cases, like in the definition.
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ x\ y\ xs{\isachardot}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }map\ f\ {\isacharparenleft}sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }map\ f\ {\isacharparenleft}sep\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}a{\isachardot}\ map\ f\ {\isacharparenleft}sep\ a\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
+\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}a\ v{\isachardot}\ map\ f\ {\isacharparenleft}sep\ a\ {\isacharbrackleft}v{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharbrackleft}v{\isacharbrackright}{\isacharparenright}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ auto\ \isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+With the \cmd{fun} command, you can define about 80\% of the
+ functions that occur in practice. The rest of this tutorial explains
+ the remaining 20\%.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{fun vs.\ function%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The \cmd{fun} command provides a
+ convenient shorthand notation for simple function definitions. In
+ this mode, Isabelle tries to solve all the necessary proof obligations
+ automatically. If any proof fails, the definition is
+ rejected. This can either mean that the definition is indeed faulty,
+ or that the default proof procedures are just not smart enough (or
+ rather: not designed) to handle the definition.
+
+ By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
+ solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
+
+\end{isamarkuptext}
+
+
+\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
+\cmd{fun} \isa{f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}\\%
+\cmd{where}\\%
+\hspace*{2ex}{\it equations}\\%
+\hspace*{2ex}\vdots\vspace*{6pt}
+\end{minipage}\right]
+\quad\equiv\quad
+\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
+\cmd{function} \isa{{\isacharparenleft}}\cmd{sequential}\isa{{\isacharparenright}\ f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}\\%
+\cmd{where}\\%
+\hspace*{2ex}{\it equations}\\%
+\hspace*{2ex}\vdots\\%
+\cmd{by} \isa{pat{\isacharunderscore}completeness\ auto}\\%
+\cmd{termination by} \isa{lexicographic{\isacharunderscore}order}\vspace{6pt}
+\end{minipage}
+\right]\]
+
+\begin{isamarkuptext}
+ \vspace*{1em}
+ \noindent Some details have now become explicit:
+
+ \begin{enumerate}
+ \item The \cmd{sequential} option enables the preprocessing of
+ pattern overlaps which we already saw. Without this option, the equations
+ must already be disjoint and complete. The automatic completion only
+ works with constructor patterns.
+
+ \item A function definition produces a proof obligation which
+ expresses completeness and compatibility of patterns (we talk about
+ this later). The combination of the methods \isa{pat{\isacharunderscore}completeness} and
+ \isa{auto} is used to solve this proof obligation.
+
+ \item A termination proof follows the definition, started by the
+ \cmd{termination} command. This will be explained in \S\ref{termination}.
+ \end{enumerate}
+ Whenever a \cmd{fun} command fails, it is usually a good idea to
+ expand the syntax to the more verbose \cmd{function} form, to see
+ what is actually going on.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Termination%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{termination}
+ The method \isa{lexicographic{\isacharunderscore}order} is the default method for
+ termination proofs. It can prove termination of a
+ certain class of functions by searching for a suitable lexicographic
+ combination of size measures. Of course, not all functions have such
+ a simple termination argument. For them, we can specify the termination
+ relation manually.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{The {\tt relation} method%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Consider the following function, which sums up natural numbers up to
+ \isa{N}, using a counter \isa{i}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ sum\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}sum\ i\ N\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isachargreater}\ N\ then\ {\isadigit{0}}\ else\ i\ {\isacharplus}\ sum\ {\isacharparenleft}Suc\ i{\isacharparenright}\ N{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ pat{\isacharunderscore}completeness\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent The \isa{lexicographic{\isacharunderscore}order} method fails on this example, because none of the
+ arguments decreases in the recursive call, with respect to the standard size ordering.
+ To prove termination manually, we must provide a custom wellfounded relation.
+
+ The termination argument for \isa{sum} is based on the fact that
+ the \emph{difference} between \isa{i} and \isa{N} gets
+ smaller in every step, and that the recursion stops when \isa{i}
+ is greater than \isa{N}. Phrased differently, the expression
+ \isa{N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i} always decreases.
+
+ We can use this expression as a measure function suitable to prove termination.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{termination}\isamarkupfalse%
+\ sum\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}%
+\begin{isamarkuptxt}%
+The \cmd{termination} command sets up the termination goal for the
+ specified function \isa{sum}. If the function name is omitted, it
+ implicitly refers to the last function definition.
+
+ The \isa{relation} method takes a relation of
+ type \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}, where \isa{{\isacharprime}a} is the argument type of
+ the function. If the function has multiple curried arguments, then
+ these are packed together into a tuple, as it happened in the above
+ example.
+
+ The predefined function \isa{{\isachardoublequote}measure\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set{\isachardoublequote}} constructs a
+ wellfounded relation from a mapping into the natural numbers (a
+ \emph{measure function}).
+
+ After the invocation of \isa{relation}, we must prove that (a)
+ the relation we supplied is wellfounded, and (b) that the arguments
+ of recursive calls indeed decrease with respect to the
+ relation:
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ wf\ {\isacharparenleft}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}i\ N{\isachardot}\ {\isasymnot}\ N\ {\isacharless}\ i\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharparenleft}Suc\ i{\isacharcomma}\ N{\isacharparenright}{\isacharcomma}\ i{\isacharcomma}\ N{\isacharparenright}\ {\isasymin}\ measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}%
+\end{isabelle}
+
+ These goals are all solved by \isa{auto}:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ auto\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Let us complicate the function a little, by adding some more
+ recursive calls:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ foo\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}foo\ i\ N\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isachargreater}\ N\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ then\ {\isacharparenleft}if\ N\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ foo\ {\isadigit{0}}\ {\isacharparenleft}N\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ i\ {\isacharplus}\ foo\ {\isacharparenleft}Suc\ i{\isacharparenright}\ N{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ pat{\isacharunderscore}completeness\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+When \isa{i} has reached \isa{N}, it starts at zero again
+ and \isa{N} is decremented.
+ This corresponds to a nested
+ loop where one index counts up and the other down. Termination can
+ be proved using a lexicographic combination of two measures, namely
+ the value of \isa{N} and the above difference. The \isa{measures} combinator generalizes \isa{measure} by taking a
+ list of measure functions.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{termination}\isamarkupfalse%
+\ \isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measures\ {\isacharbrackleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N{\isacharcomma}\ {\isasymlambda}{\isacharparenleft}i{\isacharcomma}N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharbrackright}{\isachardoublequoteclose}{\isacharparenright}\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsubsection{How \isa{lexicographic{\isacharunderscore}order} works%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+To see how the automatic termination proofs work, let's look at an
+ example where it fails\footnote{For a detailed discussion of the
+ termination prover, see \cite{bulwahnKN07}}:
+
+\end{isamarkuptext}
+\cmd{fun} \isa{fails\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ {\isasymRightarrow}\ nat\ list\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
+\cmd{where}\\%
+\hspace*{2ex}\isa{{\isachardoublequote}fails\ a\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ a{\isachardoublequote}}\\%
+|\hspace*{1.5ex}\isa{{\isachardoublequote}fails\ a\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ fails\ {\isacharparenleft}x\ {\isacharplus}\ a{\isacharparenright}\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}{\isachardoublequote}}\\
+\begin{isamarkuptext}
+
+\noindent Isabelle responds with the following error:
+
+\begin{isabelle}
+*** Unfinished subgoals:\newline
+*** (a, 1, <):\newline
+*** \ 1.~\isa{{\isasymAnd}x{\isachardot}\ x\ {\isacharequal}\ {\isadigit{0}}}\newline
+*** (a, 1, <=):\newline
+*** \ 1.~False\newline
+*** (a, 2, <):\newline
+*** \ 1.~False\newline
+*** Calls:\newline
+*** a) \isa{{\isacharparenleft}a{\isacharcomma}\ x\ {\isacharhash}\ xs{\isacharparenright}\ {\isacharminus}{\isacharminus}{\isachargreater}{\isachargreater}\ {\isacharparenleft}x\ {\isacharplus}\ a{\isacharcomma}\ x\ {\isacharhash}\ xs{\isacharparenright}}\newline
+*** Measures:\newline
+*** 1) \isa{{\isasymlambda}x{\isachardot}\ size\ {\isacharparenleft}fst\ x{\isacharparenright}}\newline
+*** 2) \isa{{\isasymlambda}x{\isachardot}\ size\ {\isacharparenleft}snd\ x{\isacharparenright}}\newline
+*** Result matrix:\newline
+*** \ \ \ \ 1\ \ 2 \newline
+*** a: ? <= \newline
+*** Could not find lexicographic termination order.\newline
+*** At command "fun".\newline
+\end{isabelle}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The key to this error message is the matrix at the bottom. The rows
+ of that matrix correspond to the different recursive calls (In our
+ case, there is just one). The columns are the function's arguments
+ (expressed through different measure functions, which map the
+ argument tuple to a natural number).
+
+ The contents of the matrix summarize what is known about argument
+ descents: The second argument has a weak descent (\isa{{\isacharless}{\isacharequal}}) at the
+ recursive call, and for the first argument nothing could be proved,
+ which is expressed by \isa{{\isacharquery}}. In general, there are the values
+ \isa{{\isacharless}}, \isa{{\isacharless}{\isacharequal}} and \isa{{\isacharquery}}.
+
+ For the failed proof attempts, the unfinished subgoals are also
+ printed. Looking at these will often point to a missing lemma.
+
+% As a more real example, here is quicksort:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Mutual Recursion%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+If two or more functions call one another mutually, they have to be defined
+ in one step. Here are \isa{even} and \isa{odd}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ even\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ odd\ \ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}even\ {\isadigit{0}}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}odd\ {\isadigit{0}}\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ odd\ n{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}odd\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ even\ n{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ pat{\isacharunderscore}completeness\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+To eliminate the mutual dependencies, Isabelle internally
+ creates a single function operating on the sum
+ type \isa{nat\ {\isacharplus}\ nat}. Then, \isa{even} and \isa{odd} are
+ defined as projections. Consequently, termination has to be proved
+ simultaneously for both functions, by specifying a measure on the
+ sum type:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{termination}\isamarkupfalse%
+\ \isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ case\ x\ of\ Inl\ n\ {\isasymRightarrow}\ n\ {\isacharbar}\ Inr\ n\ {\isasymRightarrow}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+We could also have used \isa{lexicographic{\isacharunderscore}order}, which
+ supports mutual recursive termination proofs to a certain extent.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Induction for mutual recursion%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+When functions are mutually recursive, proving properties about them
+ generally requires simultaneous induction. The induction rule \isa{even{\isacharunderscore}odd{\isachardot}induct}
+ generated from the above definition reflects this.
+
+ Let us prove something about \isa{even} and \isa{odd}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ even{\isacharunderscore}odd{\isacharunderscore}mod{\isadigit{2}}{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+We apply simultaneous induction, specifying the induction variable
+ for both goals, separated by \cmd{and}:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}induct\ n\ \isakeyword{and}\ n\ rule{\isacharcolon}\ even{\isacharunderscore}odd{\isachardot}induct{\isacharparenright}%
+\begin{isamarkuptxt}%
+We get four subgoals, which correspond to the clauses in the
+ definition of \isa{even} and \isa{odd}:
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ even\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ odd\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}\isanewline
+\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}\ {\isasymLongrightarrow}\ even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
+\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ odd\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}%
+\end{isabelle}
+ Simplification solves the first two goals, leaving us with two
+ statements about the \isa{mod} operation to prove:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ simp{\isacharunderscore}all%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}%
+\end{isabelle}
+
+ \noindent These can be handled by Isabelle's arithmetic decision procedures.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ arith\isanewline
+\isacommand{apply}\isamarkupfalse%
+\ arith\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+In proofs like this, the simultaneous induction is really essential:
+ Even if we are just interested in one of the results, the other
+ one is necessary to strengthen the induction hypothesis. If we leave
+ out the statement about \isa{odd} and just write \isa{True} instead,
+ the same proof fails:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ failed{\isacharunderscore}attempt{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ {\isachardoublequoteopen}True{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}induct\ n\ rule{\isacharcolon}\ even{\isacharunderscore}odd{\isachardot}induct{\isacharparenright}%
+\begin{isamarkuptxt}%
+\noindent Now the third subgoal is a dead end, since we have no
+ useful induction hypothesis available:
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ even\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
+\ {\isadigit{2}}{\isachardot}\ True\isanewline
+\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ True\ {\isasymLongrightarrow}\ even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
+\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ True%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{oops}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsection{General pattern matching%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{genpats}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Avoiding automatic pattern splitting%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Up to now, we used pattern matching only on datatypes, and the
+ patterns were always disjoint and complete, and if they weren't,
+ they were made disjoint automatically like in the definition of
+ \isa{sep} in \S\ref{patmatch}.
+
+ This automatic splitting can significantly increase the number of
+ equations involved, and this is not always desirable. The following
+ example shows the problem:
+
+ Suppose we are modeling incomplete knowledge about the world by a
+ three-valued datatype, which has values \isa{T}, \isa{F}
+ and \isa{X} for true, false and uncertain propositions, respectively.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ P{\isadigit{3}}\ {\isacharequal}\ T\ {\isacharbar}\ F\ {\isacharbar}\ X%
+\begin{isamarkuptext}%
+\noindent Then the conjunction of such values can be defined as follows:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ And\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}And\ T\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And\ p\ T\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And\ p\ F\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And\ F\ p\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And\ X\ X\ {\isacharequal}\ X{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+This definition is useful, because the equations can directly be used
+ as simplification rules. But the patterns overlap: For example,
+ the expression \isa{And\ T\ T} is matched by both the first and
+ the second equation. By default, Isabelle makes the patterns disjoint by
+ splitting them up, producing instances:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{thm}\isamarkupfalse%
+\ And{\isachardot}simps%
+\begin{isamarkuptext}%
+\isa{And\ T\ {\isacharquery}p\ {\isacharequal}\ {\isacharquery}p\isasep\isanewline%
+And\ F\ T\ {\isacharequal}\ F\isasep\isanewline%
+And\ X\ T\ {\isacharequal}\ X\isasep\isanewline%
+And\ F\ F\ {\isacharequal}\ F\isasep\isanewline%
+And\ X\ F\ {\isacharequal}\ F\isasep\isanewline%
+And\ F\ X\ {\isacharequal}\ F\isasep\isanewline%
+And\ X\ X\ {\isacharequal}\ X}
+
+ \vspace*{1em}
+ \noindent There are several problems with this:
+
+ \begin{enumerate}
+ \item If the datatype has many constructors, there can be an
+ explosion of equations. For \isa{And}, we get seven instead of
+ five equations, which can be tolerated, but this is just a small
+ example.
+
+ \item Since splitting makes the equations \qt{less general}, they
+ do not always match in rewriting. While the term \isa{And\ x\ F}
+ can be simplified to \isa{F} with the original equations, a
+ (manual) case split on \isa{x} is now necessary.
+
+ \item The splitting also concerns the induction rule \isa{And{\isachardot}induct}. Instead of five premises it now has seven, which
+ means that our induction proofs will have more cases.
+
+ \item In general, it increases clarity if we get the same definition
+ back which we put in.
+ \end{enumerate}
+
+ If we do not want the automatic splitting, we can switch it off by
+ leaving out the \cmd{sequential} option. However, we will have to
+ prove that our pattern matching is consistent\footnote{This prevents
+ us from defining something like \isa{f\ x\ {\isacharequal}\ True} and \isa{f\ x\ {\isacharequal}\ False} simultaneously.}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ And{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}And{\isadigit{2}}\ T\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ p\ T\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ p\ F\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ F\ p\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ X\ X\ {\isacharequal}\ X{\isachardoublequoteclose}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+\noindent Now let's look at the proof obligations generated by a
+ function definition. In this case, they are:
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ {\isasymlbrakk}{\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ \ }{\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ p{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ x\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ }{\isasymLongrightarrow}\ P\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
+\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
+\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
+\ {\isadigit{5}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
+\ {\isadigit{6}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ X\isanewline
+\ {\isadigit{7}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
+\ {\isadigit{8}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
+\ {\isadigit{9}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
+\ {\isadigit{1}}{\isadigit{0}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ X%
+\end{isabelle}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
+
+ The first subgoal expresses the completeness of the patterns. It has
+ the form of an elimination rule and states that every \isa{x} of
+ the function's input type must match at least one of the patterns\footnote{Completeness could
+ be equivalently stated as a disjunction of existential statements:
+\isa{{\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ F{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ p{\isacharparenright}{\isacharparenright}\ {\isasymor}\ x\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}}, and you can use the method \isa{atomize{\isacharunderscore}elim} to get that form instead.}. If the patterns just involve
+ datatypes, we can solve it with the \isa{pat{\isacharunderscore}completeness}
+ method:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ pat{\isacharunderscore}completeness%
+\begin{isamarkuptxt}%
+The remaining subgoals express \emph{pattern compatibility}. We do
+ allow that an input value matches multiple patterns, but in this
+ case, the result (i.e.~the right hand sides of the equations) must
+ also be equal. For each pair of two patterns, there is one such
+ subgoal. Usually this needs injectivity of the constructors, which
+ is used automatically by \isa{auto}.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{by}\isamarkupfalse%
+\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsubsection{Non-constructor patterns%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Most of Isabelle's basic types take the form of inductive datatypes,
+ and usually pattern matching works on the constructors of such types.
+ However, this need not be always the case, and the \cmd{function}
+ command handles other kind of patterns, too.
+
+ One well-known instance of non-constructor patterns are
+ so-called \emph{$n+k$-patterns}, which are a little controversial in
+ the functional programming world. Here is the initial fibonacci
+ example with $n+k$-patterns:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ fib{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ fib{\isadigit{2}}\ n\ {\isacharplus}\ fib{\isadigit{2}}\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+This kind of matching is again justified by the proof of pattern
+ completeness and compatibility.
+ The proof obligation for pattern completeness states that every natural number is
+ either \isa{{\isadigit{0}}}, \isa{{\isadigit{1}}} or \isa{n\ {\isacharplus}\ {\isadigit{2}}}:
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ {\isasymlbrakk}x\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ x\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}n{\isachardot}\ x\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
+\ {\isadigit{3}}{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
+\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\isanewline
+\ {\isadigit{5}}{\isachardot}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
+\ {\isadigit{6}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ {\isadigit{1}}\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\isanewline
+\ {\isadigit{7}}{\isachardot}\ {\isasymAnd}n\ na{\isachardot}\isanewline
+\isaindent{\ {\isadigit{7}}{\isachardot}\ \ \ \ }n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ na\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\isanewline
+\isaindent{\ {\isadigit{7}}{\isachardot}\ \ \ \ }fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ na\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ na{\isacharparenright}%
+\end{isabelle}
+
+ This is an arithmetic triviality, but unfortunately the
+ \isa{arith} method cannot handle this specific form of an
+ elimination rule. However, we can use the method \isa{atomize{\isacharunderscore}elim} to do an ad-hoc conversion to a disjunction of
+ existentials, which can then be solved by the arithmetic decision procedure.
+ Pattern compatibility and termination are automatic as usual.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ atomize{\isacharunderscore}elim\isanewline
+\isacommand{apply}\isamarkupfalse%
+\ arith\isanewline
+\isacommand{apply}\isamarkupfalse%
+\ auto\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isanewline
+\isacommand{termination}\isamarkupfalse%
+%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ lexicographic{\isacharunderscore}order%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+We can stretch the notion of pattern matching even more. The
+ following function is not a sensible functional program, but a
+ perfectly valid mathematical definition:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ ev\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}ev\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}ev\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ atomize{\isacharunderscore}elim\isanewline
+\isacommand{by}\isamarkupfalse%
+\ arith{\isacharplus}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{termination}\isamarkupfalse%
+%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}relation\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}{\isacharparenright}\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+This general notion of pattern matching gives you a certain freedom
+ in writing down specifications. However, as always, such freedom should
+ be used with care:
+
+ If we leave the area of constructor
+ patterns, we have effectively departed from the world of functional
+ programming. This means that it is no longer possible to use the
+ code generator, and expect it to generate ML code for our
+ definitions. Also, such a specification might not work very well together with
+ simplification. Your mileage may vary.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Conditional equations%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The function package also supports conditional equations, which are
+ similar to guards in a language like Haskell. Here is Euclid's
+ algorithm written with conditional patterns\footnote{Note that the
+ patterns are also overlapping in the base case}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ gcd\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}gcd\ x\ {\isadigit{0}}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}gcd\ {\isadigit{0}}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}y\ {\isacharminus}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}{\isasymnot}\ x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}x\ {\isacharminus}\ y{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}atomize{\isacharunderscore}elim{\isacharcomma}\ auto{\isacharcomma}\ arith{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{termination}\isamarkupfalse%
+%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ lexicographic{\isacharunderscore}order%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+By now, you can probably guess what the proof obligations for the
+ pattern completeness and compatibility look like.
+
+ Again, functions with conditional patterns are not supported by the
+ code generator.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Pattern matching on strings%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+As strings (as lists of characters) are normal datatypes, pattern
+ matching on them is possible, but somewhat problematic. Consider the
+ following definition:
+
+\end{isamarkuptext}
+\noindent\cmd{fun} \isa{check\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}string\ {\isasymRightarrow}\ bool{\isachardoublequote}}\\%
+\cmd{where}\\%
+\hspace*{2ex}\isa{{\isachardoublequote}check\ {\isacharparenleft}{\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequote}}\\%
+\isa{{\isacharbar}\ {\isachardoublequote}check\ s\ {\isacharequal}\ False{\isachardoublequote}}
+\begin{isamarkuptext}
+
+ \noindent An invocation of the above \cmd{fun} command does not
+ terminate. What is the problem? Strings are lists of characters, and
+ characters are a datatype with a lot of constructors. Splitting the
+ catch-all pattern thus leads to an explosion of cases, which cannot
+ be handled by Isabelle.
+
+ There are two things we can do here. Either we write an explicit
+ \isa{if} on the right hand side, or we can use conditional patterns:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ check\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}string\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}check\ {\isacharparenleft}{\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}s\ {\isasymnoteq}\ {\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}\ {\isasymLongrightarrow}\ check\ s\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsection{Partiality%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In HOL, all functions are total. A function \isa{f} applied to
+ \isa{x} always has the value \isa{f\ x}, and there is no notion
+ of undefinedness.
+ This is why we have to do termination
+ proofs when defining functions: The proof justifies that the
+ function can be defined by wellfounded recursion.
+
+ However, the \cmd{function} package does support partiality to a
+ certain extent. Let's look at the following function which looks
+ for a zero of a given function f.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}findzero\ f\ n\ {\isacharequal}\ {\isacharparenleft}if\ f\ n\ {\isacharequal}\ {\isadigit{0}}\ then\ n\ else\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ pat{\isacharunderscore}completeness\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent Clearly, any attempt of a termination proof must fail. And without
+ that, we do not get the usual rules \isa{findzero{\isachardot}simps} and
+ \isa{findzero{\isachardot}induct}. So what was the definition good for at all?%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Domain predicates%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The trick is that Isabelle has not only defined the function \isa{findzero}, but also
+ a predicate \isa{findzero{\isacharunderscore}dom} that characterizes the values where the function
+ terminates: the \emph{domain} of the function. If we treat a
+ partial function just as a total function with an additional domain
+ predicate, we can derive simplification and
+ induction rules as we do for total functions. They are guarded
+ by domain conditions and are called \isa{psimps} and \isa{pinduct}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
+findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
+findzero\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isacharquery}n\ else\ findzero\ {\isacharquery}f\ {\isacharparenleft}Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}%
+\end{isabelle}\end{minipage}
+ \hfill(\isa{findzero{\isachardot}psimps})
+ \vspace{1em}
+
+ \noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
+{\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}{\isacharcomma}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}{\isacharparenright}{\isacharsemicolon}\isanewline
+\isaindent{\ }{\isasymAnd}f\ n{\isachardot}\ {\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ f\ n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isacharquery}P\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ f\ n{\isasymrbrakk}\isanewline
+{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}%
+\end{isabelle}\end{minipage}
+ \hfill(\isa{findzero{\isachardot}pinduct})%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Remember that all we
+ are doing here is use some tricks to make a total function appear
+ as if it was partial. We can still write the term \isa{findzero\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ {\isadigit{1}}{\isacharparenright}\ {\isadigit{0}}} and like any other term of type \isa{nat} it is equal
+ to some natural number, although we might not be able to find out
+ which one. The function is \emph{underdefined}.
+
+ But it is defined enough to prove something interesting about it. We
+ can prove that if \isa{findzero\ f\ n}
+ terminates, it indeed returns a zero of \isa{f}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ findzero{\isacharunderscore}zero{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ n{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+\noindent We apply induction as usual, but using the partial induction
+ rule:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}induct\ f\ n\ rule{\isacharcolon}\ findzero{\isachardot}pinduct{\isacharparenright}%
+\begin{isamarkuptxt}%
+\noindent This gives the following subgoals:
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}f\ n{\isachardot}\ {\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ f\ n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isasymrbrakk}\isanewline
+\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}f\ n{\isachardot}\ }{\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ n{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}%
+\end{isabelle}
+
+ \noindent The hypothesis in our lemma was used to satisfy the first premise in
+ the induction rule. However, we also get \isa{findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}} as a local assumption in the induction step. This
+ allows to unfold \isa{findzero\ f\ n} using the \isa{psimps}
+ rule, and the rest is trivial. Since the \isa{psimps} rules carry the
+ \isa{{\isacharbrackleft}simp{\isacharbrackright}} attribute by default, we just need a single step:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Proofs about partial functions are often not harder than for total
+ functions. Fig.~\ref{findzero_isar} shows a slightly more
+ complicated proof written in Isar. It is verbose enough to show how
+ partiality comes into play: From the partial induction, we get an
+ additional domain condition hypothesis. Observe how this condition
+ is applied when calls to \isa{findzero} are unfolded.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+\isacommand{lemma}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ x\ {\isasymin}\ {\isacharbraceleft}n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}induct\ rule{\isacharcolon}\ findzero{\isachardot}pinduct{\isacharparenright}\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ f\ n\ \isacommand{assume}\isamarkupfalse%
+\ dom{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ IH{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharsemicolon}\ x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharbraceright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ x{\isacharunderscore}range{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ \isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}f\ n\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{with}\isamarkupfalse%
+\ dom\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}findzero\ f\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \ \ \isacommand{with}\isamarkupfalse%
+\ x{\isacharunderscore}range\ \isacommand{show}\isamarkupfalse%
+\ False\ \isacommand{by}\isamarkupfalse%
+\ auto\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\ \ \isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ x{\isacharunderscore}range\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isacharequal}\ n\ {\isasymor}\ x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ auto\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isachardoublequoteopen}f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{with}\isamarkupfalse%
+\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{with}\isamarkupfalse%
+\ dom\ \isakeyword{and}\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \ \ \isacommand{with}\isamarkupfalse%
+\ IH\ \isakeyword{and}\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}\vspace{6pt}\hrule
+\caption{A proof about a partial function}\label{findzero_isar}
+\end{figure}
+%
+\isamarkupsubsection{Partial termination proofs%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Now that we have proved some interesting properties about our
+ function, we should turn to the domain predicate and see if it is
+ actually true for some values. Otherwise we would have just proved
+ lemmas with \isa{False} as a premise.
+
+ Essentially, we need some introduction rules for \isa{findzero{\isacharunderscore}dom}. The function package can prove such domain
+ introduction rules automatically. But since they are not used very
+ often (they are almost never needed if the function is total), this
+ functionality is disabled by default for efficiency reasons. So we have to go
+ back and ask for them explicitly by passing the \isa{{\isacharparenleft}domintros{\isacharparenright}} option to the function package:
+
+\vspace{1ex}
+\noindent\cmd{function} \isa{{\isacharparenleft}domintros{\isacharparenright}\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
+\cmd{where}\isanewline%
+\ \ \ldots\\
+
+ \noindent Now the package has proved an introduction rule for \isa{findzero{\isacharunderscore}dom}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{thm}\isamarkupfalse%
+\ findzero{\isachardot}domintros%
+\begin{isamarkuptext}%
+\begin{isabelle}%
+{\isacharparenleft}{\isadigit{0}}\ {\isacharless}\ {\isacharquery}f\ {\isacharquery}n\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}%
+\end{isabelle}
+
+ Domain introduction rules allow to show that a given value lies in the
+ domain of a function, if the arguments of all recursive calls
+ are in the domain as well. They allow to do a \qt{single step} in a
+ termination proof. Usually, you want to combine them with a suitable
+ induction principle.
+
+ Since our function increases its argument at recursive calls, we
+ need an induction principle which works \qt{backwards}. We will use
+ \isa{inc{\isacharunderscore}induct}, which allows to do induction from a fixed number
+ \qt{downwards}:
+
+ \begin{center}\isa{{\isasymlbrakk}{\isacharquery}i\ {\isasymle}\ {\isacharquery}j{\isacharsemicolon}\ {\isacharquery}P\ {\isacharquery}j{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}i\ {\isacharless}\ {\isacharquery}j{\isacharsemicolon}\ {\isacharquery}P\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}i}\hfill(\isa{inc{\isacharunderscore}induct})\end{center}
+
+ Figure \ref{findzero_term} gives a detailed Isar proof of the fact
+ that \isa{findzero} terminates if there is a zero which is greater
+ or equal to \isa{n}. First we derive two useful rules which will
+ solve the base case and the step case of the induction. The
+ induction is then straightforward, except for the unusual induction
+ principle.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+\isacommand{lemma}\isamarkupfalse%
+\ findzero{\isacharunderscore}termination{\isacharcolon}\isanewline
+\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isasymge}\ n{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{shows}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\ \isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ base{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ findzero{\isachardot}domintros{\isacharparenright}\ {\isacharparenleft}simp\ add{\isacharcolon}{\isacharbackquoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isacharbackquoteclose}{\isacharparenright}\isanewline
+\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ step{\isacharcolon}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ Suc\ i{\isacharparenright}\ \isanewline
+\ \ \ \ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ findzero{\isachardot}domintros{\isacharparenright}\ simp\isanewline
+\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ {\isacharbackquoteopen}x\ {\isasymge}\ n{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
+\ {\isacharquery}thesis\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isacharparenleft}induct\ rule{\isacharcolon}inc{\isacharunderscore}induct{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ base{\isacharparenright}\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ i\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ Suc\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{thus}\isamarkupfalse%
+\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ i{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ step{\isacharparenright}\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}\vspace{6pt}\hrule
+\caption{Termination proof for \isa{findzero}}\label{findzero_term}
+\end{figure}
+%
+\begin{isamarkuptext}%
+Again, the proof given in Fig.~\ref{findzero_term} has a lot of
+ detail in order to explain the principles. Using more automation, we
+ can also have a short proof:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ findzero{\isacharunderscore}termination{\isacharunderscore}short{\isacharcolon}\isanewline
+\ \ \isakeyword{assumes}\ zero{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isachargreater}{\isacharequal}\ n{\isachardoublequoteclose}\ \isanewline
+\ \ \isakeyword{assumes}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{shows}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{using}\isamarkupfalse%
+\ zero\isanewline
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ rule{\isacharcolon}inc{\isacharunderscore}induct{\isacharparenright}\ {\isacharparenleft}auto\ intro{\isacharcolon}\ findzero{\isachardot}domintros{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent It is simple to combine the partial correctness result with the
+ termination lemma:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ findzero{\isacharunderscore}total{\isacharunderscore}correctness{\isacharcolon}\isanewline
+\ \ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}blast\ intro{\isacharcolon}\ findzero{\isacharunderscore}zero\ findzero{\isacharunderscore}termination{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsubsection{Definition of the domain predicate%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Sometimes it is useful to know what the definition of the domain
+ predicate looks like. Actually, \isa{findzero{\isacharunderscore}dom} is just an
+ abbreviation:
+
+ \begin{isabelle}%
+findzero{\isacharunderscore}dom\ {\isasymequiv}\ accp\ findzero{\isacharunderscore}rel%
+\end{isabelle}
+
+ The domain predicate is the \emph{accessible part} of a relation \isa{findzero{\isacharunderscore}rel}, which was also created internally by the function
+ package. \isa{findzero{\isacharunderscore}rel} is just a normal
+ inductive predicate, so we can inspect its definition by
+ looking at the introduction rules \isa{findzero{\isacharunderscore}rel{\isachardot}intros}.
+ In our case there is just a single rule:
+
+ \begin{isabelle}%
+{\isacharquery}f\ {\isacharquery}n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}rel\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ Suc\ {\isacharquery}n{\isacharparenright}\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}%
+\end{isabelle}
+
+ The predicate \isa{findzero{\isacharunderscore}rel}
+ describes the \emph{recursion relation} of the function
+ definition. The recursion relation is a binary relation on
+ the arguments of the function that relates each argument to its
+ recursive calls. In general, there is one introduction rule for each
+ recursive call.
+
+ The predicate \isa{findzero{\isacharunderscore}dom} is the accessible part of
+ that relation. An argument belongs to the accessible part, if it can
+ be reached in a finite number of steps (cf.~its definition in \isa{Wellfounded{\isachardot}thy}).
+
+ Since the domain predicate is just an abbreviation, you can use
+ lemmas for \isa{accp} and \isa{findzero{\isacharunderscore}rel} directly. Some
+ lemmas which are occasionally useful are \isa{accpI}, \isa{accp{\isacharunderscore}downward}, and of course the introduction and elimination rules
+ for the recursion relation \isa{findzero{\isachardot}intros} and \isa{findzero{\isachardot}cases}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{A Useful Special Case: Tail recursion%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The domain predicate is our trick that allows us to model partiality
+ in a world of total functions. The downside of this is that we have
+ to carry it around all the time. The termination proof above allowed
+ us to replace the abstract \isa{findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}} by the more
+ concrete \isa{n\ {\isasymle}\ x\ {\isasymand}\ f\ x\ {\isacharequal}\ {\isadigit{0}}}, but the condition is still
+ there and can only be discharged for special cases.
+ In particular, the domain predicate guards the unfolding of our
+ function, since it is there as a condition in the \isa{psimp}
+ rules.
+
+ Now there is an important special case: We can actually get rid
+ of the condition in the simplification rules, \emph{if the function
+ is tail-recursive}. The reason is that for all tail-recursive
+ equations there is a total function satisfying them, even if they
+ are non-terminating.
+
+% A function is tail recursive, if each call to the function is either
+% equal
+%
+% So the outer form of the
+%
+%if it can be written in the following
+% form:
+% {term[display] "f x = (if COND x then BASE x else f (LOOP x))"}
+
+
+ The function package internally does the right construction and can
+ derive the unconditional simp rules, if we ask it to do so. Luckily,
+ our \isa{findzero} function is tail-recursive, so we can just go
+ back and add another option to the \cmd{function} command:
+
+\vspace{1ex}
+\noindent\cmd{function} \isa{{\isacharparenleft}domintros{\isacharcomma}\ tailrec{\isacharparenright}\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
+\cmd{where}\isanewline%
+\ \ \ldots\\%
+
+
+ \noindent Now, we actually get unconditional simplification rules, even
+ though the function is partial:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{thm}\isamarkupfalse%
+\ findzero{\isachardot}simps%
+\begin{isamarkuptext}%
+\begin{isabelle}%
+findzero\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isacharquery}n\ else\ findzero\ {\isacharquery}f\ {\isacharparenleft}Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}%
+\end{isabelle}
+
+ \noindent Of course these would make the simplifier loop, so we better remove
+ them from the simpset:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{declare}\isamarkupfalse%
+\ findzero{\isachardot}simps{\isacharbrackleft}simp\ del{\isacharbrackright}%
+\begin{isamarkuptext}%
+Getting rid of the domain conditions in the simplification rules is
+ not only useful because it simplifies proofs. It is also required in
+ order to use Isabelle's code generator to generate ML code
+ from a function definition.
+ Since the code generator only works with equations, it cannot be
+ used with \isa{psimp} rules. Thus, in order to generate code for
+ partial functions, they must be defined as a tail recursion.
+ Luckily, many functions have a relatively natural tail recursive
+ definition.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Nested recursion%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Recursive calls which are nested in one another frequently cause
+ complications, since their termination proof can depend on a partial
+ correctness property of the function itself.
+
+ As a small example, we define the \qt{nested zero} function:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ nz\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}nz\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}nz\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ nz\ {\isacharparenleft}nz\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ pat{\isacharunderscore}completeness\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+If we attempt to prove termination using the identity measure on
+ naturals, this fails:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{termination}\isamarkupfalse%
+\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}n{\isachardot}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\isanewline
+\ \ \isacommand{apply}\isamarkupfalse%
+\ auto%
+\begin{isamarkuptxt}%
+We get stuck with the subgoal
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ nz{\isacharunderscore}dom\ n\ {\isasymLongrightarrow}\ nz\ n\ {\isacharless}\ Suc\ n%
+\end{isabelle}
+
+ Of course this statement is true, since we know that \isa{nz} is
+ the zero function. And in fact we have no problem proving this
+ property by induction.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{lemma}\isamarkupfalse%
+\ nz{\isacharunderscore}is{\isacharunderscore}zero{\isacharcolon}\ {\isachardoublequoteopen}nz{\isacharunderscore}dom\ n\ {\isasymLongrightarrow}\ nz\ n\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}induct\ rule{\isacharcolon}nz{\isachardot}pinduct{\isacharparenright}\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+We formulate this as a partial correctness lemma with the condition
+ \isa{nz{\isacharunderscore}dom\ n}. This allows us to prove it with the \isa{pinduct} rule before we have proved termination. With this lemma,
+ the termination proof works as expected:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{termination}\isamarkupfalse%
+\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}n{\isachardot}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\ {\isacharparenleft}auto\ simp{\isacharcolon}\ nz{\isacharunderscore}is{\isacharunderscore}zero{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+As a general strategy, one should prove the statements needed for
+ termination as a partial property first. Then they can be used to do
+ the termination proof. This also works for less trivial
+ examples. Figure \ref{f91} defines the 91-function, a well-known
+ challenge problem due to John McCarthy, and proves its termination.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+\isacommand{function}\isamarkupfalse%
+\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}\ n\ {\isacharequal}\ {\isacharparenleft}if\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n\ then\ n\ {\isacharminus}\ {\isadigit{1}}{\isadigit{0}}\ else\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ pat{\isacharunderscore}completeness\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}estimate{\isacharcolon}\ \isanewline
+\ \ \isakeyword{assumes}\ trm{\isacharcolon}\ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}dom\ n{\isachardoublequoteclose}\ \isanewline
+\ \ \isakeyword{shows}\ {\isachardoublequoteopen}n\ {\isacharless}\ f{\isadigit{9}}{\isadigit{1}}\ n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{using}\isamarkupfalse%
+\ trm\ \isacommand{by}\isamarkupfalse%
+\ induct\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{termination}\isamarkupfalse%
+\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{let}\isamarkupfalse%
+\ {\isacharquery}R\ {\isacharequal}\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isacharminus}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}wf\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
+\isanewline
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ n\ {\isacharcolon}{\isacharcolon}\ nat\ \isacommand{assume}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n{\isachardoublequoteclose}\ %
+\isamarkupcmt{Assumptions for both calls%
+}
+\isanewline
+\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharcomma}\ n{\isacharparenright}\ {\isasymin}\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\ %
+\isamarkupcmt{Inner call%
+}
+\isanewline
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ inner{\isacharunderscore}trm{\isacharcolon}\ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}dom\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\ %
+\isamarkupcmt{Outer call%
+}
+\isanewline
+\ \ \isacommand{with}\isamarkupfalse%
+\ f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}estimate\ \isacommand{have}\isamarkupfalse%
+\ {\isachardoublequoteopen}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}\ {\isacharless}\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{with}\isamarkupfalse%
+\ {\isacharbackquoteopen}{\isasymnot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}{\isacharparenleft}f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isacharcomma}\ n{\isacharparenright}\ {\isasymin}\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}
+\vspace{6pt}\hrule
+\caption{McCarthy's 91-function}\label{f91}
+\end{figure}
+%
+\isamarkupsection{Higher-Order Recursion%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Higher-order recursion occurs when recursive calls
+ are passed as arguments to higher-order combinators such as \isa{map}, \isa{filter} etc.
+ As an example, imagine a datatype of n-ary trees:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ {\isacharprime}a\ tree\ {\isacharequal}\ \isanewline
+\ \ Leaf\ {\isacharprime}a\ \isanewline
+{\isacharbar}\ Branch\ {\isachardoublequoteopen}{\isacharprime}a\ tree\ list{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent We can define a function which swaps the left and right subtrees recursively, using the
+ list functions \isa{rev} and \isa{map}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ mirror\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ tree\ {\isasymRightarrow}\ {\isacharprime}a\ tree{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}mirror\ {\isacharparenleft}Leaf\ n{\isacharparenright}\ {\isacharequal}\ Leaf\ n{\isachardoublequoteclose}\isanewline
+{\isacharbar}\ {\isachardoublequoteopen}mirror\ {\isacharparenleft}Branch\ l{\isacharparenright}\ {\isacharequal}\ Branch\ {\isacharparenleft}rev\ {\isacharparenleft}map\ mirror\ l{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+Although the definition is accepted without problems, let us look at the termination proof:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{termination}\isamarkupfalse%
+%
+\isadelimproof
+\ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+%
+\begin{isamarkuptxt}%
+As usual, we have to give a wellfounded relation, such that the
+ arguments of the recursive calls get smaller. But what exactly are
+ the arguments of the recursive calls when mirror is given as an
+ argument to \isa{map}? Isabelle gives us the
+ subgoals
+
+ \begin{isabelle}%
+\ {\isadigit{1}}{\isachardot}\ wf\ {\isacharquery}R\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}l\ x{\isachardot}\ x\ {\isasymin}\ set\ l\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ Branch\ l{\isacharparenright}\ {\isasymin}\ {\isacharquery}R%
+\end{isabelle}
+
+ So the system seems to know that \isa{map} only
+ applies the recursive call \isa{mirror} to elements
+ of \isa{l}, which is essential for the termination proof.
+
+ This knowledge about \isa{map} is encoded in so-called congruence rules,
+ which are special theorems known to the \cmd{function} command. The
+ rule for \isa{map} is
+
+ \begin{isabelle}%
+{\isasymlbrakk}{\isacharquery}xs\ {\isacharequal}\ {\isacharquery}ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ {\isacharquery}ys\ {\isasymLongrightarrow}\ {\isacharquery}f\ x\ {\isacharequal}\ {\isacharquery}g\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ map\ {\isacharquery}f\ {\isacharquery}xs\ {\isacharequal}\ map\ {\isacharquery}g\ {\isacharquery}ys%
+\end{isabelle}
+
+ You can read this in the following way: Two applications of \isa{map} are equal, if the list arguments are equal and the functions
+ coincide on the elements of the list. This means that for the value
+ \isa{map\ f\ l} we only have to know how \isa{f} behaves on
+ the elements of \isa{l}.
+
+ Usually, one such congruence rule is
+ needed for each higher-order construct that is used when defining
+ new functions. In fact, even basic functions like \isa{If} and \isa{Let} are handled by this mechanism. The congruence
+ rule for \isa{If} states that the \isa{then} branch is only
+ relevant if the condition is true, and the \isa{else} branch only if it
+ is false:
+
+ \begin{isabelle}%
+{\isasymlbrakk}{\isacharquery}b\ {\isacharequal}\ {\isacharquery}c{\isacharsemicolon}\ {\isacharquery}c\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}u{\isacharsemicolon}\ {\isasymnot}\ {\isacharquery}c\ {\isasymLongrightarrow}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}v{\isasymrbrakk}\isanewline
+{\isasymLongrightarrow}\ {\isacharparenleft}if\ {\isacharquery}b\ then\ {\isacharquery}x\ else\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}c\ then\ {\isacharquery}u\ else\ {\isacharquery}v{\isacharparenright}%
+\end{isabelle}
+
+ Congruence rules can be added to the
+ function package by giving them the \isa{fundef{\isacharunderscore}cong} attribute.
+
+ The constructs that are predefined in Isabelle, usually
+ come with the respective congruence rules.
+ But if you define your own higher-order functions, you may have to
+ state and prove the required congruence rules yourself, if you want to use your
+ functions in recursive definitions.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsubsection{Congruence Rules and Evaluation Order%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Higher order logic differs from functional programming languages in
+ that it has no built-in notion of evaluation order. A program is
+ just a set of equations, and it is not specified how they must be
+ evaluated.
+
+ However for the purpose of function definition, we must talk about
+ evaluation order implicitly, when we reason about termination.
+ Congruence rules express that a certain evaluation order is
+ consistent with the logical definition.
+
+ Consider the following function.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{function}\isamarkupfalse%
+\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}f\ n\ {\isacharequal}\ {\isacharparenleft}n\ {\isacharequal}\ {\isadigit{0}}\ {\isasymor}\ f\ {\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+For this definition, the termination proof fails. The default configuration
+ specifies no congruence rule for disjunction. We have to add a
+ congruence rule that specifies left-to-right evaluation order:
+
+ \vspace{1ex}
+ \noindent \isa{{\isasymlbrakk}{\isacharquery}P\ {\isacharequal}\ {\isacharquery}P{\isacharprime}{\isacharsemicolon}\ {\isasymnot}\ {\isacharquery}P{\isacharprime}\ {\isasymLongrightarrow}\ {\isacharquery}Q\ {\isacharequal}\ {\isacharquery}Q{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharquery}P\ {\isasymor}\ {\isacharquery}Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}P{\isacharprime}\ {\isasymor}\ {\isacharquery}Q{\isacharprime}{\isacharparenright}}\hfill(\isa{disj{\isacharunderscore}cong})
+ \vspace{1ex}
+
+ Now the definition works without problems. Note how the termination
+ proof depends on the extra condition that we get from the congruence
+ rule.
+
+ However, as evaluation is not a hard-wired concept, we
+ could just turn everything around by declaring a different
+ congruence rule. Then we can make the reverse definition:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ disj{\isacharunderscore}cong{\isadigit{2}}{\isacharbrackleft}fundef{\isacharunderscore}cong{\isacharbrackright}{\isacharcolon}\ \isanewline
+\ \ {\isachardoublequoteopen}{\isacharparenleft}{\isasymnot}\ Q{\isacharprime}\ {\isasymLongrightarrow}\ P\ {\isacharequal}\ P{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}Q\ {\isacharequal}\ Q{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymor}\ Q{\isacharprime}{\isacharparenright}{\isachardoublequoteclose}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ blast%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{fun}\isamarkupfalse%
+\ f{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isachardoublequoteopen}f{\isacharprime}\ n\ {\isacharequal}\ {\isacharparenleft}f{\isacharprime}\ {\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}\ {\isasymor}\ n\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}%
+\begin{isamarkuptext}%
+\noindent These examples show that, in general, there is no \qt{best} set of
+ congruence rules.
+
+ However, such tweaking should rarely be necessary in
+ practice, as most of the time, the default set of congruence rules
+ works well.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/Thy/document/session.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,6 @@
+\input{Functions.tex}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/conclusion.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,7 @@
+\section{Conclusion}
+
+\fixme{}
+
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/functions.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,91 @@
+
+\documentclass[a4paper,fleqn]{article}
+
+\usepackage{latexsym,graphicx}
+\usepackage[refpage]{nomencl}
+\usepackage{../iman,../extra,../isar,../proof}
+\usepackage{../isabelle,../isabellesym}
+\usepackage{style}
+\usepackage{mathpartir}
+\usepackage{amsthm}
+\usepackage{../pdfsetup}
+
+\newcommand{\cmd}[1]{\isacommand{#1}}
+
+\newcommand{\isasymINFIX}{\cmd{infix}}
+\newcommand{\isasymLOCALE}{\cmd{locale}}
+\newcommand{\isasymINCLUDES}{\cmd{includes}}
+\newcommand{\isasymDATATYPE}{\cmd{datatype}}
+\newcommand{\isasymAXCLASS}{\cmd{axclass}}
+\newcommand{\isasymDEFINES}{\cmd{defines}}
+\newcommand{\isasymNOTES}{\cmd{notes}}
+\newcommand{\isasymCLASS}{\cmd{class}}
+\newcommand{\isasymINSTANCE}{\cmd{instance}}
+\newcommand{\isasymLEMMA}{\cmd{lemma}}
+\newcommand{\isasymPROOF}{\cmd{proof}}
+\newcommand{\isasymQED}{\cmd{qed}}
+\newcommand{\isasymFIX}{\cmd{fix}}
+\newcommand{\isasymASSUME}{\cmd{assume}}
+\newcommand{\isasymSHOW}{\cmd{show}}
+\newcommand{\isasymNOTE}{\cmd{note}}
+\newcommand{\isasymCODEGEN}{\cmd{code\_gen}}
+\newcommand{\isasymPRINTCODETHMS}{\cmd{print\_codethms}}
+\newcommand{\isasymFUN}{\cmd{fun}}
+\newcommand{\isasymFUNCTION}{\cmd{function}}
+\newcommand{\isasymPRIMREC}{\cmd{primrec}}
+\newcommand{\isasymRECDEF}{\cmd{recdef}}
+
+\newcommand{\qt}[1]{``#1''}
+\newcommand{\qtt}[1]{"{}{#1}"{}}
+\newcommand{\qn}[1]{\emph{#1}}
+\newcommand{\strong}[1]{{\bfseries #1}}
+\newcommand{\fixme}[1][!]{\strong{FIXME: #1}}
+
+\newtheorem{exercise}{Exercise}{\bf}{\itshape}
+%\newtheorem*{thmstar}{Theorem}{\bf}{\itshape}
+
+\hyphenation{Isabelle}
+\hyphenation{Isar}
+
+\isadroptag{theory}
+\title{Defining Recursive Functions in Isabelle/HOL}
+\author{Alexander Krauss}
+
+\isabellestyle{tt}
+\renewcommand{\isastyletxt}{\isastyletext}% use same formatting for txt and text
+
+\begin{document}
+
+\date{\ \\}
+\maketitle
+
+\begin{abstract}
+ This tutorial describes the use of the new \emph{function} package,
+ which provides general recursive function definitions for Isabelle/HOL.
+ We start with very simple examples and then gradually move on to more
+ advanced topics such as manual termination proofs, nested recursion,
+ partiality, tail recursion and congruence rules.
+\end{abstract}
+
+%\thispagestyle{empty}\clearpage
+
+%\pagenumbering{roman}
+%\clearfirst
+
+\input{intro.tex}
+\input{Thy/document/Functions.tex}
+%\input{conclusion.tex}
+
+\begingroup
+%\tocentry{\bibname}
+\bibliographystyle{plain} \small\raggedright\frenchspacing
+\bibliography{../manual}
+\endgroup
+
+\end{document}
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/intro.tex Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,55 @@
+\section{Introduction}
+
+Starting from Isabelle 2007, new facilities for recursive
+function definitions~\cite{krauss2006} are available. They provide
+better support for general recursive definitions than previous
+packages. But despite all tool support, function definitions can
+sometimes be a difficult thing.
+
+This tutorial is an example-guided introduction to the practical use
+of the package and related tools. It should help you get started with
+defining functions quickly. For the more difficult definitions we will
+discuss what problems can arise, and how they can be solved.
+
+We assume that you have mastered the fundamentals of Isabelle/HOL
+and are able to write basic specifications and proofs. To start out
+with Isabelle in general, consult the Isabelle/HOL tutorial
+\cite{isa-tutorial}.
+
+
+
+\paragraph{Structure of this tutorial.}
+Section 2 introduces the syntax and basic operation of the \cmd{fun}
+command, which provides full automation with reasonable default
+behavior. The impatient reader can stop after that
+section, and consult the remaining sections only when needed.
+Section 3 introduces the more verbose \cmd{function} command which
+gives fine-grained control. This form should be used
+whenever the short form fails.
+After that we discuss more specialized issues:
+termination, mutual, nested and higher-order recursion, partiality, pattern matching
+and others.
+
+
+\paragraph{Some background.}
+Following the LCF tradition, the package is realized as a definitional
+extension: Recursive definitions are internally transformed into a
+non-recursive form, such that the function can be defined using
+standard definition facilities. Then the recursive specification is
+derived from the primitive definition. This is a complex task, but it
+is fully automated and mostly transparent to the user. Definitional
+extensions are valuable because they are conservative by construction:
+The \qt{new} concept of general wellfounded recursion is completely reduced
+to existing principles.
+
+
+
+
+The new \cmd{function} command, and its short form \cmd{fun} have mostly
+replaced the traditional \cmd{recdef} command \cite{slind-tfl}. They solve
+a few of technical issues around \cmd{recdef}, and allow definitions
+which were not previously possible.
+
+
+
+
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+MacVec 128 128 getinterval astore pop
+end %. AltsysDict
+%%EndResource
+%%EndProlog
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+%%DocumentNeededResources: font Symbol
+%%+ font ZapfHumanist601BT-Bold
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Binary file doc-src/Functions/isabelle_isar.pdf has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/mathpartir.sty Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,421 @@
+% Mathpartir --- Math Paragraph for Typesetting Inference Rules
+%
+% Copyright (C) 2001, 2002, 2003, 2004, 2005 Didier Rémy
+%
+% Author : Didier Remy
+% Version : 1.2.0
+% Bug Reports : to author
+% Web Site : http://pauillac.inria.fr/~remy/latex/
+%
+% Mathpartir is free software; you can redistribute it and/or modify
+% it under the terms of the GNU General Public License as published by
+% the Free Software Foundation; either version 2, or (at your option)
+% any later version.
+%
+% Mathpartir is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+% GNU General Public License for more details
+% (http://pauillac.inria.fr/~remy/license/GPL).
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% File mathpartir.sty (LaTeX macros)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{mathpartir}
+ [2005/12/20 version 1.2.0 Math Paragraph for Typesetting Inference Rules]
+
+%%
+
+%% Identification
+%% Preliminary declarations
+
+\RequirePackage {keyval}
+
+%% Options
+%% More declarations
+
+%% PART I: Typesetting maths in paragraphe mode
+
+\newdimen \mpr@tmpdim
+
+% To ensure hevea \hva compatibility, \hva should expands to nothing
+% in mathpar or in inferrule
+\let \mpr@hva \empty
+
+%% normal paragraph parametters, should rather be taken dynamically
+\def \mpr@savepar {%
+ \edef \MathparNormalpar
+ {\noexpand \lineskiplimit \the\lineskiplimit
+ \noexpand \lineskip \the\lineskip}%
+ }
+
+\def \mpr@rulelineskip {\lineskiplimit=0.3em\lineskip=0.2em plus 0.1em}
+\def \mpr@lesslineskip {\lineskiplimit=0.6em\lineskip=0.5em plus 0.2em}
+\def \mpr@lineskip {\lineskiplimit=1.2em\lineskip=1.2em plus 0.2em}
+\let \MathparLineskip \mpr@lineskip
+\def \mpr@paroptions {\MathparLineskip}
+\let \mpr@prebindings \relax
+
+\newskip \mpr@andskip \mpr@andskip 2em plus 0.5fil minus 0.5em
+
+\def \mpr@goodbreakand
+ {\hskip -\mpr@andskip \penalty -1000\hskip \mpr@andskip}
+\def \mpr@and {\hskip \mpr@andskip}
+\def \mpr@andcr {\penalty 50\mpr@and}
+\def \mpr@cr {\penalty -10000\mpr@and}
+\def \mpr@eqno #1{\mpr@andcr #1\hskip 0em plus -1fil \penalty 10}
+
+\def \mpr@bindings {%
+ \let \and \mpr@andcr
+ \let \par \mpr@andcr
+ \let \\\mpr@cr
+ \let \eqno \mpr@eqno
+ \let \hva \mpr@hva
+ }
+\let \MathparBindings \mpr@bindings
+
+% \@ifundefined {ignorespacesafterend}
+% {\def \ignorespacesafterend {\aftergroup \ignorespaces}
+
+\newenvironment{mathpar}[1][]
+ {$$\mpr@savepar \parskip 0em \hsize \linewidth \centering
+ \vbox \bgroup \mpr@prebindings \mpr@paroptions #1\ifmmode $\else
+ \noindent $\displaystyle\fi
+ \MathparBindings}
+ {\unskip \ifmmode $\fi\egroup $$\ignorespacesafterend}
+
+% \def \math@mathpar #1{\setbox0 \hbox {$\displaystyle #1$}\ifnum
+% \wd0 < \hsize $$\box0$$\else \bmathpar #1\emathpar \fi}
+
+%%% HOV BOXES
+
+\def \mathvbox@ #1{\hbox \bgroup \mpr@normallineskip
+ \vbox \bgroup \tabskip 0em \let \\ \cr
+ \halign \bgroup \hfil $##$\hfil\cr #1\crcr \egroup \egroup
+ \egroup}
+
+\def \mathhvbox@ #1{\setbox0 \hbox {\let \\\qquad $#1$}\ifnum \wd0 < \hsize
+ \box0\else \mathvbox {#1}\fi}
+
+
+%% Part II -- operations on lists
+
+\newtoks \mpr@lista
+\newtoks \mpr@listb
+
+\long \def\mpr@cons #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
+{#2}\edef #2{\the \mpr@lista \the \mpr@listb}}
+
+\long \def\mpr@snoc #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
+{#2}\edef #2{\the \mpr@listb\the\mpr@lista}}
+
+\long \def \mpr@concat#1=#2\mpr@to#3{\mpr@lista \expandafter {#2}\mpr@listb
+\expandafter {#3}\edef #1{\the \mpr@listb\the\mpr@lista}}
+
+\def \mpr@head #1\mpr@to #2{\expandafter \mpr@head@ #1\mpr@head@ #1#2}
+\long \def \mpr@head@ #1#2\mpr@head@ #3#4{\def #4{#1}\def#3{#2}}
+
+\def \mpr@flatten #1\mpr@to #2{\expandafter \mpr@flatten@ #1\mpr@flatten@ #1#2}
+\long \def \mpr@flatten@ \\#1\\#2\mpr@flatten@ #3#4{\def #4{#1}\def #3{\\#2}}
+
+\def \mpr@makelist #1\mpr@to #2{\def \mpr@all {#1}%
+ \mpr@lista {\\}\mpr@listb \expandafter {\mpr@all}\edef \mpr@all {\the
+ \mpr@lista \the \mpr@listb \the \mpr@lista}\let #2\empty
+ \def \mpr@stripof ##1##2\mpr@stripend{\def \mpr@stripped{##2}}\loop
+ \mpr@flatten \mpr@all \mpr@to \mpr@one
+ \expandafter \mpr@snoc \mpr@one \mpr@to #2\expandafter \mpr@stripof
+ \mpr@all \mpr@stripend
+ \ifx \mpr@stripped \empty \let \mpr@isempty 0\else \let \mpr@isempty 1\fi
+ \ifx 1\mpr@isempty
+ \repeat
+}
+
+\def \mpr@rev #1\mpr@to #2{\let \mpr@tmp \empty
+ \def \\##1{\mpr@cons ##1\mpr@to \mpr@tmp}#1\let #2\mpr@tmp}
+
+%% Part III -- Type inference rules
+
+\newif \if@premisse
+\newbox \mpr@hlist
+\newbox \mpr@vlist
+\newif \ifmpr@center \mpr@centertrue
+\def \mpr@htovlist {%
+ \setbox \mpr@hlist
+ \hbox {\strut
+ \ifmpr@center \hskip -0.5\wd\mpr@hlist\fi
+ \unhbox \mpr@hlist}%
+ \setbox \mpr@vlist
+ \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
+ \else \unvbox \mpr@vlist \box \mpr@hlist
+ \fi}%
+}
+% OLD version
+% \def \mpr@htovlist {%
+% \setbox \mpr@hlist
+% \hbox {\strut \hskip -0.5\wd\mpr@hlist \unhbox \mpr@hlist}%
+% \setbox \mpr@vlist
+% \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
+% \else \unvbox \mpr@vlist \box \mpr@hlist
+% \fi}%
+% }
+
+\def \mpr@item #1{$\displaystyle #1$}
+\def \mpr@sep{2em}
+\def \mpr@blank { }
+\def \mpr@hovbox #1#2{\hbox
+ \bgroup
+ \ifx #1T\@premissetrue
+ \else \ifx #1B\@premissefalse
+ \else
+ \PackageError{mathpartir}
+ {Premisse orientation should either be T or B}
+ {Fatal error in Package}%
+ \fi \fi
+ \def \@test {#2}\ifx \@test \mpr@blank\else
+ \setbox \mpr@hlist \hbox {}%
+ \setbox \mpr@vlist \vbox {}%
+ \if@premisse \let \snoc \mpr@cons \else \let \snoc \mpr@snoc \fi
+ \let \@hvlist \empty \let \@rev \empty
+ \mpr@tmpdim 0em
+ \expandafter \mpr@makelist #2\mpr@to \mpr@flat
+ \if@premisse \mpr@rev \mpr@flat \mpr@to \@rev \else \let \@rev \mpr@flat \fi
+ \def \\##1{%
+ \def \@test {##1}\ifx \@test \empty
+ \mpr@htovlist
+ \mpr@tmpdim 0em %%% last bug fix not extensively checked
+ \else
+ \setbox0 \hbox{\mpr@item {##1}}\relax
+ \advance \mpr@tmpdim by \wd0
+ %\mpr@tmpdim 1.02\mpr@tmpdim
+ \ifnum \mpr@tmpdim < \hsize
+ \ifnum \wd\mpr@hlist > 0
+ \if@premisse
+ \setbox \mpr@hlist
+ \hbox {\unhbox0 \hskip \mpr@sep \unhbox \mpr@hlist}%
+ \else
+ \setbox \mpr@hlist
+ \hbox {\unhbox \mpr@hlist \hskip \mpr@sep \unhbox0}%
+ \fi
+ \else
+ \setbox \mpr@hlist \hbox {\unhbox0}%
+ \fi
+ \else
+ \ifnum \wd \mpr@hlist > 0
+ \mpr@htovlist
+ \mpr@tmpdim \wd0
+ \fi
+ \setbox \mpr@hlist \hbox {\unhbox0}%
+ \fi
+ \advance \mpr@tmpdim by \mpr@sep
+ \fi
+ }%
+ \@rev
+ \mpr@htovlist
+ \ifmpr@center \hskip \wd\mpr@vlist\fi \box \mpr@vlist
+ \fi
+ \egroup
+}
+
+%%% INFERENCE RULES
+
+\@ifundefined{@@over}{%
+ \let\@@over\over % fallback if amsmath is not loaded
+ \let\@@overwithdelims\overwithdelims
+ \let\@@atop\atop \let\@@atopwithdelims\atopwithdelims
+ \let\@@above\above \let\@@abovewithdelims\abovewithdelims
+ }{}
+
+%% The default
+
+\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
+ $\displaystyle {#1\mpr@over #2}$}}
+\let \mpr@fraction \mpr@@fraction
+
+%% A generic solution to arrow
+
+\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
+ \def \mpr@tail{#1}%
+ \def \mpr@body{#2}%
+ \def \mpr@head{#3}%
+ \setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
+ \setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
+ \setbox3=\hbox{$\mkern -3mu \mpr@body\mkern -3mu$}%
+ \dimen0=\dp1\advance\dimen0 by \ht3\relax\dp1\dimen0\relax
+ \dimen0=\ht2\advance\dimen0 by \dp3\relax\ht2\dimen0\relax
+ \setbox0=\hbox {$\box1 \@@atop \box2$}%
+ \dimen0=\wd0\box0
+ \box0 \hskip -\dimen0\relax
+ \hbox to \dimen0 {$%
+ \mathrel{\mpr@tail}\joinrel
+ \xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
+ $}}}
+
+%% Old stuff should be removed in next version
+\def \mpr@@reduce #1#2{\hbox
+ {$\lower 0.01pt \mpr@@fraction {#1}{#2}\mkern -15mu\rightarrow$}}
+\def \mpr@@rewrite #1#2#3{\hbox
+ {$\lower 0.01pt \mpr@@fraction {#2}{#3}\mkern -8mu#1$}}
+\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
+
+\def \mpr@empty {}
+\def \mpr@inferrule
+ {\bgroup
+ \ifnum \linewidth<\hsize \hsize \linewidth\fi
+ \mpr@rulelineskip
+ \let \and \qquad
+ \let \hva \mpr@hva
+ \let \@rulename \mpr@empty
+ \let \@rule@options \mpr@empty
+ \let \mpr@over \@@over
+ \mpr@inferrule@}
+\newcommand {\mpr@inferrule@}[3][]
+ {\everymath={\displaystyle}%
+ \def \@test {#2}\ifx \empty \@test
+ \setbox0 \hbox {$\vcenter {\mpr@hovbox{B}{#3}}$}%
+ \else
+ \def \@test {#3}\ifx \empty \@test
+ \setbox0 \hbox {$\vcenter {\mpr@hovbox{T}{#2}}$}%
+ \else
+ \setbox0 \mpr@fraction {\mpr@hovbox{T}{#2}}{\mpr@hovbox{B}{#3}}%
+ \fi \fi
+ \def \@test {#1}\ifx \@test\empty \box0
+ \else \vbox
+%%% Suggestion de Francois pour les etiquettes longues
+%%% {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
+ {\hbox {\RefTirName {#1}}\box0}\fi
+ \egroup}
+
+\def \mpr@vdotfil #1{\vbox to #1{\leaders \hbox{$\cdot$} \vfil}}
+
+% They are two forms
+% \inferrule [label]{[premisses}{conclusions}
+% or
+% \inferrule* [options]{[premisses}{conclusions}
+%
+% Premisses and conclusions are lists of elements separated by \\
+% Each \\ produces a break, attempting horizontal breaks if possible,
+% and vertical breaks if needed.
+%
+% An empty element obtained by \\\\ produces a vertical break in all cases.
+%
+% The former rule is aligned on the fraction bar.
+% The optional label appears on top of the rule
+% The second form to be used in a derivation tree is aligned on the last
+% line of its conclusion
+%
+% The second form can be parameterized, using the key=val interface. The
+% folloiwng keys are recognized:
+%
+% width set the width of the rule to val
+% narrower set the width of the rule to val\hsize
+% before execute val at the beginning/left
+% lab put a label [Val] on top of the rule
+% lskip add negative skip on the right
+% left put a left label [Val]
+% Left put a left label [Val], ignoring its width
+% right put a right label [Val]
+% Right put a right label [Val], ignoring its width
+% leftskip skip negative space on the left-hand side
+% rightskip skip negative space on the right-hand side
+% vdots lift the rule by val and fill vertical space with dots
+% after execute val at the end/right
+%
+% Note that most options must come in this order to avoid strange
+% typesetting (in particular leftskip must preceed left and Left and
+% rightskip must follow Right or right; vdots must come last
+% or be only followed by rightskip.
+%
+
+%% Keys that make sence in all kinds of rules
+\def \mprset #1{\setkeys{mprset}{#1}}
+\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
+\define@key {mprset}{center}[]{\mpr@centertrue}
+\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
+\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
+\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
+
+\newbox \mpr@right
+\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
+\define@key {mpr}{center}[]{\mpr@centertrue}
+\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
+\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
+\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
+\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
+ \advance \hsize by -\wd0\box0}
+\define@key {mpr}{width}{\hsize #1}
+\define@key {mpr}{sep}{\def\mpr@sep{#1}}
+\define@key {mpr}{before}{#1}
+\define@key {mpr}{lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
+\define@key {mpr}{Lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
+\define@key {mpr}{narrower}{\hsize #1\hsize}
+\define@key {mpr}{leftskip}{\hskip -#1}
+\define@key {mpr}{reduce}[]{\let \mpr@fraction \mpr@@reduce}
+\define@key {mpr}{rightskip}
+ {\setbox \mpr@right \hbox {\unhbox \mpr@right \hskip -#1}}
+\define@key {mpr}{LEFT}{\setbox0 \hbox {$#1$}\relax
+ \advance \hsize by -\wd0\box0}
+\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
+ \advance \hsize by -\wd0\box0}
+\define@key {mpr}{Left}{\llap{$\TirName {#1}\;$}}
+\define@key {mpr}{right}
+ {\setbox0 \hbox {$\;\TirName {#1}$}\relax \advance \hsize by -\wd0
+ \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
+\define@key {mpr}{RIGHT}
+ {\setbox0 \hbox {$#1$}\relax \advance \hsize by -\wd0
+ \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
+\define@key {mpr}{Right}
+ {\setbox \mpr@right \hbox {\unhbox \mpr@right \rlap {$\;\TirName {#1}$}}}
+\define@key {mpr}{vdots}{\def \mpr@vdots {\@@atop \mpr@vdotfil{#1}}}
+\define@key {mpr}{after}{\edef \mpr@after {\mpr@after #1}}
+
+\newdimen \rule@dimen
+\newcommand \mpr@inferstar@ [3][]{\setbox0
+ \hbox {\let \mpr@rulename \mpr@empty \let \mpr@vdots \relax
+ \setbox \mpr@right \hbox{}%
+ $\setkeys{mpr}{#1}%
+ \ifx \mpr@rulename \mpr@empty \mpr@inferrule {#2}{#3}\else
+ \mpr@inferrule [{\mpr@rulename}]{#2}{#3}\fi
+ \box \mpr@right \mpr@vdots$}
+ \setbox1 \hbox {\strut}
+ \rule@dimen \dp0 \advance \rule@dimen by -\dp1
+ \raise \rule@dimen \box0}
+
+\def \mpr@infer {\@ifnextchar *{\mpr@inferstar}{\mpr@inferrule}}
+\newcommand \mpr@err@skipargs[3][]{}
+\def \mpr@inferstar*{\ifmmode
+ \let \@do \mpr@inferstar@
+ \else
+ \let \@do \mpr@err@skipargs
+ \PackageError {mathpartir}
+ {\string\inferrule* can only be used in math mode}{}%
+ \fi \@do}
+
+
+%%% Exports
+
+% Envirnonment mathpar
+
+\let \inferrule \mpr@infer
+
+% make a short name \infer is not already defined
+\@ifundefined {infer}{\let \infer \mpr@infer}{}
+
+\def \TirNameStyle #1{\small \textsc{#1}}
+\def \tir@name #1{\hbox {\small \TirNameStyle{#1}}}
+\let \TirName \tir@name
+\let \DefTirName \TirName
+\let \RefTirName \TirName
+
+%%% Other Exports
+
+% \let \listcons \mpr@cons
+% \let \listsnoc \mpr@snoc
+% \let \listhead \mpr@head
+% \let \listmake \mpr@makelist
+
+
+
+
+\endinput
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/style.sty Tue Mar 03 11:00:51 2009 +0100
@@ -0,0 +1,46 @@
+%% toc
+\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
+\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
+
+%% references
+\newcommand{\secref}[1]{\S\ref{#1}}
+\newcommand{\chref}[1]{chapter~\ref{#1}}
+\newcommand{\figref}[1]{figure~\ref{#1}}
+
+%% math
+\newcommand{\text}[1]{\mbox{#1}}
+\newcommand{\isasymvartheta}{\isamath{\theta}}
+\newcommand{\isactrlvec}[1]{\emph{$\overline{#1}$}}
+
+\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
+
+\pagestyle{headings}
+\sloppy
+\binperiod
+\underscoreon
+
+\renewcommand{\isadigit}[1]{\isamath{#1}}
+
+\newenvironment{mldecls}{\par\noindent\begingroup\footnotesize\def\isanewline{\\}\begin{tabular}{l}}{\end{tabular}\smallskip\endgroup}
+
+\isafoldtag{FIXME}
+\isakeeptag{mlref}
+\renewcommand{\isatagmlref}{\subsection*{\makebox[0pt][r]{\fbox{\ML}~~}Reference}\begingroup\def\isastyletext{\rm}\small}
+\renewcommand{\endisatagmlref}{\endgroup}
+
+\newcommand{\isasymGUESS}{\isakeyword{guess}}
+\newcommand{\isasymOBTAIN}{\isakeyword{obtain}}
+\newcommand{\isasymTHEORY}{\isakeyword{theory}}
+\newcommand{\isasymUSES}{\isakeyword{uses}}
+\newcommand{\isasymEND}{\isakeyword{end}}
+\newcommand{\isasymCONSTS}{\isakeyword{consts}}
+\newcommand{\isasymDEFS}{\isakeyword{defs}}
+\newcommand{\isasymTHEOREM}{\isakeyword{theorem}}
+\newcommand{\isasymDEFINITION}{\isakeyword{definition}}
+
+\isabellestyle{it}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "implementation"
+%%% End:
--- a/doc-src/IsarAdvanced/Classes/IsaMakefile Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,33 +0,0 @@
-
-## targets
-
-default: Thy
-images:
-test: Thy
-
-all: images test
-
-
-## global settings
-
-SRC = $(ISABELLE_HOME)/src
-OUT = $(ISABELLE_OUTPUT)
-LOG = $(OUT)/log
-
-USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
-
-
-## Thy
-
-THY = $(LOG)/HOL-Thy.gz
-
-Thy: $(THY)
-
-$(THY): Thy/ROOT.ML Thy/Setup.thy Thy/Classes.thy
- @$(USEDIR) HOL Thy
-
-
-## clean
-
-clean:
- @rm -f $(THY)
--- a/doc-src/IsarAdvanced/Classes/Makefile Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,38 +0,0 @@
-#
-# $Id$
-#
-
-## targets
-
-default: dvi
-
-
-## dependencies
-
-include ../Makefile.in
-
-NAME = classes
-
-FILES = $(NAME).tex classes.tex Thy/document/Classes.tex \
- style.sty ../../iman.sty ../../extra.sty ../../isar.sty \
- ../../isabelle.sty ../../isabellesym.sty ../../pdfsetup.sty \
- ../../manual.bib ../../proof.sty
-
-dvi: $(NAME).dvi
-
-$(NAME).dvi: $(FILES) isabelle_isar.eps
- $(LATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(LATEX) $(NAME)
- $(LATEX) $(NAME)
-
-pdf: $(NAME).pdf
-
-$(NAME).pdf: $(FILES) isabelle_isar.pdf
- $(PDFLATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(FIXBOOKMARKS) $(NAME).out
- $(PDFLATEX) $(NAME)
- $(PDFLATEX) $(NAME)
--- a/doc-src/IsarAdvanced/Classes/Thy/Classes.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,635 +0,0 @@
-theory Classes
-imports Main Setup
-begin
-
-chapter {* Haskell-style classes with Isabelle/Isar *}
-
-section {* Introduction *}
-
-text {*
- Type classes were introduces by Wadler and Blott \cite{wadler89how}
- into the Haskell language, to allow for a reasonable implementation
- of overloading\footnote{throughout this tutorial, we are referring
- to classical Haskell 1.0 type classes, not considering
- later additions in expressiveness}.
- As a canonical example, a polymorphic equality function
- @{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} which is overloaded on different
- types for @{text "\<alpha>"}, which is achieved by splitting introduction
- of the @{text eq} function from its overloaded definitions by means
- of @{text class} and @{text instance} declarations:
-
- \begin{quote}
-
- \noindent@{text "class eq where"}\footnote{syntax here is a kind of isabellized Haskell} \\
- \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
-
- \medskip\noindent@{text "instance nat \<Colon> eq where"} \\
- \hspace*{2ex}@{text "eq 0 0 = True"} \\
- \hspace*{2ex}@{text "eq 0 _ = False"} \\
- \hspace*{2ex}@{text "eq _ 0 = False"} \\
- \hspace*{2ex}@{text "eq (Suc n) (Suc m) = eq n m"}
-
- \medskip\noindent@{text "instance (\<alpha>\<Colon>eq, \<beta>\<Colon>eq) pair \<Colon> eq where"} \\
- \hspace*{2ex}@{text "eq (x1, y1) (x2, y2) = eq x1 x2 \<and> eq y1 y2"}
-
- \medskip\noindent@{text "class ord extends eq where"} \\
- \hspace*{2ex}@{text "less_eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
- \hspace*{2ex}@{text "less \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
-
- \end{quote}
-
- \noindent Type variables are annotated with (finitely many) classes;
- these annotations are assertions that a particular polymorphic type
- provides definitions for overloaded functions.
-
- Indeed, type classes not only allow for simple overloading
- but form a generic calculus, an instance of order-sorted
- algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
-
- From a software engeneering point of view, type classes
- roughly correspond to interfaces in object-oriented languages like Java;
- so, it is naturally desirable that type classes do not only
- provide functions (class parameters) but also state specifications
- implementations must obey. For example, the @{text "class eq"}
- above could be given the following specification, demanding that
- @{text "class eq"} is an equivalence relation obeying reflexivity,
- symmetry and transitivity:
-
- \begin{quote}
-
- \noindent@{text "class eq where"} \\
- \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
- @{text "satisfying"} \\
- \hspace*{2ex}@{text "refl: eq x x"} \\
- \hspace*{2ex}@{text "sym: eq x y \<longleftrightarrow> eq x y"} \\
- \hspace*{2ex}@{text "trans: eq x y \<and> eq y z \<longrightarrow> eq x z"}
-
- \end{quote}
-
- \noindent From a theoretic point of view, type classes are lightweight
- modules; Haskell type classes may be emulated by
- SML functors \cite{classes_modules}.
- Isabelle/Isar offers a discipline of type classes which brings
- all those aspects together:
-
- \begin{enumerate}
- \item specifying abstract parameters together with
- corresponding specifications,
- \item instantiating those abstract parameters by a particular
- type
- \item in connection with a ``less ad-hoc'' approach to overloading,
- \item with a direct link to the Isabelle module system
- (aka locales \cite{kammueller-locales}).
- \end{enumerate}
-
- \noindent Isar type classes also directly support code generation
- in a Haskell like fashion.
-
- This tutorial demonstrates common elements of structured specifications
- and abstract reasoning with type classes by the algebraic hierarchy of
- semigroups, monoids and groups. Our background theory is that of
- Isabelle/HOL \cite{isa-tutorial}, for which some
- familiarity is assumed.
-
- Here we merely present the look-and-feel for end users.
- Internally, those are mapped to more primitive Isabelle concepts.
- See \cite{Haftmann-Wenzel:2006:classes} for more detail.
-*}
-
-section {* A simple algebra example \label{sec:example} *}
-
-subsection {* Class definition *}
-
-text {*
- Depending on an arbitrary type @{text "\<alpha>"}, class @{text
- "semigroup"} introduces a binary operator @{text "(\<otimes>)"} that is
- assumed to be associative:
-*}
-
-class %quote semigroup =
- fixes mult :: "\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" (infixl "\<otimes>" 70)
- assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
-
-text {*
- \noindent This @{command class} specification consists of two
- parts: the \qn{operational} part names the class parameter
- (@{element "fixes"}), the \qn{logical} part specifies properties on them
- (@{element "assumes"}). The local @{element "fixes"} and
- @{element "assumes"} are lifted to the theory toplevel,
- yielding the global
- parameter @{term [source] "mult \<Colon> \<alpha>\<Colon>semigroup \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"} and the
- global theorem @{fact "semigroup.assoc:"}~@{prop [source] "\<And>x y
- z \<Colon> \<alpha>\<Colon>semigroup. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"}.
-*}
-
-
-subsection {* Class instantiation \label{sec:class_inst} *}
-
-text {*
- The concrete type @{typ int} is made a @{class semigroup}
- instance by providing a suitable definition for the class parameter
- @{text "(\<otimes>)"} and a proof for the specification of @{fact assoc}.
- This is accomplished by the @{command instantiation} target:
-*}
-
-instantiation %quote int :: semigroup
-begin
-
-definition %quote
- mult_int_def: "i \<otimes> j = i + (j\<Colon>int)"
-
-instance %quote proof
- fix i j k :: int have "(i + j) + k = i + (j + k)" by simp
- then show "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
- unfolding mult_int_def .
-qed
-
-end %quote
-
-text {*
- \noindent @{command instantiation} allows to define class parameters
- at a particular instance using common specification tools (here,
- @{command definition}). The concluding @{command instance}
- opens a proof that the given parameters actually conform
- to the class specification. Note that the first proof step
- is the @{method default} method,
- which for such instance proofs maps to the @{method intro_classes} method.
- This boils down an instance judgement to the relevant primitive
- proof goals and should conveniently always be the first method applied
- in an instantiation proof.
-
- From now on, the type-checker will consider @{typ int}
- as a @{class semigroup} automatically, i.e.\ any general results
- are immediately available on concrete instances.
-
- \medskip Another instance of @{class semigroup} are the natural numbers:
-*}
-
-instantiation %quote nat :: semigroup
-begin
-
-primrec %quote mult_nat where
- "(0\<Colon>nat) \<otimes> n = n"
- | "Suc m \<otimes> n = Suc (m \<otimes> n)"
-
-instance %quote proof
- fix m n q :: nat
- show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
- by (induct m) auto
-qed
-
-end %quote
-
-text {*
- \noindent Note the occurence of the name @{text mult_nat}
- in the primrec declaration; by default, the local name of
- a class operation @{text f} to instantiate on type constructor
- @{text \<kappa>} are mangled as @{text f_\<kappa>}. In case of uncertainty,
- these names may be inspected using the @{command "print_context"} command
- or the corresponding ProofGeneral button.
-*}
-
-subsection {* Lifting and parametric types *}
-
-text {*
- Overloaded definitions giving on class instantiation
- may include recursion over the syntactic structure of types.
- As a canonical example, we model product semigroups
- using our simple algebra:
-*}
-
-instantiation %quote * :: (semigroup, semigroup) semigroup
-begin
-
-definition %quote
- mult_prod_def: "p\<^isub>1 \<otimes> p\<^isub>2 = (fst p\<^isub>1 \<otimes> fst p\<^isub>2, snd p\<^isub>1 \<otimes> snd p\<^isub>2)"
-
-instance %quote proof
- fix p\<^isub>1 p\<^isub>2 p\<^isub>3 :: "\<alpha>\<Colon>semigroup \<times> \<beta>\<Colon>semigroup"
- show "p\<^isub>1 \<otimes> p\<^isub>2 \<otimes> p\<^isub>3 = p\<^isub>1 \<otimes> (p\<^isub>2 \<otimes> p\<^isub>3)"
- unfolding mult_prod_def by (simp add: assoc)
-qed
-
-end %quote
-
-text {*
- \noindent Associativity from product semigroups is
- established using
- the definition of @{text "(\<otimes>)"} on products and the hypothetical
- associativity of the type components; these hypotheses
- are facts due to the @{class semigroup} constraints imposed
- on the type components by the @{command instance} proposition.
- Indeed, this pattern often occurs with parametric types
- and type classes.
-*}
-
-
-subsection {* Subclassing *}
-
-text {*
- We define a subclass @{text monoidl} (a semigroup with a left-hand neutral)
- by extending @{class semigroup}
- with one additional parameter @{text neutral} together
- with its property:
-*}
-
-class %quote monoidl = semigroup +
- fixes neutral :: "\<alpha>" ("\<one>")
- assumes neutl: "\<one> \<otimes> x = x"
-
-text {*
- \noindent Again, we prove some instances, by
- providing suitable parameter definitions and proofs for the
- additional specifications. Observe that instantiations
- for types with the same arity may be simultaneous:
-*}
-
-instantiation %quote nat and int :: monoidl
-begin
-
-definition %quote
- neutral_nat_def: "\<one> = (0\<Colon>nat)"
-
-definition %quote
- neutral_int_def: "\<one> = (0\<Colon>int)"
-
-instance %quote proof
- fix n :: nat
- show "\<one> \<otimes> n = n"
- unfolding neutral_nat_def by simp
-next
- fix k :: int
- show "\<one> \<otimes> k = k"
- unfolding neutral_int_def mult_int_def by simp
-qed
-
-end %quote
-
-instantiation %quote * :: (monoidl, monoidl) monoidl
-begin
-
-definition %quote
- neutral_prod_def: "\<one> = (\<one>, \<one>)"
-
-instance %quote proof
- fix p :: "\<alpha>\<Colon>monoidl \<times> \<beta>\<Colon>monoidl"
- show "\<one> \<otimes> p = p"
- unfolding neutral_prod_def mult_prod_def by (simp add: neutl)
-qed
-
-end %quote
-
-text {*
- \noindent Fully-fledged monoids are modelled by another subclass
- which does not add new parameters but tightens the specification:
-*}
-
-class %quote monoid = monoidl +
- assumes neutr: "x \<otimes> \<one> = x"
-
-instantiation %quote nat and int :: monoid
-begin
-
-instance %quote proof
- fix n :: nat
- show "n \<otimes> \<one> = n"
- unfolding neutral_nat_def by (induct n) simp_all
-next
- fix k :: int
- show "k \<otimes> \<one> = k"
- unfolding neutral_int_def mult_int_def by simp
-qed
-
-end %quote
-
-instantiation %quote * :: (monoid, monoid) monoid
-begin
-
-instance %quote proof
- fix p :: "\<alpha>\<Colon>monoid \<times> \<beta>\<Colon>monoid"
- show "p \<otimes> \<one> = p"
- unfolding neutral_prod_def mult_prod_def by (simp add: neutr)
-qed
-
-end %quote
-
-text {*
- \noindent To finish our small algebra example, we add a @{text group} class
- with a corresponding instance:
-*}
-
-class %quote group = monoidl +
- fixes inverse :: "\<alpha> \<Rightarrow> \<alpha>" ("(_\<div>)" [1000] 999)
- assumes invl: "x\<div> \<otimes> x = \<one>"
-
-instantiation %quote int :: group
-begin
-
-definition %quote
- inverse_int_def: "i\<div> = - (i\<Colon>int)"
-
-instance %quote proof
- fix i :: int
- have "-i + i = 0" by simp
- then show "i\<div> \<otimes> i = \<one>"
- unfolding mult_int_def neutral_int_def inverse_int_def .
-qed
-
-end %quote
-
-
-section {* Type classes as locales *}
-
-subsection {* A look behind the scene *}
-
-text {*
- The example above gives an impression how Isar type classes work
- in practice. As stated in the introduction, classes also provide
- a link to Isar's locale system. Indeed, the logical core of a class
- is nothing else than a locale:
-*}
-
-class %quote idem =
- fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
- assumes idem: "f (f x) = f x"
-
-text {*
- \noindent essentially introduces the locale
-*} setup %invisible {* Sign.add_path "foo" *}
-
-locale %quote idem =
- fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
- assumes idem: "f (f x) = f x"
-
-text {* \noindent together with corresponding constant(s): *}
-
-consts %quote f :: "\<alpha> \<Rightarrow> \<alpha>"
-
-text {*
- \noindent The connection to the type system is done by means
- of a primitive axclass
-*} setup %invisible {* Sign.add_path "foo" *}
-
-axclass %quote idem < type
- idem: "f (f x) = f x" setup %invisible {* Sign.parent_path *}
-
-text {* \noindent together with a corresponding interpretation: *}
-
-interpretation %quote idem_class:
- idem "f \<Colon> (\<alpha>\<Colon>idem) \<Rightarrow> \<alpha>"
-proof qed (rule idem)
-
-text {*
- \noindent This gives you at hand the full power of the Isabelle module system;
- conclusions in locale @{text idem} are implicitly propagated
- to class @{text idem}.
-*} setup %invisible {* Sign.parent_path *}
-
-subsection {* Abstract reasoning *}
-
-text {*
- Isabelle locales enable reasoning at a general level, while results
- are implicitly transferred to all instances. For example, we can
- now establish the @{text "left_cancel"} lemma for groups, which
- states that the function @{text "(x \<otimes>)"} is injective:
-*}
-
-lemma %quote (in group) left_cancel: "x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"
-proof
- assume "x \<otimes> y = x \<otimes> z"
- then have "x\<div> \<otimes> (x \<otimes> y) = x\<div> \<otimes> (x \<otimes> z)" by simp
- then have "(x\<div> \<otimes> x) \<otimes> y = (x\<div> \<otimes> x) \<otimes> z" using assoc by simp
- then show "y = z" using neutl and invl by simp
-next
- assume "y = z"
- then show "x \<otimes> y = x \<otimes> z" by simp
-qed
-
-text {*
- \noindent Here the \qt{@{keyword "in"} @{class group}} target specification
- indicates that the result is recorded within that context for later
- use. This local theorem is also lifted to the global one @{fact
- "group.left_cancel:"} @{prop [source] "\<And>x y z \<Colon> \<alpha>\<Colon>group. x \<otimes> y = x \<otimes>
- z \<longleftrightarrow> y = z"}. Since type @{text "int"} has been made an instance of
- @{text "group"} before, we may refer to that fact as well: @{prop
- [source] "\<And>x y z \<Colon> int. x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"}.
-*}
-
-
-subsection {* Derived definitions *}
-
-text {*
- Isabelle locales support a concept of local definitions
- in locales:
-*}
-
-primrec %quote (in monoid) pow_nat :: "nat \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
- "pow_nat 0 x = \<one>"
- | "pow_nat (Suc n) x = x \<otimes> pow_nat n x"
-
-text {*
- \noindent If the locale @{text group} is also a class, this local
- definition is propagated onto a global definition of
- @{term [source] "pow_nat \<Colon> nat \<Rightarrow> \<alpha>\<Colon>monoid \<Rightarrow> \<alpha>\<Colon>monoid"}
- with corresponding theorems
-
- @{thm pow_nat.simps [no_vars]}.
-
- \noindent As you can see from this example, for local
- definitions you may use any specification tool
- which works together with locales (e.g. \cite{krauss2006}).
-*}
-
-
-subsection {* A functor analogy *}
-
-text {*
- We introduced Isar classes by analogy to type classes
- functional programming; if we reconsider this in the
- context of what has been said about type classes and locales,
- we can drive this analogy further by stating that type
- classes essentially correspond to functors which have
- a canonical interpretation as type classes.
- Anyway, there is also the possibility of other interpretations.
- For example, also @{text list}s form a monoid with
- @{text append} and @{term "[]"} as operations, but it
- seems inappropriate to apply to lists
- the same operations as for genuinely algebraic types.
- In such a case, we simply can do a particular interpretation
- of monoids for lists:
-*}
-
-interpretation %quote list_monoid!: monoid append "[]"
- proof qed auto
-
-text {*
- \noindent This enables us to apply facts on monoids
- to lists, e.g. @{thm list_monoid.neutl [no_vars]}.
-
- When using this interpretation pattern, it may also
- be appropriate to map derived definitions accordingly:
-*}
-
-primrec %quote replicate :: "nat \<Rightarrow> \<alpha> list \<Rightarrow> \<alpha> list" where
- "replicate 0 _ = []"
- | "replicate (Suc n) xs = xs @ replicate n xs"
-
-interpretation %quote list_monoid!: monoid append "[]" where
- "monoid.pow_nat append [] = replicate"
-proof -
- interpret monoid append "[]" ..
- show "monoid.pow_nat append [] = replicate"
- proof
- fix n
- show "monoid.pow_nat append [] n = replicate n"
- by (induct n) auto
- qed
-qed intro_locales
-
-
-subsection {* Additional subclass relations *}
-
-text {*
- Any @{text "group"} is also a @{text "monoid"}; this
- can be made explicit by claiming an additional
- subclass relation,
- together with a proof of the logical difference:
-*}
-
-subclass %quote (in group) monoid
-proof
- fix x
- from invl have "x\<div> \<otimes> x = \<one>" by simp
- with assoc [symmetric] neutl invl have "x\<div> \<otimes> (x \<otimes> \<one>) = x\<div> \<otimes> x" by simp
- with left_cancel show "x \<otimes> \<one> = x" by simp
-qed
-
-text {*
- \noindent The logical proof is carried out on the locale level.
- Afterwards it is propagated
- to the type system, making @{text group} an instance of
- @{text monoid} by adding an additional edge
- to the graph of subclass relations
- (cf.\ \figref{fig:subclass}).
-
- \begin{figure}[htbp]
- \begin{center}
- \small
- \unitlength 0.6mm
- \begin{picture}(40,60)(0,0)
- \put(20,60){\makebox(0,0){@{text semigroup}}}
- \put(20,40){\makebox(0,0){@{text monoidl}}}
- \put(00,20){\makebox(0,0){@{text monoid}}}
- \put(40,00){\makebox(0,0){@{text group}}}
- \put(20,55){\vector(0,-1){10}}
- \put(15,35){\vector(-1,-1){10}}
- \put(25,35){\vector(1,-3){10}}
- \end{picture}
- \hspace{8em}
- \begin{picture}(40,60)(0,0)
- \put(20,60){\makebox(0,0){@{text semigroup}}}
- \put(20,40){\makebox(0,0){@{text monoidl}}}
- \put(00,20){\makebox(0,0){@{text monoid}}}
- \put(40,00){\makebox(0,0){@{text group}}}
- \put(20,55){\vector(0,-1){10}}
- \put(15,35){\vector(-1,-1){10}}
- \put(05,15){\vector(3,-1){30}}
- \end{picture}
- \caption{Subclass relationship of monoids and groups:
- before and after establishing the relationship
- @{text "group \<subseteq> monoid"}; transitive edges are left out.}
- \label{fig:subclass}
- \end{center}
- \end{figure}
-7
- For illustration, a derived definition
- in @{text group} which uses @{text pow_nat}:
-*}
-
-definition %quote (in group) pow_int :: "int \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
- "pow_int k x = (if k >= 0
- then pow_nat (nat k) x
- else (pow_nat (nat (- k)) x)\<div>)"
-
-text {*
- \noindent yields the global definition of
- @{term [source] "pow_int \<Colon> int \<Rightarrow> \<alpha>\<Colon>group \<Rightarrow> \<alpha>\<Colon>group"}
- with the corresponding theorem @{thm pow_int_def [no_vars]}.
-*}
-
-subsection {* A note on syntax *}
-
-text {*
- As a commodity, class context syntax allows to refer
- to local class operations and their global counterparts
- uniformly; type inference resolves ambiguities. For example:
-*}
-
-context %quote semigroup
-begin
-
-term %quote "x \<otimes> y" -- {* example 1 *}
-term %quote "(x\<Colon>nat) \<otimes> y" -- {* example 2 *}
-
-end %quote
-
-term %quote "x \<otimes> y" -- {* example 3 *}
-
-text {*
- \noindent Here in example 1, the term refers to the local class operation
- @{text "mult [\<alpha>]"}, whereas in example 2 the type constraint
- enforces the global class operation @{text "mult [nat]"}.
- In the global context in example 3, the reference is
- to the polymorphic global class operation @{text "mult [?\<alpha> \<Colon> semigroup]"}.
-*}
-
-section {* Further issues *}
-
-subsection {* Type classes and code generation *}
-
-text {*
- Turning back to the first motivation for type classes,
- namely overloading, it is obvious that overloading
- stemming from @{command class} statements and
- @{command instantiation}
- targets naturally maps to Haskell type classes.
- The code generator framework \cite{isabelle-codegen}
- takes this into account. Concerning target languages
- lacking type classes (e.g.~SML), type classes
- are implemented by explicit dictionary construction.
- As example, let's go back to the power function:
-*}
-
-definition %quote example :: int where
- "example = pow_int 10 (-2)"
-
-text {*
- \noindent This maps to Haskell as:
-*}
-
-text %quote {*@{code_stmts example (Haskell)}*}
-
-text {*
- \noindent The whole code in SML with explicit dictionary passing:
-*}
-
-text %quote {*@{code_stmts example (SML)}*}
-
-subsection {* Inspecting the type class universe *}
-
-text {*
- To facilitate orientation in complex subclass structures,
- two diagnostics commands are provided:
-
- \begin{description}
-
- \item[@{command "print_classes"}] print a list of all classes
- together with associated operations etc.
-
- \item[@{command "class_deps"}] visualizes the subclass relation
- between all classes as a Hasse diagram.
-
- \end{description}
-*}
-
-end
--- a/doc-src/IsarAdvanced/Classes/Thy/ROOT.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,6 +0,0 @@
-
-(* $Id$ *)
-
-no_document use_thy "Setup";
-
-use_thy "Classes";
--- a/doc-src/IsarAdvanced/Classes/Thy/Setup.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,34 +0,0 @@
-theory Setup
-imports Main Code_Integer
-uses
- "../../../antiquote_setup"
- "../../../more_antiquote"
-begin
-
-ML {* Code_Target.code_width := 74 *}
-
-syntax
- "_alpha" :: "type" ("\<alpha>")
- "_alpha_ofsort" :: "sort \<Rightarrow> type" ("\<alpha>()\<Colon>_" [0] 1000)
- "_beta" :: "type" ("\<beta>")
- "_beta_ofsort" :: "sort \<Rightarrow> type" ("\<beta>()\<Colon>_" [0] 1000)
-
-parse_ast_translation {*
- let
- fun alpha_ast_tr [] = Syntax.Variable "'a"
- | alpha_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
- fun alpha_ofsort_ast_tr [ast] =
- Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'a", ast]
- | alpha_ofsort_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
- fun beta_ast_tr [] = Syntax.Variable "'b"
- | beta_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
- fun beta_ofsort_ast_tr [ast] =
- Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'b", ast]
- | beta_ofsort_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
- in [
- ("_alpha", alpha_ast_tr), ("_alpha_ofsort", alpha_ofsort_ast_tr),
- ("_beta", beta_ast_tr), ("_beta_ofsort", beta_ofsort_ast_tr)
- ] end
-*}
-
-end
\ No newline at end of file
--- a/doc-src/IsarAdvanced/Classes/Thy/document/Classes.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1346 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Classes}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Classes\isanewline
-\isakeyword{imports}\ Main\ Setup\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupchapter{Haskell-style classes with Isabelle/Isar%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Introduction%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Type classes were introduces by Wadler and Blott \cite{wadler89how}
- into the Haskell language, to allow for a reasonable implementation
- of overloading\footnote{throughout this tutorial, we are referring
- to classical Haskell 1.0 type classes, not considering
- later additions in expressiveness}.
- As a canonical example, a polymorphic equality function
- \isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} which is overloaded on different
- types for \isa{{\isasymalpha}}, which is achieved by splitting introduction
- of the \isa{eq} function from its overloaded definitions by means
- of \isa{class} and \isa{instance} declarations:
-
- \begin{quote}
-
- \noindent\isa{class\ eq\ where}\footnote{syntax here is a kind of isabellized Haskell} \\
- \hspace*{2ex}\isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool}
-
- \medskip\noindent\isa{instance\ nat\ {\isasymColon}\ eq\ where} \\
- \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isadigit{0}}\ {\isacharequal}\ True} \\
- \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isacharunderscore}\ {\isacharequal}\ False} \\
- \hspace*{2ex}\isa{eq\ {\isacharunderscore}\ {\isadigit{0}}\ {\isacharequal}\ False} \\
- \hspace*{2ex}\isa{eq\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharparenleft}Suc\ m{\isacharparenright}\ {\isacharequal}\ eq\ n\ m}
-
- \medskip\noindent\isa{instance\ {\isacharparenleft}{\isasymalpha}{\isasymColon}eq{\isacharcomma}\ {\isasymbeta}{\isasymColon}eq{\isacharparenright}\ pair\ {\isasymColon}\ eq\ where} \\
- \hspace*{2ex}\isa{eq\ {\isacharparenleft}x{\isadigit{1}}{\isacharcomma}\ y{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ eq\ x{\isadigit{1}}\ x{\isadigit{2}}\ {\isasymand}\ eq\ y{\isadigit{1}}\ y{\isadigit{2}}}
-
- \medskip\noindent\isa{class\ ord\ extends\ eq\ where} \\
- \hspace*{2ex}\isa{less{\isacharunderscore}eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} \\
- \hspace*{2ex}\isa{less\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool}
-
- \end{quote}
-
- \noindent Type variables are annotated with (finitely many) classes;
- these annotations are assertions that a particular polymorphic type
- provides definitions for overloaded functions.
-
- Indeed, type classes not only allow for simple overloading
- but form a generic calculus, an instance of order-sorted
- algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
-
- From a software engeneering point of view, type classes
- roughly correspond to interfaces in object-oriented languages like Java;
- so, it is naturally desirable that type classes do not only
- provide functions (class parameters) but also state specifications
- implementations must obey. For example, the \isa{class\ eq}
- above could be given the following specification, demanding that
- \isa{class\ eq} is an equivalence relation obeying reflexivity,
- symmetry and transitivity:
-
- \begin{quote}
-
- \noindent\isa{class\ eq\ where} \\
- \hspace*{2ex}\isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} \\
- \isa{satisfying} \\
- \hspace*{2ex}\isa{refl{\isacharcolon}\ eq\ x\ x} \\
- \hspace*{2ex}\isa{sym{\isacharcolon}\ eq\ x\ y\ {\isasymlongleftrightarrow}\ eq\ x\ y} \\
- \hspace*{2ex}\isa{trans{\isacharcolon}\ eq\ x\ y\ {\isasymand}\ eq\ y\ z\ {\isasymlongrightarrow}\ eq\ x\ z}
-
- \end{quote}
-
- \noindent From a theoretic point of view, type classes are lightweight
- modules; Haskell type classes may be emulated by
- SML functors \cite{classes_modules}.
- Isabelle/Isar offers a discipline of type classes which brings
- all those aspects together:
-
- \begin{enumerate}
- \item specifying abstract parameters together with
- corresponding specifications,
- \item instantiating those abstract parameters by a particular
- type
- \item in connection with a ``less ad-hoc'' approach to overloading,
- \item with a direct link to the Isabelle module system
- (aka locales \cite{kammueller-locales}).
- \end{enumerate}
-
- \noindent Isar type classes also directly support code generation
- in a Haskell like fashion.
-
- This tutorial demonstrates common elements of structured specifications
- and abstract reasoning with type classes by the algebraic hierarchy of
- semigroups, monoids and groups. Our background theory is that of
- Isabelle/HOL \cite{isa-tutorial}, for which some
- familiarity is assumed.
-
- Here we merely present the look-and-feel for end users.
- Internally, those are mapped to more primitive Isabelle concepts.
- See \cite{Haftmann-Wenzel:2006:classes} for more detail.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{A simple algebra example \label{sec:example}%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Class definition%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Depending on an arbitrary type \isa{{\isasymalpha}}, class \isa{semigroup} introduces a binary operator \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} that is
- assumed to be associative:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{class}\isamarkupfalse%
-\ semigroup\ {\isacharequal}\isanewline
-\ \ \isakeyword{fixes}\ mult\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymotimes}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} specification consists of two
- parts: the \qn{operational} part names the class parameter
- (\hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}}), the \qn{logical} part specifies properties on them
- (\hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}}). The local \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}} and
- \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}} are lifted to the theory toplevel,
- yielding the global
- parameter \isa{{\isachardoublequote}mult\ {\isasymColon}\ {\isasymalpha}{\isasymColon}semigroup\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequote}} and the
- global theorem \hyperlink{fact.semigroup.assoc:}{\mbox{\isa{semigroup{\isachardot}assoc{\isacharcolon}}}}~\isa{{\isachardoublequote}{\isasymAnd}x\ y\ z\ {\isasymColon}\ {\isasymalpha}{\isasymColon}semigroup{\isachardot}\ {\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequote}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Class instantiation \label{sec:class_inst}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The concrete type \isa{int} is made a \isa{semigroup}
- instance by providing a suitable definition for the class parameter
- \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} and a proof for the specification of \hyperlink{fact.assoc}{\mbox{\isa{assoc}}}.
- This is accomplished by the \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} target:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{instantiation}\isamarkupfalse%
-\ int\ {\isacharcolon}{\isacharcolon}\ semigroup\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ mult{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymotimes}\ j\ {\isacharequal}\ i\ {\isacharplus}\ {\isacharparenleft}j{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ i\ j\ k\ {\isacharcolon}{\isacharcolon}\ int\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isacharplus}\ j{\isacharparenright}\ {\isacharplus}\ k\ {\isacharequal}\ i\ {\isacharplus}\ {\isacharparenleft}j\ {\isacharplus}\ k{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isasymotimes}\ j{\isacharparenright}\ {\isasymotimes}\ k\ {\isacharequal}\ i\ {\isasymotimes}\ {\isacharparenleft}j\ {\isasymotimes}\ k{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} allows to define class parameters
- at a particular instance using common specification tools (here,
- \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}). The concluding \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}
- opens a proof that the given parameters actually conform
- to the class specification. Note that the first proof step
- is the \hyperlink{method.default}{\mbox{\isa{default}}} method,
- which for such instance proofs maps to the \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}} method.
- This boils down an instance judgement to the relevant primitive
- proof goals and should conveniently always be the first method applied
- in an instantiation proof.
-
- From now on, the type-checker will consider \isa{int}
- as a \isa{semigroup} automatically, i.e.\ any general results
- are immediately available on concrete instances.
-
- \medskip Another instance of \isa{semigroup} are the natural numbers:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{instantiation}\isamarkupfalse%
-\ nat\ {\isacharcolon}{\isacharcolon}\ semigroup\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{primrec}\isamarkupfalse%
-\ mult{\isacharunderscore}nat\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}Suc\ m\ {\isasymotimes}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}m\ {\isasymotimes}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ m\ n\ q\ {\isacharcolon}{\isacharcolon}\ nat\ \isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ {\isasymotimes}\ n\ {\isasymotimes}\ q\ {\isacharequal}\ m\ {\isasymotimes}\ {\isacharparenleft}n\ {\isasymotimes}\ q{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ m{\isacharparenright}\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Note the occurence of the name \isa{mult{\isacharunderscore}nat}
- in the primrec declaration; by default, the local name of
- a class operation \isa{f} to instantiate on type constructor
- \isa{{\isasymkappa}} are mangled as \isa{f{\isacharunderscore}{\isasymkappa}}. In case of uncertainty,
- these names may be inspected using the \hyperlink{command.print-context}{\mbox{\isa{\isacommand{print{\isacharunderscore}context}}}} command
- or the corresponding ProofGeneral button.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Lifting and parametric types%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Overloaded definitions giving on class instantiation
- may include recursion over the syntactic structure of types.
- As a canonical example, we model product semigroups
- using our simple algebra:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{instantiation}\isamarkupfalse%
-\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ mult{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{2}}\ {\isacharequal}\ {\isacharparenleft}fst\ p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ fst\ p\isactrlisub {\isadigit{2}}{\isacharcomma}\ snd\ p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ snd\ p\isactrlisub {\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ p\isactrlisub {\isadigit{1}}\ p\isactrlisub {\isadigit{2}}\ p\isactrlisub {\isadigit{3}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}semigroup\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}semigroup{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{2}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{3}}\ {\isacharequal}\ p\isactrlisub {\isadigit{1}}\ {\isasymotimes}\ {\isacharparenleft}p\isactrlisub {\isadigit{2}}\ {\isasymotimes}\ p\isactrlisub {\isadigit{3}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ \ \ \ \ \ \isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Associativity from product semigroups is
- established using
- the definition of \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} on products and the hypothetical
- associativity of the type components; these hypotheses
- are facts due to the \isa{semigroup} constraints imposed
- on the type components by the \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} proposition.
- Indeed, this pattern often occurs with parametric types
- and type classes.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Subclassing%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-We define a subclass \isa{monoidl} (a semigroup with a left-hand neutral)
- by extending \isa{semigroup}
- with one additional parameter \isa{neutral} together
- with its property:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{class}\isamarkupfalse%
-\ monoidl\ {\isacharequal}\ semigroup\ {\isacharplus}\isanewline
-\ \ \isakeyword{fixes}\ neutral\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ neutl{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Again, we prove some instances, by
- providing suitable parameter definitions and proofs for the
- additional specifications. Observe that instantiations
- for types with the same arity may be simultaneous:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{instantiation}\isamarkupfalse%
-\ nat\ \isakeyword{and}\ int\ {\isacharcolon}{\isacharcolon}\ monoidl\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ neutral{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ n\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ neutral{\isacharunderscore}nat{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ k\ {\isacharcolon}{\isacharcolon}\ int\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ k\ {\isacharequal}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ neutral{\isacharunderscore}int{\isacharunderscore}def\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}monoidl{\isacharcomma}\ monoidl{\isacharparenright}\ monoidl\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ neutral{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isasymone}{\isacharcomma}\ {\isasymone}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ p\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}monoidl\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}monoidl{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ neutral{\isacharunderscore}prod{\isacharunderscore}def\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutl{\isacharparenright}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Fully-fledged monoids are modelled by another subclass
- which does not add new parameters but tightens the specification:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{class}\isamarkupfalse%
-\ monoid\ {\isacharequal}\ monoidl\ {\isacharplus}\isanewline
-\ \ \isakeyword{assumes}\ neutr{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ nat\ \isakeyword{and}\ int\ {\isacharcolon}{\isacharcolon}\ monoid\ \isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ n\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ neutral{\isacharunderscore}nat{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ n{\isacharparenright}\ simp{\isacharunderscore}all\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ k\ {\isacharcolon}{\isacharcolon}\ int\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}k\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ k{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ neutral{\isacharunderscore}int{\isacharunderscore}def\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}monoid{\isacharcomma}\ monoid{\isacharparenright}\ monoid\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\ \isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ p\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}monoid\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}monoid{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}p\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ neutral{\isacharunderscore}prod{\isacharunderscore}def\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutr{\isacharparenright}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent To finish our small algebra example, we add a \isa{group} class
- with a corresponding instance:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{class}\isamarkupfalse%
-\ group\ {\isacharequal}\ monoidl\ {\isacharplus}\isanewline
-\ \ \isakeyword{fixes}\ inverse\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}{\isachardoublequoteopen}{\isacharparenleft}{\isacharunderscore}{\isasymdiv}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ invl{\isacharcolon}\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ int\ {\isacharcolon}{\isacharcolon}\ group\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ inverse{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}i{\isasymdiv}\ {\isacharequal}\ {\isacharminus}\ {\isacharparenleft}i{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ i\ {\isacharcolon}{\isacharcolon}\ int\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharminus}i\ {\isacharplus}\ i\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}i{\isasymdiv}\ {\isasymotimes}\ i\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ mult{\isacharunderscore}int{\isacharunderscore}def\ neutral{\isacharunderscore}int{\isacharunderscore}def\ inverse{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isamarkupsection{Type classes as locales%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{A look behind the scene%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The example above gives an impression how Isar type classes work
- in practice. As stated in the introduction, classes also provide
- a link to Isar's locale system. Indeed, the logical core of a class
- is nothing else than a locale:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{class}\isamarkupfalse%
-\ idem\ {\isacharequal}\isanewline
-\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent essentially introduces the locale%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadeliminvisible
-\ %
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{setup}\isamarkupfalse%
-\ {\isacharverbatimopen}\ Sign{\isachardot}add{\isacharunderscore}path\ {\isachardoublequote}foo{\isachardoublequote}\ {\isacharverbatimclose}%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-\isanewline
-%
-\isadelimquote
-\isanewline
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{locale}\isamarkupfalse%
-\ idem\ {\isacharequal}\isanewline
-\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent together with corresponding constant(s):%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{consts}\isamarkupfalse%
-\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The connection to the type system is done by means
- of a primitive axclass%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadeliminvisible
-\ %
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{setup}\isamarkupfalse%
-\ {\isacharverbatimopen}\ Sign{\isachardot}add{\isacharunderscore}path\ {\isachardoublequote}foo{\isachardoublequote}\ {\isacharverbatimclose}%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-\isanewline
-%
-\isadelimquote
-\isanewline
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{axclass}\isamarkupfalse%
-\ idem\ {\isacharless}\ type\isanewline
-\ \ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isadeliminvisible
-\ %
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{setup}\isamarkupfalse%
-\ {\isacharverbatimopen}\ Sign{\isachardot}parent{\isacharunderscore}path\ {\isacharverbatimclose}%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\begin{isamarkuptext}%
-\noindent together with a corresponding interpretation:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{interpretation}\isamarkupfalse%
-\ idem{\isacharunderscore}class{\isacharcolon}\isanewline
-\ \ idem\ {\isachardoublequoteopen}f\ {\isasymColon}\ {\isacharparenleft}{\isasymalpha}{\isasymColon}idem{\isacharparenright}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ \isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}rule\ idem{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This gives you at hand the full power of the Isabelle module system;
- conclusions in locale \isa{idem} are implicitly propagated
- to class \isa{idem}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadeliminvisible
-\ %
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{setup}\isamarkupfalse%
-\ {\isacharverbatimopen}\ Sign{\isachardot}parent{\isacharunderscore}path\ {\isacharverbatimclose}%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isamarkupsubsection{Abstract reasoning%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Isabelle locales enable reasoning at a general level, while results
- are implicitly transferred to all instances. For example, we can
- now establish the \isa{left{\isacharunderscore}cancel} lemma for groups, which
- states that the function \isa{{\isacharparenleft}x\ {\isasymotimes}{\isacharparenright}} is injective:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ left{\isacharunderscore}cancel{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z\ {\isasymlongleftrightarrow}\ y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isacharequal}\ x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasymdiv}\ {\isasymotimes}\ x{\isacharparenright}\ {\isasymotimes}\ y\ {\isacharequal}\ {\isacharparenleft}x{\isasymdiv}\ {\isasymotimes}\ x{\isacharparenright}\ {\isasymotimes}\ z{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ assoc\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
-\ neutl\ \isakeyword{and}\ invl\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Here the \qt{\hyperlink{keyword.in}{\mbox{\isa{\isakeyword{in}}}} \isa{group}} target specification
- indicates that the result is recorded within that context for later
- use. This local theorem is also lifted to the global one \hyperlink{fact.group.left-cancel:}{\mbox{\isa{group{\isachardot}left{\isacharunderscore}cancel{\isacharcolon}}}} \isa{{\isachardoublequote}{\isasymAnd}x\ y\ z\ {\isasymColon}\ {\isasymalpha}{\isasymColon}group{\isachardot}\ x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z\ {\isasymlongleftrightarrow}\ y\ {\isacharequal}\ z{\isachardoublequote}}. Since type \isa{int} has been made an instance of
- \isa{group} before, we may refer to that fact as well: \isa{{\isachardoublequote}{\isasymAnd}x\ y\ z\ {\isasymColon}\ int{\isachardot}\ x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z\ {\isasymlongleftrightarrow}\ y\ {\isacharequal}\ z{\isachardoublequote}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Derived definitions%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Isabelle locales support a concept of local definitions
- in locales:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{primrec}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ monoid{\isacharparenright}\ pow{\isacharunderscore}nat\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pow{\isacharunderscore}nat\ {\isadigit{0}}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}pow{\isacharunderscore}nat\ {\isacharparenleft}Suc\ n{\isacharparenright}\ x\ {\isacharequal}\ x\ {\isasymotimes}\ pow{\isacharunderscore}nat\ n\ x{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent If the locale \isa{group} is also a class, this local
- definition is propagated onto a global definition of
- \isa{{\isachardoublequote}pow{\isacharunderscore}nat\ {\isasymColon}\ nat\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}monoid\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}monoid{\isachardoublequote}}
- with corresponding theorems
-
- \isa{pow{\isacharunderscore}nat\ {\isadigit{0}}\ x\ {\isacharequal}\ {\isasymone}\isasep\isanewline%
-pow{\isacharunderscore}nat\ {\isacharparenleft}Suc\ n{\isacharparenright}\ x\ {\isacharequal}\ x\ {\isasymotimes}\ pow{\isacharunderscore}nat\ n\ x}.
-
- \noindent As you can see from this example, for local
- definitions you may use any specification tool
- which works together with locales (e.g. \cite{krauss2006}).%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{A functor analogy%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-We introduced Isar classes by analogy to type classes
- functional programming; if we reconsider this in the
- context of what has been said about type classes and locales,
- we can drive this analogy further by stating that type
- classes essentially correspond to functors which have
- a canonical interpretation as type classes.
- Anyway, there is also the possibility of other interpretations.
- For example, also \isa{list}s form a monoid with
- \isa{append} and \isa{{\isacharbrackleft}{\isacharbrackright}} as operations, but it
- seems inappropriate to apply to lists
- the same operations as for genuinely algebraic types.
- In such a case, we simply can do a particular interpretation
- of monoids for lists:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{interpretation}\isamarkupfalse%
-\ list{\isacharunderscore}monoid{\isacharbang}{\isacharcolon}\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ \isacommand{qed}\isamarkupfalse%
-\ auto%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This enables us to apply facts on monoids
- to lists, e.g. \isa{{\isacharbrackleft}{\isacharbrackright}\ {\isacharat}\ x\ {\isacharequal}\ x}.
-
- When using this interpretation pattern, it may also
- be appropriate to map derived definitions accordingly:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{primrec}\isamarkupfalse%
-\ replicate\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isasymalpha}\ list\ {\isasymRightarrow}\ {\isasymalpha}\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}replicate\ {\isadigit{0}}\ {\isacharunderscore}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}replicate\ {\isacharparenleft}Suc\ n{\isacharparenright}\ xs\ {\isacharequal}\ xs\ {\isacharat}\ replicate\ n\ xs{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{interpretation}\isamarkupfalse%
-\ list{\isacharunderscore}monoid{\isacharbang}{\isacharcolon}\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ replicate{\isachardoublequoteclose}\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}\isanewline
-\ \ \isacommand{interpret}\isamarkupfalse%
-\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ replicate{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ n\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ n\ {\isacharequal}\ replicate\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ n{\isacharparenright}\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ intro{\isacharunderscore}locales%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isamarkupsubsection{Additional subclass relations%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Any \isa{group} is also a \isa{monoid}; this
- can be made explicit by claiming an additional
- subclass relation,
- together with a proof of the logical difference:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{subclass}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ monoid\isanewline
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ x\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ invl\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ assoc\ {\isacharbrackleft}symmetric{\isacharbrackright}\ neutl\ invl\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ {\isasymone}{\isacharparenright}\ {\isacharequal}\ x{\isasymdiv}\ {\isasymotimes}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ left{\isacharunderscore}cancel\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The logical proof is carried out on the locale level.
- Afterwards it is propagated
- to the type system, making \isa{group} an instance of
- \isa{monoid} by adding an additional edge
- to the graph of subclass relations
- (cf.\ \figref{fig:subclass}).
-
- \begin{figure}[htbp]
- \begin{center}
- \small
- \unitlength 0.6mm
- \begin{picture}(40,60)(0,0)
- \put(20,60){\makebox(0,0){\isa{semigroup}}}
- \put(20,40){\makebox(0,0){\isa{monoidl}}}
- \put(00,20){\makebox(0,0){\isa{monoid}}}
- \put(40,00){\makebox(0,0){\isa{group}}}
- \put(20,55){\vector(0,-1){10}}
- \put(15,35){\vector(-1,-1){10}}
- \put(25,35){\vector(1,-3){10}}
- \end{picture}
- \hspace{8em}
- \begin{picture}(40,60)(0,0)
- \put(20,60){\makebox(0,0){\isa{semigroup}}}
- \put(20,40){\makebox(0,0){\isa{monoidl}}}
- \put(00,20){\makebox(0,0){\isa{monoid}}}
- \put(40,00){\makebox(0,0){\isa{group}}}
- \put(20,55){\vector(0,-1){10}}
- \put(15,35){\vector(-1,-1){10}}
- \put(05,15){\vector(3,-1){30}}
- \end{picture}
- \caption{Subclass relationship of monoids and groups:
- before and after establishing the relationship
- \isa{group\ {\isasymsubseteq}\ monoid}; transitive edges left out.}
- \label{fig:subclass}
- \end{center}
- \end{figure}
-7
- For illustration, a derived definition
- in \isa{group} which uses \isa{pow{\isacharunderscore}nat}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{definition}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ pow{\isacharunderscore}int\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}int\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pow{\isacharunderscore}int\ k\ x\ {\isacharequal}\ {\isacharparenleft}if\ k\ {\isachargreater}{\isacharequal}\ {\isadigit{0}}\isanewline
-\ \ \ \ then\ pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ k{\isacharparenright}\ x\isanewline
-\ \ \ \ else\ {\isacharparenleft}pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ {\isacharparenleft}{\isacharminus}\ k{\isacharparenright}{\isacharparenright}\ x{\isacharparenright}{\isasymdiv}{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent yields the global definition of
- \isa{{\isachardoublequote}pow{\isacharunderscore}int\ {\isasymColon}\ int\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}group\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}group{\isachardoublequote}}
- with the corresponding theorem \isa{pow{\isacharunderscore}int\ k\ x\ {\isacharequal}\ {\isacharparenleft}if\ {\isadigit{0}}\ {\isasymle}\ k\ then\ pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ k{\isacharparenright}\ x\ else\ {\isacharparenleft}pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ {\isacharparenleft}{\isacharminus}\ k{\isacharparenright}{\isacharparenright}\ x{\isacharparenright}{\isasymdiv}{\isacharparenright}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{A note on syntax%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-As a commodity, class context syntax allows to refer
- to local class operations and their global counterparts
- uniformly; type inference resolves ambiguities. For example:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{context}\isamarkupfalse%
-\ semigroup\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{term}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
-\isamarkupcmt{example 1%
-}
-\isanewline
-\isacommand{term}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasymColon}nat{\isacharparenright}\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
-\isamarkupcmt{example 2%
-}
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{term}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
-\isamarkupcmt{example 3%
-}
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Here in example 1, the term refers to the local class operation
- \isa{mult\ {\isacharbrackleft}{\isasymalpha}{\isacharbrackright}}, whereas in example 2 the type constraint
- enforces the global class operation \isa{mult\ {\isacharbrackleft}nat{\isacharbrackright}}.
- In the global context in example 3, the reference is
- to the polymorphic global class operation \isa{mult\ {\isacharbrackleft}{\isacharquery}{\isasymalpha}\ {\isasymColon}\ semigroup{\isacharbrackright}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Further issues%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Type classes and code generation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Turning back to the first motivation for type classes,
- namely overloading, it is obvious that overloading
- stemming from \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} statements and
- \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}}
- targets naturally maps to Haskell type classes.
- The code generator framework \cite{isabelle-codegen}
- takes this into account. Concerning target languages
- lacking type classes (e.g.~SML), type classes
- are implemented by explicit dictionary construction.
- As example, let's go back to the power function:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{definition}\isamarkupfalse%
-\ example\ {\isacharcolon}{\isacharcolon}\ int\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}example\ {\isacharequal}\ pow{\isacharunderscore}int\ {\isadigit{1}}{\isadigit{0}}\ {\isacharparenleft}{\isacharminus}{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This maps to Haskell as:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}module Example where {\char123}\\
-\hspace*{0pt}\\
-\hspace*{0pt}\\
-\hspace*{0pt}data Nat = Zero{\char95}nat | Suc Nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}nat{\char95}aux ::~Integer -> Nat -> Nat;\\
-\hspace*{0pt}nat{\char95}aux i n = (if i <= 0 then n else nat{\char95}aux (i - 1) (Suc n));\\
-\hspace*{0pt}\\
-\hspace*{0pt}nat ::~Integer -> Nat;\\
-\hspace*{0pt}nat i = nat{\char95}aux i Zero{\char95}nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}class Semigroup a where {\char123}\\
-\hspace*{0pt} ~mult ::~a -> a -> a;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}class (Semigroup a) => Monoidl a where {\char123}\\
-\hspace*{0pt} ~neutral ::~a;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}class (Monoidl a) => Monoid a where {\char123}\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}class (Monoid a) => Group a where {\char123}\\
-\hspace*{0pt} ~inverse ::~a -> a;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}inverse{\char95}int ::~Integer -> Integer;\\
-\hspace*{0pt}inverse{\char95}int i = negate i;\\
-\hspace*{0pt}\\
-\hspace*{0pt}neutral{\char95}int ::~Integer;\\
-\hspace*{0pt}neutral{\char95}int = 0;\\
-\hspace*{0pt}\\
-\hspace*{0pt}mult{\char95}int ::~Integer -> Integer -> Integer;\\
-\hspace*{0pt}mult{\char95}int i j = i + j;\\
-\hspace*{0pt}\\
-\hspace*{0pt}instance Semigroup Integer where {\char123}\\
-\hspace*{0pt} ~mult = mult{\char95}int;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}instance Monoidl Integer where {\char123}\\
-\hspace*{0pt} ~neutral = neutral{\char95}int;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}instance Monoid Integer where {\char123}\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}instance Group Integer where {\char123}\\
-\hspace*{0pt} ~inverse = inverse{\char95}int;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}pow{\char95}nat ::~forall a.~(Monoid a) => Nat -> a -> a;\\
-\hspace*{0pt}pow{\char95}nat Zero{\char95}nat x = neutral;\\
-\hspace*{0pt}pow{\char95}nat (Suc n) x = mult x (pow{\char95}nat n x);\\
-\hspace*{0pt}\\
-\hspace*{0pt}pow{\char95}int ::~forall a.~(Group a) => Integer -> a -> a;\\
-\hspace*{0pt}pow{\char95}int k x =\\
-\hspace*{0pt} ~(if 0 <= k then pow{\char95}nat (nat k) x\\
-\hspace*{0pt} ~~~else inverse (pow{\char95}nat (nat (negate k)) x));\\
-\hspace*{0pt}\\
-\hspace*{0pt}example ::~Integer;\\
-\hspace*{0pt}example = pow{\char95}int 10 (-2);\\
-\hspace*{0pt}\\
-\hspace*{0pt}{\char125}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The whole code in SML with explicit dictionary passing:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun nat{\char95}aux i n =\\
-\hspace*{0pt} ~(if IntInf.<= (i,~(0 :~IntInf.int)) then n\\
-\hspace*{0pt} ~~~else nat{\char95}aux (IntInf.- (i,~(1 :~IntInf.int))) (Suc n));\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun nat i = nat{\char95}aux i Zero{\char95}nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a semigroup = {\char123}mult :~'a -> 'a -> 'a{\char125};\\
-\hspace*{0pt}fun mult (A{\char95}:'a semigroup) = {\char35}mult A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a monoidl =\\
-\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}semigroup{\char95}monoidl :~'a semigroup,~neutral :~'a{\char125};\\
-\hspace*{0pt}fun semigroup{\char95}monoidl (A{\char95}:'a monoidl) = {\char35}Classes{\char95}{\char95}semigroup{\char95}monoidl A{\char95};\\
-\hspace*{0pt}fun neutral (A{\char95}:'a monoidl) = {\char35}neutral A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a monoid = {\char123}Classes{\char95}{\char95}monoidl{\char95}monoid :~'a monoidl{\char125};\\
-\hspace*{0pt}fun monoidl{\char95}monoid (A{\char95}:'a monoid) = {\char35}Classes{\char95}{\char95}monoidl{\char95}monoid A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a group = {\char123}Classes{\char95}{\char95}monoid{\char95}group :~'a monoid,~inverse :~'a -> 'a{\char125};\\
-\hspace*{0pt}fun monoid{\char95}group (A{\char95}:'a group) = {\char35}Classes{\char95}{\char95}monoid{\char95}group A{\char95};\\
-\hspace*{0pt}fun inverse (A{\char95}:'a group) = {\char35}inverse A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun inverse{\char95}int i = IntInf.{\char126}~i;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val neutral{\char95}int :~IntInf.int = (0 :~IntInf.int)\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun mult{\char95}int i j = IntInf.+ (i,~j);\\
-\hspace*{0pt}\\
-\hspace*{0pt}val semigroup{\char95}int = {\char123}mult = mult{\char95}int{\char125}~:~IntInf.int semigroup;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val monoidl{\char95}int =\\
-\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}semigroup{\char95}monoidl = semigroup{\char95}int,~neutral = neutral{\char95}int{\char125}~:\\
-\hspace*{0pt} ~IntInf.int monoidl;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val monoid{\char95}int = {\char123}Classes{\char95}{\char95}monoidl{\char95}monoid = monoidl{\char95}int{\char125}~:\\
-\hspace*{0pt} ~IntInf.int monoid;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val group{\char95}int =\\
-\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}monoid{\char95}group = monoid{\char95}int,~inverse = inverse{\char95}int{\char125}~:\\
-\hspace*{0pt} ~IntInf.int group;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun pow{\char95}nat A{\char95}~Zero{\char95}nat x = neutral (monoidl{\char95}monoid A{\char95})\\
-\hspace*{0pt} ~| pow{\char95}nat A{\char95}~(Suc n) x =\\
-\hspace*{0pt} ~~~mult ((semigroup{\char95}monoidl o monoidl{\char95}monoid) A{\char95}) x (pow{\char95}nat A{\char95}~n x);\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun pow{\char95}int A{\char95}~k x =\\
-\hspace*{0pt} ~(if IntInf.<= ((0 :~IntInf.int),~k)\\
-\hspace*{0pt} ~~~then pow{\char95}nat (monoid{\char95}group A{\char95}) (nat k) x\\
-\hspace*{0pt} ~~~else inverse A{\char95}~(pow{\char95}nat (monoid{\char95}group A{\char95}) (nat (IntInf.{\char126}~k)) x));\\
-\hspace*{0pt}\\
-\hspace*{0pt}val example :~IntInf.int =\\
-\hspace*{0pt} ~pow{\char95}int group{\char95}int (10 :~IntInf.int) ({\char126}2 :~IntInf.int)\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isamarkupsubsection{Inspecting the type class universe%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-To facilitate orientation in complex subclass structures,
- two diagnostics commands are provided:
-
- \begin{description}
-
- \item[\hyperlink{command.print-classes}{\mbox{\isa{\isacommand{print{\isacharunderscore}classes}}}}] print a list of all classes
- together with associated operations etc.
-
- \item[\hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isacharunderscore}deps}}}}] visualizes the subclass relation
- between all classes as a Hasse diagram.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Classes/classes.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,50 +0,0 @@
-
-\documentclass[12pt,a4paper,fleqn]{report}
-\usepackage{latexsym,graphicx}
-\usepackage[refpage]{nomencl}
-\usepackage{../../iman,../../extra,../../isar,../../proof}
-\usepackage{../../isabelle,../../isabellesym}
-\usepackage{style}
-\usepackage{../../pdfsetup}
-
-
-\hyphenation{Isabelle}
-\hyphenation{Isar}
-\isadroptag{theory}
-
-\title{\includegraphics[scale=0.5]{isabelle_isar}
- \\[4ex] Haskell-style type classes with Isabelle/Isar}
-\author{\emph{Florian Haftmann}}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
- This tutorial introduces the look-and-feel of Isar type classes
- to the end-user; Isar type classes are a convenient mechanism
- for organizing specifications, overcoming some drawbacks
- of raw axiomatic type classes. Essentially, they combine
- an operational aspect (in the manner of Haskell) with
- a logical aspect, both managed uniformly.
-\end{abstract}
-
-\thispagestyle{empty}\clearpage
-
-\pagenumbering{roman}
-\clearfirst
-
-\input{Thy/document/Classes.tex}
-
-\begingroup
-\bibliographystyle{plain} \small\raggedright\frenchspacing
-\bibliography{../../manual}
-\endgroup
-
-\end{document}
-
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
--- a/doc-src/IsarAdvanced/Classes/style.sty Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,43 +0,0 @@
-
-%% toc
-\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
-\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
-
-%% references
-\newcommand{\secref}[1]{\S\ref{#1}}
-\newcommand{\figref}[1]{figure~\ref{#1}}
-
-%% logical markup
-\newcommand{\strong}[1]{{\bfseries {#1}}}
-\newcommand{\qn}[1]{\emph{#1}}
-
-%% typographic conventions
-\newcommand{\qt}[1]{``{#1}''}
-
-%% verbatim text
-\newcommand{\isatypewriter}{\fontsize{9pt}{0pt}\tt\renewcommand{\baselinestretch}{1}\setlength{\baselineskip}{9pt}}
-
-%% quoted segments
-\makeatletter
-\isakeeptag{quote}
-\newenvironment{quotesegment}{\begin{quote}\isa@parindent\parindent\parindent0pt\isa@parskip\parskip\parskip0pt}{\end{quote}}
-\renewcommand{\isatagquote}{\begin{quotesegment}}
-\renewcommand{\endisatagquote}{\end{quotesegment}}
-\makeatother
-
-%% presentation
-\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
-
-\pagestyle{headings}
-\binperiod
-\underscoreoff
-
-\renewcommand{\isadigit}[1]{\isamath{#1}}
-
-\isabellestyle{it}
-
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "implementation"
-%%% End:
--- a/doc-src/IsarAdvanced/Codegen/IsaMakefile Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,33 +0,0 @@
-
-## targets
-
-default: Thy
-images:
-test: Thy
-
-all: images test
-
-
-## global settings
-
-SRC = $(ISABELLE_HOME)/src
-OUT = $(ISABELLE_OUTPUT)
-LOG = $(OUT)/log
-
-USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
-
-
-## Thy
-
-THY = $(LOG)/HOL-Thy.gz
-
-Thy: $(THY)
-
-$(THY): Thy/ROOT.ML Thy/*.thy ../../antiquote_setup.ML
- @$(USEDIR) HOL Thy
-
-
-## clean
-
-clean:
- @rm -f $(THY)
--- a/doc-src/IsarAdvanced/Codegen/Makefile Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,38 +0,0 @@
-#
-# $Id$
-#
-
-## targets
-
-default: dvi
-
-
-## dependencies
-
-include ../Makefile.in
-
-NAME = codegen
-
-FILES = $(NAME).tex Thy/document/*.tex \
- style.sty ../../iman.sty ../../extra.sty ../../isar.sty \
- ../../isabelle.sty ../../isabellesym.sty ../../pdfsetup.sty \
- ../../manual.bib ../../proof.sty
-
-dvi: $(NAME).dvi
-
-$(NAME).dvi: $(FILES) isabelle_isar.eps codegen_process.ps
- $(LATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(LATEX) $(NAME)
- $(LATEX) $(NAME)
-
-pdf: $(NAME).pdf
-
-$(NAME).pdf: $(FILES) isabelle_isar.pdf codegen_process.pdf
- $(PDFLATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(FIXBOOKMARKS) $(NAME).out
- $(PDFLATEX) $(NAME)
- $(PDFLATEX) $(NAME)
--- a/doc-src/IsarAdvanced/Codegen/Thy/Adaption.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,361 +0,0 @@
-theory Adaption
-imports Setup
-begin
-
-setup %invisible {* Code_Target.extend_target ("\<SML>", ("SML", K I)) *}
-
-section {* Adaption to target languages \label{sec:adaption} *}
-
-subsection {* Adapting code generation *}
-
-text {*
- The aspects of code generation introduced so far have two aspects
- in common:
-
- \begin{itemize}
- \item They act uniformly, without reference to a specific
- target language.
- \item They are \emph{safe} in the sense that as long as you trust
- the code generator meta theory and implementation, you cannot
- produce programs that yield results which are not derivable
- in the logic.
- \end{itemize}
-
- \noindent In this section we will introduce means to \emph{adapt} the serialiser
- to a specific target language, i.e.~to print program fragments
- in a way which accommodates \qt{already existing} ingredients of
- a target language environment, for three reasons:
-
- \begin{itemize}
- \item improving readability and aesthetics of generated code
- \item gaining efficiency
- \item interface with language parts which have no direct counterpart
- in @{text "HOL"} (say, imperative data structures)
- \end{itemize}
-
- \noindent Generally, you should avoid using those features yourself
- \emph{at any cost}:
-
- \begin{itemize}
- \item The safe configuration methods act uniformly on every target language,
- whereas for adaption you have to treat each target language separate.
- \item Application is extremely tedious since there is no abstraction
- which would allow for a static check, making it easy to produce garbage.
- \item More or less subtle errors can be introduced unconsciously.
- \end{itemize}
-
- \noindent However, even if you ought refrain from setting up adaption
- yourself, already the @{text "HOL"} comes with some reasonable default
- adaptions (say, using target language list syntax). There also some
- common adaption cases which you can setup by importing particular
- library theories. In order to understand these, we provide some clues here;
- these however are not supposed to replace a careful study of the sources.
-*}
-
-subsection {* The adaption principle *}
-
-text {*
- The following figure illustrates what \qt{adaption} is conceptually
- supposed to be:
-
- \begin{figure}[here]
- \begin{tikzpicture}[scale = 0.5]
- \tikzstyle water=[color = blue, thick]
- \tikzstyle ice=[color = black, very thick, cap = round, join = round, fill = white]
- \tikzstyle process=[color = green, semithick, ->]
- \tikzstyle adaption=[color = red, semithick, ->]
- \tikzstyle target=[color = black]
- \foreach \x in {0, ..., 24}
- \draw[style=water] (\x, 0.25) sin + (0.25, 0.25) cos + (0.25, -0.25) sin
- + (0.25, -0.25) cos + (0.25, 0.25);
- \draw[style=ice] (1, 0) --
- (3, 6) node[above, fill=white] {logic} -- (5, 0) -- cycle;
- \draw[style=ice] (9, 0) --
- (11, 6) node[above, fill=white] {intermediate language} -- (13, 0) -- cycle;
- \draw[style=ice] (15, -6) --
- (19, 6) node[above, fill=white] {target language} -- (23, -6) -- cycle;
- \draw[style=process]
- (3.5, 3) .. controls (7, 5) .. node[fill=white] {translation} (10.5, 3);
- \draw[style=process]
- (11.5, 3) .. controls (15, 5) .. node[fill=white] (serialisation) {serialisation} (18.5, 3);
- \node (adaption) at (11, -2) [style=adaption] {adaption};
- \node at (19, 3) [rotate=90] {generated};
- \node at (19.5, -5) {language};
- \node at (19.5, -3) {library};
- \node (includes) at (19.5, -1) {includes};
- \node (reserved) at (16.5, -3) [rotate=72] {reserved}; % proper 71.57
- \draw[style=process]
- (includes) -- (serialisation);
- \draw[style=process]
- (reserved) -- (serialisation);
- \draw[style=adaption]
- (adaption) -- (serialisation);
- \draw[style=adaption]
- (adaption) -- (includes);
- \draw[style=adaption]
- (adaption) -- (reserved);
- \end{tikzpicture}
- \caption{The adaption principle}
- \label{fig:adaption}
- \end{figure}
-
- \noindent In the tame view, code generation acts as broker between
- @{text logic}, @{text "intermediate language"} and
- @{text "target language"} by means of @{text translation} and
- @{text serialisation}; for the latter, the serialiser has to observe
- the structure of the @{text language} itself plus some @{text reserved}
- keywords which have to be avoided for generated code.
- However, if you consider @{text adaption} mechanisms, the code generated
- by the serializer is just the tip of the iceberg:
-
- \begin{itemize}
- \item @{text serialisation} can be \emph{parametrised} such that
- logical entities are mapped to target-specific ones
- (e.g. target-specific list syntax,
- see also \secref{sec:adaption_mechanisms})
- \item Such parametrisations can involve references to a
- target-specific standard @{text library} (e.g. using
- the @{text Haskell} @{verbatim Maybe} type instead
- of the @{text HOL} @{type "option"} type);
- if such are used, the corresponding identifiers
- (in our example, @{verbatim Maybe}, @{verbatim Nothing}
- and @{verbatim Just}) also have to be considered @{text reserved}.
- \item Even more, the user can enrich the library of the
- target-language by providing code snippets
- (\qt{@{text "includes"}}) which are prepended to
- any generated code (see \secref{sec:include}); this typically
- also involves further @{text reserved} identifiers.
- \end{itemize}
-
- \noindent As figure \ref{fig:adaption} illustrates, all these adaption mechanisms
- have to act consistently; it is at the discretion of the user
- to take care for this.
-*}
-
-subsection {* Common adaption patterns *}
-
-text {*
- The @{theory HOL} @{theory Main} theory already provides a code
- generator setup
- which should be suitable for most applications. Common extensions
- and modifications are available by certain theories of the @{text HOL}
- library; beside being useful in applications, they may serve
- as a tutorial for customising the code generator setup (see below
- \secref{sec:adaption_mechanisms}).
-
- \begin{description}
-
- \item[@{theory "Code_Integer"}] represents @{text HOL} integers by big
- integer literals in target languages.
- \item[@{theory "Code_Char"}] represents @{text HOL} characters by
- character literals in target languages.
- \item[@{theory "Code_Char_chr"}] like @{text "Code_Char"},
- but also offers treatment of character codes; includes
- @{theory "Code_Char"}.
- \item[@{theory "Efficient_Nat"}] \label{eff_nat} implements natural numbers by integers,
- which in general will result in higher efficiency; pattern
- matching with @{term "0\<Colon>nat"} / @{const "Suc"}
- is eliminated; includes @{theory "Code_Integer"}
- and @{theory "Code_Index"}.
- \item[@{theory "Code_Index"}] provides an additional datatype
- @{typ index} which is mapped to target-language built-in integers.
- Useful for code setups which involve e.g. indexing of
- target-language arrays.
- \item[@{theory "Code_Message"}] provides an additional datatype
- @{typ message_string} which is isomorphic to strings;
- @{typ message_string}s are mapped to target-language strings.
- Useful for code setups which involve e.g. printing (error) messages.
-
- \end{description}
-
- \begin{warn}
- When importing any of these theories, they should form the last
- items in an import list. Since these theories adapt the
- code generator setup in a non-conservative fashion,
- strange effects may occur otherwise.
- \end{warn}
-*}
-
-
-subsection {* Parametrising serialisation \label{sec:adaption_mechanisms} *}
-
-text {*
- Consider the following function and its corresponding
- SML code:
-*}
-
-primrec %quote in_interval :: "nat \<times> nat \<Rightarrow> nat \<Rightarrow> bool" where
- "in_interval (k, l) n \<longleftrightarrow> k \<le> n \<and> n \<le> l"
-(*<*)
-code_type %invisible bool
- (SML)
-code_const %invisible True and False and "op \<and>" and Not
- (SML and and and)
-(*>*)
-text %quote {*@{code_stmts in_interval (SML)}*}
-
-text {*
- \noindent Though this is correct code, it is a little bit unsatisfactory:
- boolean values and operators are materialised as distinguished
- entities with have nothing to do with the SML-built-in notion
- of \qt{bool}. This results in less readable code;
- additionally, eager evaluation may cause programs to
- loop or break which would perfectly terminate when
- the existing SML @{verbatim "bool"} would be used. To map
- the HOL @{typ bool} on SML @{verbatim "bool"}, we may use
- \qn{custom serialisations}:
-*}
-
-code_type %quotett bool
- (SML "bool")
-code_const %quotett True and False and "op \<and>"
- (SML "true" and "false" and "_ andalso _")
-
-text {*
- \noindent The @{command code_type} command takes a type constructor
- as arguments together with a list of custom serialisations.
- Each custom serialisation starts with a target language
- identifier followed by an expression, which during
- code serialisation is inserted whenever the type constructor
- would occur. For constants, @{command code_const} implements
- the corresponding mechanism. Each ``@{verbatim "_"}'' in
- a serialisation expression is treated as a placeholder
- for the type constructor's (the constant's) arguments.
-*}
-
-text %quote {*@{code_stmts in_interval (SML)}*}
-
-text {*
- \noindent This still is not perfect: the parentheses
- around the \qt{andalso} expression are superfluous.
- Though the serialiser
- by no means attempts to imitate the rich Isabelle syntax
- framework, it provides some common idioms, notably
- associative infixes with precedences which may be used here:
-*}
-
-code_const %quotett "op \<and>"
- (SML infixl 1 "andalso")
-
-text %quote {*@{code_stmts in_interval (SML)}*}
-
-text {*
- \noindent The attentive reader may ask how we assert that no generated
- code will accidentally overwrite. For this reason the serialiser has
- an internal table of identifiers which have to be avoided to be used
- for new declarations. Initially, this table typically contains the
- keywords of the target language. It can be extended manually, thus avoiding
- accidental overwrites, using the @{command "code_reserved"} command:
-*}
-
-code_reserved %quote "\<SML>" bool true false andalso
-
-text {*
- \noindent Next, we try to map HOL pairs to SML pairs, using the
- infix ``@{verbatim "*"}'' type constructor and parentheses:
-*}
-(*<*)
-code_type %invisible *
- (SML)
-code_const %invisible Pair
- (SML)
-(*>*)
-code_type %quotett *
- (SML infix 2 "*")
-code_const %quotett Pair
- (SML "!((_),/ (_))")
-
-text {*
- \noindent The initial bang ``@{verbatim "!"}'' tells the serialiser
- never to put
- parentheses around the whole expression (they are already present),
- while the parentheses around argument place holders
- tell not to put parentheses around the arguments.
- The slash ``@{verbatim "/"}'' (followed by arbitrary white space)
- inserts a space which may be used as a break if necessary
- during pretty printing.
-
- These examples give a glimpse what mechanisms
- custom serialisations provide; however their usage
- requires careful thinking in order not to introduce
- inconsistencies -- or, in other words:
- custom serialisations are completely axiomatic.
-
- A further noteworthy details is that any special
- character in a custom serialisation may be quoted
- using ``@{verbatim "'"}''; thus, in
- ``@{verbatim "fn '_ => _"}'' the first
- ``@{verbatim "_"}'' is a proper underscore while the
- second ``@{verbatim "_"}'' is a placeholder.
-*}
-
-
-subsection {* @{text Haskell} serialisation *}
-
-text {*
- For convenience, the default
- @{text HOL} setup for @{text Haskell} maps the @{class eq} class to
- its counterpart in @{text Haskell}, giving custom serialisations
- for the class @{class eq} (by command @{command code_class}) and its operation
- @{const HOL.eq}
-*}
-
-code_class %quotett eq
- (Haskell "Eq")
-
-code_const %quotett "op ="
- (Haskell infixl 4 "==")
-
-text {*
- \noindent A problem now occurs whenever a type which
- is an instance of @{class eq} in @{text HOL} is mapped
- on a @{text Haskell}-built-in type which is also an instance
- of @{text Haskell} @{text Eq}:
-*}
-
-typedecl %quote bar
-
-instantiation %quote bar :: eq
-begin
-
-definition %quote "eq_class.eq (x\<Colon>bar) y \<longleftrightarrow> x = y"
-
-instance %quote by default (simp add: eq_bar_def)
-
-end %quote
-
-code_type %quotett bar
- (Haskell "Integer")
-
-text {*
- \noindent The code generator would produce
- an additional instance, which of course is rejected by the @{text Haskell}
- compiler.
- To suppress this additional instance, use
- @{text "code_instance"}:
-*}
-
-code_instance %quotett bar :: eq
- (Haskell -)
-
-
-subsection {* Enhancing the target language context \label{sec:include} *}
-
-text {*
- In rare cases it is necessary to \emph{enrich} the context of a
- target language; this is accomplished using the @{command "code_include"}
- command:
-*}
-
-code_include %quotett Haskell "Errno"
-{*errno i = error ("Error number: " ++ show i)*}
-
-code_reserved %quotett Haskell Errno
-
-text {*
- \noindent Such named @{text include}s are then prepended to every generated code.
- Inspect such code in order to find out how @{command "code_include"} behaves
- with respect to a particular target language.
-*}
-
-end
--- a/doc-src/IsarAdvanced/Codegen/Thy/Codegen.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2 +0,0 @@
-
-end
--- a/doc-src/IsarAdvanced/Codegen/Thy/Further.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,113 +0,0 @@
-theory Further
-imports Setup
-begin
-
-section {* Further issues \label{sec:further} *}
-
-subsection {* Further reading *}
-
-text {*
- Do dive deeper into the issue of code generation, you should visit
- the Isabelle/Isar Reference Manual \cite{isabelle-isar-ref} which
- contains exhaustive syntax diagrams.
-*}
-
-subsection {* Modules *}
-
-text {*
- When invoking the @{command export_code} command it is possible to leave
- out the @{keyword "module_name"} part; then code is distributed over
- different modules, where the module name space roughly is induced
- by the @{text Isabelle} theory name space.
-
- Then sometimes the awkward situation occurs that dependencies between
- definitions introduce cyclic dependencies between modules, which in the
- @{text Haskell} world leaves you to the mercy of the @{text Haskell} implementation
- you are using, while for @{text SML}/@{text OCaml} code generation is not possible.
-
- A solution is to declare module names explicitly.
- Let use assume the three cyclically dependent
- modules are named \emph{A}, \emph{B} and \emph{C}.
- Then, by stating
-*}
-
-code_modulename %quote SML
- A ABC
- B ABC
- C ABC
-
-text {*
- we explicitly map all those modules on \emph{ABC},
- resulting in an ad-hoc merge of this three modules
- at serialisation time.
-*}
-
-subsection {* Evaluation oracle *}
-
-text {*
- Code generation may also be used to \emph{evaluate} expressions
- (using @{text SML} as target language of course).
- For instance, the @{command value} allows to reduce an expression to a
- normal form with respect to the underlying code equations:
-*}
-
-value %quote "42 / (12 :: rat)"
-
-text {*
- \noindent will display @{term "7 / (2 :: rat)"}.
-
- The @{method eval} method tries to reduce a goal by code generation to @{term True}
- and solves it in that case, but fails otherwise:
-*}
-
-lemma %quote "42 / (12 :: rat) = 7 / 2"
- by %quote eval
-
-text {*
- \noindent The soundness of the @{method eval} method depends crucially
- on the correctness of the code generator; this is one of the reasons
- why you should not use adaption (see \secref{sec:adaption}) frivolously.
-*}
-
-subsection {* Code antiquotation *}
-
-text {*
- In scenarios involving techniques like reflection it is quite common
- that code generated from a theory forms the basis for implementing
- a proof procedure in @{text SML}. To facilitate interfacing of generated code
- with system code, the code generator provides a @{text code} antiquotation:
-*}
-
-datatype %quote form = T | F | And form form | Or form form
-
-ML %quote {*
- fun eval_form @{code T} = true
- | eval_form @{code F} = false
- | eval_form (@{code And} (p, q)) =
- eval_form p andalso eval_form q
- | eval_form (@{code Or} (p, q)) =
- eval_form p orelse eval_form q;
-*}
-
-text {*
- \noindent @{text code} takes as argument the name of a constant; after the
- whole @{text SML} is read, the necessary code is generated transparently
- and the corresponding constant names are inserted. This technique also
- allows to use pattern matching on constructors stemming from compiled
- @{text datatypes}.
-
- For a less simplistic example, theory @{theory Ferrack} is
- a good reference.
-*}
-
-subsection {* Imperative data structures *}
-
-text {*
- If you consider imperative data structures as inevitable for a specific
- application, you should consider
- \emph{Imperative Functional Programming with Isabelle/HOL}
- (\cite{bulwahn-et-al:2008:imperative});
- the framework described there is available in theory @{theory Imperative_HOL}.
-*}
-
-end
--- a/doc-src/IsarAdvanced/Codegen/Thy/Introduction.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,206 +0,0 @@
-theory Introduction
-imports Setup
-begin
-
-chapter {* Code generation from @{text "Isabelle/HOL"} theories *}
-
-section {* Introduction and Overview *}
-
-text {*
- This tutorial introduces a generic code generator for the
- @{text Isabelle} system.
- Generic in the sense that the
- \qn{target language} for which code shall ultimately be
- generated is not fixed but may be an arbitrary state-of-the-art
- functional programming language (currently, the implementation
- supports @{text SML} \cite{SML}, @{text OCaml} \cite{OCaml} and @{text Haskell}
- \cite{haskell-revised-report}).
-
- Conceptually the code generator framework is part
- of Isabelle's @{theory Pure} meta logic framework; the logic
- @{theory HOL} which is an extension of @{theory Pure}
- already comes with a reasonable framework setup and thus provides
- a good working horse for raising code-generation-driven
- applications. So, we assume some familiarity and experience
- with the ingredients of the @{theory HOL} distribution theories.
- (see also \cite{isa-tutorial}).
-
- The code generator aims to be usable with no further ado
- in most cases while allowing for detailed customisation.
- This manifests in the structure of this tutorial: after a short
- conceptual introduction with an example (\secref{sec:intro}),
- we discuss the generic customisation facilities (\secref{sec:program}).
- A further section (\secref{sec:adaption}) is dedicated to the matter of
- \qn{adaption} to specific target language environments. After some
- further issues (\secref{sec:further}) we conclude with an overview
- of some ML programming interfaces (\secref{sec:ml}).
-
- \begin{warn}
- Ultimately, the code generator which this tutorial deals with
- is supposed to replace the existing code generator
- by Stefan Berghofer \cite{Berghofer-Nipkow:2002}.
- So, for the moment, there are two distinct code generators
- in Isabelle. In case of ambiguity, we will refer to the framework
- described here as @{text "generic code generator"}, to the
- other as @{text "SML code generator"}.
- Also note that while the framework itself is
- object-logic independent, only @{theory HOL} provides a reasonable
- framework setup.
- \end{warn}
-
-*}
-
-subsection {* Code generation via shallow embedding \label{sec:intro} *}
-
-text {*
- The key concept for understanding @{text Isabelle}'s code generation is
- \emph{shallow embedding}, i.e.~logical entities like constants, types and
- classes are identified with corresponding concepts in the target language.
-
- Inside @{theory HOL}, the @{command datatype} and
- @{command definition}/@{command primrec}/@{command fun} declarations form
- the core of a functional programming language. The default code generator setup
- allows to turn those into functional programs immediately.
- This means that \qt{naive} code generation can proceed without further ado.
- For example, here a simple \qt{implementation} of amortised queues:
-*}
-
-datatype %quote 'a queue = AQueue "'a list" "'a list"
-
-definition %quote empty :: "'a queue" where
- "empty = AQueue [] []"
-
-primrec %quote enqueue :: "'a \<Rightarrow> 'a queue \<Rightarrow> 'a queue" where
- "enqueue x (AQueue xs ys) = AQueue (x # xs) ys"
-
-fun %quote dequeue :: "'a queue \<Rightarrow> 'a option \<times> 'a queue" where
- "dequeue (AQueue [] []) = (None, AQueue [] [])"
- | "dequeue (AQueue xs (y # ys)) = (Some y, AQueue xs ys)"
- | "dequeue (AQueue xs []) =
- (case rev xs of y # ys \<Rightarrow> (Some y, AQueue [] ys))"
-
-text {* \noindent Then we can generate code e.g.~for @{text SML} as follows: *}
-
-export_code %quote empty dequeue enqueue in SML
- module_name Example file "examples/example.ML"
-
-text {* \noindent resulting in the following code: *}
-
-text %quote {*@{code_stmts empty enqueue dequeue (SML)}*}
-
-text {*
- \noindent The @{command export_code} command takes a space-separated list of
- constants for which code shall be generated; anything else needed for those
- is added implicitly. Then follows a target language identifier
- (@{text SML}, @{text OCaml} or @{text Haskell}) and a freely chosen module name.
- A file name denotes the destination to store the generated code. Note that
- the semantics of the destination depends on the target language: for
- @{text SML} and @{text OCaml} it denotes a \emph{file}, for @{text Haskell}
- it denotes a \emph{directory} where a file named as the module name
- (with extension @{text ".hs"}) is written:
-*}
-
-export_code %quote empty dequeue enqueue in Haskell
- module_name Example file "examples/"
-
-text {*
- \noindent This is how the corresponding code in @{text Haskell} looks like:
-*}
-
-text %quote {*@{code_stmts empty enqueue dequeue (Haskell)}*}
-
-text {*
- \noindent This demonstrates the basic usage of the @{command export_code} command;
- for more details see \secref{sec:further}.
-*}
-
-subsection {* Code generator architecture \label{sec:concept} *}
-
-text {*
- What you have seen so far should be already enough in a lot of cases. If you
- are content with this, you can quit reading here. Anyway, in order to customise
- and adapt the code generator, it is inevitable to gain some understanding
- how it works.
-
- \begin{figure}[h]
- \begin{tikzpicture}[x = 4.2cm, y = 1cm]
- \tikzstyle entity=[rounded corners, draw, thick, color = black, fill = white];
- \tikzstyle process=[ellipse, draw, thick, color = green, fill = white];
- \tikzstyle process_arrow=[->, semithick, color = green];
- \node (HOL) at (0, 4) [style=entity] {@{text "Isabelle/HOL"} theory};
- \node (eqn) at (2, 2) [style=entity] {code equations};
- \node (iml) at (2, 0) [style=entity] {intermediate language};
- \node (seri) at (1, 0) [style=process] {serialisation};
- \node (SML) at (0, 3) [style=entity] {@{text SML}};
- \node (OCaml) at (0, 2) [style=entity] {@{text OCaml}};
- \node (further) at (0, 1) [style=entity] {@{text "\<dots>"}};
- \node (Haskell) at (0, 0) [style=entity] {@{text Haskell}};
- \draw [style=process_arrow] (HOL) .. controls (2, 4) ..
- node [style=process, near start] {selection}
- node [style=process, near end] {preprocessing}
- (eqn);
- \draw [style=process_arrow] (eqn) -- node (transl) [style=process] {translation} (iml);
- \draw [style=process_arrow] (iml) -- (seri);
- \draw [style=process_arrow] (seri) -- (SML);
- \draw [style=process_arrow] (seri) -- (OCaml);
- \draw [style=process_arrow, dashed] (seri) -- (further);
- \draw [style=process_arrow] (seri) -- (Haskell);
- \end{tikzpicture}
- \caption{Code generator architecture}
- \label{fig:arch}
- \end{figure}
-
- The code generator employs a notion of executability
- for three foundational executable ingredients known
- from functional programming:
- \emph{code equations}, \emph{datatypes}, and
- \emph{type classes}. A code equation as a first approximation
- is a theorem of the form @{text "f t\<^isub>1 t\<^isub>2 \<dots> t\<^isub>n \<equiv> t"}
- (an equation headed by a constant @{text f} with arguments
- @{text "t\<^isub>1 t\<^isub>2 \<dots> t\<^isub>n"} and right hand side @{text t}).
- Code generation aims to turn code equations
- into a functional program. This is achieved by three major
- components which operate sequentially, i.e. the result of one is
- the input
- of the next in the chain, see diagram \ref{fig:arch}:
-
- \begin{itemize}
-
- \item Out of the vast collection of theorems proven in a
- \qn{theory}, a reasonable subset modelling
- code equations is \qn{selected}.
-
- \item On those selected theorems, certain
- transformations are carried out
- (\qn{preprocessing}). Their purpose is to turn theorems
- representing non- or badly executable
- specifications into equivalent but executable counterparts.
- The result is a structured collection of \qn{code theorems}.
-
- \item Before the selected code equations are continued with,
- they can be \qn{preprocessed}, i.e. subjected to theorem
- transformations. This \qn{preprocessor} is an interface which
- allows to apply
- the full expressiveness of ML-based theorem transformations
- to code generation; motivating examples are shown below, see
- \secref{sec:preproc}.
- The result of the preprocessing step is a structured collection
- of code equations.
-
- \item These code equations are \qn{translated} to a program
- in an abstract intermediate language. Think of it as a kind
- of \qt{Mini-Haskell} with four \qn{statements}: @{text data}
- (for datatypes), @{text fun} (stemming from code equations),
- also @{text class} and @{text inst} (for type classes).
-
- \item Finally, the abstract program is \qn{serialised} into concrete
- source code of a target language.
-
- \end{itemize}
-
- \noindent From these steps, only the two last are carried out outside the logic; by
- keeping this layer as thin as possible, the amount of code to trust is
- kept to a minimum.
-*}
-
-end
--- a/doc-src/IsarAdvanced/Codegen/Thy/ML.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,177 +0,0 @@
-theory "ML"
-imports Setup
-begin
-
-section {* ML system interfaces \label{sec:ml} *}
-
-text {*
- Since the code generator framework not only aims to provide
- a nice Isar interface but also to form a base for
- code-generation-based applications, here a short
- description of the most important ML interfaces.
-*}
-
-subsection {* Executable theory content: @{text Code} *}
-
-text {*
- This Pure module implements the core notions of
- executable content of a theory.
-*}
-
-subsubsection {* Managing executable content *}
-
-text %mlref {*
- \begin{mldecls}
- @{index_ML Code.add_eqn: "thm -> theory -> theory"} \\
- @{index_ML Code.del_eqn: "thm -> theory -> theory"} \\
- @{index_ML Code.add_eqnl: "string * (thm * bool) list lazy -> theory -> theory"} \\
- @{index_ML Code.map_pre: "(simpset -> simpset) -> theory -> theory"} \\
- @{index_ML Code.map_post: "(simpset -> simpset) -> theory -> theory"} \\
- @{index_ML Code.add_functrans: "string * (theory -> (thm * bool) list -> (thm * bool) list option)
- -> theory -> theory"} \\
- @{index_ML Code.del_functrans: "string -> theory -> theory"} \\
- @{index_ML Code.add_datatype: "(string * typ) list -> theory -> theory"} \\
- @{index_ML Code.get_datatype: "theory -> string
- -> (string * sort) list * (string * typ list) list"} \\
- @{index_ML Code.get_datatype_of_constr: "theory -> string -> string option"}
- \end{mldecls}
-
- \begin{description}
-
- \item @{ML Code.add_eqn}~@{text "thm"}~@{text "thy"} adds function
- theorem @{text "thm"} to executable content.
-
- \item @{ML Code.del_eqn}~@{text "thm"}~@{text "thy"} removes function
- theorem @{text "thm"} from executable content, if present.
-
- \item @{ML Code.add_eqnl}~@{text "(const, lthms)"}~@{text "thy"} adds
- suspended code equations @{text lthms} for constant
- @{text const} to executable content.
-
- \item @{ML Code.map_pre}~@{text "f"}~@{text "thy"} changes
- the preprocessor simpset.
-
- \item @{ML Code.add_functrans}~@{text "(name, f)"}~@{text "thy"} adds
- function transformer @{text f} (named @{text name}) to executable content;
- @{text f} is a transformer of the code equations belonging
- to a certain function definition, depending on the
- current theory context. Returning @{text NONE} indicates that no
- transformation took place; otherwise, the whole process will be iterated
- with the new code equations.
-
- \item @{ML Code.del_functrans}~@{text "name"}~@{text "thy"} removes
- function transformer named @{text name} from executable content.
-
- \item @{ML Code.add_datatype}~@{text cs}~@{text thy} adds
- a datatype to executable content, with generation
- set @{text cs}.
-
- \item @{ML Code.get_datatype_of_constr}~@{text "thy"}~@{text "const"}
- returns type constructor corresponding to
- constructor @{text const}; returns @{text NONE}
- if @{text const} is no constructor.
-
- \end{description}
-*}
-
-subsection {* Auxiliary *}
-
-text %mlref {*
- \begin{mldecls}
- @{index_ML Code_Unit.read_const: "theory -> string -> string"} \\
- @{index_ML Code_Unit.head_eqn: "theory -> thm -> string * ((string * sort) list * typ)"} \\
- @{index_ML Code_Unit.rewrite_eqn: "simpset -> thm -> thm"} \\
- \end{mldecls}
-
- \begin{description}
-
- \item @{ML Code_Unit.read_const}~@{text thy}~@{text s}
- reads a constant as a concrete term expression @{text s}.
-
- \item @{ML Code_Unit.head_eqn}~@{text thy}~@{text thm}
- extracts the constant and its type from a code equation @{text thm}.
-
- \item @{ML Code_Unit.rewrite_eqn}~@{text ss}~@{text thm}
- rewrites a code equation @{text thm} with a simpset @{text ss};
- only arguments and right hand side are rewritten,
- not the head of the code equation.
-
- \end{description}
-
-*}
-
-subsection {* Implementing code generator applications *}
-
-text {*
- Implementing code generator applications on top
- of the framework set out so far usually not only
- involves using those primitive interfaces
- but also storing code-dependent data and various
- other things.
-*}
-
-subsubsection {* Data depending on the theory's executable content *}
-
-text {*
- Due to incrementality of code generation, changes in the
- theory's executable content have to be propagated in a
- certain fashion. Additionally, such changes may occur
- not only during theory extension but also during theory
- merge, which is a little bit nasty from an implementation
- point of view. The framework provides a solution
- to this technical challenge by providing a functorial
- data slot @{ML_functor CodeDataFun}; on instantiation
- of this functor, the following types and operations
- are required:
-
- \medskip
- \begin{tabular}{l}
- @{text "type T"} \\
- @{text "val empty: T"} \\
- @{text "val purge: theory \<rightarrow> string list option \<rightarrow> T \<rightarrow> T"}
- \end{tabular}
-
- \begin{description}
-
- \item @{text T} the type of data to store.
-
- \item @{text empty} initial (empty) data.
-
- \item @{text purge}~@{text thy}~@{text consts} propagates changes in executable content;
- @{text consts} indicates the kind
- of change: @{ML NONE} stands for a fundamental change
- which invalidates any existing code, @{text "SOME consts"}
- hints that executable content for constants @{text consts}
- has changed.
-
- \end{description}
-
- \noindent An instance of @{ML_functor CodeDataFun} provides the following
- interface:
-
- \medskip
- \begin{tabular}{l}
- @{text "get: theory \<rightarrow> T"} \\
- @{text "change: theory \<rightarrow> (T \<rightarrow> T) \<rightarrow> T"} \\
- @{text "change_yield: theory \<rightarrow> (T \<rightarrow> 'a * T) \<rightarrow> 'a * T"}
- \end{tabular}
-
- \begin{description}
-
- \item @{text get} retrieval of the current data.
-
- \item @{text change} update of current data (cached!)
- by giving a continuation.
-
- \item @{text change_yield} update with side result.
-
- \end{description}
-*}
-
-text {*
- \bigskip
-
- \emph{Happy proving, happy hacking!}
-*}
-
-end
--- a/doc-src/IsarAdvanced/Codegen/Thy/Program.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,534 +0,0 @@
-theory Program
-imports Introduction
-begin
-
-section {* Turning Theories into Programs \label{sec:program} *}
-
-subsection {* The @{text "Isabelle/HOL"} default setup *}
-
-text {*
- We have already seen how by default equations stemming from
- @{command definition}/@{command primrec}/@{command fun}
- statements are used for code generation. This default behaviour
- can be changed, e.g. by providing different code equations.
- All kinds of customisation shown in this section is \emph{safe}
- in the sense that the user does not have to worry about
- correctness -- all programs generatable that way are partially
- correct.
-*}
-
-subsection {* Selecting code equations *}
-
-text {*
- Coming back to our introductory example, we
- could provide an alternative code equations for @{const dequeue}
- explicitly:
-*}
-
-lemma %quote [code]:
- "dequeue (AQueue xs []) =
- (if xs = [] then (None, AQueue [] [])
- else dequeue (AQueue [] (rev xs)))"
- "dequeue (AQueue xs (y # ys)) =
- (Some y, AQueue xs ys)"
- by (cases xs, simp_all) (cases "rev xs", simp_all)
-
-text {*
- \noindent The annotation @{text "[code]"} is an @{text Isar}
- @{text attribute} which states that the given theorems should be
- considered as code equations for a @{text fun} statement --
- the corresponding constant is determined syntactically. The resulting code:
-*}
-
-text %quote {*@{code_stmts dequeue (consts) dequeue (Haskell)}*}
-
-text {*
- \noindent You may note that the equality test @{term "xs = []"} has been
- replaced by the predicate @{term "null xs"}. This is due to the default
- setup in the \qn{preprocessor} to be discussed further below (\secref{sec:preproc}).
-
- Changing the default constructor set of datatypes is also
- possible. See \secref{sec:datatypes} for an example.
-
- As told in \secref{sec:concept}, code generation is based
- on a structured collection of code theorems.
- For explorative purpose, this collection
- may be inspected using the @{command code_thms} command:
-*}
-
-code_thms %quote dequeue
-
-text {*
- \noindent prints a table with \emph{all} code equations
- for @{const dequeue}, including
- \emph{all} code equations those equations depend
- on recursively.
-
- Similarly, the @{command code_deps} command shows a graph
- visualising dependencies between code equations.
-*}
-
-subsection {* @{text class} and @{text instantiation} *}
-
-text {*
- Concerning type classes and code generation, let us examine an example
- from abstract algebra:
-*}
-
-class %quote semigroup =
- fixes mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<otimes>" 70)
- assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
-
-class %quote monoid = semigroup +
- fixes neutral :: 'a ("\<one>")
- assumes neutl: "\<one> \<otimes> x = x"
- and neutr: "x \<otimes> \<one> = x"
-
-instantiation %quote nat :: monoid
-begin
-
-primrec %quote mult_nat where
- "0 \<otimes> n = (0\<Colon>nat)"
- | "Suc m \<otimes> n = n + m \<otimes> n"
-
-definition %quote neutral_nat where
- "\<one> = Suc 0"
-
-lemma %quote add_mult_distrib:
- fixes n m q :: nat
- shows "(n + m) \<otimes> q = n \<otimes> q + m \<otimes> q"
- by (induct n) simp_all
-
-instance %quote proof
- fix m n q :: nat
- show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
- by (induct m) (simp_all add: add_mult_distrib)
- show "\<one> \<otimes> n = n"
- by (simp add: neutral_nat_def)
- show "m \<otimes> \<one> = m"
- by (induct m) (simp_all add: neutral_nat_def)
-qed
-
-end %quote
-
-text {*
- \noindent We define the natural operation of the natural numbers
- on monoids:
-*}
-
-primrec %quote (in monoid) pow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" where
- "pow 0 a = \<one>"
- | "pow (Suc n) a = a \<otimes> pow n a"
-
-text {*
- \noindent This we use to define the discrete exponentiation function:
-*}
-
-definition %quote bexp :: "nat \<Rightarrow> nat" where
- "bexp n = pow n (Suc (Suc 0))"
-
-text {*
- \noindent The corresponding code:
-*}
-
-text %quote {*@{code_stmts bexp (Haskell)}*}
-
-text {*
- \noindent This is a convenient place to show how explicit dictionary construction
- manifests in generated code (here, the same example in @{text SML}):
-*}
-
-text %quote {*@{code_stmts bexp (SML)}*}
-
-text {*
- \noindent Note the parameters with trailing underscore (@{verbatim "A_"})
- which are the dictionary parameters.
-*}
-
-subsection {* The preprocessor \label{sec:preproc} *}
-
-text {*
- Before selected function theorems are turned into abstract
- code, a chain of definitional transformation steps is carried
- out: \emph{preprocessing}. In essence, the preprocessor
- consists of two components: a \emph{simpset} and \emph{function transformers}.
-
- The \emph{simpset} allows to employ the full generality of the Isabelle
- simplifier. Due to the interpretation of theorems
- as code equations, rewrites are applied to the right
- hand side and the arguments of the left hand side of an
- equation, but never to the constant heading the left hand side.
- An important special case are \emph{inline theorems} which may be
- declared and undeclared using the
- \emph{code inline} or \emph{code inline del} attribute respectively.
-
- Some common applications:
-*}
-
-text_raw {*
- \begin{itemize}
-*}
-
-text {*
- \item replacing non-executable constructs by executable ones:
-*}
-
-lemma %quote [code inline]:
- "x \<in> set xs \<longleftrightarrow> x mem xs" by (induct xs) simp_all
-
-text {*
- \item eliminating superfluous constants:
-*}
-
-lemma %quote [code inline]:
- "1 = Suc 0" by simp
-
-text {*
- \item replacing executable but inconvenient constructs:
-*}
-
-lemma %quote [code inline]:
- "xs = [] \<longleftrightarrow> List.null xs" by (induct xs) simp_all
-
-text_raw {*
- \end{itemize}
-*}
-
-text {*
- \noindent \emph{Function transformers} provide a very general interface,
- transforming a list of function theorems to another
- list of function theorems, provided that neither the heading
- constant nor its type change. The @{term "0\<Colon>nat"} / @{const Suc}
- pattern elimination implemented in
- theory @{text Efficient_Nat} (see \secref{eff_nat}) uses this
- interface.
-
- \noindent The current setup of the preprocessor may be inspected using
- the @{command print_codesetup} command.
- @{command code_thms} provides a convenient
- mechanism to inspect the impact of a preprocessor setup
- on code equations.
-
- \begin{warn}
- The attribute \emph{code unfold}
- associated with the @{text "SML code generator"} also applies to
- the @{text "generic code generator"}:
- \emph{code unfold} implies \emph{code inline}.
- \end{warn}
-*}
-
-subsection {* Datatypes \label{sec:datatypes} *}
-
-text {*
- Conceptually, any datatype is spanned by a set of
- \emph{constructors} of type @{text "\<tau> = \<dots> \<Rightarrow> \<kappa> \<alpha>\<^isub>1 \<dots> \<alpha>\<^isub>n"} where @{text
- "{\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>n}"} is exactly the set of \emph{all} type variables in
- @{text "\<tau>"}. The HOL datatype package by default registers any new
- datatype in the table of datatypes, which may be inspected using the
- @{command print_codesetup} command.
-
- In some cases, it is appropriate to alter or extend this table. As
- an example, we will develop an alternative representation of the
- queue example given in \secref{sec:intro}. The amortised
- representation is convenient for generating code but exposes its
- \qt{implementation} details, which may be cumbersome when proving
- theorems about it. Therefore, here a simple, straightforward
- representation of queues:
-*}
-
-datatype %quote 'a queue = Queue "'a list"
-
-definition %quote empty :: "'a queue" where
- "empty = Queue []"
-
-primrec %quote enqueue :: "'a \<Rightarrow> 'a queue \<Rightarrow> 'a queue" where
- "enqueue x (Queue xs) = Queue (xs @ [x])"
-
-fun %quote dequeue :: "'a queue \<Rightarrow> 'a option \<times> 'a queue" where
- "dequeue (Queue []) = (None, Queue [])"
- | "dequeue (Queue (x # xs)) = (Some x, Queue xs)"
-
-text {*
- \noindent This we can use directly for proving; for executing,
- we provide an alternative characterisation:
-*}
-
-definition %quote AQueue :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a queue" where
- "AQueue xs ys = Queue (ys @ rev xs)"
-
-code_datatype %quote AQueue
-
-text {*
- \noindent Here we define a \qt{constructor} @{const "AQueue"} which
- is defined in terms of @{text "Queue"} and interprets its arguments
- according to what the \emph{content} of an amortised queue is supposed
- to be. Equipped with this, we are able to prove the following equations
- for our primitive queue operations which \qt{implement} the simple
- queues in an amortised fashion:
-*}
-
-lemma %quote empty_AQueue [code]:
- "empty = AQueue [] []"
- unfolding AQueue_def empty_def by simp
-
-lemma %quote enqueue_AQueue [code]:
- "enqueue x (AQueue xs ys) = AQueue (x # xs) ys"
- unfolding AQueue_def by simp
-
-lemma %quote dequeue_AQueue [code]:
- "dequeue (AQueue xs []) =
- (if xs = [] then (None, AQueue [] [])
- else dequeue (AQueue [] (rev xs)))"
- "dequeue (AQueue xs (y # ys)) = (Some y, AQueue xs ys)"
- unfolding AQueue_def by simp_all
-
-text {*
- \noindent For completeness, we provide a substitute for the
- @{text case} combinator on queues:
-*}
-
-definition %quote
- aqueue_case_def: "aqueue_case = queue_case"
-
-lemma %quote aqueue_case [code, code inline]:
- "queue_case = aqueue_case"
- unfolding aqueue_case_def ..
-
-lemma %quote case_AQueue [code]:
- "aqueue_case f (AQueue xs ys) = f (ys @ rev xs)"
- unfolding aqueue_case_def AQueue_def by simp
-
-text {*
- \noindent The resulting code looks as expected:
-*}
-
-text %quote {*@{code_stmts empty enqueue dequeue (SML)}*}
-
-text {*
- \noindent From this example, it can be glimpsed that using own
- constructor sets is a little delicate since it changes the set of
- valid patterns for values of that type. Without going into much
- detail, here some practical hints:
-
- \begin{itemize}
-
- \item When changing the constructor set for datatypes, take care
- to provide an alternative for the @{text case} combinator
- (e.g.~by replacing it using the preprocessor).
-
- \item Values in the target language need not to be normalised --
- different values in the target language may represent the same
- value in the logic.
-
- \item Usually, a good methodology to deal with the subtleties of
- pattern matching is to see the type as an abstract type: provide
- a set of operations which operate on the concrete representation
- of the type, and derive further operations by combinations of
- these primitive ones, without relying on a particular
- representation.
-
- \end{itemize}
-*}
-
-
-subsection {* Equality and wellsortedness *}
-
-text {*
- Surely you have already noticed how equality is treated
- by the code generator:
-*}
-
-primrec %quote collect_duplicates :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
- "collect_duplicates xs ys [] = xs"
- | "collect_duplicates xs ys (z#zs) = (if z \<in> set xs
- then if z \<in> set ys
- then collect_duplicates xs ys zs
- else collect_duplicates xs (z#ys) zs
- else collect_duplicates (z#xs) (z#ys) zs)"
-
-text {*
- \noindent The membership test during preprocessing is rewritten,
- resulting in @{const List.member}, which itself
- performs an explicit equality check.
-*}
-
-text %quote {*@{code_stmts collect_duplicates (SML)}*}
-
-text {*
- \noindent Obviously, polymorphic equality is implemented the Haskell
- way using a type class. How is this achieved? HOL introduces
- an explicit class @{class eq} with a corresponding operation
- @{const eq_class.eq} such that @{thm eq [no_vars]}.
- The preprocessing framework does the rest by propagating the
- @{class eq} constraints through all dependent code equations.
- For datatypes, instances of @{class eq} are implicitly derived
- when possible. For other types, you may instantiate @{text eq}
- manually like any other type class.
-
- Though this @{text eq} class is designed to get rarely in
- the way, a subtlety
- enters the stage when definitions of overloaded constants
- are dependent on operational equality. For example, let
- us define a lexicographic ordering on tuples
- (also see theory @{theory Product_ord}):
-*}
-
-instantiation %quote "*" :: (order, order) order
-begin
-
-definition %quote [code del]:
- "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
-
-definition %quote [code del]:
- "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
-
-instance %quote proof
-qed (auto simp: less_eq_prod_def less_prod_def intro: order_less_trans)
-
-end %quote
-
-lemma %quote order_prod [code]:
- "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
- "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
- by (simp_all add: less_prod_def less_eq_prod_def)
-
-text {*
- \noindent Then code generation will fail. Why? The definition
- of @{term "op \<le>"} depends on equality on both arguments,
- which are polymorphic and impose an additional @{class eq}
- class constraint, which the preprocessor does not propagate
- (for technical reasons).
-
- The solution is to add @{class eq} explicitly to the first sort arguments in the
- code theorems:
-*}
-
-lemma %quote order_prod_code [code]:
- "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
- "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
- by (simp_all add: less_prod_def less_eq_prod_def)
-
-text {*
- \noindent Then code generation succeeds:
-*}
-
-text %quote {*@{code_stmts "op \<le> \<Colon> _ \<times> _ \<Rightarrow> _ \<times> _ \<Rightarrow> bool" (SML)}*}
-
-text {*
- In some cases, the automatically derived code equations
- for equality on a particular type may not be appropriate.
- As example, watch the following datatype representing
- monomorphic parametric types (where type constructors
- are referred to by natural numbers):
-*}
-
-datatype %quote monotype = Mono nat "monotype list"
-(*<*)
-lemma monotype_eq:
- "eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<equiv>
- eq_class.eq tyco1 tyco2 \<and> eq_class.eq typargs1 typargs2" by (simp add: eq)
-(*>*)
-
-text {*
- \noindent Then code generation for SML would fail with a message
- that the generated code contains illegal mutual dependencies:
- the theorem @{thm monotype_eq [no_vars]} already requires the
- instance @{text "monotype \<Colon> eq"}, which itself requires
- @{thm monotype_eq [no_vars]}; Haskell has no problem with mutually
- recursive @{text instance} and @{text function} definitions,
- but the SML serialiser does not support this.
-
- In such cases, you have to provide your own equality equations
- involving auxiliary constants. In our case,
- @{const [show_types] list_all2} can do the job:
-*}
-
-lemma %quote monotype_eq_list_all2 [code]:
- "eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<longleftrightarrow>
- eq_class.eq tyco1 tyco2 \<and> list_all2 eq_class.eq typargs1 typargs2"
- by (simp add: eq list_all2_eq [symmetric])
-
-text {*
- \noindent does not depend on instance @{text "monotype \<Colon> eq"}:
-*}
-
-text %quote {*@{code_stmts "eq_class.eq :: monotype \<Rightarrow> monotype \<Rightarrow> bool" (SML)}*}
-
-
-subsection {* Explicit partiality *}
-
-text {*
- Partiality usually enters the game by partial patterns, as
- in the following example, again for amortised queues:
-*}
-
-definition %quote strict_dequeue :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
- "strict_dequeue q = (case dequeue q
- of (Some x, q') \<Rightarrow> (x, q'))"
-
-lemma %quote strict_dequeue_AQueue [code]:
- "strict_dequeue (AQueue xs (y # ys)) = (y, AQueue xs ys)"
- "strict_dequeue (AQueue xs []) =
- (case rev xs of y # ys \<Rightarrow> (y, AQueue [] ys))"
- by (simp_all add: strict_dequeue_def dequeue_AQueue split: list.splits)
-
-text {*
- \noindent In the corresponding code, there is no equation
- for the pattern @{term "AQueue [] []"}:
-*}
-
-text %quote {*@{code_stmts strict_dequeue (consts) strict_dequeue (Haskell)}*}
-
-text {*
- \noindent In some cases it is desirable to have this
- pseudo-\qt{partiality} more explicitly, e.g.~as follows:
-*}
-
-axiomatization %quote empty_queue :: 'a
-
-definition %quote strict_dequeue' :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
- "strict_dequeue' q = (case dequeue q of (Some x, q') \<Rightarrow> (x, q') | _ \<Rightarrow> empty_queue)"
-
-lemma %quote strict_dequeue'_AQueue [code]:
- "strict_dequeue' (AQueue xs []) = (if xs = [] then empty_queue
- else strict_dequeue' (AQueue [] (rev xs)))"
- "strict_dequeue' (AQueue xs (y # ys)) =
- (y, AQueue xs ys)"
- by (simp_all add: strict_dequeue'_def dequeue_AQueue split: list.splits)
-
-text {*
- Observe that on the right hand side of the definition of @{const
- "strict_dequeue'"} the constant @{const empty_queue} occurs
- which is unspecified.
-
- Normally, if constants without any code equations occur in a
- program, the code generator complains (since in most cases this is
- not what the user expects). But such constants can also be thought
- of as function definitions with no equations which always fail,
- since there is never a successful pattern match on the left hand
- side. In order to categorise a constant into that category
- explicitly, use @{command "code_abort"}:
-*}
-
-code_abort %quote empty_queue
-
-text {*
- \noindent Then the code generator will just insert an error or
- exception at the appropriate position:
-*}
-
-text %quote {*@{code_stmts strict_dequeue' (consts) empty_queue strict_dequeue' (Haskell)}*}
-
-text {*
- \noindent This feature however is rarely needed in practice.
- Note also that the @{text HOL} default setup already declares
- @{const undefined} as @{command "code_abort"}, which is most
- likely to be used in such situations.
-*}
-
-end
-
\ No newline at end of file
--- a/doc-src/IsarAdvanced/Codegen/Thy/ROOT.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,11 +0,0 @@
-
-(* $Id$ *)
-
-no_document use_thy "Setup";
-no_document use_thys ["Efficient_Nat"];
-
-use_thy "Introduction";
-use_thy "Program";
-use_thy "Adaption";
-use_thy "Further";
-use_thy "ML";
--- a/doc-src/IsarAdvanced/Codegen/Thy/Setup.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,12 +0,0 @@
-theory Setup
-imports Complex_Main
-uses "../../../antiquote_setup.ML" "../../../more_antiquote.ML"
-begin
-
-ML {* no_document use_thys
- ["Efficient_Nat", "Code_Char_chr", "Product_ord", "~~/src/HOL/Imperative_HOL/Imperative_HOL",
- "~~/src/HOL/Decision_Procs/Ferrack"] *}
-
-ML_val {* Code_Target.code_width := 74 *}
-
-end
--- a/doc-src/IsarAdvanced/Codegen/Thy/document/Adaption.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,679 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Adaption}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Adaption\isanewline
-\isakeyword{imports}\ Setup\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isadeliminvisible
-\isanewline
-%
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{setup}\isamarkupfalse%
-\ {\isacharverbatimopen}\ Code{\isacharunderscore}Target{\isachardot}extend{\isacharunderscore}target\ {\isacharparenleft}{\isachardoublequote}{\isasymSML}{\isachardoublequote}{\isacharcomma}\ {\isacharparenleft}{\isachardoublequote}SML{\isachardoublequote}{\isacharcomma}\ K\ I{\isacharparenright}{\isacharparenright}\ {\isacharverbatimclose}%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isamarkupsection{Adaption to target languages \label{sec:adaption}%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Adapting code generation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The aspects of code generation introduced so far have two aspects
- in common:
-
- \begin{itemize}
- \item They act uniformly, without reference to a specific
- target language.
- \item They are \emph{safe} in the sense that as long as you trust
- the code generator meta theory and implementation, you cannot
- produce programs that yield results which are not derivable
- in the logic.
- \end{itemize}
-
- \noindent In this section we will introduce means to \emph{adapt} the serialiser
- to a specific target language, i.e.~to print program fragments
- in a way which accommodates \qt{already existing} ingredients of
- a target language environment, for three reasons:
-
- \begin{itemize}
- \item improving readability and aesthetics of generated code
- \item gaining efficiency
- \item interface with language parts which have no direct counterpart
- in \isa{HOL} (say, imperative data structures)
- \end{itemize}
-
- \noindent Generally, you should avoid using those features yourself
- \emph{at any cost}:
-
- \begin{itemize}
- \item The safe configuration methods act uniformly on every target language,
- whereas for adaption you have to treat each target language separate.
- \item Application is extremely tedious since there is no abstraction
- which would allow for a static check, making it easy to produce garbage.
- \item More or less subtle errors can be introduced unconsciously.
- \end{itemize}
-
- \noindent However, even if you ought refrain from setting up adaption
- yourself, already the \isa{HOL} comes with some reasonable default
- adaptions (say, using target language list syntax). There also some
- common adaption cases which you can setup by importing particular
- library theories. In order to understand these, we provide some clues here;
- these however are not supposed to replace a careful study of the sources.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{The adaption principle%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The following figure illustrates what \qt{adaption} is conceptually
- supposed to be:
-
- \begin{figure}[here]
- \begin{tikzpicture}[scale = 0.5]
- \tikzstyle water=[color = blue, thick]
- \tikzstyle ice=[color = black, very thick, cap = round, join = round, fill = white]
- \tikzstyle process=[color = green, semithick, ->]
- \tikzstyle adaption=[color = red, semithick, ->]
- \tikzstyle target=[color = black]
- \foreach \x in {0, ..., 24}
- \draw[style=water] (\x, 0.25) sin + (0.25, 0.25) cos + (0.25, -0.25) sin
- + (0.25, -0.25) cos + (0.25, 0.25);
- \draw[style=ice] (1, 0) --
- (3, 6) node[above, fill=white] {logic} -- (5, 0) -- cycle;
- \draw[style=ice] (9, 0) --
- (11, 6) node[above, fill=white] {intermediate language} -- (13, 0) -- cycle;
- \draw[style=ice] (15, -6) --
- (19, 6) node[above, fill=white] {target language} -- (23, -6) -- cycle;
- \draw[style=process]
- (3.5, 3) .. controls (7, 5) .. node[fill=white] {translation} (10.5, 3);
- \draw[style=process]
- (11.5, 3) .. controls (15, 5) .. node[fill=white] (serialisation) {serialisation} (18.5, 3);
- \node (adaption) at (11, -2) [style=adaption] {adaption};
- \node at (19, 3) [rotate=90] {generated};
- \node at (19.5, -5) {language};
- \node at (19.5, -3) {library};
- \node (includes) at (19.5, -1) {includes};
- \node (reserved) at (16.5, -3) [rotate=72] {reserved}; % proper 71.57
- \draw[style=process]
- (includes) -- (serialisation);
- \draw[style=process]
- (reserved) -- (serialisation);
- \draw[style=adaption]
- (adaption) -- (serialisation);
- \draw[style=adaption]
- (adaption) -- (includes);
- \draw[style=adaption]
- (adaption) -- (reserved);
- \end{tikzpicture}
- \caption{The adaption principle}
- \label{fig:adaption}
- \end{figure}
-
- \noindent In the tame view, code generation acts as broker between
- \isa{logic}, \isa{intermediate\ language} and
- \isa{target\ language} by means of \isa{translation} and
- \isa{serialisation}; for the latter, the serialiser has to observe
- the structure of the \isa{language} itself plus some \isa{reserved}
- keywords which have to be avoided for generated code.
- However, if you consider \isa{adaption} mechanisms, the code generated
- by the serializer is just the tip of the iceberg:
-
- \begin{itemize}
- \item \isa{serialisation} can be \emph{parametrised} such that
- logical entities are mapped to target-specific ones
- (e.g. target-specific list syntax,
- see also \secref{sec:adaption_mechanisms})
- \item Such parametrisations can involve references to a
- target-specific standard \isa{library} (e.g. using
- the \isa{Haskell} \verb|Maybe| type instead
- of the \isa{HOL} \isa{option} type);
- if such are used, the corresponding identifiers
- (in our example, \verb|Maybe|, \verb|Nothing|
- and \verb|Just|) also have to be considered \isa{reserved}.
- \item Even more, the user can enrich the library of the
- target-language by providing code snippets
- (\qt{\isa{includes}}) which are prepended to
- any generated code (see \secref{sec:include}); this typically
- also involves further \isa{reserved} identifiers.
- \end{itemize}
-
- \noindent As figure \ref{fig:adaption} illustrates, all these adaption mechanisms
- have to act consistently; it is at the discretion of the user
- to take care for this.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Common adaption patterns%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} \hyperlink{theory.Main}{\mbox{\isa{Main}}} theory already provides a code
- generator setup
- which should be suitable for most applications. Common extensions
- and modifications are available by certain theories of the \isa{HOL}
- library; beside being useful in applications, they may serve
- as a tutorial for customising the code generator setup (see below
- \secref{sec:adaption_mechanisms}).
-
- \begin{description}
-
- \item[\hyperlink{theory.Code-Integer}{\mbox{\isa{Code{\isacharunderscore}Integer}}}] represents \isa{HOL} integers by big
- integer literals in target languages.
- \item[\hyperlink{theory.Code-Char}{\mbox{\isa{Code{\isacharunderscore}Char}}}] represents \isa{HOL} characters by
- character literals in target languages.
- \item[\hyperlink{theory.Code-Char-chr}{\mbox{\isa{Code{\isacharunderscore}Char{\isacharunderscore}chr}}}] like \isa{Code{\isacharunderscore}Char},
- but also offers treatment of character codes; includes
- \hyperlink{theory.Code-Char}{\mbox{\isa{Code{\isacharunderscore}Char}}}.
- \item[\hyperlink{theory.Efficient-Nat}{\mbox{\isa{Efficient{\isacharunderscore}Nat}}}] \label{eff_nat} implements natural numbers by integers,
- which in general will result in higher efficiency; pattern
- matching with \isa{{\isadigit{0}}} / \isa{Suc}
- is eliminated; includes \hyperlink{theory.Code-Integer}{\mbox{\isa{Code{\isacharunderscore}Integer}}}
- and \hyperlink{theory.Code-Index}{\mbox{\isa{Code{\isacharunderscore}Index}}}.
- \item[\hyperlink{theory.Code-Index}{\mbox{\isa{Code{\isacharunderscore}Index}}}] provides an additional datatype
- \isa{index} which is mapped to target-language built-in integers.
- Useful for code setups which involve e.g. indexing of
- target-language arrays.
- \item[\hyperlink{theory.Code-Message}{\mbox{\isa{Code{\isacharunderscore}Message}}}] provides an additional datatype
- \isa{message{\isacharunderscore}string} which is isomorphic to strings;
- \isa{message{\isacharunderscore}string}s are mapped to target-language strings.
- Useful for code setups which involve e.g. printing (error) messages.
-
- \end{description}
-
- \begin{warn}
- When importing any of these theories, they should form the last
- items in an import list. Since these theories adapt the
- code generator setup in a non-conservative fashion,
- strange effects may occur otherwise.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Parametrising serialisation \label{sec:adaption_mechanisms}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Consider the following function and its corresponding
- SML code:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{primrec}\isamarkupfalse%
-\ in{\isacharunderscore}interval\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymtimes}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}in{\isacharunderscore}interval\ {\isacharparenleft}k{\isacharcomma}\ l{\isacharparenright}\ n\ {\isasymlongleftrightarrow}\ k\ {\isasymle}\ n\ {\isasymand}\ n\ {\isasymle}\ l{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isataginvisible
-%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype boola = True | False;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun anda x True = x\\
-\hspace*{0pt} ~| anda x False = False\\
-\hspace*{0pt} ~| anda True x = x\\
-\hspace*{0pt} ~| anda False x = False;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun less{\char95}nat m (Suc n) = less{\char95}eq{\char95}nat m n\\
-\hspace*{0pt} ~| less{\char95}nat n Zero{\char95}nat = False\\
-\hspace*{0pt}and less{\char95}eq{\char95}nat (Suc m) n = less{\char95}nat m n\\
-\hspace*{0pt} ~| less{\char95}eq{\char95}nat Zero{\char95}nat n = True;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun in{\char95}interval (k,~l) n = anda (less{\char95}eq{\char95}nat k n) (less{\char95}eq{\char95}nat n l);\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Though this is correct code, it is a little bit unsatisfactory:
- boolean values and operators are materialised as distinguished
- entities with have nothing to do with the SML-built-in notion
- of \qt{bool}. This results in less readable code;
- additionally, eager evaluation may cause programs to
- loop or break which would perfectly terminate when
- the existing SML \verb|bool| would be used. To map
- the HOL \isa{bool} on SML \verb|bool|, we may use
- \qn{custom serialisations}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isatagquotett
-\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
-\ bool\isanewline
-\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}bool{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ True\ \isakeyword{and}\ False\ \isakeyword{and}\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}true{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}false{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharunderscore}\ andalso\ {\isacharunderscore}{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagquotett
-{\isafoldquotett}%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\begin{isamarkuptext}%
-\noindent The \hyperlink{command.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} command takes a type constructor
- as arguments together with a list of custom serialisations.
- Each custom serialisation starts with a target language
- identifier followed by an expression, which during
- code serialisation is inserted whenever the type constructor
- would occur. For constants, \hyperlink{command.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} implements
- the corresponding mechanism. Each ``\verb|_|'' in
- a serialisation expression is treated as a placeholder
- for the type constructor's (the constant's) arguments.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun less{\char95}nat m (Suc n) = less{\char95}eq{\char95}nat m n\\
-\hspace*{0pt} ~| less{\char95}nat n Zero{\char95}nat = false\\
-\hspace*{0pt}and less{\char95}eq{\char95}nat (Suc m) n = less{\char95}nat m n\\
-\hspace*{0pt} ~| less{\char95}eq{\char95}nat Zero{\char95}nat n = true;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun in{\char95}interval (k,~l) n = (less{\char95}eq{\char95}nat k n) andalso (less{\char95}eq{\char95}nat n l);\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This still is not perfect: the parentheses
- around the \qt{andalso} expression are superfluous.
- Though the serialiser
- by no means attempts to imitate the rich Isabelle syntax
- framework, it provides some common idioms, notably
- associative infixes with precedences which may be used here:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isatagquotett
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharparenleft}SML\ \isakeyword{infixl}\ {\isadigit{1}}\ {\isachardoublequoteopen}andalso{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagquotett
-{\isafoldquotett}%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun less{\char95}nat m (Suc n) = less{\char95}eq{\char95}nat m n\\
-\hspace*{0pt} ~| less{\char95}nat n Zero{\char95}nat = false\\
-\hspace*{0pt}and less{\char95}eq{\char95}nat (Suc m) n = less{\char95}nat m n\\
-\hspace*{0pt} ~| less{\char95}eq{\char95}nat Zero{\char95}nat n = true;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun in{\char95}interval (k,~l) n = less{\char95}eq{\char95}nat k n andalso less{\char95}eq{\char95}nat n l;\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The attentive reader may ask how we assert that no generated
- code will accidentally overwrite. For this reason the serialiser has
- an internal table of identifiers which have to be avoided to be used
- for new declarations. Initially, this table typically contains the
- keywords of the target language. It can be extended manually, thus avoiding
- accidental overwrites, using the \hyperlink{command.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} command:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{code{\isacharunderscore}reserved}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymSML}{\isachardoublequoteclose}\ bool\ true\ false\ andalso%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Next, we try to map HOL pairs to SML pairs, using the
- infix ``\verb|*|'' type constructor and parentheses:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isataginvisible
-%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isatagquotett
-\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
-\ {\isacharasterisk}\isanewline
-\ \ {\isacharparenleft}SML\ \isakeyword{infix}\ {\isadigit{2}}\ {\isachardoublequoteopen}{\isacharasterisk}{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ Pair\isanewline
-\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}{\isacharbang}{\isacharparenleft}{\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharcomma}{\isacharslash}\ {\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagquotett
-{\isafoldquotett}%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\begin{isamarkuptext}%
-\noindent The initial bang ``\verb|!|'' tells the serialiser
- never to put
- parentheses around the whole expression (they are already present),
- while the parentheses around argument place holders
- tell not to put parentheses around the arguments.
- The slash ``\verb|/|'' (followed by arbitrary white space)
- inserts a space which may be used as a break if necessary
- during pretty printing.
-
- These examples give a glimpse what mechanisms
- custom serialisations provide; however their usage
- requires careful thinking in order not to introduce
- inconsistencies -- or, in other words:
- custom serialisations are completely axiomatic.
-
- A further noteworthy details is that any special
- character in a custom serialisation may be quoted
- using ``\verb|'|''; thus, in
- ``\verb|fn '_ => _|'' the first
- ``\verb|_|'' is a proper underscore while the
- second ``\verb|_|'' is a placeholder.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{\isa{Haskell} serialisation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-For convenience, the default
- \isa{HOL} setup for \isa{Haskell} maps the \isa{eq} class to
- its counterpart in \isa{Haskell}, giving custom serialisations
- for the class \isa{eq} (by command \hyperlink{command.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}) and its operation
- \isa{eq{\isacharunderscore}class{\isachardot}eq}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isatagquotett
-\isacommand{code{\isacharunderscore}class}\isamarkupfalse%
-\ eq\isanewline
-\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Eq{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ {\isacharequal}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharparenleft}Haskell\ \isakeyword{infixl}\ {\isadigit{4}}\ {\isachardoublequoteopen}{\isacharequal}{\isacharequal}{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagquotett
-{\isafoldquotett}%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\begin{isamarkuptext}%
-\noindent A problem now occurs whenever a type which
- is an instance of \isa{eq} in \isa{HOL} is mapped
- on a \isa{Haskell}-built-in type which is also an instance
- of \isa{Haskell} \isa{Eq}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{typedecl}\isamarkupfalse%
-\ bar\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}x{\isasymColon}bar{\isacharparenright}\ y\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{by}\isamarkupfalse%
-\ default\ {\isacharparenleft}simp\ add{\isacharcolon}\ eq{\isacharunderscore}bar{\isacharunderscore}def{\isacharparenright}\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
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-\endisatagquote
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-\isadelimquote
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-\endisadelimquote
-\isanewline
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-\isadelimquotett
-\isanewline
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-\endisadelimquotett
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-\isatagquotett
-\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
-\ bar\isanewline
-\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Integer{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagquotett
-{\isafoldquotett}%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\begin{isamarkuptext}%
-\noindent The code generator would produce
- an additional instance, which of course is rejected by the \isa{Haskell}
- compiler.
- To suppress this additional instance, use
- \isa{code{\isacharunderscore}instance}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isatagquotett
-\isacommand{code{\isacharunderscore}instance}\isamarkupfalse%
-\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
-\ \ {\isacharparenleft}Haskell\ {\isacharminus}{\isacharparenright}%
-\endisatagquotett
-{\isafoldquotett}%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isamarkupsubsection{Enhancing the target language context \label{sec:include}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In rare cases it is necessary to \emph{enrich} the context of a
- target language; this is accomplished using the \hyperlink{command.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}
- command:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\isatagquotett
-\isacommand{code{\isacharunderscore}include}\isamarkupfalse%
-\ Haskell\ {\isachardoublequoteopen}Errno{\isachardoublequoteclose}\isanewline
-{\isacharverbatimopen}errno\ i\ {\isacharequal}\ error\ {\isacharparenleft}{\isachardoublequote}Error\ number{\isacharcolon}\ {\isachardoublequote}\ {\isacharplus}{\isacharplus}\ show\ i{\isacharparenright}{\isacharverbatimclose}\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}reserved}\isamarkupfalse%
-\ Haskell\ Errno%
-\endisatagquotett
-{\isafoldquotett}%
-%
-\isadelimquotett
-%
-\endisadelimquotett
-%
-\begin{isamarkuptext}%
-\noindent Such named \isa{include}s are then prepended to every generated code.
- Inspect such code in order to find out how \hyperlink{command.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} behaves
- with respect to a particular target language.%
-\end{isamarkuptext}%
-\isamarkuptrue%
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Codegen/Thy/document/Codegen.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1690 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Codegen}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
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-\isatagtheory
-%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\isadelimML
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-\endisadelimML
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-\isatagML
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-\endisatagML
-{\isafoldML}%
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-\isadelimML
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-\endisadelimML
-%
-\isamarkupchapter{Code generation from Isabelle theories%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Introduction%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Motivation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Executing formal specifications as programs is a well-established
- topic in the theorem proving community. With increasing
- application of theorem proving systems in the area of
- software development and verification, its relevance manifests
- for running test cases and rapid prototyping. In logical
- calculi like constructive type theory,
- a notion of executability is implicit due to the nature
- of the calculus. In contrast, specifications in Isabelle
- can be highly non-executable. In order to bridge
- the gap between logic and executable specifications,
- an explicit non-trivial transformation has to be applied:
- code generation.
-
- This tutorial introduces a generic code generator for the
- Isabelle system \cite{isa-tutorial}.
- Generic in the sense that the
- \qn{target language} for which code shall ultimately be
- generated is not fixed but may be an arbitrary state-of-the-art
- functional programming language (currently, the implementation
- supports SML \cite{SML}, OCaml \cite{OCaml} and Haskell
- \cite{haskell-revised-report}).
- We aim to provide a
- versatile environment
- suitable for software development and verification,
- structuring the process
- of code generation into a small set of orthogonal principles
- while achieving a big coverage of application areas
- with maximum flexibility.
-
- Conceptually the code generator framework is part
- of Isabelle's \isa{Pure} meta logic; the object logic
- \isa{HOL} which is an extension of \isa{Pure}
- already comes with a reasonable framework setup and thus provides
- a good working horse for raising code-generation-driven
- applications. So, we assume some familiarity and experience
- with the ingredients of the \isa{HOL} \emph{Main} theory
- (see also \cite{isa-tutorial}).%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Overview%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The code generator aims to be usable with no further ado
- in most cases while allowing for detailed customization.
- This manifests in the structure of this tutorial:
- we start with a generic example \secref{sec:example}
- and introduce code generation concepts \secref{sec:concept}.
- Section
- \secref{sec:basics} explains how to use the framework naively,
- presuming a reasonable default setup. Then, section
- \secref{sec:advanced} deals with advanced topics,
- introducing further aspects of the code generator framework
- in a motivation-driven manner. Last, section \secref{sec:ml}
- introduces the framework's internal programming interfaces.
-
- \begin{warn}
- Ultimately, the code generator which this tutorial deals with
- is supposed to replace the already established code generator
- by Stefan Berghofer \cite{Berghofer-Nipkow:2002}.
- So, for the moment, there are two distinct code generators
- in Isabelle.
- Also note that while the framework itself is
- object-logic independent, only \isa{HOL} provides a reasonable
- framework setup.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{An example: a simple theory of search trees \label{sec:example}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-When writing executable specifications using \isa{HOL},
- it is convenient to use
- three existing packages: the datatype package for defining
- datatypes, the function package for (recursive) functions,
- and the class package for overloaded definitions.
-
- We develope a small theory of search trees; trees are represented
- as a datatype with key type \isa{{\isacharprime}a} and value type \isa{{\isacharprime}b}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree\ {\isacharequal}\ Leaf\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}linorder{\isachardoublequoteclose}\ {\isacharprime}b\isanewline
-\ \ {\isacharbar}\ Branch\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharprime}a{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent Note that we have constrained the type of keys
- to the class of total orders, \isa{linorder}.
-
- We define \isa{find} and \isa{update} functions:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\isanewline
-\ \ find\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharprime}a{\isasymColon}linorder{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ option{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}find\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\ it\ {\isacharequal}\ {\isacharparenleft}if\ it\ {\isacharequal}\ key\ then\ Some\ val\ else\ None{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}find\ {\isacharparenleft}Branch\ t{\isadigit{1}}\ key\ t{\isadigit{2}}{\isacharparenright}\ it\ {\isacharequal}\ {\isacharparenleft}if\ it\ {\isasymle}\ key\ then\ find\ t{\isadigit{1}}\ it\ else\ find\ t{\isadigit{2}}\ it{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\isanewline
-\ \ update\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}linorder\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ searchtree{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}\isanewline
-\ \ \ \ if\ it\ {\isacharequal}\ key\ then\ Leaf\ key\ entry\isanewline
-\ \ \ \ \ \ else\ if\ it\ {\isasymle}\ key\isanewline
-\ \ \ \ \ \ then\ Branch\ {\isacharparenleft}Leaf\ it\ entry{\isacharparenright}\ it\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\isanewline
-\ \ \ \ \ \ else\ Branch\ {\isacharparenleft}Leaf\ key\ val{\isacharparenright}\ it\ {\isacharparenleft}Leaf\ it\ entry{\isacharparenright}\isanewline
-\ \ \ {\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ {\isacharparenleft}Branch\ t{\isadigit{1}}\ key\ t{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}\isanewline
-\ \ \ \ if\ it\ {\isasymle}\ key\isanewline
-\ \ \ \ \ \ then\ {\isacharparenleft}Branch\ {\isacharparenleft}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ t{\isadigit{1}}{\isacharparenright}\ key\ t{\isadigit{2}}{\isacharparenright}\isanewline
-\ \ \ \ \ \ else\ {\isacharparenleft}Branch\ t{\isadigit{1}}\ key\ {\isacharparenleft}update\ {\isacharparenleft}it{\isacharcomma}\ entry{\isacharparenright}\ t{\isadigit{2}}{\isacharparenright}{\isacharparenright}\isanewline
-\ \ \ {\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent For testing purpose, we define a small example
- using natural numbers \isa{nat} (which are a \isa{linorder})
- as keys and list of nats as values:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ example\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}nat{\isacharcomma}\ nat\ list{\isacharparenright}\ searchtree{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}example\ {\isacharequal}\ update\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isacharcomma}\ {\isacharbrackleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharbrackright}{\isacharparenright}\ {\isacharparenleft}update\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isacharcomma}\ {\isacharbrackleft}Suc\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isacharbrackright}{\isacharparenright}\isanewline
-\ \ \ \ {\isacharparenleft}update\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ {\isacharbrackleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharbrackright}{\isacharparenright}\ {\isacharparenleft}Leaf\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent Then we generate code%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ example\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}tree{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent which looks like:
- \lstsml{Thy/examples/tree.ML}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Code generation concepts and process \label{sec:concept}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\begin{figure}[h]
- \centering
- \includegraphics[width=0.7\textwidth]{codegen_process}
- \caption{code generator -- processing overview}
- \label{fig:process}
- \end{figure}
-
- The code generator employs a notion of executability
- for three foundational executable ingredients known
- from functional programming:
- \emph{defining equations}, \emph{datatypes}, and
- \emph{type classes}. A defining equation as a first approximation
- is a theorem of the form \isa{f\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n\ {\isasymequiv}\ t}
- (an equation headed by a constant \isa{f} with arguments
- \isa{t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n} and right hand side \isa{t}).
- Code generation aims to turn defining equations
- into a functional program by running through
- a process (see figure \ref{fig:process}):
-
- \begin{itemize}
-
- \item Out of the vast collection of theorems proven in a
- \qn{theory}, a reasonable subset modeling
- defining equations is \qn{selected}.
-
- \item On those selected theorems, certain
- transformations are carried out
- (\qn{preprocessing}). Their purpose is to turn theorems
- representing non- or badly executable
- specifications into equivalent but executable counterparts.
- The result is a structured collection of \qn{code theorems}.
-
- \item These \qn{code theorems} then are \qn{translated}
- into an Haskell-like intermediate
- language.
-
- \item Finally, out of the intermediate language the final
- code in the desired \qn{target language} is \qn{serialized}.
-
- \end{itemize}
-
- From these steps, only the two last are carried out
- outside the logic; by keeping this layer as
- thin as possible, the amount of code to trust is
- kept to a minimum.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Basics \label{sec:basics}%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Invoking the code generator%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Thanks to a reasonable setup of the \isa{HOL} theories, in
- most cases code generation proceeds without further ado:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\isanewline
-\ \ fac\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}fac\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}fac\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ Suc\ n\ {\isacharasterisk}\ fac\ n{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent This executable specification is now turned to SML code:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ fac\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}fac{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent The \isa{{\isasymEXPORTCODE}} command takes a space-separated list of
- constants together with \qn{serialization directives}
- These start with a \qn{target language}
- identifier, followed by a file specification
- where to write the generated code to.
-
- Internally, the defining equations for all selected
- constants are taken, including any transitively required
- constants, datatypes and classes, resulting in the following
- code:
-
- \lstsml{Thy/examples/fac.ML}
-
- The code generator will complain when a required
- ingredient does not provide a executable counterpart,
- e.g.~generating code
- for constants not yielding
- a defining equation (e.g.~the Hilbert choice
- operation \isa{SOME}):%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ pick{\isacharunderscore}some\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pick{\isacharunderscore}some\ xs\ {\isacharequal}\ {\isacharparenleft}SOME\ x{\isachardot}\ x\ {\isasymin}\ set\ xs{\isacharparenright}{\isachardoublequoteclose}%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-\isanewline
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ pick{\isacharunderscore}some\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}fail{\isacharunderscore}const{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent will fail.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Theorem selection%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The list of all defining equations in a theory may be inspected
- using the \isa{{\isasymPRINTCODESETUP}} command:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{print{\isacharunderscore}codesetup}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-\noindent which displays a table of constant with corresponding
- defining equations (the additional stuff displayed
- shall not bother us for the moment).
-
- The typical \isa{HOL} tools are already set up in a way that
- function definitions introduced by \isa{{\isasymDEFINITION}},
- \isa{{\isasymPRIMREC}}, \isa{{\isasymFUN}},
- \isa{{\isasymFUNCTION}}, \isa{{\isasymCONSTDEFS}},
- \isa{{\isasymRECDEF}} are implicitly propagated
- to this defining equation table. Specific theorems may be
- selected using an attribute: \emph{code func}. As example,
- a weight selector function:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\isanewline
-\ \ pick\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}nat\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ list\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pick\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ n\ {\isacharequal}\ {\isacharparenleft}let\ {\isacharparenleft}k{\isacharcomma}\ v{\isacharparenright}\ {\isacharequal}\ x\ in\isanewline
-\ \ \ \ if\ n\ {\isacharless}\ k\ then\ v\ else\ pick\ xs\ {\isacharparenleft}n\ {\isacharminus}\ k{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent We want to eliminate the explicit destruction
- of \isa{x} to \isa{{\isacharparenleft}k{\isacharcomma}\ v{\isacharparenright}}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}pick\ {\isacharparenleft}{\isacharparenleft}k{\isacharcomma}\ v{\isacharparenright}{\isacharhash}xs{\isacharparenright}\ n\ {\isacharequal}\ {\isacharparenleft}if\ n\ {\isacharless}\ k\ then\ v\ else\ pick\ xs\ {\isacharparenleft}n\ {\isacharminus}\ k{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ pick\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}pick{\isadigit{1}}{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent This theorem now is used for generating code:
-
- \lstsml{Thy/examples/pick1.ML}
-
- \noindent The policy is that \emph{default equations} stemming from
- \isa{{\isasymDEFINITION}},
- \isa{{\isasymPRIMREC}}, \isa{{\isasymFUN}},
- \isa{{\isasymFUNCTION}}, \isa{{\isasymCONSTDEFS}},
- \isa{{\isasymRECDEF}} statements are discarded as soon as an
- equation is explicitly selected by means of \emph{code func}.
- Further applications of \emph{code func} add theorems incrementally,
- but syntactic redundancies are implicitly dropped. For example,
- using a modified version of the \isa{fac} function
- as defining equation, the then redundant (since
- syntactically subsumed) original defining equations
- are dropped.
-
- \begin{warn}
- The attributes \emph{code} and \emph{code del}
- associated with the existing code generator also apply to
- the new one: \emph{code} implies \emph{code func},
- and \emph{code del} implies \emph{code func del}.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Type classes%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Type classes enter the game via the Isar class package.
- For a short introduction how to use it, see \cite{isabelle-classes};
- here we just illustrate its impact on code generation.
-
- In a target language, type classes may be represented
- natively (as in the case of Haskell). For languages
- like SML, they are implemented using \emph{dictionaries}.
- Our following example specifies a class \qt{null},
- assigning to each of its inhabitants a \qt{null} value:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{class}\isamarkupfalse%
-\ null\ {\isacharequal}\ type\ {\isacharplus}\isanewline
-\ \ \isakeyword{fixes}\ null\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\isanewline
-\isanewline
-\isacommand{primrec}\isamarkupfalse%
-\isanewline
-\ \ head\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}null\ list\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}head\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ null{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}head\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent We provide some instances for our \isa{null}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{instantiation}\isamarkupfalse%
-\ option\ \isakeyword{and}\ list\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharparenright}\ null\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ {\isachardoublequoteopen}null\ {\isacharequal}\ None{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ {\isachardoublequoteopen}null\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\begin{isamarkuptext}%
-\noindent Constructing a dummy example:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ {\isachardoublequoteopen}dummy\ {\isacharequal}\ head\ {\isacharbrackleft}Some\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharcomma}\ None{\isacharbrackright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Type classes offer a suitable occasion to introduce
- the Haskell serializer. Its usage is almost the same
- as SML, but, in accordance with conventions
- some Haskell systems enforce, each module ends
- up in a single file. The module hierarchy is reflected in
- the file system, with root directory given as file specification.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ dummy\ \isakeyword{in}\ Haskell\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lsthaskell{Thy/examples/Codegen.hs}
- \noindent (we have left out all other modules).
-
- \medskip
-
- The whole code in SML with explicit dictionary passing:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ dummy\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}class{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/class.ML}
-
- \medskip
-
- \noindent or in OCaml:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ dummy\ \isakeyword{in}\ OCaml\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}class{\isachardot}ocaml{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/class.ocaml}
-
- \medskip The explicit association of constants
- to classes can be inspected using the \isa{{\isasymPRINTCLASSES}}
- command.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Recipes and advanced topics \label{sec:advanced}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In this tutorial, we do not attempt to give an exhaustive
- description of the code generator framework; instead,
- we cast a light on advanced topics by introducing
- them together with practically motivated examples. Concerning
- further reading, see
-
- \begin{itemize}
-
- \item the Isabelle/Isar Reference Manual \cite{isabelle-isar-ref}
- for exhaustive syntax diagrams.
- \item or \cite{Haftmann-Nipkow:2007:codegen} which deals with foundational issues
- of the code generator framework.
-
- \end{itemize}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Library theories \label{sec:library}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The \isa{HOL} \isa{Main} theory already provides a code
- generator setup
- which should be suitable for most applications. Common extensions
- and modifications are available by certain theories of the \isa{HOL}
- library; beside being useful in applications, they may serve
- as a tutorial for customizing the code generator setup.
-
- \begin{description}
-
- \item[\isa{Code{\isacharunderscore}Integer}] represents \isa{HOL} integers by big
- integer literals in target languages.
- \item[\isa{Code{\isacharunderscore}Char}] represents \isa{HOL} characters by
- character literals in target languages.
- \item[\isa{Code{\isacharunderscore}Char{\isacharunderscore}chr}] like \isa{Code{\isacharunderscore}Char},
- but also offers treatment of character codes; includes
- \isa{Code{\isacharunderscore}Integer}.
- \item[\isa{Efficient{\isacharunderscore}Nat}] \label{eff_nat} implements natural numbers by integers,
- which in general will result in higher efficency; pattern
- matching with \isa{{\isadigit{0}}} / \isa{Suc}
- is eliminated; includes \isa{Code{\isacharunderscore}Integer}.
- \item[\isa{Code{\isacharunderscore}Index}] provides an additional datatype
- \isa{index} which is mapped to target-language built-in integers.
- Useful for code setups which involve e.g. indexing of
- target-language arrays.
- \item[\isa{Code{\isacharunderscore}Message}] provides an additional datatype
- \isa{message{\isacharunderscore}string} which is isomorphic to strings;
- \isa{message{\isacharunderscore}string}s are mapped to target-language strings.
- Useful for code setups which involve e.g. printing (error) messages.
-
- \end{description}
-
- \begin{warn}
- When importing any of these theories, they should form the last
- items in an import list. Since these theories adapt the
- code generator setup in a non-conservative fashion,
- strange effects may occur otherwise.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Preprocessing%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Before selected function theorems are turned into abstract
- code, a chain of definitional transformation steps is carried
- out: \emph{preprocessing}. In essence, the preprocessor
- consists of two components: a \emph{simpset} and \emph{function transformers}.
-
- The \emph{simpset} allows to employ the full generality of the Isabelle
- simplifier. Due to the interpretation of theorems
- as defining equations, rewrites are applied to the right
- hand side and the arguments of the left hand side of an
- equation, but never to the constant heading the left hand side.
- An important special case are \emph{inline theorems} which may be
- declared an undeclared using the
- \emph{code inline} or \emph{code inline del} attribute respectively.
- Some common applications:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{itemize}
-%
-\begin{isamarkuptext}%
-\item replacing non-executable constructs by executable ones:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ set\ xs\ {\isasymlongleftrightarrow}\ x\ mem\ xs{\isachardoublequoteclose}%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\item eliminating superfluous constants:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\item replacing executable but inconvenient constructs:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\ \ \isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ \ \ {\isachardoublequoteopen}xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymlongleftrightarrow}\ List{\isachardot}null\ xs{\isachardoublequoteclose}%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\end{itemize}
-%
-\begin{isamarkuptext}%
-\emph{Function transformers} provide a very general interface,
- transforming a list of function theorems to another
- list of function theorems, provided that neither the heading
- constant nor its type change. The \isa{{\isadigit{0}}} / \isa{Suc}
- pattern elimination implemented in
- theory \isa{Efficient{\isacharunderscore}Nat} (see \secref{eff_nat}) uses this
- interface.
-
- \noindent The current setup of the preprocessor may be inspected using
- the \isa{{\isasymPRINTCODESETUP}} command.
-
- \begin{warn}
- The attribute \emph{code unfold}
- associated with the existing code generator also applies to
- the new one: \emph{code unfold} implies \emph{code inline}.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Concerning operational equality%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Surely you have already noticed how equality is treated
- by the code generator:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\isanewline
-\ \ collect{\isacharunderscore}duplicates\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ xs{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharparenleft}z{\isacharhash}zs{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ z\ {\isasymin}\ set\ xs\isanewline
-\ \ \ \ \ \ then\ if\ z\ {\isasymin}\ set\ ys\isanewline
-\ \ \ \ \ \ \ \ then\ collect{\isacharunderscore}duplicates\ xs\ ys\ zs\isanewline
-\ \ \ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ xs\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs\isanewline
-\ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ {\isacharparenleft}z{\isacharhash}xs{\isacharparenright}\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The membership test during preprocessing is rewritten,
- resulting in \isa{op\ mem}, which itself
- performs an explicit equality check.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ collect{\isacharunderscore}duplicates\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}collect{\isacharunderscore}duplicates{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/collect_duplicates.ML}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Obviously, polymorphic equality is implemented the Haskell
- way using a type class. How is this achieved? HOL introduces
- an explicit class \isa{eq} with a corresponding operation
- \isa{eq{\isacharunderscore}class{\isachardot}eq} such that \isa{eq{\isacharunderscore}class{\isachardot}eq\ x\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharequal}\ y{\isacharparenright}}.
- The preprocessing framework does the rest.
- For datatypes, instances of \isa{eq} are implicitly derived
- when possible. For other types, you may instantiate \isa{eq}
- manually like any other type class.
-
- Though this \isa{eq} class is designed to get rarely in
- the way, a subtlety
- enters the stage when definitions of overloaded constants
- are dependent on operational equality. For example, let
- us define a lexicographic ordering on tuples:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{instantiation}\isamarkupfalse%
-\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}ord{\isacharcomma}\ ord{\isacharparenright}\ ord\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ {\isacharbrackleft}code\ func\ del{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}p{\isadigit{1}}\ {\isacharless}\ p{\isadigit{2}}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}let\ {\isacharparenleft}x{\isadigit{1}}{\isacharcomma}\ y{\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{1}}{\isacharsemicolon}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{2}}\ in\isanewline
-\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ {\isacharbrackleft}code\ func\ del{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}p{\isadigit{1}}\ {\isasymle}\ p{\isadigit{2}}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}let\ {\isacharparenleft}x{\isadigit{1}}{\isacharcomma}\ y{\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{1}}{\isacharsemicolon}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ p{\isadigit{2}}\ in\isanewline
-\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ord{\isacharunderscore}prod\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}ord{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}ord{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp{\isacharunderscore}all%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Then code generation will fail. Why? The definition
- of \isa{op\ {\isasymle}} depends on equality on both arguments,
- which are polymorphic and impose an additional \isa{eq}
- class constraint, thus violating the type discipline
- for class operations.
-
- The solution is to add \isa{eq} explicitly to the first sort arguments in the
- code theorems:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ ord{\isacharunderscore}prod{\isacharunderscore}code\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}ord{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}ord{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}ord{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ {\isacharparenleft}x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ ord{\isacharunderscore}prod\ \isacommand{by}\isamarkupfalse%
-\ rule{\isacharplus}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent Then code generation succeeds:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}eq{\isacharcomma}\ ord{\isacharbraceright}\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}ord\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}lexicographic{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/lexicographic.ML}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In general, code theorems for overloaded constants may have more
- restrictive sort constraints than the underlying instance relation
- between class and type constructor as long as the whole system of
- constraints is coregular; code theorems violating coregularity
- are rejected immediately. Consequently, it might be necessary
- to delete disturbing theorems in the code theorem table,
- as we have done here with the original definitions \isa{less{\isacharunderscore}prod{\isacharunderscore}def}
- and \isa{less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def}.
-
- In some cases, the automatically derived defining equations
- for equality on a particular type may not be appropriate.
- As example, watch the following datatype representing
- monomorphic parametric types (where type constructors
- are referred to by natural numbers):%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ monotype\ {\isacharequal}\ Mono\ nat\ {\isachardoublequoteopen}monotype\ list{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Then code generation for SML would fail with a message
- that the generated code conains illegal mutual dependencies:
- the theorem \isa{Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}\ {\isacharequal}\ Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}\ {\isasymequiv}\ tyco{\isadigit{1}}\ {\isacharequal}\ tyco{\isadigit{2}}\ {\isasymand}\ typargs{\isadigit{1}}\ {\isacharequal}\ typargs{\isadigit{2}}} already requires the
- instance \isa{monotype\ {\isasymColon}\ eq}, which itself requires
- \isa{Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}\ {\isacharequal}\ Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}\ {\isasymequiv}\ tyco{\isadigit{1}}\ {\isacharequal}\ tyco{\isadigit{2}}\ {\isasymand}\ typargs{\isadigit{1}}\ {\isacharequal}\ typargs{\isadigit{2}}}; Haskell has no problem with mutually
- recursive \isa{instance} and \isa{function} definitions,
- but the SML serializer does not support this.
-
- In such cases, you have to provide you own equality equations
- involving auxiliary constants. In our case,
- \isa{list{\isacharunderscore}all{\isadigit{2}}} can do the job:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ monotype{\isacharunderscore}eq{\isacharunderscore}list{\isacharunderscore}all{\isadigit{2}}\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}\ {\isacharequal}\ Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ tyco{\isadigit{1}}\ {\isacharequal}\ tyco{\isadigit{2}}\ {\isasymand}\ list{\isacharunderscore}all{\isadigit{2}}\ {\isacharparenleft}op\ {\isacharequal}{\isacharparenright}\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ list{\isacharunderscore}all{\isadigit{2}}{\isacharunderscore}eq\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-does not depend on instance \isa{monotype\ {\isasymColon}\ eq}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ {\isacharequal}\ {\isacharcolon}{\isacharcolon}\ monotype\ {\isasymRightarrow}\ monotype\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}monotype{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/monotype.ML}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Programs as sets of theorems%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-As told in \secref{sec:concept}, code generation is based
- on a structured collection of code theorems.
- For explorative purpose, this collection
- may be inspected using the \isa{{\isasymCODETHMS}} command:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{code{\isacharunderscore}thms}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ mod\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent prints a table with \emph{all} defining equations
- for \isa{op\ mod}, including
- \emph{all} defining equations those equations depend
- on recursivly. \isa{{\isasymCODETHMS}} provides a convenient
- mechanism to inspect the impact of a preprocessor setup
- on defining equations.
-
- Similarly, the \isa{{\isasymCODEDEPS}} command shows a graph
- visualizing dependencies between defining equations.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Constructor sets for datatypes%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Conceptually, any datatype is spanned by a set of
- \emph{constructors} of type \isa{{\isasymtau}\ {\isacharequal}\ {\isasymdots}\ {\isasymRightarrow}\ {\isasymkappa}\ {\isasymalpha}\isactrlisub {\isadigit{1}}\ {\isasymdots}\ {\isasymalpha}\isactrlisub n}
- where \isa{{\isacharbraceleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub n{\isacharbraceright}} is excactly the set of \emph{all}
- type variables in \isa{{\isasymtau}}. The HOL datatype package
- by default registers any new datatype in the table
- of datatypes, which may be inspected using
- the \isa{{\isasymPRINTCODESETUP}} command.
-
- In some cases, it may be convenient to alter or
- extend this table; as an example, we will develope an alternative
- representation of natural numbers as binary digits, whose
- size does increase logarithmically with its value, not linear
- \footnote{Indeed, the \isa{Efficient{\isacharunderscore}Nat} theory (see \ref{eff_nat})
- does something similar}. First, the digit representation:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ Dig{\isadigit{0}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ n\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ Dig{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent We will use these two ">digits"< to represent natural numbers
- in binary digits, e.g.:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isadigit{4}}{\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{2}}\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}Dig{\isadigit{1}}\ {\isacharparenleft}Dig{\isadigit{0}}\ {\isacharparenleft}Dig{\isadigit{1}}\ {\isacharparenleft}Dig{\isadigit{0}}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ Dig{\isadigit{0}}{\isacharunderscore}def\ Dig{\isadigit{1}}{\isacharunderscore}def{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent Of course we also have to provide proper code equations for
- the operations, e.g. \isa{op\ {\isacharplus}}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ plus{\isacharunderscore}Dig\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharplus}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isacharplus}\ {\isadigit{0}}\ {\isacharequal}\ m{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharplus}\ Dig{\isadigit{0}}\ n\ {\isacharequal}\ Dig{\isadigit{1}}\ n{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ m\ {\isacharplus}\ {\isadigit{1}}\ {\isacharequal}\ Dig{\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharplus}\ Dig{\isadigit{1}}\ n\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ m\ {\isacharplus}\ {\isadigit{1}}\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}m\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ m\ {\isacharplus}\ Dig{\isadigit{0}}\ n\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{0}}\ m\ {\isacharplus}\ Dig{\isadigit{1}}\ n\ {\isacharequal}\ Dig{\isadigit{1}}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ m\ {\isacharplus}\ Dig{\isadigit{0}}\ n\ {\isacharequal}\ Dig{\isadigit{1}}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}Dig{\isadigit{1}}\ m\ {\isacharplus}\ Dig{\isadigit{1}}\ n\ {\isacharequal}\ Dig{\isadigit{0}}\ {\isacharparenleft}m\ {\isacharplus}\ n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ Dig{\isadigit{0}}{\isacharunderscore}def\ Dig{\isadigit{1}}{\isacharunderscore}def{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent We then instruct the code generator to view \isa{{\isadigit{0}}},
- \isa{{\isadigit{1}}}, \isa{Dig{\isadigit{0}}} and \isa{Dig{\isadigit{1}}} as
- datatype constructors:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isasymColon}nat{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isasymColon}nat{\isachardoublequoteclose}\ Dig{\isadigit{0}}\ Dig{\isadigit{1}}%
-\begin{isamarkuptext}%
-\noindent For the former constructor \isa{Suc}, we provide a code
- equation and remove some parts of the default code generator setup
- which are an obstacle here:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ Suc{\isacharunderscore}Dig\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}Suc\ n\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ One{\isacharunderscore}nat{\isacharunderscore}def\ {\isacharbrackleft}code\ inline\ del{\isacharbrackright}\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ add{\isacharunderscore}Suc{\isacharunderscore}shift\ {\isacharbrackleft}code\ func\ del{\isacharbrackright}%
-\begin{isamarkuptext}%
-\noindent This yields the following code:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ {\isacharplus}\ {\isasymColon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}nat{\isacharunderscore}binary{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/nat_binary.ML}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\medskip
-
- From this example, it can be easily glimpsed that using own constructor sets
- is a little delicate since it changes the set of valid patterns for values
- of that type. Without going into much detail, here some practical hints:
-
- \begin{itemize}
- \item When changing the constuctor set for datatypes, take care to
- provide an alternative for the \isa{case} combinator (e.g. by replacing
- it using the preprocessor).
- \item Values in the target language need not to be normalized -- different
- values in the target language may represent the same value in the
- logic (e.g. \isa{Dig{\isadigit{1}}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}}).
- \item Usually, a good methodology to deal with the subleties of pattern
- matching is to see the type as an abstract type: provide a set
- of operations which operate on the concrete representation of the type,
- and derive further operations by combinations of these primitive ones,
- without relying on a particular representation.
- \end{itemize}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isataginvisible
-\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isacharcolon}{\isacharcolon}nat{\isachardoublequoteclose}\ Suc\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ plus{\isacharunderscore}Dig\ {\isacharbrackleft}code\ func\ del{\isacharbrackright}\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ One{\isacharunderscore}nat{\isacharunderscore}def\ {\isacharbrackleft}code\ inline{\isacharbrackright}\isanewline
-\isacommand{declare}\isamarkupfalse%
-\ add{\isacharunderscore}Suc{\isacharunderscore}shift\ {\isacharbrackleft}code\ func{\isacharbrackright}\ \isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ func{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isacharplus}\ n\ {\isacharequal}\ {\isacharparenleft}n\ {\isasymColon}\ nat{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisataginvisible
-{\isafoldinvisible}%
-%
-\isadeliminvisible
-%
-\endisadeliminvisible
-%
-\isamarkupsubsection{Customizing serialization%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Basics%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Consider the following function and its corresponding
- SML code:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\isanewline
-\ \ in{\isacharunderscore}interval\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymtimes}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}in{\isacharunderscore}interval\ {\isacharparenleft}k{\isacharcomma}\ l{\isacharparenright}\ n\ {\isasymlongleftrightarrow}\ k\ {\isasymle}\ n\ {\isasymand}\ n\ {\isasymle}\ l{\isachardoublequoteclose}%
-\isadelimtt
-%
-\endisadelimtt
-%
-\isatagtt
-%
-\endisatagtt
-{\isafoldtt}%
-%
-\isadelimtt
-%
-\endisadelimtt
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ in{\isacharunderscore}interval\ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}bool{\isacharunderscore}literal{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/bool_literal.ML}
-
- \noindent Though this is correct code, it is a little bit unsatisfactory:
- boolean values and operators are materialized as distinguished
- entities with have nothing to do with the SML-builtin notion
- of \qt{bool}. This results in less readable code;
- additionally, eager evaluation may cause programs to
- loop or break which would perfectly terminate when
- the existing SML \qt{bool} would be used. To map
- the HOL \qt{bool} on SML \qt{bool}, we may use
- \qn{custom serializations}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\isatagtt
-\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
-\ bool\isanewline
-\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}bool{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ True\ \isakeyword{and}\ False\ \isakeyword{and}\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}true{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}false{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}{\isacharunderscore}\ andalso\ {\isacharunderscore}{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagtt
-{\isafoldtt}%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\begin{isamarkuptext}%
-The \isa{{\isasymCODETYPE}} commad takes a type constructor
- as arguments together with a list of custom serializations.
- Each custom serialization starts with a target language
- identifier followed by an expression, which during
- code serialization is inserted whenever the type constructor
- would occur. For constants, \isa{{\isasymCODECONST}} implements
- the corresponding mechanism. Each ``\verb|_|'' in
- a serialization expression is treated as a placeholder
- for the type constructor's (the constant's) arguments.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ in{\isacharunderscore}interval\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}bool{\isacharunderscore}mlbool{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/bool_mlbool.ML}
-
- \noindent This still is not perfect: the parentheses
- around the \qt{andalso} expression are superfluous.
- Though the serializer
- by no means attempts to imitate the rich Isabelle syntax
- framework, it provides some common idioms, notably
- associative infixes with precedences which may be used here:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\isatagtt
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ {\isasymand}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharparenleft}SML\ \isakeyword{infixl}\ {\isadigit{1}}\ {\isachardoublequoteopen}andalso{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagtt
-{\isafoldtt}%
-%
-\isadelimtt
-%
-\endisadelimtt
-\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ in{\isacharunderscore}interval\ \ \isakeyword{in}\ SML\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}bool{\isacharunderscore}infix{\isachardot}ML{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\lstsml{Thy/examples/bool_infix.ML}
-
- \medskip
-
- Next, we try to map HOL pairs to SML pairs, using the
- infix ``\verb|*|'' type constructor and parentheses:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\isatagtt
-\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
-\ {\isacharasterisk}\isanewline
-\ \ {\isacharparenleft}SML\ \isakeyword{infix}\ {\isadigit{2}}\ {\isachardoublequoteopen}{\isacharasterisk}{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ Pair\isanewline
-\ \ {\isacharparenleft}SML\ {\isachardoublequoteopen}{\isacharbang}{\isacharparenleft}{\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharcomma}{\isacharslash}\ {\isacharparenleft}{\isacharunderscore}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagtt
-{\isafoldtt}%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\begin{isamarkuptext}%
-The initial bang ``\verb|!|'' tells the serializer to never put
- parentheses around the whole expression (they are already present),
- while the parentheses around argument place holders
- tell not to put parentheses around the arguments.
- The slash ``\verb|/|'' (followed by arbitrary white space)
- inserts a space which may be used as a break if necessary
- during pretty printing.
-
- These examples give a glimpse what mechanisms
- custom serializations provide; however their usage
- requires careful thinking in order not to introduce
- inconsistencies -- or, in other words:
- custom serializations are completely axiomatic.
-
- A further noteworthy details is that any special
- character in a custom serialization may be quoted
- using ``\verb|'|''; thus, in
- ``\verb|fn '_ => _|'' the first
- ``\verb|_|'' is a proper underscore while the
- second ``\verb|_|'' is a placeholder.
-
- The HOL theories provide further
- examples for custom serializations.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Haskell serialization%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-For convenience, the default
- HOL setup for Haskell maps the \isa{eq} class to
- its counterpart in Haskell, giving custom serializations
- for the class (\isa{{\isasymCODECLASS}}) and its operation:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\isatagtt
-\isacommand{code{\isacharunderscore}class}\isamarkupfalse%
-\ eq\isanewline
-\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Eq{\isachardoublequoteclose}\ \isakeyword{where}\ {\isachardoublequoteopen}op\ {\isacharequal}{\isachardoublequoteclose}\ {\isasymequiv}\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharequal}{\isacharequal}{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}const}\isamarkupfalse%
-\ {\isachardoublequoteopen}op\ {\isacharequal}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharparenleft}Haskell\ \isakeyword{infixl}\ {\isadigit{4}}\ {\isachardoublequoteopen}{\isacharequal}{\isacharequal}{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagtt
-{\isafoldtt}%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\begin{isamarkuptext}%
-A problem now occurs whenever a type which
- is an instance of \isa{eq} in HOL is mapped
- on a Haskell-builtin type which is also an instance
- of Haskell \isa{Eq}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{typedecl}\isamarkupfalse%
-\ bar\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isachardoublequoteopen}eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}x{\isasymColon}bar{\isacharparenright}\ y\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ default\ {\isacharparenleft}simp\ add{\isacharcolon}\ eq{\isacharunderscore}bar{\isacharunderscore}def{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-%
-\isadelimtt
-\isanewline
-%
-\endisadelimtt
-%
-\isatagtt
-\isacommand{code{\isacharunderscore}type}\isamarkupfalse%
-\ bar\isanewline
-\ \ {\isacharparenleft}Haskell\ {\isachardoublequoteopen}Integer{\isachardoublequoteclose}{\isacharparenright}%
-\endisatagtt
-{\isafoldtt}%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\begin{isamarkuptext}%
-The code generator would produce
- an additional instance, which of course is rejected.
- To suppress this additional instance, use
- \isa{{\isasymCODEINSTANCE}}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\isatagtt
-\isacommand{code{\isacharunderscore}instance}\isamarkupfalse%
-\ bar\ {\isacharcolon}{\isacharcolon}\ eq\isanewline
-\ \ {\isacharparenleft}Haskell\ {\isacharminus}{\isacharparenright}%
-\endisatagtt
-{\isafoldtt}%
-%
-\isadelimtt
-%
-\endisadelimtt
-%
-\isamarkupsubsubsection{Pretty printing%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The serializer provides ML interfaces to set up
- pretty serializations for expressions like lists, numerals
- and characters; these are
- monolithic stubs and should only be used with the
- theories introduced in \secref{sec:library}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Cyclic module dependencies%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Sometimes the awkward situation occurs that dependencies
- between definitions introduce cyclic dependencies
- between modules, which in the Haskell world leaves
- you to the mercy of the Haskell implementation you are using,
- while for SML code generation is not possible.
-
- A solution is to declare module names explicitly.
- Let use assume the three cyclically dependent
- modules are named \emph{A}, \emph{B} and \emph{C}.
- Then, by stating%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{code{\isacharunderscore}modulename}\isamarkupfalse%
-\ SML\isanewline
-\ \ A\ ABC\isanewline
-\ \ B\ ABC\isanewline
-\ \ C\ ABC%
-\begin{isamarkuptext}%
-we explicitly map all those modules on \emph{ABC},
- resulting in an ad-hoc merge of this three modules
- at serialization time.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Incremental code generation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Code generation is \emph{incremental}: theorems
- and abstract intermediate code are cached and extended on demand.
- The cache may be partially or fully dropped if the underlying
- executable content of the theory changes.
- Implementation of caching is supposed to transparently
- hid away the details from the user. Anyway, caching
- reaches the surface by using a slightly more general form
- of the \isa{{\isasymCODETHMS}}, \isa{{\isasymCODEDEPS}}
- and \isa{{\isasymEXPORTCODE}} commands: the list of constants
- may be omitted. Then, all constants with code theorems
- in the current cache are referred to.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{ML interfaces \label{sec:ml}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Since the code generator framework not only aims to provide
- a nice Isar interface but also to form a base for
- code-generation-based applications, here a short
- description of the most important ML interfaces.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Executable theory content: \isa{Code}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-This Pure module implements the core notions of
- executable content of a theory.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Managing executable content%
-}
-\isamarkuptrue%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isatagmlref
-%
-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexml{Code.add\_func}\verb|Code.add_func: thm -> theory -> theory| \\
- \indexml{Code.del\_func}\verb|Code.del_func: thm -> theory -> theory| \\
- \indexml{Code.add\_funcl}\verb|Code.add_funcl: string * thm list Susp.T -> theory -> theory| \\
- \indexml{Code.map\_pre}\verb|Code.map_pre: (MetaSimplifier.simpset -> MetaSimplifier.simpset) -> theory -> theory| \\
- \indexml{Code.map\_post}\verb|Code.map_post: (MetaSimplifier.simpset -> MetaSimplifier.simpset) -> theory -> theory| \\
- \indexml{Code.add\_functrans}\verb|Code.add_functrans: string * (theory -> thm list -> thm list option)|\isasep\isanewline%
-\verb| -> theory -> theory| \\
- \indexml{Code.del\_functrans}\verb|Code.del_functrans: string -> theory -> theory| \\
- \indexml{Code.add\_datatype}\verb|Code.add_datatype: (string * typ) list -> theory -> theory| \\
- \indexml{Code.get\_datatype}\verb|Code.get_datatype: theory -> string|\isasep\isanewline%
-\verb| -> (string * sort) list * (string * typ list) list| \\
- \indexml{Code.get\_datatype\_of\_constr}\verb|Code.get_datatype_of_constr: theory -> string -> string option|
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Code.add_func|~\isa{thm}~\isa{thy} adds function
- theorem \isa{thm} to executable content.
-
- \item \verb|Code.del_func|~\isa{thm}~\isa{thy} removes function
- theorem \isa{thm} from executable content, if present.
-
- \item \verb|Code.add_funcl|~\isa{{\isacharparenleft}const{\isacharcomma}\ lthms{\isacharparenright}}~\isa{thy} adds
- suspended defining equations \isa{lthms} for constant
- \isa{const} to executable content.
-
- \item \verb|Code.map_pre|~\isa{f}~\isa{thy} changes
- the preprocessor simpset.
-
- \item \verb|Code.add_functrans|~\isa{{\isacharparenleft}name{\isacharcomma}\ f{\isacharparenright}}~\isa{thy} adds
- function transformer \isa{f} (named \isa{name}) to executable content;
- \isa{f} is a transformer of the defining equations belonging
- to a certain function definition, depending on the
- current theory context. Returning \isa{NONE} indicates that no
- transformation took place; otherwise, the whole process will be iterated
- with the new defining equations.
-
- \item \verb|Code.del_functrans|~\isa{name}~\isa{thy} removes
- function transformer named \isa{name} from executable content.
-
- \item \verb|Code.add_datatype|~\isa{cs}~\isa{thy} adds
- a datatype to executable content, with generation
- set \isa{cs}.
-
- \item \verb|Code.get_datatype_of_constr|~\isa{thy}~\isa{const}
- returns type constructor corresponding to
- constructor \isa{const}; returns \isa{NONE}
- if \isa{const} is no constructor.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagmlref
-{\isafoldmlref}%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isamarkupsubsection{Auxiliary%
-}
-\isamarkuptrue%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isatagmlref
-%
-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexml{Code\_Unit.read\_const}\verb|Code_Unit.read_const: theory -> string -> string| \\
- \indexml{Code\_Unit.head\_func}\verb|Code_Unit.head_func: thm -> string * ((string * sort) list * typ)| \\
- \indexml{Code\_Unit.rewrite\_func}\verb|Code_Unit.rewrite_func: MetaSimplifier.simpset -> thm -> thm| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Code_Unit.read_const|~\isa{thy}~\isa{s}
- reads a constant as a concrete term expression \isa{s}.
-
- \item \verb|Code_Unit.head_func|~\isa{thm}
- extracts the constant and its type from a defining equation \isa{thm}.
-
- \item \verb|Code_Unit.rewrite_func|~\isa{ss}~\isa{thm}
- rewrites a defining equation \isa{thm} with a simpset \isa{ss};
- only arguments and right hand side are rewritten,
- not the head of the defining equation.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagmlref
-{\isafoldmlref}%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isamarkupsubsection{Implementing code generator applications%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Implementing code generator applications on top
- of the framework set out so far usually not only
- involves using those primitive interfaces
- but also storing code-dependent data and various
- other things.
-
- \begin{warn}
- Some interfaces discussed here have not reached
- a final state yet.
- Changes likely to occur in future.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Data depending on the theory's executable content%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Due to incrementality of code generation, changes in the
- theory's executable content have to be propagated in a
- certain fashion. Additionally, such changes may occur
- not only during theory extension but also during theory
- merge, which is a little bit nasty from an implementation
- point of view. The framework provides a solution
- to this technical challenge by providing a functorial
- data slot \verb|CodeDataFun|; on instantiation
- of this functor, the following types and operations
- are required:
-
- \medskip
- \begin{tabular}{l}
- \isa{type\ T} \\
- \isa{val\ empty{\isacharcolon}\ T} \\
- \isa{val\ merge{\isacharcolon}\ Pretty{\isachardot}pp\ {\isasymrightarrow}\ T\ {\isacharasterisk}\ T\ {\isasymrightarrow}\ T} \\
- \isa{val\ purge{\isacharcolon}\ theory\ option\ {\isasymrightarrow}\ CodeUnit{\isachardot}const\ list\ option\ {\isasymrightarrow}\ T\ {\isasymrightarrow}\ T}
- \end{tabular}
-
- \begin{description}
-
- \item \isa{T} the type of data to store.
-
- \item \isa{empty} initial (empty) data.
-
- \item \isa{merge} merging two data slots.
-
- \item \isa{purge}~\isa{thy}~\isa{consts} propagates changes in executable content;
- if possible, the current theory context is handed over
- as argument \isa{thy} (if there is no current theory context (e.g.~during
- theory merge, \verb|NONE|); \isa{consts} indicates the kind
- of change: \verb|NONE| stands for a fundamental change
- which invalidates any existing code, \isa{SOME\ consts}
- hints that executable content for constants \isa{consts}
- has changed.
-
- \end{description}
-
- An instance of \verb|CodeDataFun| provides the following
- interface:
-
- \medskip
- \begin{tabular}{l}
- \isa{get{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} \\
- \isa{change{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ T{\isacharparenright}\ {\isasymrightarrow}\ T} \\
- \isa{change{\isacharunderscore}yield{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T{\isacharparenright}\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T}
- \end{tabular}
-
- \begin{description}
-
- \item \isa{get} retrieval of the current data.
-
- \item \isa{change} update of current data (cached!)
- by giving a continuation.
-
- \item \isa{change{\isacharunderscore}yield} update with side result.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\emph{Happy proving, happy hacking!}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Codegen/Thy/document/Further.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,234 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Further}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Further\isanewline
-\isakeyword{imports}\ Setup\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupsection{Further issues \label{sec:further}%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{Further reading%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Do dive deeper into the issue of code generation, you should visit
- the Isabelle/Isar Reference Manual \cite{isabelle-isar-ref} which
- contains exhaustive syntax diagrams.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Modules%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-When invoking the \hyperlink{command.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} command it is possible to leave
- out the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} part; then code is distributed over
- different modules, where the module name space roughly is induced
- by the \isa{Isabelle} theory name space.
-
- Then sometimes the awkward situation occurs that dependencies between
- definitions introduce cyclic dependencies between modules, which in the
- \isa{Haskell} world leaves you to the mercy of the \isa{Haskell} implementation
- you are using, while for \isa{SML}/\isa{OCaml} code generation is not possible.
-
- A solution is to declare module names explicitly.
- Let use assume the three cyclically dependent
- modules are named \emph{A}, \emph{B} and \emph{C}.
- Then, by stating%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{code{\isacharunderscore}modulename}\isamarkupfalse%
-\ SML\isanewline
-\ \ A\ ABC\isanewline
-\ \ B\ ABC\isanewline
-\ \ C\ ABC%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-we explicitly map all those modules on \emph{ABC},
- resulting in an ad-hoc merge of this three modules
- at serialisation time.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Evaluation oracle%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Code generation may also be used to \emph{evaluate} expressions
- (using \isa{SML} as target language of course).
- For instance, the \hyperlink{command.value}{\mbox{\isa{\isacommand{value}}}} allows to reduce an expression to a
- normal form with respect to the underlying code equations:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{value}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{2}}\ {\isacharslash}\ {\isacharparenleft}{\isadigit{1}}{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ rat{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent will display \isa{{\isadigit{7}}\ {\isacharslash}\ {\isadigit{2}}}.
-
- The \hyperlink{method.eval}{\mbox{\isa{eval}}} method tries to reduce a goal by code generation to \isa{True}
- and solves it in that case, but fails otherwise:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{2}}\ {\isacharslash}\ {\isacharparenleft}{\isadigit{1}}{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ rat{\isacharparenright}\ {\isacharequal}\ {\isadigit{7}}\ {\isacharslash}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ eval%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The soundness of the \hyperlink{method.eval}{\mbox{\isa{eval}}} method depends crucially
- on the correctness of the code generator; this is one of the reasons
- why you should not use adaption (see \secref{sec:adaption}) frivolously.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Code antiquotation%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In scenarios involving techniques like reflection it is quite common
- that code generated from a theory forms the basis for implementing
- a proof procedure in \isa{SML}. To facilitate interfacing of generated code
- with system code, the code generator provides a \isa{code} antiquotation:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{datatype}\isamarkupfalse%
-\ form\ {\isacharequal}\ T\ {\isacharbar}\ F\ {\isacharbar}\ And\ form\ form\ {\isacharbar}\ Or\ form\ form\isanewline
-\isanewline
-\isacommand{ML}\isamarkupfalse%
-\ {\isacharverbatimopen}\isanewline
-\ \ fun\ eval{\isacharunderscore}form\ %
-\isaantiq
-code\ T%
-\endisaantiq
-\ {\isacharequal}\ true\isanewline
-\ \ \ \ {\isacharbar}\ eval{\isacharunderscore}form\ %
-\isaantiq
-code\ F%
-\endisaantiq
-\ {\isacharequal}\ false\isanewline
-\ \ \ \ {\isacharbar}\ eval{\isacharunderscore}form\ {\isacharparenleft}%
-\isaantiq
-code\ And%
-\endisaantiq
-\ {\isacharparenleft}p{\isacharcomma}\ q{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ \ \ \ eval{\isacharunderscore}form\ p\ andalso\ eval{\isacharunderscore}form\ q\isanewline
-\ \ \ \ {\isacharbar}\ eval{\isacharunderscore}form\ {\isacharparenleft}%
-\isaantiq
-code\ Or%
-\endisaantiq
-\ {\isacharparenleft}p{\isacharcomma}\ q{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ \ \ \ eval{\isacharunderscore}form\ p\ orelse\ eval{\isacharunderscore}form\ q{\isacharsemicolon}\isanewline
-{\isacharverbatimclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent \isa{code} takes as argument the name of a constant; after the
- whole \isa{SML} is read, the necessary code is generated transparently
- and the corresponding constant names are inserted. This technique also
- allows to use pattern matching on constructors stemming from compiled
- \isa{datatypes}.
-
- For a less simplistic example, theory \hyperlink{theory.Ferrack}{\mbox{\isa{Ferrack}}} is
- a good reference.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Imperative data structures%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-If you consider imperative data structures as inevitable for a specific
- application, you should consider
- \emph{Imperative Functional Programming with Isabelle/HOL}
- (\cite{bulwahn-et-al:2008:imperative});
- the framework described there is available in theory \hyperlink{theory.Imperative-HOL}{\mbox{\isa{Imperative{\isacharunderscore}HOL}}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Codegen/Thy/document/Introduction.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,390 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Introduction}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Introduction\isanewline
-\isakeyword{imports}\ Setup\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupchapter{Code generation from \isa{Isabelle{\isacharslash}HOL} theories%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Introduction and Overview%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-This tutorial introduces a generic code generator for the
- \isa{Isabelle} system.
- Generic in the sense that the
- \qn{target language} for which code shall ultimately be
- generated is not fixed but may be an arbitrary state-of-the-art
- functional programming language (currently, the implementation
- supports \isa{SML} \cite{SML}, \isa{OCaml} \cite{OCaml} and \isa{Haskell}
- \cite{haskell-revised-report}).
-
- Conceptually the code generator framework is part
- of Isabelle's \hyperlink{theory.Pure}{\mbox{\isa{Pure}}} meta logic framework; the logic
- \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} which is an extension of \hyperlink{theory.Pure}{\mbox{\isa{Pure}}}
- already comes with a reasonable framework setup and thus provides
- a good working horse for raising code-generation-driven
- applications. So, we assume some familiarity and experience
- with the ingredients of the \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} distribution theories.
- (see also \cite{isa-tutorial}).
-
- The code generator aims to be usable with no further ado
- in most cases while allowing for detailed customisation.
- This manifests in the structure of this tutorial: after a short
- conceptual introduction with an example (\secref{sec:intro}),
- we discuss the generic customisation facilities (\secref{sec:program}).
- A further section (\secref{sec:adaption}) is dedicated to the matter of
- \qn{adaption} to specific target language environments. After some
- further issues (\secref{sec:further}) we conclude with an overview
- of some ML programming interfaces (\secref{sec:ml}).
-
- \begin{warn}
- Ultimately, the code generator which this tutorial deals with
- is supposed to replace the existing code generator
- by Stefan Berghofer \cite{Berghofer-Nipkow:2002}.
- So, for the moment, there are two distinct code generators
- in Isabelle. In case of ambiguity, we will refer to the framework
- described here as \isa{generic\ code\ generator}, to the
- other as \isa{SML\ code\ generator}.
- Also note that while the framework itself is
- object-logic independent, only \hyperlink{theory.HOL}{\mbox{\isa{HOL}}} provides a reasonable
- framework setup.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Code generation via shallow embedding \label{sec:intro}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The key concept for understanding \isa{Isabelle}'s code generation is
- \emph{shallow embedding}, i.e.~logical entities like constants, types and
- classes are identified with corresponding concepts in the target language.
-
- Inside \hyperlink{theory.HOL}{\mbox{\isa{HOL}}}, the \hyperlink{command.datatype}{\mbox{\isa{\isacommand{datatype}}}} and
- \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}/\hyperlink{command.primrec}{\mbox{\isa{\isacommand{primrec}}}}/\hyperlink{command.fun}{\mbox{\isa{\isacommand{fun}}}} declarations form
- the core of a functional programming language. The default code generator setup
- allows to turn those into functional programs immediately.
- This means that \qt{naive} code generation can proceed without further ado.
- For example, here a simple \qt{implementation} of amortised queues:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{datatype}\isamarkupfalse%
-\ {\isacharprime}a\ queue\ {\isacharequal}\ AQueue\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ empty\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{primrec}\isamarkupfalse%
-\ enqueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ AQueue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}\ ys{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ option\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}case\ rev\ xs\ of\ y\ {\isacharhash}\ ys\ {\isasymRightarrow}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ ys{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Then we can generate code e.g.~for \isa{SML} as follows:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ empty\ dequeue\ enqueue\ \isakeyword{in}\ SML\isanewline
-\ \ \isakeyword{module{\isacharunderscore}name}\ Example\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}example{\isachardot}ML{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent resulting in the following code:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun foldl f a [] = a\\
-\hspace*{0pt} ~| foldl f a (x ::~xs) = foldl f (f a x) xs;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun rev xs = foldl (fn xsa => fn x => x ::~xsa) [] xs;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun list{\char95}case f1 f2 (a ::~lista) = f2 a lista\\
-\hspace*{0pt} ~| list{\char95}case f1 f2 [] = f1;\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype 'a queue = AQueue of 'a list * 'a list;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val empty :~'a queue = AQueue ([],~[])\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun dequeue (AQueue ([],~[])) = (NONE,~AQueue ([],~[]))\\
-\hspace*{0pt} ~| dequeue (AQueue (xs,~y ::~ys)) = (SOME y,~AQueue (xs,~ys))\\
-\hspace*{0pt} ~| dequeue (AQueue (v ::~va,~[])) =\\
-\hspace*{0pt} ~~~let\\
-\hspace*{0pt} ~~~~~val y ::~ys = rev (v ::~va);\\
-\hspace*{0pt} ~~~in\\
-\hspace*{0pt} ~~~~~(SOME y,~AQueue ([],~ys))\\
-\hspace*{0pt} ~~~end;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun enqueue x (AQueue (xs,~ys)) = AQueue (x ::~xs,~ys);\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The \hyperlink{command.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} command takes a space-separated list of
- constants for which code shall be generated; anything else needed for those
- is added implicitly. Then follows a target language identifier
- (\isa{SML}, \isa{OCaml} or \isa{Haskell}) and a freely chosen module name.
- A file name denotes the destination to store the generated code. Note that
- the semantics of the destination depends on the target language: for
- \isa{SML} and \isa{OCaml} it denotes a \emph{file}, for \isa{Haskell}
- it denotes a \emph{directory} where a file named as the module name
- (with extension \isa{{\isachardot}hs}) is written:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{export{\isacharunderscore}code}\isamarkupfalse%
-\ empty\ dequeue\ enqueue\ \isakeyword{in}\ Haskell\isanewline
-\ \ \isakeyword{module{\isacharunderscore}name}\ Example\ \isakeyword{file}\ {\isachardoublequoteopen}examples{\isacharslash}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This is how the corresponding code in \isa{Haskell} looks like:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}module Example where {\char123}\\
-\hspace*{0pt}\\
-\hspace*{0pt}\\
-\hspace*{0pt}foldla ::~forall a b.~(a -> b -> a) -> a -> [b] -> a;\\
-\hspace*{0pt}foldla f a [] = a;\\
-\hspace*{0pt}foldla f a (x :~xs) = foldla f (f a x) xs;\\
-\hspace*{0pt}\\
-\hspace*{0pt}rev ::~forall a.~[a] -> [a];\\
-\hspace*{0pt}rev xs = foldla ({\char92}~xsa x -> x :~xsa) [] xs;\\
-\hspace*{0pt}\\
-\hspace*{0pt}list{\char95}case ::~forall t a.~t -> (a -> [a] -> t) -> [a] -> t;\\
-\hspace*{0pt}list{\char95}case f1 f2 (a :~list) = f2 a list;\\
-\hspace*{0pt}list{\char95}case f1 f2 [] = f1;\\
-\hspace*{0pt}\\
-\hspace*{0pt}data Queue a = AQueue [a] [a];\\
-\hspace*{0pt}\\
-\hspace*{0pt}empty ::~forall a.~Queue a;\\
-\hspace*{0pt}empty = AQueue [] [];\\
-\hspace*{0pt}\\
-\hspace*{0pt}dequeue ::~forall a.~Queue a -> (Maybe a,~Queue a);\\
-\hspace*{0pt}dequeue (AQueue [] []) = (Nothing,~AQueue [] []);\\
-\hspace*{0pt}dequeue (AQueue xs (y :~ys)) = (Just y,~AQueue xs ys);\\
-\hspace*{0pt}dequeue (AQueue (v :~va) []) =\\
-\hspace*{0pt} ~let {\char123}\\
-\hspace*{0pt} ~~~(y :~ys) = rev (v :~va);\\
-\hspace*{0pt} ~{\char125}~in (Just y,~AQueue [] ys);\\
-\hspace*{0pt}\\
-\hspace*{0pt}enqueue ::~forall a.~a -> Queue a -> Queue a;\\
-\hspace*{0pt}enqueue x (AQueue xs ys) = AQueue (x :~xs) ys;\\
-\hspace*{0pt}\\
-\hspace*{0pt}{\char125}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This demonstrates the basic usage of the \hyperlink{command.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} command;
- for more details see \secref{sec:further}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Code generator architecture \label{sec:concept}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-What you have seen so far should be already enough in a lot of cases. If you
- are content with this, you can quit reading here. Anyway, in order to customise
- and adapt the code generator, it is inevitable to gain some understanding
- how it works.
-
- \begin{figure}[h]
- \begin{tikzpicture}[x = 4.2cm, y = 1cm]
- \tikzstyle entity=[rounded corners, draw, thick, color = black, fill = white];
- \tikzstyle process=[ellipse, draw, thick, color = green, fill = white];
- \tikzstyle process_arrow=[->, semithick, color = green];
- \node (HOL) at (0, 4) [style=entity] {\isa{Isabelle{\isacharslash}HOL} theory};
- \node (eqn) at (2, 2) [style=entity] {code equations};
- \node (iml) at (2, 0) [style=entity] {intermediate language};
- \node (seri) at (1, 0) [style=process] {serialisation};
- \node (SML) at (0, 3) [style=entity] {\isa{SML}};
- \node (OCaml) at (0, 2) [style=entity] {\isa{OCaml}};
- \node (further) at (0, 1) [style=entity] {\isa{{\isasymdots}}};
- \node (Haskell) at (0, 0) [style=entity] {\isa{Haskell}};
- \draw [style=process_arrow] (HOL) .. controls (2, 4) ..
- node [style=process, near start] {selection}
- node [style=process, near end] {preprocessing}
- (eqn);
- \draw [style=process_arrow] (eqn) -- node (transl) [style=process] {translation} (iml);
- \draw [style=process_arrow] (iml) -- (seri);
- \draw [style=process_arrow] (seri) -- (SML);
- \draw [style=process_arrow] (seri) -- (OCaml);
- \draw [style=process_arrow, dashed] (seri) -- (further);
- \draw [style=process_arrow] (seri) -- (Haskell);
- \end{tikzpicture}
- \caption{Code generator architecture}
- \label{fig:arch}
- \end{figure}
-
- The code generator employs a notion of executability
- for three foundational executable ingredients known
- from functional programming:
- \emph{code equations}, \emph{datatypes}, and
- \emph{type classes}. A code equation as a first approximation
- is a theorem of the form \isa{f\ t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n\ {\isasymequiv}\ t}
- (an equation headed by a constant \isa{f} with arguments
- \isa{t\isactrlisub {\isadigit{1}}\ t\isactrlisub {\isadigit{2}}\ {\isasymdots}\ t\isactrlisub n} and right hand side \isa{t}).
- Code generation aims to turn code equations
- into a functional program. This is achieved by three major
- components which operate sequentially, i.e. the result of one is
- the input
- of the next in the chain, see diagram \ref{fig:arch}:
-
- \begin{itemize}
-
- \item Out of the vast collection of theorems proven in a
- \qn{theory}, a reasonable subset modelling
- code equations is \qn{selected}.
-
- \item On those selected theorems, certain
- transformations are carried out
- (\qn{preprocessing}). Their purpose is to turn theorems
- representing non- or badly executable
- specifications into equivalent but executable counterparts.
- The result is a structured collection of \qn{code theorems}.
-
- \item Before the selected code equations are continued with,
- they can be \qn{preprocessed}, i.e. subjected to theorem
- transformations. This \qn{preprocessor} is an interface which
- allows to apply
- the full expressiveness of ML-based theorem transformations
- to code generation; motivating examples are shown below, see
- \secref{sec:preproc}.
- The result of the preprocessing step is a structured collection
- of code equations.
-
- \item These code equations are \qn{translated} to a program
- in an abstract intermediate language. Think of it as a kind
- of \qt{Mini-Haskell} with four \qn{statements}: \isa{data}
- (for datatypes), \isa{fun} (stemming from code equations),
- also \isa{class} and \isa{inst} (for type classes).
-
- \item Finally, the abstract program is \qn{serialised} into concrete
- source code of a target language.
-
- \end{itemize}
-
- \noindent From these steps, only the two last are carried out outside the logic; by
- keeping this layer as thin as possible, the amount of code to trust is
- kept to a minimum.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Codegen/Thy/document/ML.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,255 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{ML}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ {\isachardoublequoteopen}ML{\isachardoublequoteclose}\isanewline
-\isakeyword{imports}\ Setup\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupsection{ML system interfaces \label{sec:ml}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Since the code generator framework not only aims to provide
- a nice Isar interface but also to form a base for
- code-generation-based applications, here a short
- description of the most important ML interfaces.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Executable theory content: \isa{Code}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-This Pure module implements the core notions of
- executable content of a theory.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Managing executable content%
-}
-\isamarkuptrue%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isatagmlref
-%
-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexdef{}{ML}{Code.add\_eqn}\verb|Code.add_eqn: thm -> theory -> theory| \\
- \indexdef{}{ML}{Code.del\_eqn}\verb|Code.del_eqn: thm -> theory -> theory| \\
- \indexdef{}{ML}{Code.add\_eqnl}\verb|Code.add_eqnl: string * (thm * bool) list lazy -> theory -> theory| \\
- \indexdef{}{ML}{Code.map\_pre}\verb|Code.map_pre: (simpset -> simpset) -> theory -> theory| \\
- \indexdef{}{ML}{Code.map\_post}\verb|Code.map_post: (simpset -> simpset) -> theory -> theory| \\
- \indexdef{}{ML}{Code.add\_functrans}\verb|Code.add_functrans: string * (theory -> (thm * bool) list -> (thm * bool) list option)|\isasep\isanewline%
-\verb| -> theory -> theory| \\
- \indexdef{}{ML}{Code.del\_functrans}\verb|Code.del_functrans: string -> theory -> theory| \\
- \indexdef{}{ML}{Code.add\_datatype}\verb|Code.add_datatype: (string * typ) list -> theory -> theory| \\
- \indexdef{}{ML}{Code.get\_datatype}\verb|Code.get_datatype: theory -> string|\isasep\isanewline%
-\verb| -> (string * sort) list * (string * typ list) list| \\
- \indexdef{}{ML}{Code.get\_datatype\_of\_constr}\verb|Code.get_datatype_of_constr: theory -> string -> string option|
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Code.add_eqn|~\isa{thm}~\isa{thy} adds function
- theorem \isa{thm} to executable content.
-
- \item \verb|Code.del_eqn|~\isa{thm}~\isa{thy} removes function
- theorem \isa{thm} from executable content, if present.
-
- \item \verb|Code.add_eqnl|~\isa{{\isacharparenleft}const{\isacharcomma}\ lthms{\isacharparenright}}~\isa{thy} adds
- suspended code equations \isa{lthms} for constant
- \isa{const} to executable content.
-
- \item \verb|Code.map_pre|~\isa{f}~\isa{thy} changes
- the preprocessor simpset.
-
- \item \verb|Code.add_functrans|~\isa{{\isacharparenleft}name{\isacharcomma}\ f{\isacharparenright}}~\isa{thy} adds
- function transformer \isa{f} (named \isa{name}) to executable content;
- \isa{f} is a transformer of the code equations belonging
- to a certain function definition, depending on the
- current theory context. Returning \isa{NONE} indicates that no
- transformation took place; otherwise, the whole process will be iterated
- with the new code equations.
-
- \item \verb|Code.del_functrans|~\isa{name}~\isa{thy} removes
- function transformer named \isa{name} from executable content.
-
- \item \verb|Code.add_datatype|~\isa{cs}~\isa{thy} adds
- a datatype to executable content, with generation
- set \isa{cs}.
-
- \item \verb|Code.get_datatype_of_constr|~\isa{thy}~\isa{const}
- returns type constructor corresponding to
- constructor \isa{const}; returns \isa{NONE}
- if \isa{const} is no constructor.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagmlref
-{\isafoldmlref}%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isamarkupsubsection{Auxiliary%
-}
-\isamarkuptrue%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isatagmlref
-%
-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexdef{}{ML}{Code\_Unit.read\_const}\verb|Code_Unit.read_const: theory -> string -> string| \\
- \indexdef{}{ML}{Code\_Unit.head\_eqn}\verb|Code_Unit.head_eqn: theory -> thm -> string * ((string * sort) list * typ)| \\
- \indexdef{}{ML}{Code\_Unit.rewrite\_eqn}\verb|Code_Unit.rewrite_eqn: simpset -> thm -> thm| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Code_Unit.read_const|~\isa{thy}~\isa{s}
- reads a constant as a concrete term expression \isa{s}.
-
- \item \verb|Code_Unit.head_eqn|~\isa{thy}~\isa{thm}
- extracts the constant and its type from a code equation \isa{thm}.
-
- \item \verb|Code_Unit.rewrite_eqn|~\isa{ss}~\isa{thm}
- rewrites a code equation \isa{thm} with a simpset \isa{ss};
- only arguments and right hand side are rewritten,
- not the head of the code equation.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagmlref
-{\isafoldmlref}%
-%
-\isadelimmlref
-%
-\endisadelimmlref
-%
-\isamarkupsubsection{Implementing code generator applications%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Implementing code generator applications on top
- of the framework set out so far usually not only
- involves using those primitive interfaces
- but also storing code-dependent data and various
- other things.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Data depending on the theory's executable content%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Due to incrementality of code generation, changes in the
- theory's executable content have to be propagated in a
- certain fashion. Additionally, such changes may occur
- not only during theory extension but also during theory
- merge, which is a little bit nasty from an implementation
- point of view. The framework provides a solution
- to this technical challenge by providing a functorial
- data slot \verb|CodeDataFun|; on instantiation
- of this functor, the following types and operations
- are required:
-
- \medskip
- \begin{tabular}{l}
- \isa{type\ T} \\
- \isa{val\ empty{\isacharcolon}\ T} \\
- \isa{val\ purge{\isacharcolon}\ theory\ {\isasymrightarrow}\ string\ list\ option\ {\isasymrightarrow}\ T\ {\isasymrightarrow}\ T}
- \end{tabular}
-
- \begin{description}
-
- \item \isa{T} the type of data to store.
-
- \item \isa{empty} initial (empty) data.
-
- \item \isa{purge}~\isa{thy}~\isa{consts} propagates changes in executable content;
- \isa{consts} indicates the kind
- of change: \verb|NONE| stands for a fundamental change
- which invalidates any existing code, \isa{SOME\ consts}
- hints that executable content for constants \isa{consts}
- has changed.
-
- \end{description}
-
- \noindent An instance of \verb|CodeDataFun| provides the following
- interface:
-
- \medskip
- \begin{tabular}{l}
- \isa{get{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} \\
- \isa{change{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ T{\isacharparenright}\ {\isasymrightarrow}\ T} \\
- \isa{change{\isacharunderscore}yield{\isacharcolon}\ theory\ {\isasymrightarrow}\ {\isacharparenleft}T\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T{\isacharparenright}\ {\isasymrightarrow}\ {\isacharprime}a\ {\isacharasterisk}\ T}
- \end{tabular}
-
- \begin{description}
-
- \item \isa{get} retrieval of the current data.
-
- \item \isa{change} update of current data (cached!)
- by giving a continuation.
-
- \item \isa{change{\isacharunderscore}yield} update with side result.
-
- \end{description}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\bigskip
-
- \emph{Happy proving, happy hacking!}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Codegen/Thy/document/Program.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1250 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Program}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Program\isanewline
-\isakeyword{imports}\ Introduction\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupsection{Turning Theories into Programs \label{sec:program}%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsection{The \isa{Isabelle{\isacharslash}HOL} default setup%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-We have already seen how by default equations stemming from
- \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}/\hyperlink{command.primrec}{\mbox{\isa{\isacommand{primrec}}}}/\hyperlink{command.fun}{\mbox{\isa{\isacommand{fun}}}}
- statements are used for code generation. This default behaviour
- can be changed, e.g. by providing different code equations.
- All kinds of customisation shown in this section is \emph{safe}
- in the sense that the user does not have to worry about
- correctness -- all programs generatable that way are partially
- correct.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Selecting code equations%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Coming back to our introductory example, we
- could provide an alternative code equations for \isa{dequeue}
- explicitly:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
-\ \ \ \ \ \ \ else\ dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}cases\ xs{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}\ {\isacharparenleft}cases\ {\isachardoublequoteopen}rev\ xs{\isachardoublequoteclose}{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The annotation \isa{{\isacharbrackleft}code{\isacharbrackright}} is an \isa{Isar}
- \isa{attribute} which states that the given theorems should be
- considered as code equations for a \isa{fun} statement --
- the corresponding constant is determined syntactically. The resulting code:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}dequeue ::~forall a.~Queue a -> (Maybe a,~Queue a);\\
-\hspace*{0pt}dequeue (AQueue xs (y :~ys)) = (Just y,~AQueue xs ys);\\
-\hspace*{0pt}dequeue (AQueue xs []) =\\
-\hspace*{0pt} ~(if nulla xs then (Nothing,~AQueue [] [])\\
-\hspace*{0pt} ~~~else dequeue (AQueue [] (rev xs)));%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent You may note that the equality test \isa{xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}} has been
- replaced by the predicate \isa{null\ xs}. This is due to the default
- setup in the \qn{preprocessor} to be discussed further below (\secref{sec:preproc}).
-
- Changing the default constructor set of datatypes is also
- possible. See \secref{sec:datatypes} for an example.
-
- As told in \secref{sec:concept}, code generation is based
- on a structured collection of code theorems.
- For explorative purpose, this collection
- may be inspected using the \hyperlink{command.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} command:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{code{\isacharunderscore}thms}\isamarkupfalse%
-\ dequeue%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent prints a table with \emph{all} code equations
- for \isa{dequeue}, including
- \emph{all} code equations those equations depend
- on recursively.
-
- Similarly, the \hyperlink{command.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} command shows a graph
- visualising dependencies between code equations.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{\isa{class} and \isa{instantiation}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Concerning type classes and code generation, let us examine an example
- from abstract algebra:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{class}\isamarkupfalse%
-\ semigroup\ {\isacharequal}\isanewline
-\ \ \isakeyword{fixes}\ mult\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymotimes}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{class}\isamarkupfalse%
-\ monoid\ {\isacharequal}\ semigroup\ {\isacharplus}\isanewline
-\ \ \isakeyword{fixes}\ neutral\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\ \ \isakeyword{assumes}\ neutl{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ neutr{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instantiation}\isamarkupfalse%
-\ nat\ {\isacharcolon}{\isacharcolon}\ monoid\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{primrec}\isamarkupfalse%
-\ mult{\isacharunderscore}nat\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymotimes}\ n\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}Suc\ m\ {\isasymotimes}\ n\ {\isacharequal}\ n\ {\isacharplus}\ m\ {\isasymotimes}\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ neutral{\isacharunderscore}nat\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ add{\isacharunderscore}mult{\isacharunderscore}distrib{\isacharcolon}\isanewline
-\ \ \isakeyword{fixes}\ n\ m\ q\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharplus}\ m{\isacharparenright}\ {\isasymotimes}\ q\ {\isacharequal}\ n\ {\isasymotimes}\ q\ {\isacharplus}\ m\ {\isasymotimes}\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ n{\isacharparenright}\ simp{\isacharunderscore}all\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ m\ n\ q\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ {\isasymotimes}\ n\ {\isasymotimes}\ q\ {\isacharequal}\ m\ {\isasymotimes}\ {\isacharparenleft}n\ {\isasymotimes}\ q{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ m{\isacharparenright}\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ add{\isacharunderscore}mult{\isacharunderscore}distrib{\isacharparenright}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharparenright}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ m{\isacharparenright}\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharparenright}\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent We define the natural operation of the natural numbers
- on monoids:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{primrec}\isamarkupfalse%
-\ {\isacharparenleft}\isakeyword{in}\ monoid{\isacharparenright}\ pow\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pow\ {\isadigit{0}}\ a\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}pow\ {\isacharparenleft}Suc\ n{\isacharparenright}\ a\ {\isacharequal}\ a\ {\isasymotimes}\ pow\ n\ a{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This we use to define the discrete exponentiation function:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{definition}\isamarkupfalse%
-\ bexp\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bexp\ n\ {\isacharequal}\ pow\ n\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The corresponding code:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}module Example where {\char123}\\
-\hspace*{0pt}\\
-\hspace*{0pt}\\
-\hspace*{0pt}data Nat = Zero{\char95}nat | Suc Nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}class Semigroup a where {\char123}\\
-\hspace*{0pt} ~mult ::~a -> a -> a;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}class (Semigroup a) => Monoid a where {\char123}\\
-\hspace*{0pt} ~neutral ::~a;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}pow ::~forall a.~(Monoid a) => Nat -> a -> a;\\
-\hspace*{0pt}pow Zero{\char95}nat a = neutral;\\
-\hspace*{0pt}pow (Suc n) a = mult a (pow n a);\\
-\hspace*{0pt}\\
-\hspace*{0pt}plus{\char95}nat ::~Nat -> Nat -> Nat;\\
-\hspace*{0pt}plus{\char95}nat (Suc m) n = plus{\char95}nat m (Suc n);\\
-\hspace*{0pt}plus{\char95}nat Zero{\char95}nat n = n;\\
-\hspace*{0pt}\\
-\hspace*{0pt}neutral{\char95}nat ::~Nat;\\
-\hspace*{0pt}neutral{\char95}nat = Suc Zero{\char95}nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}mult{\char95}nat ::~Nat -> Nat -> Nat;\\
-\hspace*{0pt}mult{\char95}nat Zero{\char95}nat n = Zero{\char95}nat;\\
-\hspace*{0pt}mult{\char95}nat (Suc m) n = plus{\char95}nat n (mult{\char95}nat m n);\\
-\hspace*{0pt}\\
-\hspace*{0pt}instance Semigroup Nat where {\char123}\\
-\hspace*{0pt} ~mult = mult{\char95}nat;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}instance Monoid Nat where {\char123}\\
-\hspace*{0pt} ~neutral = neutral{\char95}nat;\\
-\hspace*{0pt}{\char125};\\
-\hspace*{0pt}\\
-\hspace*{0pt}bexp ::~Nat -> Nat;\\
-\hspace*{0pt}bexp n = pow n (Suc (Suc Zero{\char95}nat));\\
-\hspace*{0pt}\\
-\hspace*{0pt}{\char125}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This is a convenient place to show how explicit dictionary construction
- manifests in generated code (here, the same example in \isa{SML}):%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a semigroup = {\char123}mult :~'a -> 'a -> 'a{\char125};\\
-\hspace*{0pt}fun mult (A{\char95}:'a semigroup) = {\char35}mult A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a monoid = {\char123}Program{\char95}{\char95}semigroup{\char95}monoid :~'a semigroup,~neutral :~'a{\char125};\\
-\hspace*{0pt}fun semigroup{\char95}monoid (A{\char95}:'a monoid) = {\char35}Program{\char95}{\char95}semigroup{\char95}monoid A{\char95};\\
-\hspace*{0pt}fun neutral (A{\char95}:'a monoid) = {\char35}neutral A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun pow A{\char95}~Zero{\char95}nat a = neutral A{\char95}\\
-\hspace*{0pt} ~| pow A{\char95}~(Suc n) a = mult (semigroup{\char95}monoid A{\char95}) a (pow A{\char95}~n a);\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun plus{\char95}nat (Suc m) n = plus{\char95}nat m (Suc n)\\
-\hspace*{0pt} ~| plus{\char95}nat Zero{\char95}nat n = n;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val neutral{\char95}nat :~nat = Suc Zero{\char95}nat\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun mult{\char95}nat Zero{\char95}nat n = Zero{\char95}nat\\
-\hspace*{0pt} ~| mult{\char95}nat (Suc m) n = plus{\char95}nat n (mult{\char95}nat m n);\\
-\hspace*{0pt}\\
-\hspace*{0pt}val semigroup{\char95}nat = {\char123}mult = mult{\char95}nat{\char125}~:~nat semigroup;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val monoid{\char95}nat =\\
-\hspace*{0pt} ~{\char123}Program{\char95}{\char95}semigroup{\char95}monoid = semigroup{\char95}nat,~neutral = neutral{\char95}nat{\char125}~:\\
-\hspace*{0pt} ~nat monoid;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun bexp n = pow monoid{\char95}nat n (Suc (Suc Zero{\char95}nat));\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Note the parameters with trailing underscore (\verb|A_|)
- which are the dictionary parameters.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{The preprocessor \label{sec:preproc}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Before selected function theorems are turned into abstract
- code, a chain of definitional transformation steps is carried
- out: \emph{preprocessing}. In essence, the preprocessor
- consists of two components: a \emph{simpset} and \emph{function transformers}.
-
- The \emph{simpset} allows to employ the full generality of the Isabelle
- simplifier. Due to the interpretation of theorems
- as code equations, rewrites are applied to the right
- hand side and the arguments of the left hand side of an
- equation, but never to the constant heading the left hand side.
- An important special case are \emph{inline theorems} which may be
- declared and undeclared using the
- \emph{code inline} or \emph{code inline del} attribute respectively.
-
- Some common applications:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{itemize}
-%
-\begin{isamarkuptext}%
-\item replacing non-executable constructs by executable ones:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}x\ {\isasymin}\ set\ xs\ {\isasymlongleftrightarrow}\ x\ mem\ xs{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\item eliminating superfluous constants:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\item replacing executable but inconvenient constructs:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymlongleftrightarrow}\ List{\isachardot}null\ xs{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ xs{\isacharparenright}\ simp{\isacharunderscore}all%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\end{itemize}
-%
-\begin{isamarkuptext}%
-\noindent \emph{Function transformers} provide a very general interface,
- transforming a list of function theorems to another
- list of function theorems, provided that neither the heading
- constant nor its type change. The \isa{{\isadigit{0}}} / \isa{Suc}
- pattern elimination implemented in
- theory \isa{Efficient{\isacharunderscore}Nat} (see \secref{eff_nat}) uses this
- interface.
-
- \noindent The current setup of the preprocessor may be inspected using
- the \hyperlink{command.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} command.
- \hyperlink{command.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} provides a convenient
- mechanism to inspect the impact of a preprocessor setup
- on code equations.
-
- \begin{warn}
- The attribute \emph{code unfold}
- associated with the \isa{SML\ code\ generator} also applies to
- the \isa{generic\ code\ generator}:
- \emph{code unfold} implies \emph{code inline}.
- \end{warn}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Datatypes \label{sec:datatypes}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Conceptually, any datatype is spanned by a set of
- \emph{constructors} of type \isa{{\isasymtau}\ {\isacharequal}\ {\isasymdots}\ {\isasymRightarrow}\ {\isasymkappa}\ {\isasymalpha}\isactrlisub {\isadigit{1}}\ {\isasymdots}\ {\isasymalpha}\isactrlisub n} where \isa{{\isacharbraceleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub n{\isacharbraceright}} is exactly the set of \emph{all} type variables in
- \isa{{\isasymtau}}. The HOL datatype package by default registers any new
- datatype in the table of datatypes, which may be inspected using the
- \hyperlink{command.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} command.
-
- In some cases, it is appropriate to alter or extend this table. As
- an example, we will develop an alternative representation of the
- queue example given in \secref{sec:intro}. The amortised
- representation is convenient for generating code but exposes its
- \qt{implementation} details, which may be cumbersome when proving
- theorems about it. Therefore, here a simple, straightforward
- representation of queues:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{datatype}\isamarkupfalse%
-\ {\isacharprime}a\ queue\ {\isacharequal}\ Queue\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ empty\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{primrec}\isamarkupfalse%
-\ enqueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}Queue\ xs{\isacharparenright}\ {\isacharequal}\ Queue\ {\isacharparenleft}xs\ {\isacharat}\ {\isacharbrackleft}x{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ option\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}None{\isacharcomma}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ x{\isacharcomma}\ Queue\ xs{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This we can use directly for proving; for executing,
- we provide an alternative characterisation:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{definition}\isamarkupfalse%
-\ AQueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}AQueue\ xs\ ys\ {\isacharequal}\ Queue\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
-\ AQueue%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Here we define a \qt{constructor} \isa{Program{\isachardot}AQueue} which
- is defined in terms of \isa{Queue} and interprets its arguments
- according to what the \emph{content} of an amortised queue is supposed
- to be. Equipped with this, we are able to prove the following equations
- for our primitive queue operations which \qt{implement} the simple
- queues in an amortised fashion:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ empty{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ AQueue{\isacharunderscore}def\ empty{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ enqueue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ AQueue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}\ ys{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dequeue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
-\ \ \ \ else\ dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp{\isacharunderscore}all%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent For completeness, we provide a substitute for the
- \isa{case} combinator on queues:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ aqueue{\isacharunderscore}case{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}aqueue{\isacharunderscore}case\ {\isacharequal}\ queue{\isacharunderscore}case{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ aqueue{\isacharunderscore}case\ {\isacharbrackleft}code{\isacharcomma}\ code\ inline{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}queue{\isacharunderscore}case\ {\isacharequal}\ aqueue{\isacharunderscore}case{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ aqueue{\isacharunderscore}case{\isacharunderscore}def\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ case{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}aqueue{\isacharunderscore}case\ f\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{unfolding}\isamarkupfalse%
-\ aqueue{\isacharunderscore}case{\isacharunderscore}def\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The resulting code looks as expected:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun foldl f a [] = a\\
-\hspace*{0pt} ~| foldl f a (x ::~xs) = foldl f (f a x) xs;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun rev xs = foldl (fn xsa => fn x => x ::~xsa) [] xs;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun null [] = true\\
-\hspace*{0pt} ~| null (x ::~xs) = false;\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype 'a queue = AQueue of 'a list * 'a list;\\
-\hspace*{0pt}\\
-\hspace*{0pt}val empty :~'a queue = AQueue ([],~[])\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun dequeue (AQueue (xs,~y ::~ys)) = (SOME y,~AQueue (xs,~ys))\\
-\hspace*{0pt} ~| dequeue (AQueue (xs,~[])) =\\
-\hspace*{0pt} ~~~(if null xs then (NONE,~AQueue ([],~[]))\\
-\hspace*{0pt} ~~~~~else dequeue (AQueue ([],~rev xs)));\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun enqueue x (AQueue (xs,~ys)) = AQueue (x ::~xs,~ys);\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent From this example, it can be glimpsed that using own
- constructor sets is a little delicate since it changes the set of
- valid patterns for values of that type. Without going into much
- detail, here some practical hints:
-
- \begin{itemize}
-
- \item When changing the constructor set for datatypes, take care
- to provide an alternative for the \isa{case} combinator
- (e.g.~by replacing it using the preprocessor).
-
- \item Values in the target language need not to be normalised --
- different values in the target language may represent the same
- value in the logic.
-
- \item Usually, a good methodology to deal with the subtleties of
- pattern matching is to see the type as an abstract type: provide
- a set of operations which operate on the concrete representation
- of the type, and derive further operations by combinations of
- these primitive ones, without relying on a particular
- representation.
-
- \end{itemize}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Equality and wellsortedness%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Surely you have already noticed how equality is treated
- by the code generator:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{primrec}\isamarkupfalse%
-\ collect{\isacharunderscore}duplicates\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ xs{\isachardoublequoteclose}\isanewline
-\ \ {\isacharbar}\ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharparenleft}z{\isacharhash}zs{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ z\ {\isasymin}\ set\ xs\isanewline
-\ \ \ \ \ \ then\ if\ z\ {\isasymin}\ set\ ys\isanewline
-\ \ \ \ \ \ \ \ then\ collect{\isacharunderscore}duplicates\ xs\ ys\ zs\isanewline
-\ \ \ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ xs\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs\isanewline
-\ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ {\isacharparenleft}z{\isacharhash}xs{\isacharparenright}\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs{\isacharparenright}{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent The membership test during preprocessing is rewritten,
- resulting in \isa{op\ mem}, which itself
- performs an explicit equality check.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a eq = {\char123}eq :~'a -> 'a -> bool{\char125};\\
-\hspace*{0pt}fun eq (A{\char95}:'a eq) = {\char35}eq A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun eqop A{\char95}~a b = eq A{\char95}~a b;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun member A{\char95}~x [] = false\\
-\hspace*{0pt} ~| member A{\char95}~x (y ::~ys) = eqop A{\char95}~x y orelse member A{\char95}~x ys;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun collect{\char95}duplicates A{\char95}~xs ys [] = xs\\
-\hspace*{0pt} ~| collect{\char95}duplicates A{\char95}~xs ys (z ::~zs) =\\
-\hspace*{0pt} ~~~(if member A{\char95}~z xs\\
-\hspace*{0pt} ~~~~~then (if member A{\char95}~z ys then collect{\char95}duplicates A{\char95}~xs ys zs\\
-\hspace*{0pt} ~~~~~~~~~~~~else collect{\char95}duplicates A{\char95}~xs (z ::~ys) zs)\\
-\hspace*{0pt} ~~~~~else collect{\char95}duplicates A{\char95}~(z ::~xs) (z ::~ys) zs);\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Obviously, polymorphic equality is implemented the Haskell
- way using a type class. How is this achieved? HOL introduces
- an explicit class \isa{eq} with a corresponding operation
- \isa{eq{\isacharunderscore}class{\isachardot}eq} such that \isa{eq{\isacharunderscore}class{\isachardot}eq\ {\isacharequal}\ op\ {\isacharequal}}.
- The preprocessing framework does the rest by propagating the
- \isa{eq} constraints through all dependent code equations.
- For datatypes, instances of \isa{eq} are implicitly derived
- when possible. For other types, you may instantiate \isa{eq}
- manually like any other type class.
-
- Though this \isa{eq} class is designed to get rarely in
- the way, a subtlety
- enters the stage when definitions of overloaded constants
- are dependent on operational equality. For example, let
- us define a lexicographic ordering on tuples
- (also see theory \hyperlink{theory.Product-ord}{\mbox{\isa{Product{\isacharunderscore}ord}}}):%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{instantiation}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharasterisk}{\isachardoublequoteclose}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}order{\isacharcomma}\ order{\isacharparenright}\ order\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isacharbrackleft}code\ del{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}x\ {\isasymle}\ y\ {\isasymlongleftrightarrow}\ fst\ x\ {\isacharless}\ fst\ y\ {\isasymor}\ fst\ x\ {\isacharequal}\ fst\ y\ {\isasymand}\ snd\ x\ {\isasymle}\ snd\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isacharbrackleft}code\ del{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}x\ {\isacharless}\ y\ {\isasymlongleftrightarrow}\ fst\ x\ {\isacharless}\ fst\ y\ {\isasymor}\ fst\ x\ {\isacharequal}\ fst\ y\ {\isasymand}\ snd\ x\ {\isacharless}\ snd\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}auto\ simp{\isacharcolon}\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}prod{\isacharunderscore}def\ intro{\isacharcolon}\ order{\isacharunderscore}less{\isacharunderscore}trans{\isacharparenright}\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ order{\isacharunderscore}prod\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}order{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}order{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Then code generation will fail. Why? The definition
- of \isa{op\ {\isasymle}} depends on equality on both arguments,
- which are polymorphic and impose an additional \isa{eq}
- class constraint, which the preprocessor does not propagate
- (for technical reasons).
-
- The solution is to add \isa{eq} explicitly to the first sort arguments in the
- code theorems:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ order{\isacharunderscore}prod{\isacharunderscore}code\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}order{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}order{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Then code generation succeeds:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a eq = {\char123}eq :~'a -> 'a -> bool{\char125};\\
-\hspace*{0pt}fun eq (A{\char95}:'a eq) = {\char35}eq A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a ord = {\char123}less{\char95}eq :~'a -> 'a -> bool,~less :~'a -> 'a -> bool{\char125};\\
-\hspace*{0pt}fun less{\char95}eq (A{\char95}:'a ord) = {\char35}less{\char95}eq A{\char95};\\
-\hspace*{0pt}fun less (A{\char95}:'a ord) = {\char35}less A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun eqop A{\char95}~a b = eq A{\char95}~a b;\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a preorder = {\char123}Orderings{\char95}{\char95}ord{\char95}preorder :~'a ord{\char125};\\
-\hspace*{0pt}fun ord{\char95}preorder (A{\char95}:'a preorder) = {\char35}Orderings{\char95}{\char95}ord{\char95}preorder A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a order = {\char123}Orderings{\char95}{\char95}preorder{\char95}order :~'a preorder{\char125};\\
-\hspace*{0pt}fun preorder{\char95}order (A{\char95}:'a order) = {\char35}Orderings{\char95}{\char95}preorder{\char95}order A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun less{\char95}eqa (A1{\char95},~A2{\char95}) B{\char95}~(x1,~y1) (x2,~y2) =\\
-\hspace*{0pt} ~less ((ord{\char95}preorder o preorder{\char95}order) A2{\char95}) x1 x2 orelse\\
-\hspace*{0pt} ~~~eqop A1{\char95}~x1 x2 andalso\\
-\hspace*{0pt} ~~~~~less{\char95}eq ((ord{\char95}preorder o preorder{\char95}order) B{\char95}) y1 y2\\
-\hspace*{0pt} ~| less{\char95}eqa (A1{\char95},~A2{\char95}) B{\char95}~(x1,~y1) (x2,~y2) =\\
-\hspace*{0pt} ~~~less ((ord{\char95}preorder o preorder{\char95}order) A2{\char95}) x1 x2 orelse\\
-\hspace*{0pt} ~~~~~eqop A1{\char95}~x1 x2 andalso\\
-\hspace*{0pt} ~~~~~~~less{\char95}eq ((ord{\char95}preorder o preorder{\char95}order) B{\char95}) y1 y2;\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-In some cases, the automatically derived code equations
- for equality on a particular type may not be appropriate.
- As example, watch the following datatype representing
- monomorphic parametric types (where type constructors
- are referred to by natural numbers):%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{datatype}\isamarkupfalse%
-\ monotype\ {\isacharequal}\ Mono\ nat\ {\isachardoublequoteopen}monotype\ list{\isachardoublequoteclose}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent Then code generation for SML would fail with a message
- that the generated code contains illegal mutual dependencies:
- the theorem \isa{eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}{\isacharparenright}\ {\isasymequiv}\ eq{\isacharunderscore}class{\isachardot}eq\ tyco{\isadigit{1}}\ tyco{\isadigit{2}}\ {\isasymand}\ eq{\isacharunderscore}class{\isachardot}eq\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}} already requires the
- instance \isa{monotype\ {\isasymColon}\ eq}, which itself requires
- \isa{eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}{\isacharparenright}\ {\isasymequiv}\ eq{\isacharunderscore}class{\isachardot}eq\ tyco{\isadigit{1}}\ tyco{\isadigit{2}}\ {\isasymand}\ eq{\isacharunderscore}class{\isachardot}eq\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}}; Haskell has no problem with mutually
- recursive \isa{instance} and \isa{function} definitions,
- but the SML serialiser does not support this.
-
- In such cases, you have to provide your own equality equations
- involving auxiliary constants. In our case,
- \isa{list{\isacharunderscore}all{\isadigit{2}}} can do the job:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ monotype{\isacharunderscore}eq{\isacharunderscore}list{\isacharunderscore}all{\isadigit{2}}\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}eq{\isacharunderscore}class{\isachardot}eq\ {\isacharparenleft}Mono\ tyco{\isadigit{1}}\ typargs{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}Mono\ tyco{\isadigit{2}}\ typargs{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ eq{\isacharunderscore}class{\isachardot}eq\ tyco{\isadigit{1}}\ tyco{\isadigit{2}}\ {\isasymand}\ list{\isacharunderscore}all{\isadigit{2}}\ eq{\isacharunderscore}class{\isachardot}eq\ typargs{\isadigit{1}}\ typargs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp\ add{\isacharcolon}\ eq\ list{\isacharunderscore}all{\isadigit{2}}{\isacharunderscore}eq\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent does not depend on instance \isa{monotype\ {\isasymColon}\ eq}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun null [] = true\\
-\hspace*{0pt} ~| null (x ::~xs) = false;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun eq{\char95}nat (Suc a) Zero{\char95}nat = false\\
-\hspace*{0pt} ~| eq{\char95}nat Zero{\char95}nat (Suc a) = false\\
-\hspace*{0pt} ~| eq{\char95}nat (Suc nat) (Suc nat') = eq{\char95}nat nat nat'\\
-\hspace*{0pt} ~| eq{\char95}nat Zero{\char95}nat Zero{\char95}nat = true;\\
-\hspace*{0pt}\\
-\hspace*{0pt}datatype monotype = Mono of nat * monotype list;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun list{\char95}all2 p (x ::~xs) (y ::~ys) = p x y andalso list{\char95}all2 p xs ys\\
-\hspace*{0pt} ~| list{\char95}all2 p xs [] = null xs\\
-\hspace*{0pt} ~| list{\char95}all2 p [] ys = null ys;\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun eq{\char95}monotype (Mono (tyco1,~typargs1)) (Mono (tyco2,~typargs2)) =\\
-\hspace*{0pt} ~eq{\char95}nat tyco1 tyco2 andalso list{\char95}all2 eq{\char95}monotype typargs1 typargs2;\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isamarkupsubsection{Explicit partiality%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Partiality usually enters the game by partial patterns, as
- in the following example, again for amortised queues:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{definition}\isamarkupfalse%
-\ strict{\isacharunderscore}dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ q\ {\isacharequal}\ {\isacharparenleft}case\ dequeue\ q\isanewline
-\ \ \ \ of\ {\isacharparenleft}Some\ x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ strict{\isacharunderscore}dequeue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ {\isacharparenleft}case\ rev\ xs\ of\ y\ {\isacharhash}\ ys\ {\isasymRightarrow}\ {\isacharparenleft}y{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ ys{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ strict{\isacharunderscore}dequeue{\isacharunderscore}def\ dequeue{\isacharunderscore}AQueue\ split{\isacharcolon}\ list{\isachardot}splits{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent In the corresponding code, there is no equation
- for the pattern \isa{Program{\isachardot}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}strict{\char95}dequeue ::~forall a.~Queue a -> (a,~Queue a);\\
-\hspace*{0pt}strict{\char95}dequeue (AQueue xs []) =\\
-\hspace*{0pt} ~let {\char123}\\
-\hspace*{0pt} ~~~(y :~ys) = rev xs;\\
-\hspace*{0pt} ~{\char125}~in (y,~AQueue [] ys);\\
-\hspace*{0pt}strict{\char95}dequeue (AQueue xs (y :~ys)) = (y,~AQueue xs ys);%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent In some cases it is desirable to have this
- pseudo-\qt{partiality} more explicitly, e.g.~as follows:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{axiomatization}\isamarkupfalse%
-\ empty{\isacharunderscore}queue\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ q\ {\isacharequal}\ {\isacharparenleft}case\ dequeue\ q\ of\ {\isacharparenleft}Some\ x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isacharbar}\ {\isacharunderscore}\ {\isasymRightarrow}\ empty{\isacharunderscore}queue{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ strict{\isacharunderscore}dequeue{\isacharprime}{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ empty{\isacharunderscore}queue\isanewline
-\ \ \ \ \ else\ strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
-\ \ \ \ \ {\isacharparenleft}y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ strict{\isacharunderscore}dequeue{\isacharprime}{\isacharunderscore}def\ dequeue{\isacharunderscore}AQueue\ split{\isacharcolon}\ list{\isachardot}splits{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-Observe that on the right hand side of the definition of \isa{strict{\isacharunderscore}dequeue{\isacharprime}} the constant \isa{empty{\isacharunderscore}queue} occurs
- which is unspecified.
-
- Normally, if constants without any code equations occur in a
- program, the code generator complains (since in most cases this is
- not what the user expects). But such constants can also be thought
- of as function definitions with no equations which always fail,
- since there is never a successful pattern match on the left hand
- side. In order to categorise a constant into that category
- explicitly, use \hyperlink{command.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{code{\isacharunderscore}abort}\isamarkupfalse%
-\ empty{\isacharunderscore}queue%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Then the code generator will just insert an error or
- exception at the appropriate position:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}empty{\char95}queue ::~forall a.~a;\\
-\hspace*{0pt}empty{\char95}queue = error {\char34}empty{\char95}queue{\char34};\\
-\hspace*{0pt}\\
-\hspace*{0pt}strict{\char95}dequeue' ::~forall a.~Queue a -> (a,~Queue a);\\
-\hspace*{0pt}strict{\char95}dequeue' (AQueue xs (y :~ys)) = (y,~AQueue xs ys);\\
-\hspace*{0pt}strict{\char95}dequeue' (AQueue xs []) =\\
-\hspace*{0pt} ~(if nulla xs then empty{\char95}queue\\
-\hspace*{0pt} ~~~else strict{\char95}dequeue' (AQueue [] (rev xs)));%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent This feature however is rarely needed in practice.
- Note also that the \isa{HOL} default setup already declares
- \isa{undefined} as \hyperlink{command.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}, which is most
- likely to be used in such situations.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\ \end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/Codegen.hs Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,23 +0,0 @@
-module Codegen where {
-
-import qualified Nat;
-
-class Null a where {
- nulla :: a;
-};
-
-heada :: forall a. (Codegen.Null a) => [a] -> a;
-heada (x : xs) = x;
-heada [] = Codegen.nulla;
-
-null_option :: forall a. Maybe a;
-null_option = Nothing;
-
-instance Codegen.Null (Maybe a) where {
- nulla = Codegen.null_option;
-};
-
-dummy :: Maybe Nat.Nat;
-dummy = Codegen.heada [Just (Nat.Suc Nat.Zero_nat), Nothing];
-
-}
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/Example.hs Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,33 +0,0 @@
-{-# OPTIONS_GHC -fglasgow-exts #-}
-
-module Example where {
-
-
-foldla :: forall a b. (a -> b -> a) -> a -> [b] -> a;
-foldla f a [] = a;
-foldla f a (x : xs) = foldla f (f a x) xs;
-
-rev :: forall a. [a] -> [a];
-rev xs = foldla (\ xsa x -> x : xsa) [] xs;
-
-list_case :: forall t a. t -> (a -> [a] -> t) -> [a] -> t;
-list_case f1 f2 (a : list) = f2 a list;
-list_case f1 f2 [] = f1;
-
-data Queue a = AQueue [a] [a];
-
-empty :: forall a. Queue a;
-empty = AQueue [] [];
-
-dequeue :: forall a. Queue a -> (Maybe a, Queue a);
-dequeue (AQueue [] []) = (Nothing, AQueue [] []);
-dequeue (AQueue xs (y : ys)) = (Just y, AQueue xs ys);
-dequeue (AQueue (v : va) []) =
- let {
- (y : ys) = rev (v : va);
- } in (Just y, AQueue [] ys);
-
-enqueue :: forall a. a -> Queue a -> Queue a;
-enqueue x (AQueue xs ys) = AQueue (x : xs) ys;
-
-}
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/arbitrary.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,9 +0,0 @@
-structure Codegen =
-struct
-
-val arbitrary_option : 'a option = NONE;
-
-fun dummy_option [] = arbitrary_option
- | dummy_option (x :: xs) = SOME x;
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/bool_infix.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,19 +0,0 @@
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-fun less_nat m (Suc n) = less_eq_nat m n
- | less_nat n Zero_nat = false
-and less_eq_nat (Suc m) n = less_nat m n
- | less_eq_nat Zero_nat n = true;
-
-end; (*struct Nat*)
-
-structure Codegen =
-struct
-
-fun in_interval (k, l) n =
- Nat.less_eq_nat k n andalso Nat.less_eq_nat n l;
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/bool_literal.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,31 +0,0 @@
-structure HOL =
-struct
-
-datatype boola = False | True;
-
-fun anda x True = x
- | anda x False = False
- | anda True x = x
- | anda False x = False;
-
-end; (*struct HOL*)
-
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-fun less_nat m (Suc n) = less_eq_nat m n
- | less_nat n Zero_nat = HOL.False
-and less_eq_nat (Suc m) n = less_nat m n
- | less_eq_nat Zero_nat n = HOL.True;
-
-end; (*struct Nat*)
-
-structure Codegen =
-struct
-
-fun in_interval (k, l) n =
- HOL.anda (Nat.less_eq_nat k n) (Nat.less_eq_nat n l);
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/bool_mlbool.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,19 +0,0 @@
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-fun less_nat m (Suc n) = less_eq_nat m n
- | less_nat n Zero_nat = false
-and less_eq_nat (Suc m) n = less_nat m n
- | less_eq_nat Zero_nat n = true;
-
-end; (*struct Nat*)
-
-structure Codegen =
-struct
-
-fun in_interval (k, l) n =
- (Nat.less_eq_nat k n) andalso (Nat.less_eq_nat n l);
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/class.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,24 +0,0 @@
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-end; (*struct Nat*)
-
-structure Codegen =
-struct
-
-type 'a null = {null : 'a};
-fun null (A_:'a null) = #null A_;
-
-fun head A_ (x :: xs) = x
- | head A_ [] = null A_;
-
-val null_option : 'a option = NONE;
-
-fun null_optiona () = {null = null_option} : ('a option) null;
-
-val dummy : Nat.nat option =
- head (null_optiona ()) [SOME (Nat.Suc Nat.Zero_nat), NONE];
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/class.ocaml Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,24 +0,0 @@
-module Nat =
-struct
-
-type nat = Suc of nat | Zero_nat;;
-
-end;; (*struct Nat*)
-
-module Codegen =
-struct
-
-type 'a null = {null : 'a};;
-let null _A = _A.null;;
-
-let rec head _A = function x :: xs -> x
- | [] -> null _A;;
-
-let rec null_option = None;;
-
-let null_optiona () = ({null = null_option} : ('a option) null);;
-
-let rec dummy
- = head (null_optiona ()) [Some (Nat.Suc Nat.Zero_nat); None];;
-
-end;; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/collect_duplicates.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,30 +0,0 @@
-structure HOL =
-struct
-
-type 'a eq = {eq : 'a -> 'a -> bool};
-fun eq (A_:'a eq) = #eq A_;
-
-fun eqop A_ a = eq A_ a;
-
-end; (*struct HOL*)
-
-structure List =
-struct
-
-fun member A_ x (y :: ys) =
- (if HOL.eqop A_ y x then true else member A_ x ys)
- | member A_ x [] = false;
-
-end; (*struct List*)
-
-structure Codegen =
-struct
-
-fun collect_duplicates A_ xs ys (z :: zs) =
- (if List.member A_ z xs
- then (if List.member A_ z ys then collect_duplicates A_ xs ys zs
- else collect_duplicates A_ xs (z :: ys) zs)
- else collect_duplicates A_ (z :: xs) (z :: ys) zs)
- | collect_duplicates A_ xs ys [] = xs;
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/dirty_set.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,102 +0,0 @@
-structure ROOT =
-struct
-
-structure Nat =
-struct
-
-datatype nat = Zero_nat | Suc of nat;
-
-end; (*struct Nat*)
-
-structure Integer =
-struct
-
-datatype bit = B0 | B1;
-
-datatype int = Pls | Min | Bit of int * bit | Number_of_int of int;
-
-fun pred (Bit (k, B0)) = Bit (pred k, B1)
- | pred (Bit (k, B1)) = Bit (k, B0)
- | pred Min = Bit (Min, B0)
- | pred Pls = Min;
-
-fun uminus_int (Number_of_int w) = Number_of_int (uminus_int w)
- | uminus_int (Bit (k, B0)) = Bit (uminus_int k, B0)
- | uminus_int (Bit (k, B1)) = Bit (pred (uminus_int k), B1)
- | uminus_int Min = Bit (Pls, B1)
- | uminus_int Pls = Pls;
-
-fun succ (Bit (k, B0)) = Bit (k, B1)
- | succ (Bit (k, B1)) = Bit (succ k, B0)
- | succ Min = Pls
- | succ Pls = Bit (Pls, B1);
-
-fun plus_int (Number_of_int v) (Number_of_int w) =
- Number_of_int (plus_int v w)
- | plus_int k Min = pred k
- | plus_int k Pls = k
- | plus_int (Bit (k, B1)) (Bit (l, B1)) = Bit (plus_int k (succ l), B0)
- | plus_int (Bit (k, B1)) (Bit (l, B0)) = Bit (plus_int k l, B1)
- | plus_int (Bit (k, B0)) (Bit (l, b)) = Bit (plus_int k l, b)
- | plus_int Min k = pred k
- | plus_int Pls k = k;
-
-fun minus_int (Number_of_int v) (Number_of_int w) =
- Number_of_int (plus_int v (uminus_int w))
- | minus_int z w = plus_int z (uminus_int w);
-
-fun less_eq_int (Number_of_int k) (Number_of_int l) = less_eq_int k l
- | less_eq_int (Bit (k1, B1)) (Bit (k2, B0)) = less_int k1 k2
- | less_eq_int (Bit (k1, v)) (Bit (k2, B1)) = less_eq_int k1 k2
- | less_eq_int (Bit (k1, B0)) (Bit (k2, v)) = less_eq_int k1 k2
- | less_eq_int (Bit (k, v)) Min = less_eq_int k Min
- | less_eq_int (Bit (k, B1)) Pls = less_int k Pls
- | less_eq_int (Bit (k, B0)) Pls = less_eq_int k Pls
- | less_eq_int Min (Bit (k, B1)) = less_eq_int Min k
- | less_eq_int Min (Bit (k, B0)) = less_int Min k
- | less_eq_int Min Min = true
- | less_eq_int Min Pls = true
- | less_eq_int Pls (Bit (k, v)) = less_eq_int Pls k
- | less_eq_int Pls Min = false
- | less_eq_int Pls Pls = true
-and less_int (Number_of_int k) (Number_of_int l) = less_int k l
- | less_int (Bit (k1, B0)) (Bit (k2, B1)) = less_eq_int k1 k2
- | less_int (Bit (k1, B1)) (Bit (k2, v)) = less_int k1 k2
- | less_int (Bit (k1, v)) (Bit (k2, B0)) = less_int k1 k2
- | less_int (Bit (k, B1)) Min = less_int k Min
- | less_int (Bit (k, B0)) Min = less_eq_int k Min
- | less_int (Bit (k, v)) Pls = less_int k Pls
- | less_int Min (Bit (k, v)) = less_int Min k
- | less_int Min Min = false
- | less_int Min Pls = true
- | less_int Pls (Bit (k, B1)) = less_eq_int Pls k
- | less_int Pls (Bit (k, B0)) = less_int Pls k
- | less_int Pls Min = false
- | less_int Pls Pls = false;
-
-fun nat_aux n i =
- (if less_eq_int i (Number_of_int Pls) then n
- else nat_aux (Nat.Suc n)
- (minus_int i (Number_of_int (Bit (Pls, B1)))));
-
-fun nat i = nat_aux Nat.Zero_nat i;
-
-end; (*struct Integer*)
-
-structure Codegen =
-struct
-
-val dummy_set : (Nat.nat -> Nat.nat) list = Nat.Suc :: [];
-
-val foobar_set : Nat.nat list =
- Nat.Zero_nat ::
- (Nat.Suc Nat.Zero_nat ::
- (Integer.nat
- (Integer.Number_of_int
- (Integer.Bit
- (Integer.Bit (Integer.Pls, Integer.B1), Integer.B0)))
- :: []));
-
-end; (*struct Codegen*)
-
-end; (*struct ROOT*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/example.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,27 +0,0 @@
-structure Example =
-struct
-
-fun foldl f a [] = a
- | foldl f a (x :: xs) = foldl f (f a x) xs;
-
-fun rev xs = foldl (fn xsa => fn x => x :: xsa) [] xs;
-
-fun list_case f1 f2 (a :: lista) = f2 a lista
- | list_case f1 f2 [] = f1;
-
-datatype 'a queue = AQueue of 'a list * 'a list;
-
-val empty : 'a queue = AQueue ([], [])
-
-fun dequeue (AQueue ([], [])) = (NONE, AQueue ([], []))
- | dequeue (AQueue (xs, y :: ys)) = (SOME y, AQueue (xs, ys))
- | dequeue (AQueue (v :: va, [])) =
- let
- val y :: ys = rev (v :: va);
- in
- (SOME y, AQueue ([], ys))
- end;
-
-fun enqueue x (AQueue (xs, ys)) = AQueue (x :: xs, ys);
-
-end; (*struct Example*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/fac.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,22 +0,0 @@
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-val one_nat : nat = Suc Zero_nat;
-
-fun plus_nat (Suc m) n = plus_nat m (Suc n)
- | plus_nat Zero_nat n = n;
-
-fun times_nat (Suc m) n = plus_nat n (times_nat m n)
- | times_nat Zero_nat n = Zero_nat;
-
-end; (*struct Nat*)
-
-structure Codegen =
-struct
-
-fun fac (Nat.Suc n) = Nat.times_nat (Nat.Suc n) (fac n)
- | fac Nat.Zero_nat = Nat.one_nat;
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/integers.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,59 +0,0 @@
-structure ROOT =
-struct
-
-structure Integer =
-struct
-
-datatype bit = B0 | B1;
-
-datatype int = Pls | Min | Bit of int * bit | Number_of_int of int;
-
-fun pred (Bit (k, B0)) = Bit (pred k, B1)
- | pred (Bit (k, B1)) = Bit (k, B0)
- | pred Min = Bit (Min, B0)
- | pred Pls = Min;
-
-fun succ (Bit (k, B0)) = Bit (k, B1)
- | succ (Bit (k, B1)) = Bit (succ k, B0)
- | succ Min = Pls
- | succ Pls = Bit (Pls, B1);
-
-fun plus_int (Number_of_int v) (Number_of_int w) =
- Number_of_int (plus_int v w)
- | plus_int k Min = pred k
- | plus_int k Pls = k
- | plus_int (Bit (k, B1)) (Bit (l, B1)) = Bit (plus_int k (succ l), B0)
- | plus_int (Bit (k, B1)) (Bit (l, B0)) = Bit (plus_int k l, B1)
- | plus_int (Bit (k, B0)) (Bit (l, b)) = Bit (plus_int k l, b)
- | plus_int Min k = pred k
- | plus_int Pls k = k;
-
-fun uminus_int (Number_of_int w) = Number_of_int (uminus_int w)
- | uminus_int (Bit (k, B0)) = Bit (uminus_int k, B0)
- | uminus_int (Bit (k, B1)) = Bit (pred (uminus_int k), B1)
- | uminus_int Min = Bit (Pls, B1)
- | uminus_int Pls = Pls;
-
-fun times_int (Number_of_int v) (Number_of_int w) =
- Number_of_int (times_int v w)
- | times_int (Bit (k, B0)) l = Bit (times_int k l, B0)
- | times_int (Bit (k, B1)) l = plus_int (Bit (times_int k l, B0)) l
- | times_int Min k = uminus_int k
- | times_int Pls w = Pls;
-
-end; (*struct Integer*)
-
-structure Codegen =
-struct
-
-fun double_inc k =
- Integer.plus_int
- (Integer.times_int
- (Integer.Number_of_int
- (Integer.Bit (Integer.Bit (Integer.Pls, Integer.B1), Integer.B0)))
- k)
- (Integer.Number_of_int (Integer.Bit (Integer.Pls, Integer.B1)));
-
-end; (*struct Codegen*)
-
-end; (*struct ROOT*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/lexicographic.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,19 +0,0 @@
-structure HOL =
-struct
-
-type 'a eq = {eq : 'a -> 'a -> bool};
-fun eq (A_:'a eq) = #eq A_;
-
-type 'a ord = {less_eq : 'a -> 'a -> bool, less : 'a -> 'a -> bool};
-fun less_eq (A_:'a ord) = #less_eq A_;
-fun less (A_:'a ord) = #less A_;
-
-end; (*struct HOL*)
-
-structure Codegen =
-struct
-
-fun less_eq (A1_, A2_) B_ (x1, y1) (x2, y2) =
- HOL.less A2_ x1 x2 orelse HOL.eq A1_ x1 x2 andalso HOL.less_eq B_ y1 y2;
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/lookup.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,13 +0,0 @@
-structure ROOT =
-struct
-
-structure Codegen =
-struct
-
-fun lookup ((k, v) :: xs) l =
- (if ((k : string) = l) then SOME v else lookup xs l)
- | lookup [] l = NONE;
-
-end; (*struct Codegen*)
-
-end; (*struct ROOT*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/monotype.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,34 +0,0 @@
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-fun eq_nat (Suc a) Zero_nat = false
- | eq_nat Zero_nat (Suc a) = false
- | eq_nat (Suc nat) (Suc nat') = eq_nat nat nat'
- | eq_nat Zero_nat Zero_nat = true;
-
-end; (*struct Nat*)
-
-structure List =
-struct
-
-fun null (x :: xs) = false
- | null [] = true;
-
-fun list_all2 p (x :: xs) (y :: ys) = p x y andalso list_all2 p xs ys
- | list_all2 p xs [] = null xs
- | list_all2 p [] ys = null ys;
-
-end; (*struct List*)
-
-structure Codegen =
-struct
-
-datatype monotype = Mono of Nat.nat * monotype list;
-
-fun eq_monotype (Mono (tyco1, typargs1)) (Mono (tyco2, typargs2)) =
- Nat.eq_nat tyco1 tyco2 andalso
- List.list_all2 eq_monotype typargs1 typargs2;
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/nat_binary.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,17 +0,0 @@
-structure Nat =
-struct
-
-datatype nat = Dig1 of nat | Dig0 of nat | One_nat | Zero_nat;
-
-fun plus_nat (Dig1 m) (Dig1 n) = Dig0 (plus_nat (plus_nat m n) One_nat)
- | plus_nat (Dig1 m) (Dig0 n) = Dig1 (plus_nat m n)
- | plus_nat (Dig0 m) (Dig1 n) = Dig1 (plus_nat m n)
- | plus_nat (Dig0 m) (Dig0 n) = Dig0 (plus_nat m n)
- | plus_nat (Dig1 m) One_nat = Dig0 (plus_nat m One_nat)
- | plus_nat One_nat (Dig1 n) = Dig0 (plus_nat n One_nat)
- | plus_nat (Dig0 m) One_nat = Dig1 m
- | plus_nat One_nat (Dig0 n) = Dig1 n
- | plus_nat m Zero_nat = m
- | plus_nat Zero_nat n = n;
-
-end; (*struct Nat*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/pick1.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,44 +0,0 @@
-structure HOL =
-struct
-
-fun leta s f = f s;
-
-end; (*struct HOL*)
-
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-fun less_nat m (Suc n) = less_eq_nat m n
- | less_nat n Zero_nat = false
-and less_eq_nat (Suc m) n = less_nat m n
- | less_eq_nat Zero_nat n = true;
-
-fun minus_nat (Suc m) (Suc n) = minus_nat m n
- | minus_nat Zero_nat n = Zero_nat
- | minus_nat m Zero_nat = m;
-
-end; (*struct Nat*)
-
-structure Product_Type =
-struct
-
-fun split f (a, b) = f a b;
-
-end; (*struct Product_Type*)
-
-structure Codegen =
-struct
-
-fun pick ((k, v) :: xs) n =
- (if Nat.less_nat n k then v else pick xs (Nat.minus_nat n k))
- | pick (x :: xs) n =
- let
- val a = x;
- val (k, v) = a;
- in
- (if Nat.less_nat n k then v else pick xs (Nat.minus_nat n k))
- end;
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/Thy/examples/tree.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,95 +0,0 @@
-structure HOL =
-struct
-
-type 'a eq = {eq : 'a -> 'a -> bool};
-fun eq (A_:'a eq) = #eq A_;
-
-type 'a ord = {less_eq : 'a -> 'a -> bool, less : 'a -> 'a -> bool};
-fun less_eq (A_:'a ord) = #less_eq A_;
-fun less (A_:'a ord) = #less A_;
-
-fun eqop A_ a = eq A_ a;
-
-end; (*struct HOL*)
-
-structure Orderings =
-struct
-
-type 'a preorder = {Orderings__ord_preorder : 'a HOL.ord};
-fun ord_preorder (A_:'a preorder) = #Orderings__ord_preorder A_;
-
-type 'a order = {Orderings__preorder_order : 'a preorder};
-fun preorder_order (A_:'a order) = #Orderings__preorder_order A_;
-
-type 'a linorder = {Orderings__order_linorder : 'a order};
-fun order_linorder (A_:'a linorder) = #Orderings__order_linorder A_;
-
-end; (*struct Orderings*)
-
-structure Nat =
-struct
-
-datatype nat = Suc of nat | Zero_nat;
-
-fun eq_nat (Suc a) Zero_nat = false
- | eq_nat Zero_nat (Suc a) = false
- | eq_nat (Suc nat) (Suc nat') = eq_nat nat nat'
- | eq_nat Zero_nat Zero_nat = true;
-
-val eq_nata = {eq = eq_nat} : nat HOL.eq;
-
-fun less_nat m (Suc n) = less_eq_nat m n
- | less_nat n Zero_nat = false
-and less_eq_nat (Suc m) n = less_nat m n
- | less_eq_nat Zero_nat n = true;
-
-val ord_nat = {less_eq = less_eq_nat, less = less_nat} : nat HOL.ord;
-
-val preorder_nat = {Orderings__ord_preorder = ord_nat} :
- nat Orderings.preorder;
-
-val order_nat = {Orderings__preorder_order = preorder_nat} :
- nat Orderings.order;
-
-val linorder_nat = {Orderings__order_linorder = order_nat} :
- nat Orderings.linorder;
-
-end; (*struct Nat*)
-
-structure Codegen =
-struct
-
-datatype ('a, 'b) searchtree =
- Branch of ('a, 'b) searchtree * 'a * ('a, 'b) searchtree |
- Leaf of 'a * 'b;
-
-fun update (A1_, A2_) (it, entry) (Leaf (key, vala)) =
- (if HOL.eqop A1_ it key then Leaf (key, entry)
- else (if HOL.less_eq
- ((Orderings.ord_preorder o Orderings.preorder_order o
- Orderings.order_linorder)
- A2_)
- it key
- then Branch (Leaf (it, entry), it, Leaf (key, vala))
- else Branch (Leaf (key, vala), it, Leaf (it, entry))))
- | update (A1_, A2_) (it, entry) (Branch (t1, key, t2)) =
- (if HOL.less_eq
- ((Orderings.ord_preorder o Orderings.preorder_order o
- Orderings.order_linorder)
- A2_)
- it key
- then Branch (update (A1_, A2_) (it, entry) t1, key, t2)
- else Branch (t1, key, update (A1_, A2_) (it, entry) t2));
-
-val example : (Nat.nat, (Nat.nat list)) searchtree =
- update (Nat.eq_nata, Nat.linorder_nat)
- (Nat.Suc (Nat.Suc (Nat.Suc (Nat.Suc Nat.Zero_nat))),
- [Nat.Suc (Nat.Suc Nat.Zero_nat), Nat.Suc (Nat.Suc Nat.Zero_nat)])
- (update (Nat.eq_nata, Nat.linorder_nat)
- (Nat.Suc (Nat.Suc (Nat.Suc Nat.Zero_nat)),
- [Nat.Suc (Nat.Suc (Nat.Suc Nat.Zero_nat))])
- (update (Nat.eq_nata, Nat.linorder_nat)
- (Nat.Suc (Nat.Suc Nat.Zero_nat), [Nat.Suc (Nat.Suc Nat.Zero_nat)])
- (Leaf (Nat.Suc Nat.Zero_nat, []))));
-
-end; (*struct Codegen*)
--- a/doc-src/IsarAdvanced/Codegen/codegen.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,53 +0,0 @@
-
-\documentclass[12pt,a4paper,fleqn]{report}
-\usepackage{latexsym,graphicx}
-\usepackage[refpage]{nomencl}
-\usepackage{../../iman,../../extra,../../isar,../../proof}
-\usepackage{../../isabelle,../../isabellesym}
-\usepackage{style}
-\usepackage{pgf}
-\usepackage{pgflibraryshapes}
-\usepackage{tikz}
-\usepackage{../../pdfsetup}
-
-\hyphenation{Isabelle}
-\hyphenation{Isar}
-\isadroptag{theory}
-
-\title{\includegraphics[scale=0.5]{isabelle_isar}
- \\[4ex] Code generation from Isabelle/HOL theories}
-\author{\emph{Florian Haftmann}}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
- This tutorial gives an introduction to a generic code generator framework in Isabelle
- for generating executable code in functional programming languages from logical
- specifications in Isabelle/HOL.
-\end{abstract}
-
-\thispagestyle{empty}\clearpage
-
-\pagenumbering{roman}
-\clearfirst
-
-\input{Thy/document/Introduction.tex}
-\input{Thy/document/Program.tex}
-\input{Thy/document/Adaption.tex}
-\input{Thy/document/Further.tex}
-\input{Thy/document/ML.tex}
-
-\begingroup
-\bibliographystyle{plain} \small\raggedright\frenchspacing
-\bibliography{../../manual}
-\endgroup
-
-\end{document}
-
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
Binary file doc-src/IsarAdvanced/Codegen/codegen_process.pdf has changed
--- a/doc-src/IsarAdvanced/Codegen/codegen_process.ps Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,586 +0,0 @@
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--- a/doc-src/IsarAdvanced/Codegen/style.sty Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,56 +0,0 @@
-
-%% toc
-\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
-\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
-
-%% references
-\newcommand{\secref}[1]{\S\ref{#1}}
-
-%% logical markup
-\newcommand{\strong}[1]{{\bfseries {#1}}}
-\newcommand{\qn}[1]{\emph{#1}}
-
-%% typographic conventions
-\newcommand{\qt}[1]{``{#1}''}
-
-%% verbatim text
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-
-%% quoted segments
-\makeatletter
-\isakeeptag{quote}
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-\renewcommand{\endisatagquotett}{\end{quotesegment}\isabellestyle{it}\isastyle}
-
-%% a trick
-\newcommand{\isasymSML}{SML}
-
-%% presentation
-\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
-
-\pagestyle{headings}
-\binperiod
-\underscoreoff
-
-\renewcommand{\isadigit}[1]{\isamath{#1}}
-
-%% ml reference
-\newenvironment{mldecls}{\par\noindent\begingroup\footnotesize\def\isanewline{\\}\begin{tabular}{l}}{\end{tabular}\smallskip\endgroup}
-
-\isakeeptag{mlref}
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-\renewcommand{\endisatagmlref}{\endgroup}
-
-\isabellestyle{it}
-
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "implementation"
-%%% End:
--- a/doc-src/IsarAdvanced/Functions/IsaMakefile Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,33 +0,0 @@
-
-## targets
-
-default: Thy
-images:
-test: Thy
-
-all: images test
-
-
-## global settings
-
-SRC = $(ISABELLE_HOME)/src
-OUT = $(ISABELLE_OUTPUT)
-LOG = $(OUT)/log
-
-USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
-
-
-## Thy
-
-THY = $(LOG)/HOL-Thy.gz
-
-Thy: $(THY)
-
-$(THY): Thy/ROOT.ML Thy/Functions.thy
- @$(USEDIR) HOL Thy
-
-
-## clean
-
-clean:
- @rm -f $(THY)
--- a/doc-src/IsarAdvanced/Functions/Makefile Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,38 +0,0 @@
-#
-# $Id$
-#
-
-## targets
-
-default: dvi
-
-
-## dependencies
-
-include ../Makefile.in
-
-NAME = functions
-
-FILES = $(NAME).tex Thy/document/Functions.tex intro.tex conclusion.tex \
- style.sty ../../iman.sty ../../extra.sty ../../isar.sty \
- ../../isabelle.sty ../../isabellesym.sty ../../pdfsetup.sty \
- ../../manual.bib ../../proof.sty
-
-dvi: $(NAME).dvi
-
-$(NAME).dvi: $(FILES) isabelle_isar.eps
- $(LATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(LATEX) $(NAME)
- $(LATEX) $(NAME)
-
-pdf: $(NAME).pdf
-
-$(NAME).pdf: $(FILES) isabelle_isar.pdf
- $(PDFLATEX) $(NAME)
- $(BIBTEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(PDFLATEX) $(NAME)
- $(FIXBOOKMARKS) $(NAME).out
- $(PDFLATEX) $(NAME)
- $(PDFLATEX) $(NAME)
--- a/doc-src/IsarAdvanced/Functions/Thy/Functions.thy Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1264 +0,0 @@
-(* Title: doc-src/IsarAdvanced/Functions/Thy/Fundefs.thy
- Author: Alexander Krauss, TU Muenchen
-
-Tutorial for function definitions with the new "function" package.
-*)
-
-theory Functions
-imports Main
-begin
-
-section {* Function Definitions for Dummies *}
-
-text {*
- In most cases, defining a recursive function is just as simple as other definitions:
-*}
-
-fun fib :: "nat \<Rightarrow> nat"
-where
- "fib 0 = 1"
-| "fib (Suc 0) = 1"
-| "fib (Suc (Suc n)) = fib n + fib (Suc n)"
-
-text {*
- The syntax is rather self-explanatory: We introduce a function by
- giving its name, its type,
- and a set of defining recursive equations.
- If we leave out the type, the most general type will be
- inferred, which can sometimes lead to surprises: Since both @{term
- "1::nat"} and @{text "+"} are overloaded, we would end up
- with @{text "fib :: nat \<Rightarrow> 'a::{one,plus}"}.
-*}
-
-text {*
- The function always terminates, since its argument gets smaller in
- every recursive call.
- Since HOL is a logic of total functions, termination is a
- fundamental requirement to prevent inconsistencies\footnote{From the
- \qt{definition} @{text "f(n) = f(n) + 1"} we could prove
- @{text "0 = 1"} by subtracting @{text "f(n)"} on both sides.}.
- Isabelle tries to prove termination automatically when a definition
- is made. In \S\ref{termination}, we will look at cases where this
- fails and see what to do then.
-*}
-
-subsection {* Pattern matching *}
-
-text {* \label{patmatch}
- Like in functional programming, we can use pattern matching to
- define functions. At the moment we will only consider \emph{constructor
- patterns}, which only consist of datatype constructors and
- variables. Furthermore, patterns must be linear, i.e.\ all variables
- on the left hand side of an equation must be distinct. In
- \S\ref{genpats} we discuss more general pattern matching.
-
- If patterns overlap, the order of the equations is taken into
- account. The following function inserts a fixed element between any
- two elements of a list:
-*}
-
-fun sep :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- "sep a (x#y#xs) = x # a # sep a (y # xs)"
-| "sep a xs = xs"
-
-text {*
- Overlapping patterns are interpreted as \qt{increments} to what is
- already there: The second equation is only meant for the cases where
- the first one does not match. Consequently, Isabelle replaces it
- internally by the remaining cases, making the patterns disjoint:
-*}
-
-thm sep.simps
-
-text {* @{thm [display] sep.simps[no_vars]} *}
-
-text {*
- \noindent The equations from function definitions are automatically used in
- simplification:
-*}
-
-lemma "sep 0 [1, 2, 3] = [1, 0, 2, 0, 3]"
-by simp
-
-subsection {* Induction *}
-
-text {*
-
- Isabelle provides customized induction rules for recursive
- functions. These rules follow the recursive structure of the
- definition. Here is the rule @{text sep.induct} arising from the
- above definition of @{const sep}:
-
- @{thm [display] sep.induct}
-
- We have a step case for list with at least two elements, and two
- base cases for the zero- and the one-element list. Here is a simple
- proof about @{const sep} and @{const map}
-*}
-
-lemma "map f (sep x ys) = sep (f x) (map f ys)"
-apply (induct x ys rule: sep.induct)
-
-txt {*
- We get three cases, like in the definition.
-
- @{subgoals [display]}
-*}
-
-apply auto
-done
-text {*
-
- With the \cmd{fun} command, you can define about 80\% of the
- functions that occur in practice. The rest of this tutorial explains
- the remaining 20\%.
-*}
-
-
-section {* fun vs.\ function *}
-
-text {*
- The \cmd{fun} command provides a
- convenient shorthand notation for simple function definitions. In
- this mode, Isabelle tries to solve all the necessary proof obligations
- automatically. If any proof fails, the definition is
- rejected. This can either mean that the definition is indeed faulty,
- or that the default proof procedures are just not smart enough (or
- rather: not designed) to handle the definition.
-
- By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
- solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
-
-\end{isamarkuptext}
-
-
-\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
-\cmd{fun} @{text "f :: \<tau>"}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\vspace*{6pt}
-\end{minipage}\right]
-\quad\equiv\quad
-\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
-\cmd{function} @{text "("}\cmd{sequential}@{text ") f :: \<tau>"}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\\%
-\cmd{by} @{text "pat_completeness auto"}\\%
-\cmd{termination by} @{text "lexicographic_order"}\vspace{6pt}
-\end{minipage}
-\right]\]
-
-\begin{isamarkuptext}
- \vspace*{1em}
- \noindent Some details have now become explicit:
-
- \begin{enumerate}
- \item The \cmd{sequential} option enables the preprocessing of
- pattern overlaps which we already saw. Without this option, the equations
- must already be disjoint and complete. The automatic completion only
- works with constructor patterns.
-
- \item A function definition produces a proof obligation which
- expresses completeness and compatibility of patterns (we talk about
- this later). The combination of the methods @{text "pat_completeness"} and
- @{text "auto"} is used to solve this proof obligation.
-
- \item A termination proof follows the definition, started by the
- \cmd{termination} command. This will be explained in \S\ref{termination}.
- \end{enumerate}
- Whenever a \cmd{fun} command fails, it is usually a good idea to
- expand the syntax to the more verbose \cmd{function} form, to see
- what is actually going on.
- *}
-
-
-section {* Termination *}
-
-text {*\label{termination}
- The method @{text "lexicographic_order"} is the default method for
- termination proofs. It can prove termination of a
- certain class of functions by searching for a suitable lexicographic
- combination of size measures. Of course, not all functions have such
- a simple termination argument. For them, we can specify the termination
- relation manually.
-*}
-
-subsection {* The {\tt relation} method *}
-text{*
- Consider the following function, which sums up natural numbers up to
- @{text "N"}, using a counter @{text "i"}:
-*}
-
-function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
- "sum i N = (if i > N then 0 else i + sum (Suc i) N)"
-by pat_completeness auto
-
-text {*
- \noindent The @{text "lexicographic_order"} method fails on this example, because none of the
- arguments decreases in the recursive call, with respect to the standard size ordering.
- To prove termination manually, we must provide a custom wellfounded relation.
-
- The termination argument for @{text "sum"} is based on the fact that
- the \emph{difference} between @{text "i"} and @{text "N"} gets
- smaller in every step, and that the recursion stops when @{text "i"}
- is greater than @{text "N"}. Phrased differently, the expression
- @{text "N + 1 - i"} always decreases.
-
- We can use this expression as a measure function suitable to prove termination.
-*}
-
-termination sum
-apply (relation "measure (\<lambda>(i,N). N + 1 - i)")
-
-txt {*
- The \cmd{termination} command sets up the termination goal for the
- specified function @{text "sum"}. If the function name is omitted, it
- implicitly refers to the last function definition.
-
- The @{text relation} method takes a relation of
- type @{typ "('a \<times> 'a) set"}, where @{typ "'a"} is the argument type of
- the function. If the function has multiple curried arguments, then
- these are packed together into a tuple, as it happened in the above
- example.
-
- The predefined function @{term[source] "measure :: ('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set"} constructs a
- wellfounded relation from a mapping into the natural numbers (a
- \emph{measure function}).
-
- After the invocation of @{text "relation"}, we must prove that (a)
- the relation we supplied is wellfounded, and (b) that the arguments
- of recursive calls indeed decrease with respect to the
- relation:
-
- @{subgoals[display,indent=0]}
-
- These goals are all solved by @{text "auto"}:
-*}
-
-apply auto
-done
-
-text {*
- Let us complicate the function a little, by adding some more
- recursive calls:
-*}
-
-function foo :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
- "foo i N = (if i > N
- then (if N = 0 then 0 else foo 0 (N - 1))
- else i + foo (Suc i) N)"
-by pat_completeness auto
-
-text {*
- When @{text "i"} has reached @{text "N"}, it starts at zero again
- and @{text "N"} is decremented.
- This corresponds to a nested
- loop where one index counts up and the other down. Termination can
- be proved using a lexicographic combination of two measures, namely
- the value of @{text "N"} and the above difference. The @{const
- "measures"} combinator generalizes @{text "measure"} by taking a
- list of measure functions.
-*}
-
-termination
-by (relation "measures [\<lambda>(i, N). N, \<lambda>(i,N). N + 1 - i]") auto
-
-subsection {* How @{text "lexicographic_order"} works *}
-
-(*fun fails :: "nat \<Rightarrow> nat list \<Rightarrow> nat"
-where
- "fails a [] = a"
-| "fails a (x#xs) = fails (x + a) (x # xs)"
-*)
-
-text {*
- To see how the automatic termination proofs work, let's look at an
- example where it fails\footnote{For a detailed discussion of the
- termination prover, see \cite{bulwahnKN07}}:
-
-\end{isamarkuptext}
-\cmd{fun} @{text "fails :: \"nat \<Rightarrow> nat list \<Rightarrow> nat\""}\\%
-\cmd{where}\\%
-\hspace*{2ex}@{text "\"fails a [] = a\""}\\%
-|\hspace*{1.5ex}@{text "\"fails a (x#xs) = fails (x + a) (x#xs)\""}\\
-\begin{isamarkuptext}
-
-\noindent Isabelle responds with the following error:
-
-\begin{isabelle}
-*** Unfinished subgoals:\newline
-*** (a, 1, <):\newline
-*** \ 1.~@{text "\<And>x. x = 0"}\newline
-*** (a, 1, <=):\newline
-*** \ 1.~False\newline
-*** (a, 2, <):\newline
-*** \ 1.~False\newline
-*** Calls:\newline
-*** a) @{text "(a, x # xs) -->> (x + a, x # xs)"}\newline
-*** Measures:\newline
-*** 1) @{text "\<lambda>x. size (fst x)"}\newline
-*** 2) @{text "\<lambda>x. size (snd x)"}\newline
-*** Result matrix:\newline
-*** \ \ \ \ 1\ \ 2 \newline
-*** a: ? <= \newline
-*** Could not find lexicographic termination order.\newline
-*** At command "fun".\newline
-\end{isabelle}
-*}
-
-
-text {*
- The key to this error message is the matrix at the bottom. The rows
- of that matrix correspond to the different recursive calls (In our
- case, there is just one). The columns are the function's arguments
- (expressed through different measure functions, which map the
- argument tuple to a natural number).
-
- The contents of the matrix summarize what is known about argument
- descents: The second argument has a weak descent (@{text "<="}) at the
- recursive call, and for the first argument nothing could be proved,
- which is expressed by @{text "?"}. In general, there are the values
- @{text "<"}, @{text "<="} and @{text "?"}.
-
- For the failed proof attempts, the unfinished subgoals are also
- printed. Looking at these will often point to a missing lemma.
-
-% As a more real example, here is quicksort:
-*}
-(*
-function qs :: "nat list \<Rightarrow> nat list"
-where
- "qs [] = []"
-| "qs (x#xs) = qs [y\<in>xs. y < x] @ x # qs [y\<in>xs. y \<ge> x]"
-by pat_completeness auto
-
-termination apply lexicographic_order
-
-text {* If we try @{text "lexicographic_order"} method, we get the
- following error *}
-termination by (lexicographic_order simp:l2)
-
-lemma l: "x \<le> y \<Longrightarrow> x < Suc y" by arith
-
-function
-
-*)
-
-section {* Mutual Recursion *}
-
-text {*
- If two or more functions call one another mutually, they have to be defined
- in one step. Here are @{text "even"} and @{text "odd"}:
-*}
-
-function even :: "nat \<Rightarrow> bool"
- and odd :: "nat \<Rightarrow> bool"
-where
- "even 0 = True"
-| "odd 0 = False"
-| "even (Suc n) = odd n"
-| "odd (Suc n) = even n"
-by pat_completeness auto
-
-text {*
- To eliminate the mutual dependencies, Isabelle internally
- creates a single function operating on the sum
- type @{typ "nat + nat"}. Then, @{const even} and @{const odd} are
- defined as projections. Consequently, termination has to be proved
- simultaneously for both functions, by specifying a measure on the
- sum type:
-*}
-
-termination
-by (relation "measure (\<lambda>x. case x of Inl n \<Rightarrow> n | Inr n \<Rightarrow> n)") auto
-
-text {*
- We could also have used @{text lexicographic_order}, which
- supports mutual recursive termination proofs to a certain extent.
-*}
-
-subsection {* Induction for mutual recursion *}
-
-text {*
-
- When functions are mutually recursive, proving properties about them
- generally requires simultaneous induction. The induction rule @{text "even_odd.induct"}
- generated from the above definition reflects this.
-
- Let us prove something about @{const even} and @{const odd}:
-*}
-
-lemma even_odd_mod2:
- "even n = (n mod 2 = 0)"
- "odd n = (n mod 2 = 1)"
-
-txt {*
- We apply simultaneous induction, specifying the induction variable
- for both goals, separated by \cmd{and}: *}
-
-apply (induct n and n rule: even_odd.induct)
-
-txt {*
- We get four subgoals, which correspond to the clauses in the
- definition of @{const even} and @{const odd}:
- @{subgoals[display,indent=0]}
- Simplification solves the first two goals, leaving us with two
- statements about the @{text "mod"} operation to prove:
-*}
-
-apply simp_all
-
-txt {*
- @{subgoals[display,indent=0]}
-
- \noindent These can be handled by Isabelle's arithmetic decision procedures.
-
-*}
-
-apply arith
-apply arith
-done
-
-text {*
- In proofs like this, the simultaneous induction is really essential:
- Even if we are just interested in one of the results, the other
- one is necessary to strengthen the induction hypothesis. If we leave
- out the statement about @{const odd} and just write @{term True} instead,
- the same proof fails:
-*}
-
-lemma failed_attempt:
- "even n = (n mod 2 = 0)"
- "True"
-apply (induct n rule: even_odd.induct)
-
-txt {*
- \noindent Now the third subgoal is a dead end, since we have no
- useful induction hypothesis available:
-
- @{subgoals[display,indent=0]}
-*}
-
-oops
-
-section {* General pattern matching *}
-text{*\label{genpats} *}
-
-subsection {* Avoiding automatic pattern splitting *}
-
-text {*
-
- Up to now, we used pattern matching only on datatypes, and the
- patterns were always disjoint and complete, and if they weren't,
- they were made disjoint automatically like in the definition of
- @{const "sep"} in \S\ref{patmatch}.
-
- This automatic splitting can significantly increase the number of
- equations involved, and this is not always desirable. The following
- example shows the problem:
-
- Suppose we are modeling incomplete knowledge about the world by a
- three-valued datatype, which has values @{term "T"}, @{term "F"}
- and @{term "X"} for true, false and uncertain propositions, respectively.
-*}
-
-datatype P3 = T | F | X
-
-text {* \noindent Then the conjunction of such values can be defined as follows: *}
-
-fun And :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
-where
- "And T p = p"
-| "And p T = p"
-| "And p F = F"
-| "And F p = F"
-| "And X X = X"
-
-
-text {*
- This definition is useful, because the equations can directly be used
- as simplification rules. But the patterns overlap: For example,
- the expression @{term "And T T"} is matched by both the first and
- the second equation. By default, Isabelle makes the patterns disjoint by
- splitting them up, producing instances:
-*}
-
-thm And.simps
-
-text {*
- @{thm[indent=4] And.simps}
-
- \vspace*{1em}
- \noindent There are several problems with this:
-
- \begin{enumerate}
- \item If the datatype has many constructors, there can be an
- explosion of equations. For @{const "And"}, we get seven instead of
- five equations, which can be tolerated, but this is just a small
- example.
-
- \item Since splitting makes the equations \qt{less general}, they
- do not always match in rewriting. While the term @{term "And x F"}
- can be simplified to @{term "F"} with the original equations, a
- (manual) case split on @{term "x"} is now necessary.
-
- \item The splitting also concerns the induction rule @{text
- "And.induct"}. Instead of five premises it now has seven, which
- means that our induction proofs will have more cases.
-
- \item In general, it increases clarity if we get the same definition
- back which we put in.
- \end{enumerate}
-
- If we do not want the automatic splitting, we can switch it off by
- leaving out the \cmd{sequential} option. However, we will have to
- prove that our pattern matching is consistent\footnote{This prevents
- us from defining something like @{term "f x = True"} and @{term "f x
- = False"} simultaneously.}:
-*}
-
-function And2 :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
-where
- "And2 T p = p"
-| "And2 p T = p"
-| "And2 p F = F"
-| "And2 F p = F"
-| "And2 X X = X"
-
-txt {*
- \noindent Now let's look at the proof obligations generated by a
- function definition. In this case, they are:
-
- @{subgoals[display,indent=0]}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
-
- The first subgoal expresses the completeness of the patterns. It has
- the form of an elimination rule and states that every @{term x} of
- the function's input type must match at least one of the patterns\footnote{Completeness could
- be equivalently stated as a disjunction of existential statements:
-@{term "(\<exists>p. x = (T, p)) \<or> (\<exists>p. x = (p, T)) \<or> (\<exists>p. x = (p, F)) \<or>
- (\<exists>p. x = (F, p)) \<or> (x = (X, X))"}, and you can use the method @{text atomize_elim} to get that form instead.}. If the patterns just involve
- datatypes, we can solve it with the @{text "pat_completeness"}
- method:
-*}
-
-apply pat_completeness
-
-txt {*
- The remaining subgoals express \emph{pattern compatibility}. We do
- allow that an input value matches multiple patterns, but in this
- case, the result (i.e.~the right hand sides of the equations) must
- also be equal. For each pair of two patterns, there is one such
- subgoal. Usually this needs injectivity of the constructors, which
- is used automatically by @{text "auto"}.
-*}
-
-by auto
-
-
-subsection {* Non-constructor patterns *}
-
-text {*
- Most of Isabelle's basic types take the form of inductive datatypes,
- and usually pattern matching works on the constructors of such types.
- However, this need not be always the case, and the \cmd{function}
- command handles other kind of patterns, too.
-
- One well-known instance of non-constructor patterns are
- so-called \emph{$n+k$-patterns}, which are a little controversial in
- the functional programming world. Here is the initial fibonacci
- example with $n+k$-patterns:
-*}
-
-function fib2 :: "nat \<Rightarrow> nat"
-where
- "fib2 0 = 1"
-| "fib2 1 = 1"
-| "fib2 (n + 2) = fib2 n + fib2 (Suc n)"
-
-(*<*)ML_val "goals_limit := 1"(*>*)
-txt {*
- This kind of matching is again justified by the proof of pattern
- completeness and compatibility.
- The proof obligation for pattern completeness states that every natural number is
- either @{term "0::nat"}, @{term "1::nat"} or @{term "n +
- (2::nat)"}:
-
- @{subgoals[display,indent=0]}
-
- This is an arithmetic triviality, but unfortunately the
- @{text arith} method cannot handle this specific form of an
- elimination rule. However, we can use the method @{text
- "atomize_elim"} to do an ad-hoc conversion to a disjunction of
- existentials, which can then be solved by the arithmetic decision procedure.
- Pattern compatibility and termination are automatic as usual.
-*}
-(*<*)ML_val "goals_limit := 10"(*>*)
-apply atomize_elim
-apply arith
-apply auto
-done
-termination by lexicographic_order
-text {*
- We can stretch the notion of pattern matching even more. The
- following function is not a sensible functional program, but a
- perfectly valid mathematical definition:
-*}
-
-function ev :: "nat \<Rightarrow> bool"
-where
- "ev (2 * n) = True"
-| "ev (2 * n + 1) = False"
-apply atomize_elim
-by arith+
-termination by (relation "{}") simp
-
-text {*
- This general notion of pattern matching gives you a certain freedom
- in writing down specifications. However, as always, such freedom should
- be used with care:
-
- If we leave the area of constructor
- patterns, we have effectively departed from the world of functional
- programming. This means that it is no longer possible to use the
- code generator, and expect it to generate ML code for our
- definitions. Also, such a specification might not work very well together with
- simplification. Your mileage may vary.
-*}
-
-
-subsection {* Conditional equations *}
-
-text {*
- The function package also supports conditional equations, which are
- similar to guards in a language like Haskell. Here is Euclid's
- algorithm written with conditional patterns\footnote{Note that the
- patterns are also overlapping in the base case}:
-*}
-
-function gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
- "gcd x 0 = x"
-| "gcd 0 y = y"
-| "x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (Suc x) (y - x)"
-| "\<not> x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (x - y) (Suc y)"
-by (atomize_elim, auto, arith)
-termination by lexicographic_order
-
-text {*
- By now, you can probably guess what the proof obligations for the
- pattern completeness and compatibility look like.
-
- Again, functions with conditional patterns are not supported by the
- code generator.
-*}
-
-
-subsection {* Pattern matching on strings *}
-
-text {*
- As strings (as lists of characters) are normal datatypes, pattern
- matching on them is possible, but somewhat problematic. Consider the
- following definition:
-
-\end{isamarkuptext}
-\noindent\cmd{fun} @{text "check :: \"string \<Rightarrow> bool\""}\\%
-\cmd{where}\\%
-\hspace*{2ex}@{text "\"check (''good'') = True\""}\\%
-@{text "| \"check s = False\""}
-\begin{isamarkuptext}
-
- \noindent An invocation of the above \cmd{fun} command does not
- terminate. What is the problem? Strings are lists of characters, and
- characters are a datatype with a lot of constructors. Splitting the
- catch-all pattern thus leads to an explosion of cases, which cannot
- be handled by Isabelle.
-
- There are two things we can do here. Either we write an explicit
- @{text "if"} on the right hand side, or we can use conditional patterns:
-*}
-
-function check :: "string \<Rightarrow> bool"
-where
- "check (''good'') = True"
-| "s \<noteq> ''good'' \<Longrightarrow> check s = False"
-by auto
-
-
-section {* Partiality *}
-
-text {*
- In HOL, all functions are total. A function @{term "f"} applied to
- @{term "x"} always has the value @{term "f x"}, and there is no notion
- of undefinedness.
- This is why we have to do termination
- proofs when defining functions: The proof justifies that the
- function can be defined by wellfounded recursion.
-
- However, the \cmd{function} package does support partiality to a
- certain extent. Let's look at the following function which looks
- for a zero of a given function f.
-*}
-
-function (*<*)(domintros, tailrec)(*>*)findzero :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"
-where
- "findzero f n = (if f n = 0 then n else findzero f (Suc n))"
-by pat_completeness auto
-(*<*)declare findzero.simps[simp del](*>*)
-
-text {*
- \noindent Clearly, any attempt of a termination proof must fail. And without
- that, we do not get the usual rules @{text "findzero.simps"} and
- @{text "findzero.induct"}. So what was the definition good for at all?
-*}
-
-subsection {* Domain predicates *}
-
-text {*
- The trick is that Isabelle has not only defined the function @{const findzero}, but also
- a predicate @{term "findzero_dom"} that characterizes the values where the function
- terminates: the \emph{domain} of the function. If we treat a
- partial function just as a total function with an additional domain
- predicate, we can derive simplification and
- induction rules as we do for total functions. They are guarded
- by domain conditions and are called @{text psimps} and @{text
- pinduct}:
-*}
-
-text {*
- \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.psimps}\end{minipage}
- \hfill(@{text "findzero.psimps"})
- \vspace{1em}
-
- \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.pinduct}\end{minipage}
- \hfill(@{text "findzero.pinduct"})
-*}
-
-text {*
- Remember that all we
- are doing here is use some tricks to make a total function appear
- as if it was partial. We can still write the term @{term "findzero
- (\<lambda>x. 1) 0"} and like any other term of type @{typ nat} it is equal
- to some natural number, although we might not be able to find out
- which one. The function is \emph{underdefined}.
-
- But it is defined enough to prove something interesting about it. We
- can prove that if @{term "findzero f n"}
- terminates, it indeed returns a zero of @{term f}:
-*}
-
-lemma findzero_zero: "findzero_dom (f, n) \<Longrightarrow> f (findzero f n) = 0"
-
-txt {* \noindent We apply induction as usual, but using the partial induction
- rule: *}
-
-apply (induct f n rule: findzero.pinduct)
-
-txt {* \noindent This gives the following subgoals:
-
- @{subgoals[display,indent=0]}
-
- \noindent The hypothesis in our lemma was used to satisfy the first premise in
- the induction rule. However, we also get @{term
- "findzero_dom (f, n)"} as a local assumption in the induction step. This
- allows to unfold @{term "findzero f n"} using the @{text psimps}
- rule, and the rest is trivial. Since the @{text psimps} rules carry the
- @{text "[simp]"} attribute by default, we just need a single step:
- *}
-apply simp
-done
-
-text {*
- Proofs about partial functions are often not harder than for total
- functions. Fig.~\ref{findzero_isar} shows a slightly more
- complicated proof written in Isar. It is verbose enough to show how
- partiality comes into play: From the partial induction, we get an
- additional domain condition hypothesis. Observe how this condition
- is applied when calls to @{term findzero} are unfolded.
-*}
-
-text_raw {*
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-*}
-lemma "\<lbrakk>findzero_dom (f, n); x \<in> {n ..< findzero f n}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
-proof (induct rule: findzero.pinduct)
- fix f n assume dom: "findzero_dom (f, n)"
- and IH: "\<lbrakk>f n \<noteq> 0; x \<in> {Suc n ..< findzero f (Suc n)}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
- and x_range: "x \<in> {n ..< findzero f n}"
- have "f n \<noteq> 0"
- proof
- assume "f n = 0"
- with dom have "findzero f n = n" by simp
- with x_range show False by auto
- qed
-
- from x_range have "x = n \<or> x \<in> {Suc n ..< findzero f n}" by auto
- thus "f x \<noteq> 0"
- proof
- assume "x = n"
- with `f n \<noteq> 0` show ?thesis by simp
- next
- assume "x \<in> {Suc n ..< findzero f n}"
- with dom and `f n \<noteq> 0` have "x \<in> {Suc n ..< findzero f (Suc n)}" by simp
- with IH and `f n \<noteq> 0`
- show ?thesis by simp
- qed
-qed
-text_raw {*
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{A proof about a partial function}\label{findzero_isar}
-\end{figure}
-*}
-
-subsection {* Partial termination proofs *}
-
-text {*
- Now that we have proved some interesting properties about our
- function, we should turn to the domain predicate and see if it is
- actually true for some values. Otherwise we would have just proved
- lemmas with @{term False} as a premise.
-
- Essentially, we need some introduction rules for @{text
- findzero_dom}. The function package can prove such domain
- introduction rules automatically. But since they are not used very
- often (they are almost never needed if the function is total), this
- functionality is disabled by default for efficiency reasons. So we have to go
- back and ask for them explicitly by passing the @{text
- "(domintros)"} option to the function package:
-
-\vspace{1ex}
-\noindent\cmd{function} @{text "(domintros) findzero :: \"(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat\""}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\
-
- \noindent Now the package has proved an introduction rule for @{text findzero_dom}:
-*}
-
-thm findzero.domintros
-
-text {*
- @{thm[display] findzero.domintros}
-
- Domain introduction rules allow to show that a given value lies in the
- domain of a function, if the arguments of all recursive calls
- are in the domain as well. They allow to do a \qt{single step} in a
- termination proof. Usually, you want to combine them with a suitable
- induction principle.
-
- Since our function increases its argument at recursive calls, we
- need an induction principle which works \qt{backwards}. We will use
- @{text inc_induct}, which allows to do induction from a fixed number
- \qt{downwards}:
-
- \begin{center}@{thm inc_induct}\hfill(@{text "inc_induct"})\end{center}
-
- Figure \ref{findzero_term} gives a detailed Isar proof of the fact
- that @{text findzero} terminates if there is a zero which is greater
- or equal to @{term n}. First we derive two useful rules which will
- solve the base case and the step case of the induction. The
- induction is then straightforward, except for the unusual induction
- principle.
-
-*}
-
-text_raw {*
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-*}
-lemma findzero_termination:
- assumes "x \<ge> n" and "f x = 0"
- shows "findzero_dom (f, n)"
-proof -
- have base: "findzero_dom (f, x)"
- by (rule findzero.domintros) (simp add:`f x = 0`)
-
- have step: "\<And>i. findzero_dom (f, Suc i)
- \<Longrightarrow> findzero_dom (f, i)"
- by (rule findzero.domintros) simp
-
- from `x \<ge> n` show ?thesis
- proof (induct rule:inc_induct)
- show "findzero_dom (f, x)" by (rule base)
- next
- fix i assume "findzero_dom (f, Suc i)"
- thus "findzero_dom (f, i)" by (rule step)
- qed
-qed
-text_raw {*
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{Termination proof for @{text findzero}}\label{findzero_term}
-\end{figure}
-*}
-
-text {*
- Again, the proof given in Fig.~\ref{findzero_term} has a lot of
- detail in order to explain the principles. Using more automation, we
- can also have a short proof:
-*}
-
-lemma findzero_termination_short:
- assumes zero: "x >= n"
- assumes [simp]: "f x = 0"
- shows "findzero_dom (f, n)"
-using zero
-by (induct rule:inc_induct) (auto intro: findzero.domintros)
-
-text {*
- \noindent It is simple to combine the partial correctness result with the
- termination lemma:
-*}
-
-lemma findzero_total_correctness:
- "f x = 0 \<Longrightarrow> f (findzero f 0) = 0"
-by (blast intro: findzero_zero findzero_termination)
-
-subsection {* Definition of the domain predicate *}
-
-text {*
- Sometimes it is useful to know what the definition of the domain
- predicate looks like. Actually, @{text findzero_dom} is just an
- abbreviation:
-
- @{abbrev[display] findzero_dom}
-
- The domain predicate is the \emph{accessible part} of a relation @{const
- findzero_rel}, which was also created internally by the function
- package. @{const findzero_rel} is just a normal
- inductive predicate, so we can inspect its definition by
- looking at the introduction rules @{text findzero_rel.intros}.
- In our case there is just a single rule:
-
- @{thm[display] findzero_rel.intros}
-
- The predicate @{const findzero_rel}
- describes the \emph{recursion relation} of the function
- definition. The recursion relation is a binary relation on
- the arguments of the function that relates each argument to its
- recursive calls. In general, there is one introduction rule for each
- recursive call.
-
- The predicate @{term "accp findzero_rel"} is the accessible part of
- that relation. An argument belongs to the accessible part, if it can
- be reached in a finite number of steps (cf.~its definition in @{text
- "Wellfounded.thy"}).
-
- Since the domain predicate is just an abbreviation, you can use
- lemmas for @{const accp} and @{const findzero_rel} directly. Some
- lemmas which are occasionally useful are @{text accpI}, @{text
- accp_downward}, and of course the introduction and elimination rules
- for the recursion relation @{text "findzero.intros"} and @{text "findzero.cases"}.
-*}
-
-(*lemma findzero_nicer_domintros:
- "f x = 0 \<Longrightarrow> findzero_dom (f, x)"
- "findzero_dom (f, Suc x) \<Longrightarrow> findzero_dom (f, x)"
-by (rule accpI, erule findzero_rel.cases, auto)+
-*)
-
-subsection {* A Useful Special Case: Tail recursion *}
-
-text {*
- The domain predicate is our trick that allows us to model partiality
- in a world of total functions. The downside of this is that we have
- to carry it around all the time. The termination proof above allowed
- us to replace the abstract @{term "findzero_dom (f, n)"} by the more
- concrete @{term "(x \<ge> n \<and> f x = (0::nat))"}, but the condition is still
- there and can only be discharged for special cases.
- In particular, the domain predicate guards the unfolding of our
- function, since it is there as a condition in the @{text psimp}
- rules.
-
- Now there is an important special case: We can actually get rid
- of the condition in the simplification rules, \emph{if the function
- is tail-recursive}. The reason is that for all tail-recursive
- equations there is a total function satisfying them, even if they
- are non-terminating.
-
-% A function is tail recursive, if each call to the function is either
-% equal
-%
-% So the outer form of the
-%
-%if it can be written in the following
-% form:
-% {term[display] "f x = (if COND x then BASE x else f (LOOP x))"}
-
-
- The function package internally does the right construction and can
- derive the unconditional simp rules, if we ask it to do so. Luckily,
- our @{const "findzero"} function is tail-recursive, so we can just go
- back and add another option to the \cmd{function} command:
-
-\vspace{1ex}
-\noindent\cmd{function} @{text "(domintros, tailrec) findzero :: \"(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat\""}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\%
-
-
- \noindent Now, we actually get unconditional simplification rules, even
- though the function is partial:
-*}
-
-thm findzero.simps
-
-text {*
- @{thm[display] findzero.simps}
-
- \noindent Of course these would make the simplifier loop, so we better remove
- them from the simpset:
-*}
-
-declare findzero.simps[simp del]
-
-text {*
- Getting rid of the domain conditions in the simplification rules is
- not only useful because it simplifies proofs. It is also required in
- order to use Isabelle's code generator to generate ML code
- from a function definition.
- Since the code generator only works with equations, it cannot be
- used with @{text "psimp"} rules. Thus, in order to generate code for
- partial functions, they must be defined as a tail recursion.
- Luckily, many functions have a relatively natural tail recursive
- definition.
-*}
-
-section {* Nested recursion *}
-
-text {*
- Recursive calls which are nested in one another frequently cause
- complications, since their termination proof can depend on a partial
- correctness property of the function itself.
-
- As a small example, we define the \qt{nested zero} function:
-*}
-
-function nz :: "nat \<Rightarrow> nat"
-where
- "nz 0 = 0"
-| "nz (Suc n) = nz (nz n)"
-by pat_completeness auto
-
-text {*
- If we attempt to prove termination using the identity measure on
- naturals, this fails:
-*}
-
-termination
- apply (relation "measure (\<lambda>n. n)")
- apply auto
-
-txt {*
- We get stuck with the subgoal
-
- @{subgoals[display]}
-
- Of course this statement is true, since we know that @{const nz} is
- the zero function. And in fact we have no problem proving this
- property by induction.
-*}
-(*<*)oops(*>*)
-lemma nz_is_zero: "nz_dom n \<Longrightarrow> nz n = 0"
- by (induct rule:nz.pinduct) auto
-
-text {*
- We formulate this as a partial correctness lemma with the condition
- @{term "nz_dom n"}. This allows us to prove it with the @{text
- pinduct} rule before we have proved termination. With this lemma,
- the termination proof works as expected:
-*}
-
-termination
- by (relation "measure (\<lambda>n. n)") (auto simp: nz_is_zero)
-
-text {*
- As a general strategy, one should prove the statements needed for
- termination as a partial property first. Then they can be used to do
- the termination proof. This also works for less trivial
- examples. Figure \ref{f91} defines the 91-function, a well-known
- challenge problem due to John McCarthy, and proves its termination.
-*}
-
-text_raw {*
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-*}
-
-function f91 :: "nat \<Rightarrow> nat"
-where
- "f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))"
-by pat_completeness auto
-
-lemma f91_estimate:
- assumes trm: "f91_dom n"
- shows "n < f91 n + 11"
-using trm by induct auto
-
-termination
-proof
- let ?R = "measure (\<lambda>x. 101 - x)"
- show "wf ?R" ..
-
- fix n :: nat assume "\<not> 100 < n" -- "Assumptions for both calls"
-
- thus "(n + 11, n) \<in> ?R" by simp -- "Inner call"
-
- assume inner_trm: "f91_dom (n + 11)" -- "Outer call"
- with f91_estimate have "n + 11 < f91 (n + 11) + 11" .
- with `\<not> 100 < n` show "(f91 (n + 11), n) \<in> ?R" by simp
-qed
-
-text_raw {*
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}
-\vspace{6pt}\hrule
-\caption{McCarthy's 91-function}\label{f91}
-\end{figure}
-*}
-
-
-section {* Higher-Order Recursion *}
-
-text {*
- Higher-order recursion occurs when recursive calls
- are passed as arguments to higher-order combinators such as @{const
- map}, @{term filter} etc.
- As an example, imagine a datatype of n-ary trees:
-*}
-
-datatype 'a tree =
- Leaf 'a
-| Branch "'a tree list"
-
-
-text {* \noindent We can define a function which swaps the left and right subtrees recursively, using the
- list functions @{const rev} and @{const map}: *}
-
-fun mirror :: "'a tree \<Rightarrow> 'a tree"
-where
- "mirror (Leaf n) = Leaf n"
-| "mirror (Branch l) = Branch (rev (map mirror l))"
-
-text {*
- Although the definition is accepted without problems, let us look at the termination proof:
-*}
-
-termination proof
- txt {*
-
- As usual, we have to give a wellfounded relation, such that the
- arguments of the recursive calls get smaller. But what exactly are
- the arguments of the recursive calls when mirror is given as an
- argument to @{const map}? Isabelle gives us the
- subgoals
-
- @{subgoals[display,indent=0]}
-
- So the system seems to know that @{const map} only
- applies the recursive call @{term "mirror"} to elements
- of @{term "l"}, which is essential for the termination proof.
-
- This knowledge about @{const map} is encoded in so-called congruence rules,
- which are special theorems known to the \cmd{function} command. The
- rule for @{const map} is
-
- @{thm[display] map_cong}
-
- You can read this in the following way: Two applications of @{const
- map} are equal, if the list arguments are equal and the functions
- coincide on the elements of the list. This means that for the value
- @{term "map f l"} we only have to know how @{term f} behaves on
- the elements of @{term l}.
-
- Usually, one such congruence rule is
- needed for each higher-order construct that is used when defining
- new functions. In fact, even basic functions like @{const
- If} and @{const Let} are handled by this mechanism. The congruence
- rule for @{const If} states that the @{text then} branch is only
- relevant if the condition is true, and the @{text else} branch only if it
- is false:
-
- @{thm[display] if_cong}
-
- Congruence rules can be added to the
- function package by giving them the @{term fundef_cong} attribute.
-
- The constructs that are predefined in Isabelle, usually
- come with the respective congruence rules.
- But if you define your own higher-order functions, you may have to
- state and prove the required congruence rules yourself, if you want to use your
- functions in recursive definitions.
-*}
-(*<*)oops(*>*)
-
-subsection {* Congruence Rules and Evaluation Order *}
-
-text {*
- Higher order logic differs from functional programming languages in
- that it has no built-in notion of evaluation order. A program is
- just a set of equations, and it is not specified how they must be
- evaluated.
-
- However for the purpose of function definition, we must talk about
- evaluation order implicitly, when we reason about termination.
- Congruence rules express that a certain evaluation order is
- consistent with the logical definition.
-
- Consider the following function.
-*}
-
-function f :: "nat \<Rightarrow> bool"
-where
- "f n = (n = 0 \<or> f (n - 1))"
-(*<*)by pat_completeness auto(*>*)
-
-text {*
- For this definition, the termination proof fails. The default configuration
- specifies no congruence rule for disjunction. We have to add a
- congruence rule that specifies left-to-right evaluation order:
-
- \vspace{1ex}
- \noindent @{thm disj_cong}\hfill(@{text "disj_cong"})
- \vspace{1ex}
-
- Now the definition works without problems. Note how the termination
- proof depends on the extra condition that we get from the congruence
- rule.
-
- However, as evaluation is not a hard-wired concept, we
- could just turn everything around by declaring a different
- congruence rule. Then we can make the reverse definition:
-*}
-
-lemma disj_cong2[fundef_cong]:
- "(\<not> Q' \<Longrightarrow> P = P') \<Longrightarrow> (Q = Q') \<Longrightarrow> (P \<or> Q) = (P' \<or> Q')"
- by blast
-
-fun f' :: "nat \<Rightarrow> bool"
-where
- "f' n = (f' (n - 1) \<or> n = 0)"
-
-text {*
- \noindent These examples show that, in general, there is no \qt{best} set of
- congruence rules.
-
- However, such tweaking should rarely be necessary in
- practice, as most of the time, the default set of congruence rules
- works well.
-*}
-
-end
--- a/doc-src/IsarAdvanced/Functions/Thy/ROOT.ML Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,4 +0,0 @@
-
-(* $Id$ *)
-
-use_thy "Functions";
--- a/doc-src/IsarAdvanced/Functions/Thy/document/Functions.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1985 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Functions}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Functions\isanewline
-\isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupsection{Function Definitions for Dummies%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In most cases, defining a recursive function is just as simple as other definitions:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ fib\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}fib\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ fib\ n\ {\isacharplus}\ fib\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The syntax is rather self-explanatory: We introduce a function by
- giving its name, its type,
- and a set of defining recursive equations.
- If we leave out the type, the most general type will be
- inferred, which can sometimes lead to surprises: Since both \isa{{\isadigit{1}}} and \isa{{\isacharplus}} are overloaded, we would end up
- with \isa{fib\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ {\isacharprime}a{\isacharcolon}{\isacharcolon}{\isacharbraceleft}one{\isacharcomma}plus{\isacharbraceright}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The function always terminates, since its argument gets smaller in
- every recursive call.
- Since HOL is a logic of total functions, termination is a
- fundamental requirement to prevent inconsistencies\footnote{From the
- \qt{definition} \isa{f{\isacharparenleft}n{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}n{\isacharparenright}\ {\isacharplus}\ {\isadigit{1}}} we could prove
- \isa{{\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}} by subtracting \isa{f{\isacharparenleft}n{\isacharparenright}} on both sides.}.
- Isabelle tries to prove termination automatically when a definition
- is made. In \S\ref{termination}, we will look at cases where this
- fails and see what to do then.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Pattern matching%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{patmatch}
- Like in functional programming, we can use pattern matching to
- define functions. At the moment we will only consider \emph{constructor
- patterns}, which only consist of datatype constructors and
- variables. Furthermore, patterns must be linear, i.e.\ all variables
- on the left hand side of an equation must be distinct. In
- \S\ref{genpats} we discuss more general pattern matching.
-
- If patterns overlap, the order of the equations is taken into
- account. The following function inserts a fixed element between any
- two elements of a list:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ sep\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sep\ a\ {\isacharparenleft}x{\isacharhash}y{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ x\ {\isacharhash}\ a\ {\isacharhash}\ sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}sep\ a\ xs\ \ \ \ \ \ \ {\isacharequal}\ xs{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Overlapping patterns are interpreted as \qt{increments} to what is
- already there: The second equation is only meant for the cases where
- the first one does not match. Consequently, Isabelle replaces it
- internally by the remaining cases, making the patterns disjoint:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ sep{\isachardot}simps%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-sep\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}\ {\isacharequal}\ x\ {\isacharhash}\ a\ {\isacharhash}\ sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}\isasep\isanewline%
-sep\ a\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\isasep\isanewline%
-sep\ a\ {\isacharbrackleft}v{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}v{\isacharbrackright}%
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent The equations from function definitions are automatically used in
- simplification:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}sep\ {\isadigit{0}}\ {\isacharbrackleft}{\isadigit{1}}{\isacharcomma}\ {\isadigit{2}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}{\isadigit{1}}{\isacharcomma}\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{2}}{\isacharcomma}\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Induction%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Isabelle provides customized induction rules for recursive
- functions. These rules follow the recursive structure of the
- definition. Here is the rule \isa{sep{\isachardot}induct} arising from the
- above definition of \isa{sep}:
-
- \begin{isabelle}%
-{\isasymlbrakk}{\isasymAnd}a\ x\ y\ xs{\isachardot}\ {\isacharquery}P\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharquery}P\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharsemicolon}\ {\isasymAnd}a{\isachardot}\ {\isacharquery}P\ a\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\ {\isasymAnd}a\ v{\isachardot}\ {\isacharquery}P\ a\ {\isacharbrackleft}v{\isacharbrackright}{\isasymrbrakk}\isanewline
-{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}%
-\end{isabelle}
-
- We have a step case for list with at least two elements, and two
- base cases for the zero- and the one-element list. Here is a simple
- proof about \isa{sep} and \isa{map}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}map\ f\ {\isacharparenleft}sep\ x\ ys{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}map\ f\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ x\ ys\ rule{\isacharcolon}\ sep{\isachardot}induct{\isacharparenright}%
-\begin{isamarkuptxt}%
-We get three cases, like in the definition.
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ x\ y\ xs{\isachardot}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }map\ f\ {\isacharparenleft}sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }map\ f\ {\isacharparenleft}sep\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}a{\isachardot}\ map\ f\ {\isacharparenleft}sep\ a\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}a\ v{\isachardot}\ map\ f\ {\isacharparenleft}sep\ a\ {\isacharbrackleft}v{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharbrackleft}v{\isacharbrackright}{\isacharparenright}%
-\end{isabelle}%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ auto\ \isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-With the \cmd{fun} command, you can define about 80\% of the
- functions that occur in practice. The rest of this tutorial explains
- the remaining 20\%.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{fun vs.\ function%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The \cmd{fun} command provides a
- convenient shorthand notation for simple function definitions. In
- this mode, Isabelle tries to solve all the necessary proof obligations
- automatically. If any proof fails, the definition is
- rejected. This can either mean that the definition is indeed faulty,
- or that the default proof procedures are just not smart enough (or
- rather: not designed) to handle the definition.
-
- By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
- solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
-
-\end{isamarkuptext}
-
-
-\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
-\cmd{fun} \isa{f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\vspace*{6pt}
-\end{minipage}\right]
-\quad\equiv\quad
-\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
-\cmd{function} \isa{{\isacharparenleft}}\cmd{sequential}\isa{{\isacharparenright}\ f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\\%
-\cmd{by} \isa{pat{\isacharunderscore}completeness\ auto}\\%
-\cmd{termination by} \isa{lexicographic{\isacharunderscore}order}\vspace{6pt}
-\end{minipage}
-\right]\]
-
-\begin{isamarkuptext}
- \vspace*{1em}
- \noindent Some details have now become explicit:
-
- \begin{enumerate}
- \item The \cmd{sequential} option enables the preprocessing of
- pattern overlaps which we already saw. Without this option, the equations
- must already be disjoint and complete. The automatic completion only
- works with constructor patterns.
-
- \item A function definition produces a proof obligation which
- expresses completeness and compatibility of patterns (we talk about
- this later). The combination of the methods \isa{pat{\isacharunderscore}completeness} and
- \isa{auto} is used to solve this proof obligation.
-
- \item A termination proof follows the definition, started by the
- \cmd{termination} command. This will be explained in \S\ref{termination}.
- \end{enumerate}
- Whenever a \cmd{fun} command fails, it is usually a good idea to
- expand the syntax to the more verbose \cmd{function} form, to see
- what is actually going on.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Termination%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{termination}
- The method \isa{lexicographic{\isacharunderscore}order} is the default method for
- termination proofs. It can prove termination of a
- certain class of functions by searching for a suitable lexicographic
- combination of size measures. Of course, not all functions have such
- a simple termination argument. For them, we can specify the termination
- relation manually.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{The {\tt relation} method%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Consider the following function, which sums up natural numbers up to
- \isa{N}, using a counter \isa{i}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ sum\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sum\ i\ N\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isachargreater}\ N\ then\ {\isadigit{0}}\ else\ i\ {\isacharplus}\ sum\ {\isacharparenleft}Suc\ i{\isacharparenright}\ N{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent The \isa{lexicographic{\isacharunderscore}order} method fails on this example, because none of the
- arguments decreases in the recursive call, with respect to the standard size ordering.
- To prove termination manually, we must provide a custom wellfounded relation.
-
- The termination argument for \isa{sum} is based on the fact that
- the \emph{difference} between \isa{i} and \isa{N} gets
- smaller in every step, and that the recursion stops when \isa{i}
- is greater than \isa{N}. Phrased differently, the expression
- \isa{N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i} always decreases.
-
- We can use this expression as a measure function suitable to prove termination.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ sum\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}%
-\begin{isamarkuptxt}%
-The \cmd{termination} command sets up the termination goal for the
- specified function \isa{sum}. If the function name is omitted, it
- implicitly refers to the last function definition.
-
- The \isa{relation} method takes a relation of
- type \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}, where \isa{{\isacharprime}a} is the argument type of
- the function. If the function has multiple curried arguments, then
- these are packed together into a tuple, as it happened in the above
- example.
-
- The predefined function \isa{{\isachardoublequote}measure\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set{\isachardoublequote}} constructs a
- wellfounded relation from a mapping into the natural numbers (a
- \emph{measure function}).
-
- After the invocation of \isa{relation}, we must prove that (a)
- the relation we supplied is wellfounded, and (b) that the arguments
- of recursive calls indeed decrease with respect to the
- relation:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ wf\ {\isacharparenleft}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}i\ N{\isachardot}\ {\isasymnot}\ N\ {\isacharless}\ i\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharparenleft}Suc\ i{\isacharcomma}\ N{\isacharparenright}{\isacharcomma}\ i{\isacharcomma}\ N{\isacharparenright}\ {\isasymin}\ measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}%
-\end{isabelle}
-
- These goals are all solved by \isa{auto}:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Let us complicate the function a little, by adding some more
- recursive calls:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ foo\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}foo\ i\ N\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isachargreater}\ N\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ then\ {\isacharparenleft}if\ N\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ foo\ {\isadigit{0}}\ {\isacharparenleft}N\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ i\ {\isacharplus}\ foo\ {\isacharparenleft}Suc\ i{\isacharparenright}\ N{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-When \isa{i} has reached \isa{N}, it starts at zero again
- and \isa{N} is decremented.
- This corresponds to a nested
- loop where one index counts up and the other down. Termination can
- be proved using a lexicographic combination of two measures, namely
- the value of \isa{N} and the above difference. The \isa{measures} combinator generalizes \isa{measure} by taking a
- list of measure functions.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measures\ {\isacharbrackleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N{\isacharcomma}\ {\isasymlambda}{\isacharparenleft}i{\isacharcomma}N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharbrackright}{\isachardoublequoteclose}{\isacharparenright}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{How \isa{lexicographic{\isacharunderscore}order} works%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-To see how the automatic termination proofs work, let's look at an
- example where it fails\footnote{For a detailed discussion of the
- termination prover, see \cite{bulwahnKN07}}:
-
-\end{isamarkuptext}
-\cmd{fun} \isa{fails\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ {\isasymRightarrow}\ nat\ list\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
-\cmd{where}\\%
-\hspace*{2ex}\isa{{\isachardoublequote}fails\ a\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ a{\isachardoublequote}}\\%
-|\hspace*{1.5ex}\isa{{\isachardoublequote}fails\ a\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ fails\ {\isacharparenleft}x\ {\isacharplus}\ a{\isacharparenright}\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}{\isachardoublequote}}\\
-\begin{isamarkuptext}
-
-\noindent Isabelle responds with the following error:
-
-\begin{isabelle}
-*** Unfinished subgoals:\newline
-*** (a, 1, <):\newline
-*** \ 1.~\isa{{\isasymAnd}x{\isachardot}\ x\ {\isacharequal}\ {\isadigit{0}}}\newline
-*** (a, 1, <=):\newline
-*** \ 1.~False\newline
-*** (a, 2, <):\newline
-*** \ 1.~False\newline
-*** Calls:\newline
-*** a) \isa{{\isacharparenleft}a{\isacharcomma}\ x\ {\isacharhash}\ xs{\isacharparenright}\ {\isacharminus}{\isacharminus}{\isachargreater}{\isachargreater}\ {\isacharparenleft}x\ {\isacharplus}\ a{\isacharcomma}\ x\ {\isacharhash}\ xs{\isacharparenright}}\newline
-*** Measures:\newline
-*** 1) \isa{{\isasymlambda}x{\isachardot}\ size\ {\isacharparenleft}fst\ x{\isacharparenright}}\newline
-*** 2) \isa{{\isasymlambda}x{\isachardot}\ size\ {\isacharparenleft}snd\ x{\isacharparenright}}\newline
-*** Result matrix:\newline
-*** \ \ \ \ 1\ \ 2 \newline
-*** a: ? <= \newline
-*** Could not find lexicographic termination order.\newline
-*** At command "fun".\newline
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The key to this error message is the matrix at the bottom. The rows
- of that matrix correspond to the different recursive calls (In our
- case, there is just one). The columns are the function's arguments
- (expressed through different measure functions, which map the
- argument tuple to a natural number).
-
- The contents of the matrix summarize what is known about argument
- descents: The second argument has a weak descent (\isa{{\isacharless}{\isacharequal}}) at the
- recursive call, and for the first argument nothing could be proved,
- which is expressed by \isa{{\isacharquery}}. In general, there are the values
- \isa{{\isacharless}}, \isa{{\isacharless}{\isacharequal}} and \isa{{\isacharquery}}.
-
- For the failed proof attempts, the unfinished subgoals are also
- printed. Looking at these will often point to a missing lemma.
-
-% As a more real example, here is quicksort:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Mutual Recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-If two or more functions call one another mutually, they have to be defined
- in one step. Here are \isa{even} and \isa{odd}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ even\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ odd\ \ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}even\ {\isadigit{0}}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}odd\ {\isadigit{0}}\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ odd\ n{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}odd\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ even\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-To eliminate the mutual dependencies, Isabelle internally
- creates a single function operating on the sum
- type \isa{nat\ {\isacharplus}\ nat}. Then, \isa{even} and \isa{odd} are
- defined as projections. Consequently, termination has to be proved
- simultaneously for both functions, by specifying a measure on the
- sum type:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ case\ x\ of\ Inl\ n\ {\isasymRightarrow}\ n\ {\isacharbar}\ Inr\ n\ {\isasymRightarrow}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We could also have used \isa{lexicographic{\isacharunderscore}order}, which
- supports mutual recursive termination proofs to a certain extent.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Induction for mutual recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-When functions are mutually recursive, proving properties about them
- generally requires simultaneous induction. The induction rule \isa{even{\isacharunderscore}odd{\isachardot}induct}
- generated from the above definition reflects this.
-
- Let us prove something about \isa{even} and \isa{odd}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ even{\isacharunderscore}odd{\isacharunderscore}mod{\isadigit{2}}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-We apply simultaneous induction, specifying the induction variable
- for both goals, separated by \cmd{and}:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ n\ \isakeyword{and}\ n\ rule{\isacharcolon}\ even{\isacharunderscore}odd{\isachardot}induct{\isacharparenright}%
-\begin{isamarkuptxt}%
-We get four subgoals, which correspond to the clauses in the
- definition of \isa{even} and \isa{odd}:
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ even\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ odd\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}\ {\isasymLongrightarrow}\ even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ odd\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}%
-\end{isabelle}
- Simplification solves the first two goals, leaving us with two
- statements about the \isa{mod} operation to prove:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ simp{\isacharunderscore}all%
-\begin{isamarkuptxt}%
-\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}%
-\end{isabelle}
-
- \noindent These can be handled by Isabelle's arithmetic decision procedures.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-In proofs like this, the simultaneous induction is really essential:
- Even if we are just interested in one of the results, the other
- one is necessary to strengthen the induction hypothesis. If we leave
- out the statement about \isa{odd} and just write \isa{True} instead,
- the same proof fails:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ failed{\isacharunderscore}attempt{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}True{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ n\ rule{\isacharcolon}\ even{\isacharunderscore}odd{\isachardot}induct{\isacharparenright}%
-\begin{isamarkuptxt}%
-\noindent Now the third subgoal is a dead end, since we have no
- useful induction hypothesis available:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ even\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ True\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ True\ {\isasymLongrightarrow}\ even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ True%
-\end{isabelle}%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{oops}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsection{General pattern matching%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{genpats}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Avoiding automatic pattern splitting%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Up to now, we used pattern matching only on datatypes, and the
- patterns were always disjoint and complete, and if they weren't,
- they were made disjoint automatically like in the definition of
- \isa{sep} in \S\ref{patmatch}.
-
- This automatic splitting can significantly increase the number of
- equations involved, and this is not always desirable. The following
- example shows the problem:
-
- Suppose we are modeling incomplete knowledge about the world by a
- three-valued datatype, which has values \isa{T}, \isa{F}
- and \isa{X} for true, false and uncertain propositions, respectively.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ P{\isadigit{3}}\ {\isacharequal}\ T\ {\isacharbar}\ F\ {\isacharbar}\ X%
-\begin{isamarkuptext}%
-\noindent Then the conjunction of such values can be defined as follows:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ And\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}And\ T\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ p\ T\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ p\ F\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ F\ p\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ X\ X\ {\isacharequal}\ X{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-This definition is useful, because the equations can directly be used
- as simplification rules. But the patterns overlap: For example,
- the expression \isa{And\ T\ T} is matched by both the first and
- the second equation. By default, Isabelle makes the patterns disjoint by
- splitting them up, producing instances:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ And{\isachardot}simps%
-\begin{isamarkuptext}%
-\isa{And\ T\ {\isacharquery}p\ {\isacharequal}\ {\isacharquery}p\isasep\isanewline%
-And\ F\ T\ {\isacharequal}\ F\isasep\isanewline%
-And\ X\ T\ {\isacharequal}\ X\isasep\isanewline%
-And\ F\ F\ {\isacharequal}\ F\isasep\isanewline%
-And\ X\ F\ {\isacharequal}\ F\isasep\isanewline%
-And\ F\ X\ {\isacharequal}\ F\isasep\isanewline%
-And\ X\ X\ {\isacharequal}\ X}
-
- \vspace*{1em}
- \noindent There are several problems with this:
-
- \begin{enumerate}
- \item If the datatype has many constructors, there can be an
- explosion of equations. For \isa{And}, we get seven instead of
- five equations, which can be tolerated, but this is just a small
- example.
-
- \item Since splitting makes the equations \qt{less general}, they
- do not always match in rewriting. While the term \isa{And\ x\ F}
- can be simplified to \isa{F} with the original equations, a
- (manual) case split on \isa{x} is now necessary.
-
- \item The splitting also concerns the induction rule \isa{And{\isachardot}induct}. Instead of five premises it now has seven, which
- means that our induction proofs will have more cases.
-
- \item In general, it increases clarity if we get the same definition
- back which we put in.
- \end{enumerate}
-
- If we do not want the automatic splitting, we can switch it off by
- leaving out the \cmd{sequential} option. However, we will have to
- prove that our pattern matching is consistent\footnote{This prevents
- us from defining something like \isa{f\ x\ {\isacharequal}\ True} and \isa{f\ x\ {\isacharequal}\ False} simultaneously.}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ And{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}And{\isadigit{2}}\ T\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ p\ T\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ p\ F\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ F\ p\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ X\ X\ {\isacharequal}\ X{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent Now let's look at the proof obligations generated by a
- function definition. In this case, they are:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ {\isasymlbrakk}{\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ \ }{\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ p{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ x\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ }{\isasymLongrightarrow}\ P\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{5}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{6}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ X\isanewline
-\ {\isadigit{7}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
-\ {\isadigit{8}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{9}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{1}}{\isadigit{0}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ X%
-\end{isabelle}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
-
- The first subgoal expresses the completeness of the patterns. It has
- the form of an elimination rule and states that every \isa{x} of
- the function's input type must match at least one of the patterns\footnote{Completeness could
- be equivalently stated as a disjunction of existential statements:
-\isa{{\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ F{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ p{\isacharparenright}{\isacharparenright}\ {\isasymor}\ x\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}}, and you can use the method \isa{atomize{\isacharunderscore}elim} to get that form instead.}. If the patterns just involve
- datatypes, we can solve it with the \isa{pat{\isacharunderscore}completeness}
- method:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness%
-\begin{isamarkuptxt}%
-The remaining subgoals express \emph{pattern compatibility}. We do
- allow that an input value matches multiple patterns, but in this
- case, the result (i.e.~the right hand sides of the equations) must
- also be equal. For each pair of two patterns, there is one such
- subgoal. Usually this needs injectivity of the constructors, which
- is used automatically by \isa{auto}.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Non-constructor patterns%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Most of Isabelle's basic types take the form of inductive datatypes,
- and usually pattern matching works on the constructors of such types.
- However, this need not be always the case, and the \cmd{function}
- command handles other kind of patterns, too.
-
- One well-known instance of non-constructor patterns are
- so-called \emph{$n+k$-patterns}, which are a little controversial in
- the functional programming world. Here is the initial fibonacci
- example with $n+k$-patterns:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ fib{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ fib{\isadigit{2}}\ n\ {\isacharplus}\ fib{\isadigit{2}}\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-This kind of matching is again justified by the proof of pattern
- completeness and compatibility.
- The proof obligation for pattern completeness states that every natural number is
- either \isa{{\isadigit{0}}}, \isa{{\isadigit{1}}} or \isa{n\ {\isacharplus}\ {\isadigit{2}}}:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ {\isasymlbrakk}x\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ x\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}n{\isachardot}\ x\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\isanewline
-\ {\isadigit{5}}{\isachardot}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
-\ {\isadigit{6}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ {\isadigit{1}}\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\isanewline
-\ {\isadigit{7}}{\isachardot}\ {\isasymAnd}n\ na{\isachardot}\isanewline
-\isaindent{\ {\isadigit{7}}{\isachardot}\ \ \ \ }n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ na\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\isanewline
-\isaindent{\ {\isadigit{7}}{\isachardot}\ \ \ \ }fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ na\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ na{\isacharparenright}%
-\end{isabelle}
-
- This is an arithmetic triviality, but unfortunately the
- \isa{arith} method cannot handle this specific form of an
- elimination rule. However, we can use the method \isa{atomize{\isacharunderscore}elim} to do an ad-hoc conversion to a disjunction of
- existentials, which can then be solved by the arithmetic decision procedure.
- Pattern compatibility and termination are automatic as usual.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ atomize{\isacharunderscore}elim\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ lexicographic{\isacharunderscore}order%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We can stretch the notion of pattern matching even more. The
- following function is not a sensible functional program, but a
- perfectly valid mathematical definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ ev\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}ev\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}ev\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ atomize{\isacharunderscore}elim\isanewline
-\isacommand{by}\isamarkupfalse%
-\ arith{\isacharplus}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}{\isacharparenright}\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This general notion of pattern matching gives you a certain freedom
- in writing down specifications. However, as always, such freedom should
- be used with care:
-
- If we leave the area of constructor
- patterns, we have effectively departed from the world of functional
- programming. This means that it is no longer possible to use the
- code generator, and expect it to generate ML code for our
- definitions. Also, such a specification might not work very well together with
- simplification. Your mileage may vary.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Conditional equations%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The function package also supports conditional equations, which are
- similar to guards in a language like Haskell. Here is Euclid's
- algorithm written with conditional patterns\footnote{Note that the
- patterns are also overlapping in the base case}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ gcd\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}gcd\ x\ {\isadigit{0}}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}gcd\ {\isadigit{0}}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}y\ {\isacharminus}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}{\isasymnot}\ x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}x\ {\isacharminus}\ y{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}atomize{\isacharunderscore}elim{\isacharcomma}\ auto{\isacharcomma}\ arith{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ lexicographic{\isacharunderscore}order%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-By now, you can probably guess what the proof obligations for the
- pattern completeness and compatibility look like.
-
- Again, functions with conditional patterns are not supported by the
- code generator.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Pattern matching on strings%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-As strings (as lists of characters) are normal datatypes, pattern
- matching on them is possible, but somewhat problematic. Consider the
- following definition:
-
-\end{isamarkuptext}
-\noindent\cmd{fun} \isa{check\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}string\ {\isasymRightarrow}\ bool{\isachardoublequote}}\\%
-\cmd{where}\\%
-\hspace*{2ex}\isa{{\isachardoublequote}check\ {\isacharparenleft}{\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequote}}\\%
-\isa{{\isacharbar}\ {\isachardoublequote}check\ s\ {\isacharequal}\ False{\isachardoublequote}}
-\begin{isamarkuptext}
-
- \noindent An invocation of the above \cmd{fun} command does not
- terminate. What is the problem? Strings are lists of characters, and
- characters are a datatype with a lot of constructors. Splitting the
- catch-all pattern thus leads to an explosion of cases, which cannot
- be handled by Isabelle.
-
- There are two things we can do here. Either we write an explicit
- \isa{if} on the right hand side, or we can use conditional patterns:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ check\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}string\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}check\ {\isacharparenleft}{\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}s\ {\isasymnoteq}\ {\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}\ {\isasymLongrightarrow}\ check\ s\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsection{Partiality%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In HOL, all functions are total. A function \isa{f} applied to
- \isa{x} always has the value \isa{f\ x}, and there is no notion
- of undefinedness.
- This is why we have to do termination
- proofs when defining functions: The proof justifies that the
- function can be defined by wellfounded recursion.
-
- However, the \cmd{function} package does support partiality to a
- certain extent. Let's look at the following function which looks
- for a zero of a given function f.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}findzero\ f\ n\ {\isacharequal}\ {\isacharparenleft}if\ f\ n\ {\isacharequal}\ {\isadigit{0}}\ then\ n\ else\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent Clearly, any attempt of a termination proof must fail. And without
- that, we do not get the usual rules \isa{findzero{\isachardot}simps} and
- \isa{findzero{\isachardot}induct}. So what was the definition good for at all?%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Domain predicates%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The trick is that Isabelle has not only defined the function \isa{findzero}, but also
- a predicate \isa{findzero{\isacharunderscore}dom} that characterizes the values where the function
- terminates: the \emph{domain} of the function. If we treat a
- partial function just as a total function with an additional domain
- predicate, we can derive simplification and
- induction rules as we do for total functions. They are guarded
- by domain conditions and are called \isa{psimps} and \isa{pinduct}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
-findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
-findzero\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isacharquery}n\ else\ findzero\ {\isacharquery}f\ {\isacharparenleft}Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}%
-\end{isabelle}\end{minipage}
- \hfill(\isa{findzero{\isachardot}psimps})
- \vspace{1em}
-
- \noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
-{\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}{\isacharcomma}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}{\isacharparenright}{\isacharsemicolon}\isanewline
-\isaindent{\ }{\isasymAnd}f\ n{\isachardot}\ {\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ f\ n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isacharquery}P\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ f\ n{\isasymrbrakk}\isanewline
-{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}%
-\end{isabelle}\end{minipage}
- \hfill(\isa{findzero{\isachardot}pinduct})%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Remember that all we
- are doing here is use some tricks to make a total function appear
- as if it was partial. We can still write the term \isa{findzero\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ {\isadigit{1}}{\isacharparenright}\ {\isadigit{0}}} and like any other term of type \isa{nat} it is equal
- to some natural number, although we might not be able to find out
- which one. The function is \emph{underdefined}.
-
- But it is defined enough to prove something interesting about it. We
- can prove that if \isa{findzero\ f\ n}
- terminates, it indeed returns a zero of \isa{f}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}zero{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ n{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent We apply induction as usual, but using the partial induction
- rule:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ f\ n\ rule{\isacharcolon}\ findzero{\isachardot}pinduct{\isacharparenright}%
-\begin{isamarkuptxt}%
-\noindent This gives the following subgoals:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}f\ n{\isachardot}\ {\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ f\ n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isasymrbrakk}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}f\ n{\isachardot}\ }{\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ n{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}%
-\end{isabelle}
-
- \noindent The hypothesis in our lemma was used to satisfy the first premise in
- the induction rule. However, we also get \isa{findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}} as a local assumption in the induction step. This
- allows to unfold \isa{findzero\ f\ n} using the \isa{psimps}
- rule, and the rest is trivial. Since the \isa{psimps} rules carry the
- \isa{{\isacharbrackleft}simp{\isacharbrackright}} attribute by default, we just need a single step:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Proofs about partial functions are often not harder than for total
- functions. Fig.~\ref{findzero_isar} shows a slightly more
- complicated proof written in Isar. It is verbose enough to show how
- partiality comes into play: From the partial induction, we get an
- additional domain condition hypothesis. Observe how this condition
- is applied when calls to \isa{findzero} are unfolded.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ x\ {\isasymin}\ {\isacharbraceleft}n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}\ findzero{\isachardot}pinduct{\isacharparenright}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ f\ n\ \isacommand{assume}\isamarkupfalse%
-\ dom{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ IH{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharsemicolon}\ x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharbraceright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ x{\isacharunderscore}range{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}f\ n\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ dom\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero\ f\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ x{\isacharunderscore}range\ \isacommand{show}\isamarkupfalse%
-\ False\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ x{\isacharunderscore}range\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}\ n\ {\isasymor}\ x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ dom\ \isakeyword{and}\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ IH\ \isakeyword{and}\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{A proof about a partial function}\label{findzero_isar}
-\end{figure}
-%
-\isamarkupsubsection{Partial termination proofs%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Now that we have proved some interesting properties about our
- function, we should turn to the domain predicate and see if it is
- actually true for some values. Otherwise we would have just proved
- lemmas with \isa{False} as a premise.
-
- Essentially, we need some introduction rules for \isa{findzero{\isacharunderscore}dom}. The function package can prove such domain
- introduction rules automatically. But since they are not used very
- often (they are almost never needed if the function is total), this
- functionality is disabled by default for efficiency reasons. So we have to go
- back and ask for them explicitly by passing the \isa{{\isacharparenleft}domintros{\isacharparenright}} option to the function package:
-
-\vspace{1ex}
-\noindent\cmd{function} \isa{{\isacharparenleft}domintros{\isacharparenright}\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\
-
- \noindent Now the package has proved an introduction rule for \isa{findzero{\isacharunderscore}dom}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ findzero{\isachardot}domintros%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-{\isacharparenleft}{\isadigit{0}}\ {\isacharless}\ {\isacharquery}f\ {\isacharquery}n\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}%
-\end{isabelle}
-
- Domain introduction rules allow to show that a given value lies in the
- domain of a function, if the arguments of all recursive calls
- are in the domain as well. They allow to do a \qt{single step} in a
- termination proof. Usually, you want to combine them with a suitable
- induction principle.
-
- Since our function increases its argument at recursive calls, we
- need an induction principle which works \qt{backwards}. We will use
- \isa{inc{\isacharunderscore}induct}, which allows to do induction from a fixed number
- \qt{downwards}:
-
- \begin{center}\isa{{\isasymlbrakk}{\isacharquery}i\ {\isasymle}\ {\isacharquery}j{\isacharsemicolon}\ {\isacharquery}P\ {\isacharquery}j{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}i\ {\isacharless}\ {\isacharquery}j{\isacharsemicolon}\ {\isacharquery}P\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}i}\hfill(\isa{inc{\isacharunderscore}induct})\end{center}
-
- Figure \ref{findzero_term} gives a detailed Isar proof of the fact
- that \isa{findzero} terminates if there is a zero which is greater
- or equal to \isa{n}. First we derive two useful rules which will
- solve the base case and the step case of the induction. The
- induction is then straightforward, except for the unusual induction
- principle.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}termination{\isacharcolon}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isasymge}\ n{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ base{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ findzero{\isachardot}domintros{\isacharparenright}\ {\isacharparenleft}simp\ add{\isacharcolon}{\isacharbackquoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isacharbackquoteclose}{\isacharparenright}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ step{\isacharcolon}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ Suc\ i{\isacharparenright}\ \isanewline
-\ \ \ \ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ findzero{\isachardot}domintros{\isacharparenright}\ simp\isanewline
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isacharbackquoteopen}x\ {\isasymge}\ n{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}inc{\isacharunderscore}induct{\isacharparenright}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ base{\isacharparenright}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ i\ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ Suc\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ i{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ step{\isacharparenright}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{Termination proof for \isa{findzero}}\label{findzero_term}
-\end{figure}
-%
-\begin{isamarkuptext}%
-Again, the proof given in Fig.~\ref{findzero_term} has a lot of
- detail in order to explain the principles. Using more automation, we
- can also have a short proof:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}termination{\isacharunderscore}short{\isacharcolon}\isanewline
-\ \ \isakeyword{assumes}\ zero{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isachargreater}{\isacharequal}\ n{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ zero\isanewline
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}inc{\isacharunderscore}induct{\isacharparenright}\ {\isacharparenleft}auto\ intro{\isacharcolon}\ findzero{\isachardot}domintros{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent It is simple to combine the partial correctness result with the
- termination lemma:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}total{\isacharunderscore}correctness{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}blast\ intro{\isacharcolon}\ findzero{\isacharunderscore}zero\ findzero{\isacharunderscore}termination{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Definition of the domain predicate%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Sometimes it is useful to know what the definition of the domain
- predicate looks like. Actually, \isa{findzero{\isacharunderscore}dom} is just an
- abbreviation:
-
- \begin{isabelle}%
-findzero{\isacharunderscore}dom\ {\isasymequiv}\ accp\ findzero{\isacharunderscore}rel%
-\end{isabelle}
-
- The domain predicate is the \emph{accessible part} of a relation \isa{findzero{\isacharunderscore}rel}, which was also created internally by the function
- package. \isa{findzero{\isacharunderscore}rel} is just a normal
- inductive predicate, so we can inspect its definition by
- looking at the introduction rules \isa{findzero{\isacharunderscore}rel{\isachardot}intros}.
- In our case there is just a single rule:
-
- \begin{isabelle}%
-{\isacharquery}f\ {\isacharquery}n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}rel\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ Suc\ {\isacharquery}n{\isacharparenright}\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}%
-\end{isabelle}
-
- The predicate \isa{findzero{\isacharunderscore}rel}
- describes the \emph{recursion relation} of the function
- definition. The recursion relation is a binary relation on
- the arguments of the function that relates each argument to its
- recursive calls. In general, there is one introduction rule for each
- recursive call.
-
- The predicate \isa{findzero{\isacharunderscore}dom} is the accessible part of
- that relation. An argument belongs to the accessible part, if it can
- be reached in a finite number of steps (cf.~its definition in \isa{Wellfounded{\isachardot}thy}).
-
- Since the domain predicate is just an abbreviation, you can use
- lemmas for \isa{accp} and \isa{findzero{\isacharunderscore}rel} directly. Some
- lemmas which are occasionally useful are \isa{accpI}, \isa{accp{\isacharunderscore}downward}, and of course the introduction and elimination rules
- for the recursion relation \isa{findzero{\isachardot}intros} and \isa{findzero{\isachardot}cases}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{A Useful Special Case: Tail recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The domain predicate is our trick that allows us to model partiality
- in a world of total functions. The downside of this is that we have
- to carry it around all the time. The termination proof above allowed
- us to replace the abstract \isa{findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}} by the more
- concrete \isa{n\ {\isasymle}\ x\ {\isasymand}\ f\ x\ {\isacharequal}\ {\isadigit{0}}}, but the condition is still
- there and can only be discharged for special cases.
- In particular, the domain predicate guards the unfolding of our
- function, since it is there as a condition in the \isa{psimp}
- rules.
-
- Now there is an important special case: We can actually get rid
- of the condition in the simplification rules, \emph{if the function
- is tail-recursive}. The reason is that for all tail-recursive
- equations there is a total function satisfying them, even if they
- are non-terminating.
-
-% A function is tail recursive, if each call to the function is either
-% equal
-%
-% So the outer form of the
-%
-%if it can be written in the following
-% form:
-% {term[display] "f x = (if COND x then BASE x else f (LOOP x))"}
-
-
- The function package internally does the right construction and can
- derive the unconditional simp rules, if we ask it to do so. Luckily,
- our \isa{findzero} function is tail-recursive, so we can just go
- back and add another option to the \cmd{function} command:
-
-\vspace{1ex}
-\noindent\cmd{function} \isa{{\isacharparenleft}domintros{\isacharcomma}\ tailrec{\isacharparenright}\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\%
-
-
- \noindent Now, we actually get unconditional simplification rules, even
- though the function is partial:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ findzero{\isachardot}simps%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-findzero\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isacharquery}n\ else\ findzero\ {\isacharquery}f\ {\isacharparenleft}Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}%
-\end{isabelle}
-
- \noindent Of course these would make the simplifier loop, so we better remove
- them from the simpset:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{declare}\isamarkupfalse%
-\ findzero{\isachardot}simps{\isacharbrackleft}simp\ del{\isacharbrackright}%
-\begin{isamarkuptext}%
-Getting rid of the domain conditions in the simplification rules is
- not only useful because it simplifies proofs. It is also required in
- order to use Isabelle's code generator to generate ML code
- from a function definition.
- Since the code generator only works with equations, it cannot be
- used with \isa{psimp} rules. Thus, in order to generate code for
- partial functions, they must be defined as a tail recursion.
- Luckily, many functions have a relatively natural tail recursive
- definition.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Nested recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Recursive calls which are nested in one another frequently cause
- complications, since their termination proof can depend on a partial
- correctness property of the function itself.
-
- As a small example, we define the \qt{nested zero} function:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ nz\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nz\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}nz\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ nz\ {\isacharparenleft}nz\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-If we attempt to prove termination using the identity measure on
- naturals, this fails:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}n{\isachardot}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto%
-\begin{isamarkuptxt}%
-We get stuck with the subgoal
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ nz{\isacharunderscore}dom\ n\ {\isasymLongrightarrow}\ nz\ n\ {\isacharless}\ Suc\ n%
-\end{isabelle}
-
- Of course this statement is true, since we know that \isa{nz} is
- the zero function. And in fact we have no problem proving this
- property by induction.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isacommand{lemma}\isamarkupfalse%
-\ nz{\isacharunderscore}is{\isacharunderscore}zero{\isacharcolon}\ {\isachardoublequoteopen}nz{\isacharunderscore}dom\ n\ {\isasymLongrightarrow}\ nz\ n\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}nz{\isachardot}pinduct{\isacharparenright}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We formulate this as a partial correctness lemma with the condition
- \isa{nz{\isacharunderscore}dom\ n}. This allows us to prove it with the \isa{pinduct} rule before we have proved termination. With this lemma,
- the termination proof works as expected:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}n{\isachardot}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\ {\isacharparenleft}auto\ simp{\isacharcolon}\ nz{\isacharunderscore}is{\isacharunderscore}zero{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-As a general strategy, one should prove the statements needed for
- termination as a partial property first. Then they can be used to do
- the termination proof. This also works for less trivial
- examples. Figure \ref{f91} defines the 91-function, a well-known
- challenge problem due to John McCarthy, and proves its termination.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{function}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}\ n\ {\isacharequal}\ {\isacharparenleft}if\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n\ then\ n\ {\isacharminus}\ {\isadigit{1}}{\isadigit{0}}\ else\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}estimate{\isacharcolon}\ \isanewline
-\ \ \isakeyword{assumes}\ trm{\isacharcolon}\ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}dom\ n{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}n\ {\isacharless}\ f{\isadigit{9}}{\isadigit{1}}\ n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ trm\ \isacommand{by}\isamarkupfalse%
-\ induct\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}R\ {\isacharequal}\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isacharminus}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}wf\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ n\ {\isacharcolon}{\isacharcolon}\ nat\ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n{\isachardoublequoteclose}\ %
-\isamarkupcmt{Assumptions for both calls%
-}
-\isanewline
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharcomma}\ n{\isacharparenright}\ {\isasymin}\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\ %
-\isamarkupcmt{Inner call%
-}
-\isanewline
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ inner{\isacharunderscore}trm{\isacharcolon}\ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}dom\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\ %
-\isamarkupcmt{Outer call%
-}
-\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}estimate\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}\ {\isacharless}\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isacharbackquoteopen}{\isasymnot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isacharcomma}\ n{\isacharparenright}\ {\isasymin}\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}
-\vspace{6pt}\hrule
-\caption{McCarthy's 91-function}\label{f91}
-\end{figure}
-%
-\isamarkupsection{Higher-Order Recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Higher-order recursion occurs when recursive calls
- are passed as arguments to higher-order combinators such as \isa{map}, \isa{filter} etc.
- As an example, imagine a datatype of n-ary trees:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ {\isacharprime}a\ tree\ {\isacharequal}\ \isanewline
-\ \ Leaf\ {\isacharprime}a\ \isanewline
-{\isacharbar}\ Branch\ {\isachardoublequoteopen}{\isacharprime}a\ tree\ list{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent We can define a function which swaps the left and right subtrees recursively, using the
- list functions \isa{rev} and \isa{map}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ mirror\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ tree\ {\isasymRightarrow}\ {\isacharprime}a\ tree{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mirror\ {\isacharparenleft}Leaf\ n{\isacharparenright}\ {\isacharequal}\ Leaf\ n{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}mirror\ {\isacharparenleft}Branch\ l{\isacharparenright}\ {\isacharequal}\ Branch\ {\isacharparenleft}rev\ {\isacharparenleft}map\ mirror\ l{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Although the definition is accepted without problems, let us look at the termination proof:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-%
-\begin{isamarkuptxt}%
-As usual, we have to give a wellfounded relation, such that the
- arguments of the recursive calls get smaller. But what exactly are
- the arguments of the recursive calls when mirror is given as an
- argument to \isa{map}? Isabelle gives us the
- subgoals
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ wf\ {\isacharquery}R\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}l\ x{\isachardot}\ x\ {\isasymin}\ set\ l\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ Branch\ l{\isacharparenright}\ {\isasymin}\ {\isacharquery}R%
-\end{isabelle}
-
- So the system seems to know that \isa{map} only
- applies the recursive call \isa{mirror} to elements
- of \isa{l}, which is essential for the termination proof.
-
- This knowledge about \isa{map} is encoded in so-called congruence rules,
- which are special theorems known to the \cmd{function} command. The
- rule for \isa{map} is
-
- \begin{isabelle}%
-{\isasymlbrakk}{\isacharquery}xs\ {\isacharequal}\ {\isacharquery}ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ {\isacharquery}ys\ {\isasymLongrightarrow}\ {\isacharquery}f\ x\ {\isacharequal}\ {\isacharquery}g\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ map\ {\isacharquery}f\ {\isacharquery}xs\ {\isacharequal}\ map\ {\isacharquery}g\ {\isacharquery}ys%
-\end{isabelle}
-
- You can read this in the following way: Two applications of \isa{map} are equal, if the list arguments are equal and the functions
- coincide on the elements of the list. This means that for the value
- \isa{map\ f\ l} we only have to know how \isa{f} behaves on
- the elements of \isa{l}.
-
- Usually, one such congruence rule is
- needed for each higher-order construct that is used when defining
- new functions. In fact, even basic functions like \isa{If} and \isa{Let} are handled by this mechanism. The congruence
- rule for \isa{If} states that the \isa{then} branch is only
- relevant if the condition is true, and the \isa{else} branch only if it
- is false:
-
- \begin{isabelle}%
-{\isasymlbrakk}{\isacharquery}b\ {\isacharequal}\ {\isacharquery}c{\isacharsemicolon}\ {\isacharquery}c\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}u{\isacharsemicolon}\ {\isasymnot}\ {\isacharquery}c\ {\isasymLongrightarrow}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}v{\isasymrbrakk}\isanewline
-{\isasymLongrightarrow}\ {\isacharparenleft}if\ {\isacharquery}b\ then\ {\isacharquery}x\ else\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}c\ then\ {\isacharquery}u\ else\ {\isacharquery}v{\isacharparenright}%
-\end{isabelle}
-
- Congruence rules can be added to the
- function package by giving them the \isa{fundef{\isacharunderscore}cong} attribute.
-
- The constructs that are predefined in Isabelle, usually
- come with the respective congruence rules.
- But if you define your own higher-order functions, you may have to
- state and prove the required congruence rules yourself, if you want to use your
- functions in recursive definitions.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Congruence Rules and Evaluation Order%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Higher order logic differs from functional programming languages in
- that it has no built-in notion of evaluation order. A program is
- just a set of equations, and it is not specified how they must be
- evaluated.
-
- However for the purpose of function definition, we must talk about
- evaluation order implicitly, when we reason about termination.
- Congruence rules express that a certain evaluation order is
- consistent with the logical definition.
-
- Consider the following function.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}f\ n\ {\isacharequal}\ {\isacharparenleft}n\ {\isacharequal}\ {\isadigit{0}}\ {\isasymor}\ f\ {\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-For this definition, the termination proof fails. The default configuration
- specifies no congruence rule for disjunction. We have to add a
- congruence rule that specifies left-to-right evaluation order:
-
- \vspace{1ex}
- \noindent \isa{{\isasymlbrakk}{\isacharquery}P\ {\isacharequal}\ {\isacharquery}P{\isacharprime}{\isacharsemicolon}\ {\isasymnot}\ {\isacharquery}P{\isacharprime}\ {\isasymLongrightarrow}\ {\isacharquery}Q\ {\isacharequal}\ {\isacharquery}Q{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharquery}P\ {\isasymor}\ {\isacharquery}Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}P{\isacharprime}\ {\isasymor}\ {\isacharquery}Q{\isacharprime}{\isacharparenright}}\hfill(\isa{disj{\isacharunderscore}cong})
- \vspace{1ex}
-
- Now the definition works without problems. Note how the termination
- proof depends on the extra condition that we get from the congruence
- rule.
-
- However, as evaluation is not a hard-wired concept, we
- could just turn everything around by declaring a different
- congruence rule. Then we can make the reverse definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ disj{\isacharunderscore}cong{\isadigit{2}}{\isacharbrackleft}fundef{\isacharunderscore}cong{\isacharbrackright}{\isacharcolon}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\isasymnot}\ Q{\isacharprime}\ {\isasymLongrightarrow}\ P\ {\isacharequal}\ P{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}Q\ {\isacharequal}\ Q{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymor}\ Q{\isacharprime}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ f{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}f{\isacharprime}\ n\ {\isacharequal}\ {\isacharparenleft}f{\isacharprime}\ {\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}\ {\isasymor}\ n\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent These examples show that, in general, there is no \qt{best} set of
- congruence rules.
-
- However, such tweaking should rarely be necessary in
- practice, as most of the time, the default set of congruence rules
- works well.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Functions/Thy/document/session.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,6 +0,0 @@
-\input{Functions.tex}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/doc-src/IsarAdvanced/Functions/conclusion.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,7 +0,0 @@
-\section{Conclusion}
-
-\fixme{}
-
-
-
-
--- a/doc-src/IsarAdvanced/Functions/functions.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,91 +0,0 @@
-
-\documentclass[a4paper,fleqn]{article}
-
-\usepackage{latexsym,graphicx}
-\usepackage[refpage]{nomencl}
-\usepackage{../../iman,../../extra,../../isar,../../proof}
-\usepackage{../../isabelle,../../isabellesym}
-\usepackage{style}
-\usepackage{mathpartir}
-\usepackage{amsthm}
-\usepackage{../../pdfsetup}
-
-\newcommand{\cmd}[1]{\isacommand{#1}}
-
-\newcommand{\isasymINFIX}{\cmd{infix}}
-\newcommand{\isasymLOCALE}{\cmd{locale}}
-\newcommand{\isasymINCLUDES}{\cmd{includes}}
-\newcommand{\isasymDATATYPE}{\cmd{datatype}}
-\newcommand{\isasymAXCLASS}{\cmd{axclass}}
-\newcommand{\isasymDEFINES}{\cmd{defines}}
-\newcommand{\isasymNOTES}{\cmd{notes}}
-\newcommand{\isasymCLASS}{\cmd{class}}
-\newcommand{\isasymINSTANCE}{\cmd{instance}}
-\newcommand{\isasymLEMMA}{\cmd{lemma}}
-\newcommand{\isasymPROOF}{\cmd{proof}}
-\newcommand{\isasymQED}{\cmd{qed}}
-\newcommand{\isasymFIX}{\cmd{fix}}
-\newcommand{\isasymASSUME}{\cmd{assume}}
-\newcommand{\isasymSHOW}{\cmd{show}}
-\newcommand{\isasymNOTE}{\cmd{note}}
-\newcommand{\isasymCODEGEN}{\cmd{code\_gen}}
-\newcommand{\isasymPRINTCODETHMS}{\cmd{print\_codethms}}
-\newcommand{\isasymFUN}{\cmd{fun}}
-\newcommand{\isasymFUNCTION}{\cmd{function}}
-\newcommand{\isasymPRIMREC}{\cmd{primrec}}
-\newcommand{\isasymRECDEF}{\cmd{recdef}}
-
-\newcommand{\qt}[1]{``#1''}
-\newcommand{\qtt}[1]{"{}{#1}"{}}
-\newcommand{\qn}[1]{\emph{#1}}
-\newcommand{\strong}[1]{{\bfseries #1}}
-\newcommand{\fixme}[1][!]{\strong{FIXME: #1}}
-
-\newtheorem{exercise}{Exercise}{\bf}{\itshape}
-%\newtheorem*{thmstar}{Theorem}{\bf}{\itshape}
-
-\hyphenation{Isabelle}
-\hyphenation{Isar}
-
-\isadroptag{theory}
-\title{Defining Recursive Functions in Isabelle/HOL}
-\author{Alexander Krauss}
-
-\isabellestyle{tt}
-\renewcommand{\isastyletxt}{\isastyletext}% use same formatting for txt and text
-
-\begin{document}
-
-\date{\ \\}
-\maketitle
-
-\begin{abstract}
- This tutorial describes the use of the new \emph{function} package,
- which provides general recursive function definitions for Isabelle/HOL.
- We start with very simple examples and then gradually move on to more
- advanced topics such as manual termination proofs, nested recursion,
- partiality, tail recursion and congruence rules.
-\end{abstract}
-
-%\thispagestyle{empty}\clearpage
-
-%\pagenumbering{roman}
-%\clearfirst
-
-\input{intro.tex}
-\input{Thy/document/Functions.tex}
-%\input{conclusion.tex}
-
-\begingroup
-%\tocentry{\bibname}
-\bibliographystyle{plain} \small\raggedright\frenchspacing
-\bibliography{../../manual}
-\endgroup
-
-\end{document}
-
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
--- a/doc-src/IsarAdvanced/Functions/intro.tex Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,55 +0,0 @@
-\section{Introduction}
-
-Starting from Isabelle 2007, new facilities for recursive
-function definitions~\cite{krauss2006} are available. They provide
-better support for general recursive definitions than previous
-packages. But despite all tool support, function definitions can
-sometimes be a difficult thing.
-
-This tutorial is an example-guided introduction to the practical use
-of the package and related tools. It should help you get started with
-defining functions quickly. For the more difficult definitions we will
-discuss what problems can arise, and how they can be solved.
-
-We assume that you have mastered the fundamentals of Isabelle/HOL
-and are able to write basic specifications and proofs. To start out
-with Isabelle in general, consult the Isabelle/HOL tutorial
-\cite{isa-tutorial}.
-
-
-
-\paragraph{Structure of this tutorial.}
-Section 2 introduces the syntax and basic operation of the \cmd{fun}
-command, which provides full automation with reasonable default
-behavior. The impatient reader can stop after that
-section, and consult the remaining sections only when needed.
-Section 3 introduces the more verbose \cmd{function} command which
-gives fine-grained control. This form should be used
-whenever the short form fails.
-After that we discuss more specialized issues:
-termination, mutual, nested and higher-order recursion, partiality, pattern matching
-and others.
-
-
-\paragraph{Some background.}
-Following the LCF tradition, the package is realized as a definitional
-extension: Recursive definitions are internally transformed into a
-non-recursive form, such that the function can be defined using
-standard definition facilities. Then the recursive specification is
-derived from the primitive definition. This is a complex task, but it
-is fully automated and mostly transparent to the user. Definitional
-extensions are valuable because they are conservative by construction:
-The \qt{new} concept of general wellfounded recursion is completely reduced
-to existing principles.
-
-
-
-
-The new \cmd{function} command, and its short form \cmd{fun} have mostly
-replaced the traditional \cmd{recdef} command \cite{slind-tfl}. They solve
-a few of technical issues around \cmd{recdef}, and allow definitions
-which were not previously possible.
-
-
-
-
--- a/doc-src/IsarAdvanced/Functions/isabelle_isar.eps Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
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-n
-1950.8 2097.2 m
-1958.8 2092.5 1967.3 2089 1975.5 2084.9 C
-1976.7 2083.5 1976.1 2081.5 1976.7 2079.9 C
-1979.6 2081.1 1981.6 2086.8 1985.3 2084 C
-1993.4 2079.3 2001.8 2075.8 2010 2071.7 C
-2010.5 2071.5 2010.5 2071.1 2010.8 2070.8 C
-2011.2 2064.3 2010.9 2057.5 2011 2050.8 C
-2015.8 2046.9 2022.2 2046.2 2026.6 2041.7 C
-2026.5 2032.5 2026.8 2022.9 2026.4 2014.1 C
-2020.4 2008.3 2015 2002.4 2008.8 1997.1 C
-2003.8 1996.8 2000.7 2001.2 1996.1 2002.1 C
-1995.2 1996.4 1996.9 1990.5 1995.6 1984.8 C
-1989.9 1979 1984.5 1973.9 1978.8 1967.8 C
-1977.7 1968.6 1976 1967.6 1974.5 1968.3 C
-1967.4 1972.5 1960.1 1976.1 1952.7 1979.3 C
-1946.8 1976.3 1943.4 1970.7 1938.5 1966.1 C
-1933.9 1966.5 1929.4 1968.8 1925.1 1970.7 C
-1917.2 1978.2 1906 1977.9 1897.2 1983.4 C
-1893.2 1985.6 1889.4 1988.6 1885 1990.1 C
-1884.6 1990.6 1883.9 1991 1883.8 1991.6 C
-1883.7 2000.4 1884 2009.9 1883.6 2018.9 C
-1887.7 2024 1893.2 2028.8 1898 2033.8 C
-1899.1 2035.5 1900.9 2036.8 1902.5 2037.9 C
-1903.9 2037.3 1905.2 2036.6 1906.4 2035.5 C
-1906.3 2039.7 1906.5 2044.6 1906.1 2048.9 C
-1906.3 2049.6 1906.7 2050.2 1907.1 2050.8 C
-1913.4 2056 1918.5 2062.7 1924.8 2068.1 C
-1926.6 2067.9 1928 2066.9 1929.4 2066 C
-1930.2 2071 1927.7 2077.1 1930.6 2081.6 C
-1936.6 2086.9 1941.5 2092.9 1947.9 2097.9 C
-1949 2098.1 1949.9 2097.5 1950.8 2097.2 C
-[0 0 0 0.18] vc
-f
-0.4 w
-S
-n
-1975.2 2084.7 m
-1976.6 2083.4 1975.7 2081.1 1976 2079.4 C
-1979.3 2079.5 1980.9 2086.2 1984.8 2084 C
-1992.9 2078.9 2001.7 2075.6 2010 2071.2 C
-2011 2064.6 2010.2 2057.3 2010.8 2050.6 C
-2015.4 2046.9 2021.1 2045.9 2025.9 2042.4 C
-2026.5 2033.2 2026.8 2022.9 2025.6 2013.9 C
-2020.5 2008.1 2014.5 2003.1 2009.3 1997.6 C
-2004.1 1996.7 2000.7 2001.6 1995.9 2002.6 C
-1995.2 1996.7 1996.3 1990.2 1994.9 1984.6 C
-1989.8 1978.7 1983.6 1973.7 1978.4 1968 C
-1977.3 1969.3 1976 1967.6 1974.8 1968.5 C
-1967.7 1972.7 1960.4 1976.3 1952.9 1979.6 C
-1946.5 1976.9 1943.1 1970.5 1937.8 1966.1 C
-1928.3 1968.2 1920.6 1974.8 1911.6 1978.4 C
-1901.9 1979.7 1893.9 1986.6 1885 1990.6 C
-1884.3 1991 1884.3 1991.7 1884 1992.3 C
-1884.5 2001 1884.2 2011 1884.3 2019.9 C
-1890.9 2025.3 1895.9 2031.9 1902.3 2037.4 C
-1904.2 2037.9 1905.6 2034.2 1906.8 2035.7 C
-1907.4 2040.9 1905.7 2046.1 1907.3 2050.8 C
-1913.6 2056.2 1919.2 2062.6 1925.1 2067.9 C
-1926.9 2067.8 1928 2066.3 1929.6 2065.7 C
-1929.9 2070.5 1929.2 2076 1930.1 2080.8 C
-1936.5 2086.1 1941.6 2092.8 1948.4 2097.6 C
-1957.3 2093.3 1966.2 2088.8 1975.2 2084.7 C
-[0 0 0 0] vc
-f
-S
-n
-1954.8 2093.8 m
-1961.6 2090.5 1968.2 2087 1975 2084 C
-1975 2082.8 1975.6 2080.9 1974.8 2080.6 C
-1974.3 2075.2 1974.6 2069.6 1974.5 2064 C
-1977.5 2059.7 1984.5 2060 1988.9 2056.4 C
-1989.5 2055.5 1990.5 2055.3 1990.8 2054.4 C
-1991.1 2045.7 1991.4 2036.1 1990.6 2027.8 C
-1990.7 2026.6 1992 2027.3 1992.8 2027.1 C
-1997 2032.4 2002.6 2037.8 2007.6 2042.2 C
-2008.7 2042.3 2007.8 2040.6 2007.4 2040 C
-2002.3 2035.6 1997.5 2030 1992.8 2025.2 C
-1991.6 2024.7 1990.8 2024.9 1990.1 2025.4 C
-1989.4 2024.9 1988.1 2025.2 1987.2 2024.4 C
-1987.1 2025.8 1988.3 2026.5 1989.4 2026.8 C
-1989.4 2026.6 1989.3 2026.2 1989.6 2026.1 C
-1989.9 2026.2 1989.9 2026.6 1989.9 2026.8 C
-1989.8 2026.6 1990 2026.5 1990.1 2026.4 C
-1990.2 2027 1991.1 2028.3 1990.1 2028 C
-1989.9 2037.9 1990.5 2044.1 1989.6 2054.2 C
-1985.9 2058 1979.7 2057.4 1976 2061.2 C
-1974.5 2061.6 1975.2 2059.9 1974.5 2059.5 C
-1973.9 2058 1975.6 2057.8 1975 2056.6 C
-1974.5 2057.1 1974.6 2055.3 1973.6 2055.9 C
-1971.9 2059.3 1974.7 2062.1 1973.1 2065.5 C
-1973.1 2071.2 1972.9 2077 1973.3 2082.5 C
-1967.7 2085.6 1962 2088 1956.3 2090.7 C
-1953.9 2092.4 1951 2093 1948.6 2094.8 C
-1943.7 2089.9 1937.9 2084.3 1933 2079.6 C
-1931.3 2076.1 1933.2 2071.3 1932.3 2067.2 C
-1931.3 2062.9 1933.3 2060.6 1932 2057.6 C
-1932.7 2056.5 1930.9 2053.3 1933.2 2051.8 C
-1936.8 2050.1 1940.1 2046.9 1944 2046.8 C
-1946.3 2049.7 1949.3 2051.9 1952 2054.4 C
-1954.5 2054.2 1956.4 2052.3 1958.7 2051.3 C
-1960.8 2050 1963.2 2049 1965.6 2048.4 C
-1968.3 2050.8 1970.7 2054.3 1973.6 2055.4 C
-1973 2052.2 1969.7 2050.4 1967.6 2048.2 C
-1967.1 2046.7 1968.8 2046.6 1969.5 2045.8 C
-1972.8 2043.3 1980.6 2043.4 1979.3 2038.4 C
-1979.4 2038.6 1979.2 2038.7 1979.1 2038.8 C
-1978.7 2038.6 1978.9 2038.1 1978.8 2037.6 C
-1978.9 2037.9 1978.7 2038 1978.6 2038.1 C
-1978.2 2032.7 1978.4 2027.1 1978.4 2021.6 C
-1979.3 2021.1 1980 2020.2 1981.5 2020.1 C
-1983.5 2020.5 1984 2021.8 1985.1 2023.5 C
-1985.7 2024 1987.4 2023.7 1986 2022.8 C
-1984.7 2021.7 1983.3 2020.8 1983.9 2018.7 C
-1987.2 2015.9 1993 2015.4 1994.9 2011.5 C
-1992.2 2004.9 1999.3 2005.2 2002.1 2002.4 C
-2005.9 2002.7 2004.8 1997.4 2009.1 1999 C
-2011 1999.3 2010 2002.9 2012.7 2002.4 C
-2010.2 2000.7 2009.4 1996.1 2005.5 1998.5 C
-2002.1 2000.3 1999 2002.5 1995.4 2003.8 C
-1995.2 2003.6 1994.9 2003.3 1994.7 2003.1 C
-1994.3 1997 1995.6 1991.1 1994.4 1985.3 C
-1994.3 1986 1993.8 1985 1994 1985.6 C
-1993.8 1995.4 1994.4 2001.6 1993.5 2011.7 C
-1989.7 2015.5 1983.6 2014.9 1979.8 2018.7 C
-1978.3 2019.1 1979.1 2017.4 1978.4 2017 C
-1977.8 2015.5 1979.4 2015.3 1978.8 2014.1 C
-1978.4 2014.6 1978.5 2012.8 1977.4 2013.4 C
-1975.8 2016.8 1978.5 2019.6 1976.9 2023 C
-1977 2028.7 1976.7 2034.5 1977.2 2040 C
-1971.6 2043.1 1965.8 2045.6 1960.1 2048.2 C
-1957.7 2049.9 1954.8 2050.5 1952.4 2052.3 C
-1947.6 2047.4 1941.8 2041.8 1936.8 2037.2 C
-1935.2 2033.6 1937.1 2028.8 1936.1 2024.7 C
-1935.1 2020.4 1937.1 2018.1 1935.9 2015.1 C
-1936.5 2014.1 1934.7 2010.8 1937.1 2009.3 C
-1944.4 2004.8 1952 2000.9 1959.9 1997.8 C
-1963.9 1997 1963.9 2001.9 1966.8 2003.3 C
-1970.3 2006.9 1973.7 2009.9 1976.9 2012.9 C
-1977.9 2013 1977.1 2011.4 1976.7 2010.8 C
-1971.6 2006.3 1966.8 2000.7 1962 1995.9 C
-1960 1995.2 1960.1 1996.6 1958.2 1995.6 C
-1957 1997 1955.1 1998.8 1953.2 1998 C
-1951.7 1994.5 1954.1 1993.4 1952.9 1991.1 C
-1952.1 1990.5 1953.3 1990.2 1953.2 1989.6 C
-1954.2 1986.8 1950.9 1981.4 1954.4 1981.2 C
-1954.7 1981.6 1954.7 1981.7 1955.1 1982 C
-1961.9 1979.1 1967.6 1975 1974.3 1971.6 C
-1974.7 1969.8 1976.7 1969.5 1978.4 1969.7 C
-1980.3 1970 1979.3 1973.6 1982 1973.1 C
-1975.8 1962.2 1968 1975.8 1960.8 1976.7 C
-1956.9 1977.4 1953.3 1982.4 1949.1 1978.8 C
-1946 1975.8 1941.2 1971 1939.5 1969.2 C
-1938.5 1968.6 1938.9 1967.4 1937.8 1966.8 C
-1928.7 1969.4 1920.6 1974.5 1912.4 1979.1 C
-1904 1980 1896.6 1985 1889.3 1989.4 C
-1887.9 1990.4 1885.1 1990.3 1885 1992.5 C
-1885.4 2000.6 1885.2 2012.9 1885.2 2019.9 C
-1886.1 2022 1889.7 2019.5 1888.4 2022.8 C
-1889 2023.3 1889.8 2024.4 1890.3 2024 C
-1891.2 2023.5 1891.8 2028.2 1893.4 2026.6 C
-1894.2 2026.3 1893.9 2027.3 1894.4 2027.6 C
-1893.4 2027.6 1894.7 2028.3 1894.1 2028.5 C
-1894.4 2029.6 1896 2030 1896 2029.2 C
-1896.2 2029 1896.3 2029 1896.5 2029.2 C
-1896.8 2029.8 1897.3 2030 1897 2030.7 C
-1896.5 2030.7 1896.9 2031.5 1897.2 2031.6 C
-1898.3 2034 1899.5 2030.6 1899.6 2033.3 C
-1898.5 2033 1899.6 2034.4 1900.1 2034.8 C
-1901.3 2035.8 1903.2 2034.6 1902.5 2036.7 C
-1904.4 2036.9 1906.1 2032.2 1907.6 2035.5 C
-1907.5 2040.1 1907.7 2044.9 1907.3 2049.4 C
-1908 2050.2 1908.3 2051.4 1909.5 2051.6 C
-1910.1 2051.1 1911.6 2051.1 1911.4 2052.3 C
-1909.7 2052.8 1912.4 2054 1912.6 2054.7 C
-1913.4 2055.2 1913 2053.7 1913.6 2054.4 C
-1913.6 2054.5 1913.6 2055.3 1913.6 2054.7 C
-1913.7 2054.4 1913.9 2054.4 1914 2054.7 C
-1914 2054.9 1914.1 2055.3 1913.8 2055.4 C
-1913.7 2056 1915.2 2057.6 1916 2057.6 C
-1915.9 2057.3 1916.1 2057.2 1916.2 2057.1 C
-1917 2056.8 1916.7 2057.7 1917.2 2058 C
-1917 2058.3 1916.7 2058.3 1916.4 2058.3 C
-1917.1 2059 1917.3 2060.1 1918.4 2060.4 C
-1918.1 2059.2 1919.1 2060.6 1919.1 2059.5 C
-1919 2060.6 1920.6 2060.1 1919.8 2061.2 C
-1919.6 2061.2 1919.3 2061.2 1919.1 2061.2 C
-1919.6 2061.9 1921.4 2064.2 1921.5 2062.6 C
-1922.4 2062.1 1921.6 2063.9 1922.2 2064.3 C
-1922.9 2067.3 1926.1 2064.3 1925.6 2067.2 C
-1927.2 2066.8 1928.4 2064.6 1930.1 2065.2 C
-1931.8 2067.8 1931 2071.8 1930.8 2074.8 C
-1930.6 2076.4 1930.1 2078.6 1930.6 2080.4 C
-1936.6 2085.4 1941.8 2091.6 1948.1 2096.9 C
-1950.7 2096.7 1952.6 2094.8 1954.8 2093.8 C
-[0 0.33 0.33 0.99] vc
-f
-S
-n
-1989.4 2080.6 m
-1996.1 2077.3 2002.7 2073.8 2009.6 2070.8 C
-2009.6 2069.6 2010.2 2067.7 2009.3 2067.4 C
-2008.9 2062 2009.1 2056.4 2009.1 2050.8 C
-2012.3 2046.6 2019 2046.6 2023.5 2043.2 C
-2024 2042.3 2025.1 2042.1 2025.4 2041.2 C
-2025.3 2032.7 2025.6 2023.1 2025.2 2014.6 C
-2025 2015.3 2024.6 2014.2 2024.7 2014.8 C
-2024.5 2024.7 2025.1 2030.9 2024.2 2041 C
-2020.4 2044.8 2014.3 2044.2 2010.5 2048 C
-2009 2048.4 2009.8 2046.7 2009.1 2046.3 C
-2008.5 2044.8 2010.2 2044.6 2009.6 2043.4 C
-2009.1 2043.9 2009.2 2042.1 2008.1 2042.7 C
-2006.5 2046.1 2009.3 2048.9 2007.6 2052.3 C
-2007.7 2058 2007.5 2063.8 2007.9 2069.3 C
-2002.3 2072.4 1996.5 2074.8 1990.8 2077.5 C
-1988.4 2079.2 1985.6 2079.8 1983.2 2081.6 C
-1980.5 2079 1977.9 2076.5 1975.5 2074.1 C
-1975.5 2075.1 1975.5 2076.2 1975.5 2077.2 C
-1977.8 2079.3 1980.3 2081.6 1982.7 2083.7 C
-1985.3 2083.5 1987.1 2081.6 1989.4 2080.6 C
-f
-S
-n
-1930.1 2079.9 m
-1931.1 2075.6 1929.2 2071.1 1930.8 2067.2 C
-1930.3 2066.3 1930.1 2064.6 1928.7 2065.5 C
-1927.7 2066.4 1926.5 2067 1925.3 2067.4 C
-1924.5 2066.9 1925.6 2065.7 1924.4 2066 C
-1924.2 2067.2 1923.6 2065.5 1923.2 2065.7 C
-1922.3 2063.6 1917.8 2062.1 1919.6 2060.4 C
-1919.3 2060.5 1919.2 2060.3 1919.1 2060.2 C
-1919.7 2060.9 1918.2 2061 1917.6 2060.2 C
-1917 2059.6 1916.1 2058.8 1916.4 2058 C
-1915.5 2058 1917.4 2057.1 1915.7 2057.8 C
-1914.8 2057.1 1913.4 2056.2 1913.3 2054.9 C
-1913.1 2055.4 1911.3 2054.3 1910.9 2053.2 C
-1910.7 2052.9 1910.2 2052.5 1910.7 2052.3 C
-1911.1 2052.5 1910.9 2052 1910.9 2051.8 C
-1910.5 2051.2 1909.9 2052.6 1909.2 2051.8 C
-1908.2 2051.4 1907.8 2050.2 1907.1 2049.4 C
-1907.5 2044.8 1907.3 2040 1907.3 2035.2 C
-1905.3 2033 1902.8 2039.3 1902.3 2035.7 C
-1899.6 2036 1898.4 2032.5 1896.3 2030.7 C
-1895.7 2030.1 1897.5 2030 1896.3 2029.7 C
-1896.3 2030.6 1895 2029.7 1894.4 2029.2 C
-1892.9 2028.1 1894.2 2027.4 1893.6 2027.1 C
-1892.1 2027.9 1891.7 2025.6 1890.8 2024.9 C
-1891.1 2024.6 1889.1 2024.3 1888.4 2023 C
-1887.5 2022.6 1888.2 2021.9 1888.1 2021.3 C
-1886.7 2022 1885.2 2020.4 1884.8 2019.2 C
-1884.8 2010 1884.6 2000.2 1885 1991.8 C
-1886.9 1989.6 1889.9 1989.3 1892.2 1987.5 C
-1898.3 1982.7 1905.6 1980.1 1912.8 1978.6 C
-1921 1974.2 1928.8 1968.9 1937.8 1966.6 C
-1939.8 1968.3 1938.8 1968.3 1940.4 1970 C
-1945.4 1972.5 1947.6 1981.5 1954.6 1979.3 C
-1952.3 1981 1950.4 1978.4 1948.6 1977.9 C
-1945.1 1973.9 1941.1 1970.6 1938 1966.6 C
-1928.4 1968.5 1920.6 1974.8 1911.9 1978.8 C
-1907.1 1979.2 1902.6 1981.7 1898.2 1983.6 C
-1893.9 1986 1889.9 1989 1885.5 1990.8 C
-1884.9 1991.2 1884.8 1991.8 1884.5 1992.3 C
-1884.9 2001.3 1884.7 2011.1 1884.8 2019.6 C
-1890.6 2025 1896.5 2031.2 1902.3 2036.9 C
-1904.6 2037.6 1905 2033 1907.3 2035.5 C
-1907.2 2040.2 1907 2044.8 1907.1 2049.6 C
-1913.6 2055.3 1918.4 2061.5 1925.1 2067.4 C
-1927.3 2068.2 1929.6 2062.5 1930.6 2066.9 C
-1929.7 2070.7 1930.3 2076 1930.1 2080.1 C
-1935.6 2085.7 1941.9 2090.7 1947.2 2096.7 C
-1942.2 2091.1 1935.5 2085.2 1930.1 2079.9 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1930.8 2061.9 m
-1930.3 2057.8 1931.8 2053.4 1931.1 2050.4 C
-1931.3 2050.3 1931.7 2050.5 1931.6 2050.1 C
-1933 2051.1 1934.4 2049.5 1935.9 2048.7 C
-1937 2046.5 1939.5 2047.1 1941.2 2045.1 C
-1939.7 2042.6 1937.3 2041.2 1935.4 2039.3 C
-1934 2039.7 1934.5 2038.1 1933.7 2037.6 C
-1934 2033.3 1933.1 2027.9 1934.4 2024.4 C
-1934.3 2023.8 1933.9 2022.8 1933 2022.8 C
-1931.6 2023.1 1930.5 2024.4 1929.2 2024.9 C
-1928.4 2024.5 1929.8 2023.5 1928.7 2023.5 C
-1927.7 2024.1 1926.2 2022.6 1925.6 2021.6 C
-1926.9 2021.6 1924.8 2020.6 1925.6 2020.4 C
-1924.7 2021.7 1923.9 2019.6 1923.2 2019.2 C
-1923.3 2018.3 1923.8 2018.1 1923.2 2018 C
-1922.9 2017.8 1922.9 2017.5 1922.9 2017.2 C
-1922.8 2018.3 1921.3 2017.3 1920.3 2018 C
-1916.6 2019.7 1913 2022.1 1910 2024.7 C
-1910 2032.9 1910 2041.2 1910 2049.4 C
-1915.4 2055.2 1920 2058.7 1925.3 2064.8 C
-1927.2 2064 1929 2061.4 1930.8 2061.9 C
-[0 0 0 0] vc
-f
-S
-n
-1907.6 2030.4 m
-1907.5 2027.1 1906.4 2021.7 1908.5 2019.9 C
-1908.8 2020.1 1908.9 2019 1909.2 2019.6 C
-1910 2019.6 1912 2019.2 1913.1 2018.2 C
-1913.7 2016.5 1920.2 2015.7 1917.4 2012.7 C
-1918.2 2011.2 1917 2013.8 1917.2 2012 C
-1916.9 2012.3 1916 2012.4 1915.2 2012 C
-1912.5 2010.5 1916.6 2008.8 1913.6 2009.6 C
-1912.6 2009.2 1911.1 2009 1910.9 2007.6 C
-1911 1999.2 1911.8 1989.8 1911.2 1982.2 C
-1910.1 1981.1 1908.8 1982.2 1907.6 1982.2 C
-1900.8 1986.5 1893.2 1988.8 1887.2 1994.2 C
-1887.2 2002.4 1887.2 2010.7 1887.2 2018.9 C
-1892.6 2024.7 1897.2 2028.2 1902.5 2034.3 C
-1904.3 2033.3 1906.2 2032.1 1907.6 2030.4 C
-f
-S
-n
-1910.7 2025.4 m
-1912.7 2022.4 1916.7 2020.8 1919.8 2018.9 C
-1920.2 2018.7 1920.6 2018.6 1921 2018.4 C
-1925 2020 1927.4 2028.5 1932 2024.2 C
-1932.3 2025 1932.5 2023.7 1932.8 2024.4 C
-1932.8 2028 1932.8 2031.5 1932.8 2035 C
-1931.9 2033.9 1932.5 2036.3 1932.3 2036.9 C
-1933.2 2036.4 1932.5 2038.5 1933 2038.4 C
-1933.1 2040.5 1935.6 2042.2 1936.6 2043.2 C
-1936.2 2042.4 1935.1 2040.8 1933.7 2040.3 C
-1932.2 2034.4 1933.8 2029.8 1933 2023.2 C
-1931.1 2024.9 1928.4 2026.4 1926.5 2023.5 C
-1925.1 2021.6 1923 2019.8 1921.5 2018.2 C
-1917.8 2018.9 1915.2 2022.5 1911.6 2023.5 C
-1910.8 2023.8 1911.2 2024.7 1910.4 2025.2 C
-1910.9 2031.8 1910.6 2039.1 1910.7 2045.6 C
-1910.1 2048 1910.7 2045.9 1911.2 2044.8 C
-1910.6 2038.5 1911.2 2031.8 1910.7 2025.4 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1910.7 2048.9 m
-1910.3 2047.4 1911.3 2046.5 1911.6 2045.3 C
-1912.9 2045.3 1913.9 2047.1 1915.2 2045.8 C
-1915.2 2044.9 1916.6 2043.3 1917.2 2042.9 C
-1918.7 2042.9 1919.4 2044.4 1920.5 2043.2 C
-1921.2 2042.2 1921.4 2040.9 1922.4 2040.3 C
-1924.5 2040.3 1925.7 2040.9 1926.8 2039.6 C
-1927.1 2037.9 1926.8 2038.1 1927.7 2037.6 C
-1929 2037.5 1930.4 2037 1931.6 2037.2 C
-1932.3 2038.2 1933.1 2038.7 1932.8 2040.3 C
-1935 2041.8 1935.9 2043.8 1938.5 2044.8 C
-1938.6 2045 1938.3 2045.5 1938.8 2045.3 C
-1939.1 2042.9 1935.4 2044.2 1935.4 2042.2 C
-1932.1 2040.8 1932.8 2037.2 1932 2034.8 C
-1932.3 2034 1932.7 2035.4 1932.5 2034.8 C
-1931.3 2031.8 1935.5 2020.1 1928.9 2025.9 C
-1924.6 2024.7 1922.6 2014.5 1917.4 2020.4 C
-1915.5 2022.8 1912 2022.6 1910.9 2025.4 C
-1911.5 2031.9 1910.9 2038.8 1911.4 2045.3 C
-1911.1 2046.5 1910 2047.4 1910.4 2048.9 C
-1915.1 2054.4 1920.4 2058.3 1925.1 2063.8 C
-1920.8 2058.6 1914.9 2054.3 1910.7 2048.9 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1934.7 2031.9 m
-1934.6 2030.7 1934.9 2029.5 1934.4 2028.5 C
-1934 2029.5 1934.3 2031.2 1934.2 2032.6 C
-1933.8 2031.7 1934.9 2031.6 1934.7 2031.9 C
-[0.92 0.92 0 0.67] vc
-f
-S
-n
-vmrs
-1934.7 2019.4 m
-1934.1 2015.3 1935.6 2010.9 1934.9 2007.9 C
-1935.1 2007.8 1935.6 2008.1 1935.4 2007.6 C
-1936.8 2008.6 1938.2 2007 1939.7 2006.2 C
-1940.1 2004.3 1942.7 2005 1943.6 2003.8 C
-1945.1 2000.3 1954 2000.8 1950 1996.6 C
-1952.1 1993.3 1948.2 1989.2 1951.2 1985.6 C
-1953 1981.4 1948.4 1982.3 1947.9 1979.8 C
-1945.4 1979.6 1945.1 1975.5 1942.4 1975 C
-1942.4 1972.3 1938 1973.6 1938.5 1970.4 C
-1937.4 1969 1935.6 1970.1 1934.2 1970.2 C
-1927.5 1974.5 1919.8 1976.8 1913.8 1982.2 C
-1913.8 1990.4 1913.8 1998.7 1913.8 2006.9 C
-1919.3 2012.7 1923.8 2016.2 1929.2 2022.3 C
-1931.1 2021.6 1932.8 2018.9 1934.7 2019.4 C
-[0 0 0 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-2024.2 2038.1 m
-2024.1 2029.3 2024.4 2021.7 2024.7 2014.4 C
-2024.4 2013.6 2020.6 2013.4 2021.3 2011.2 C
-2020.5 2010.3 2018.4 2010.6 2018.9 2008.6 C
-2019 2008.8 2018.8 2009 2018.7 2009.1 C
-2018.2 2006.7 2015.2 2007.9 2015.3 2005.5 C
-2014.7 2004.8 2012.4 2005.1 2013.2 2003.6 C
-2012.3 2004.2 2012.8 2002.4 2012.7 2002.6 C
-2009.4 2003.3 2011.2 1998.6 2008.4 1999.2 C
-2007 1999.1 2006.1 1999.4 2005.7 2000.4 C
-2006.9 1998.5 2007.7 2000.5 2009.3 2000.2 C
-2009.2 2003.7 2012.4 2002.1 2012.9 2005.2 C
-2015.9 2005.6 2015.2 2008.6 2017.7 2008.8 C
-2018.4 2009.6 2018.3 2011.4 2019.6 2011 C
-2021.1 2011.7 2021.4 2014.8 2023.7 2015.1 C
-2023.7 2023.5 2023.9 2031.6 2023.5 2040.5 C
-2021.8 2041.7 2020.7 2043.6 2018.4 2043.9 C
-2020.8 2042.7 2025.5 2041.8 2024.2 2038.1 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-2023.5 2040 m
-2023.5 2031.1 2023.5 2023.4 2023.5 2015.1 C
-2020.2 2015 2021.8 2010.3 2018.4 2011 C
-2018.6 2007.5 2014.7 2009.3 2014.8 2006.4 C
-2011.8 2006.3 2012.2 2002.3 2009.8 2002.4 C
-2009.7 2001.5 2009.2 2000.1 2008.4 2000.2 C
-2008.7 2000.9 2009.7 2001.2 2009.3 2002.4 C
-2008.4 2004.2 2007.5 2003.1 2007.9 2005.5 C
-2007.9 2010.8 2007.7 2018.7 2008.1 2023.2 C
-2009 2024.3 2007.3 2023.4 2007.9 2024 C
-2007.7 2024.6 2007.3 2026.3 2008.6 2027.1 C
-2009.7 2026.8 2010 2027.6 2010.5 2028 C
-2010.5 2028.2 2010.5 2029.1 2010.5 2028.5 C
-2011.5 2028 2010.5 2030 2011.5 2030 C
-2014.2 2029.7 2012.9 2032.2 2014.8 2032.6 C
-2015.1 2033.6 2015.3 2033 2016 2033.3 C
-2017 2033.9 2016.6 2035.4 2017.2 2036.2 C
-2018.7 2036.4 2019.2 2039 2021.3 2038.4 C
-2021.6 2035.4 2019.7 2029.5 2021.1 2027.3 C
-2020.9 2023.5 2021.5 2018.5 2020.6 2016 C
-2020.9 2013.9 2021.5 2015.4 2022.3 2014.4 C
-2022.2 2015.1 2023.3 2014.8 2023.2 2015.6 C
-2022.7 2019.8 2023.3 2024.3 2022.8 2028.5 C
-2022.3 2028.2 2022.6 2027.6 2022.5 2027.1 C
-2022.5 2027.8 2022.5 2029.2 2022.5 2029.2 C
-2022.6 2029.2 2022.7 2029.1 2022.8 2029 C
-2023.9 2032.8 2022.6 2037 2023 2040.8 C
-2022.3 2041.2 2021.6 2041.5 2021.1 2042.2 C
-2022 2041.2 2022.9 2041.4 2023.5 2040 C
-[0 1 1 0.23] vc
-f
-S
-n
-2009.1 1997.8 m
-2003.8 1997.7 2000.1 2002.4 1995.4 2003.1 C
-1995 1999.5 1995.2 1995 1995.2 1992 C
-1995.2 1995.8 1995 1999.7 1995.4 2003.3 C
-2000.3 2002.2 2003.8 1997.9 2009.1 1997.8 C
-2012.3 2001.2 2015.6 2004.8 2018.7 2008.1 C
-2021.6 2011.2 2027.5 2013.9 2025.9 2019.9 C
-2026.1 2017.9 2025.6 2016.2 2025.4 2014.4 C
-2020.2 2008.4 2014 2003.6 2009.1 1997.8 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-2009.3 1997.8 m
-2008.7 1997.4 2007.9 1997.6 2007.2 1997.6 C
-2007.9 1997.6 2008.9 1997.4 2009.6 1997.8 C
-2014.7 2003.6 2020.8 2008.8 2025.9 2014.8 C
-2025.8 2017.7 2026.1 2014.8 2025.6 2014.1 C
-2020.4 2008.8 2014.8 2003.3 2009.3 1997.8 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-2009.6 1997.6 m
-2009 1997.1 2008.1 1997.4 2007.4 1997.3 C
-2008.1 1997.4 2009 1997.1 2009.6 1997.6 C
-2014.8 2003.7 2021.1 2008.3 2025.9 2014.4 C
-2021.1 2008.3 2014.7 2003.5 2009.6 1997.6 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-2021.8 2011.5 m
-2021.9 2012.2 2022.3 2013.5 2023.7 2013.6 C
-2023.4 2012.7 2022.8 2011.8 2021.8 2011.5 C
-[0 0.33 0.33 0.99] vc
-f
-S
-n
-2021.1 2042 m
-2022.1 2041.1 2020.9 2040.2 2020.6 2039.6 C
-2018.4 2039.5 2018.1 2036.9 2016.3 2036.4 C
-2015.8 2035.5 2015.3 2033.8 2014.8 2033.6 C
-2012.4 2033.8 2013 2030.4 2010.5 2030.2 C
-2009.6 2028.9 2009.6 2028.3 2008.4 2028 C
-2006.9 2026.7 2007.5 2024.3 2006 2023.2 C
-2006.6 2023.2 2005.7 2023.3 2005.7 2023 C
-2006.4 2022.5 2006.3 2021.1 2006.7 2020.6 C
-2006.6 2015 2006.9 2009 2006.4 2003.8 C
-2006.9 2002.5 2007.6 2001.1 2006.9 2000.7 C
-2004.6 2003.6 2003 2002.9 2000.2 2004.3 C
-1999.3 2005.8 1997.9 2006.3 1996.1 2006.7 C
-1995.7 2008.9 1996 2011.1 1995.9 2012.9 C
-1993.4 2015.1 1990.5 2016.2 1987.7 2017.7 C
-1987.1 2019.3 1991.1 2019.4 1990.4 2021.3 C
-1990.5 2021.5 1991.9 2022.3 1992 2023 C
-1994.8 2024.4 1996.2 2027.5 1998.5 2030 C
-2002.4 2033 2005.2 2037.2 2008.8 2041 C
-2010.2 2041.3 2011.6 2042 2011 2043.9 C
-2011.2 2044.8 2010.1 2045.3 2010.5 2046.3 C
-2013.8 2044.8 2017.5 2043.4 2021.1 2042 C
-[0 0.5 0.5 0.2] vc
-f
-S
-n
-2019.4 2008.8 m
-2018.9 2009.2 2019.3 2009.9 2019.6 2010.3 C
-2022.2 2011.5 2020.3 2009.1 2019.4 2008.8 C
-[0 0.33 0.33 0.99] vc
-f
-S
-n
-2018 2007.4 m
-2015.7 2006.7 2015.3 2003.6 2012.9 2002.8 C
-2013.5 2003.7 2013.5 2005.1 2015.6 2005.2 C
-2016.4 2006.1 2015.7 2007.7 2018 2007.4 C
-f
-S
-n
-vmrs
-1993.5 2008.8 m
-1993.4 2000 1993.7 1992.5 1994 1985.1 C
-1993.7 1984.3 1989.9 1984.1 1990.6 1982 C
-1989.8 1981.1 1987.7 1981.4 1988.2 1979.3 C
-1988.3 1979.6 1988.1 1979.7 1988 1979.8 C
-1987.5 1977.5 1984.5 1978.6 1984.6 1976.2 C
-1983.9 1975.5 1981.7 1975.8 1982.4 1974.3 C
-1981.6 1974.9 1982.1 1973.1 1982 1973.3 C
-1979 1973.7 1980 1968.8 1976.9 1969.7 C
-1975.9 1969.8 1975.3 1970.3 1975 1971.2 C
-1976.2 1969.2 1977 1971.2 1978.6 1970.9 C
-1978.5 1974.4 1981.7 1972.8 1982.2 1976 C
-1985.2 1976.3 1984.5 1979.3 1987 1979.6 C
-1987.7 1980.3 1987.5 1982.1 1988.9 1981.7 C
-1990.4 1982.4 1990.7 1985.5 1993 1985.8 C
-1992.9 1994.3 1993.2 2002.3 1992.8 2011.2 C
-1991.1 2012.4 1990 2014.4 1987.7 2014.6 C
-1990.1 2013.4 1994.7 2012.6 1993.5 2008.8 C
-[0 0.87 0.91 0.83] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1992.8 2010.8 m
-1992.8 2001.8 1992.8 1994.1 1992.8 1985.8 C
-1989.5 1985.7 1991.1 1981.1 1987.7 1981.7 C
-1987.9 1978.2 1983.9 1980 1984.1 1977.2 C
-1981.1 1977 1981.5 1973 1979.1 1973.1 C
-1979 1972.2 1978.5 1970.9 1977.6 1970.9 C
-1977.9 1971.6 1979 1971.9 1978.6 1973.1 C
-1977.6 1974.9 1976.8 1973.9 1977.2 1976.2 C
-1977.2 1981.5 1977 1989.4 1977.4 1994 C
-1978.3 1995 1976.6 1994.1 1977.2 1994.7 C
-1977 1995.3 1976.6 1997 1977.9 1997.8 C
-1979 1997.5 1979.3 1998.3 1979.8 1998.8 C
-1979.8 1998.9 1979.8 1999.8 1979.8 1999.2 C
-1980.8 1998.7 1979.7 2000.7 1980.8 2000.7 C
-1983.5 2000.4 1982.1 2003 1984.1 2003.3 C
-1984.4 2004.3 1984.5 2003.7 1985.3 2004 C
-1986.3 2004.6 1985.9 2006.1 1986.5 2006.9 C
-1988 2007.1 1988.4 2009.7 1990.6 2009.1 C
-1990.9 2006.1 1989 2000.2 1990.4 1998 C
-1990.2 1994.3 1990.8 1989.2 1989.9 1986.8 C
-1990.2 1984.7 1990.8 1986.2 1991.6 1985.1 C
-1991.5 1985.9 1992.6 1985.5 1992.5 1986.3 C
-1992 1990.5 1992.6 1995 1992 1999.2 C
-1991.6 1998.9 1991.9 1998.3 1991.8 1997.8 C
-1991.8 1998.5 1991.8 2000 1991.8 2000 C
-1991.9 1999.9 1992 1999.8 1992 1999.7 C
-1993.2 2003.5 1991.9 2007.7 1992.3 2011.5 C
-1991.6 2012 1990.9 2012.2 1990.4 2012.9 C
-1991.3 2011.9 1992.2 2012.1 1992.8 2010.8 C
-[0 1 1 0.23] vc
-f
-S
-n
-1978.4 1968.5 m
-1977 1969.2 1975.8 1968.2 1974.5 1969 C
-1968.3 1973 1961.6 1976 1955.1 1979.1 C
-1962 1975.9 1968.8 1972.5 1975.5 1968.8 C
-1976.5 1968.8 1977.6 1968.8 1978.6 1968.8 C
-1981.7 1972.1 1984.8 1975.7 1988 1978.8 C
-1990.9 1981.9 1996.8 1984.6 1995.2 1990.6 C
-1995.3 1988.6 1994.9 1986.9 1994.7 1985.1 C
-1989.5 1979.1 1983.3 1974.3 1978.4 1968.5 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1978.4 1968.3 m
-1977.9 1968.7 1977.1 1968.5 1976.4 1968.5 C
-1977.3 1968.8 1978.1 1967.9 1978.8 1968.5 C
-1984 1974.3 1990.1 1979.5 1995.2 1985.6 C
-1995.1 1988.4 1995.3 1985.6 1994.9 1984.8 C
-1989.5 1979.4 1983.9 1973.8 1978.4 1968.3 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1978.6 1968 m
-1977.9 1968 1977.4 1968.6 1978.4 1968 C
-1983.9 1973.9 1990.1 1979.1 1995.2 1985.1 C
-1990.2 1979 1983.8 1974.1 1978.6 1968 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1991.1 1982.2 m
-1991.2 1982.9 1991.6 1984.2 1993 1984.4 C
-1992.6 1983.5 1992.1 1982.5 1991.1 1982.2 C
-[0 0.33 0.33 0.99] vc
-f
-S
-n
-1990.4 2012.7 m
-1991.4 2011.8 1990.2 2010.9 1989.9 2010.3 C
-1987.7 2010.2 1987.4 2007.6 1985.6 2007.2 C
-1985.1 2006.2 1984.6 2004.5 1984.1 2004.3 C
-1981.7 2004.5 1982.3 2001.2 1979.8 2000.9 C
-1978.8 1999.6 1978.8 1999.1 1977.6 1998.8 C
-1976.1 1997.4 1976.7 1995 1975.2 1994 C
-1975.8 1994 1975 1994 1975 1993.7 C
-1975.7 1993.2 1975.6 1991.8 1976 1991.3 C
-1975.9 1985.7 1976.1 1979.7 1975.7 1974.5 C
-1976.2 1973.3 1976.9 1971.8 1976.2 1971.4 C
-1973.9 1974.3 1972.2 1973.6 1969.5 1975 C
-1967.9 1977.5 1963.8 1977.1 1961.8 1980 C
-1959 1980 1957.6 1983 1954.8 1982.9 C
-1953.8 1984.2 1954.8 1985.7 1955.1 1987.2 C
-1956.2 1989.5 1959.7 1990.1 1959.9 1991.8 C
-1965.9 1998 1971.8 2005.2 1978.1 2011.7 C
-1979.5 2012 1980.9 2012.7 1980.3 2014.6 C
-1980.5 2015.6 1979.4 2016 1979.8 2017 C
-1983 2015.6 1986.8 2014.1 1990.4 2012.7 C
-[0 0.5 0.5 0.2] vc
-f
-S
-n
-1988.7 1979.6 m
-1988.2 1979.9 1988.6 1980.6 1988.9 1981 C
-1991.4 1982.2 1989.6 1979.9 1988.7 1979.6 C
-[0 0.33 0.33 0.99] vc
-f
-S
-n
-1987.2 1978.1 m
-1985 1977.5 1984.6 1974.3 1982.2 1973.6 C
-1982.7 1974.5 1982.8 1975.8 1984.8 1976 C
-1985.7 1976.9 1985 1978.4 1987.2 1978.1 C
-f
-S
-n
-1975.5 2084 m
-1975.5 2082 1975.3 2080 1975.7 2078.2 C
-1978.8 2079 1980.9 2085.5 1984.8 2083.5 C
-1993 2078.7 2001.6 2075 2010 2070.8 C
-2010.1 2064 2009.9 2057.2 2010.3 2050.6 C
-2014.8 2046.2 2020.9 2045.7 2025.6 2042 C
-2026.1 2035.1 2025.8 2028 2025.9 2021.1 C
-2025.8 2027.8 2026.1 2034.6 2025.6 2041.2 C
-2022.2 2044.9 2017.6 2046.8 2012.9 2048 C
-2012.5 2049.5 2010.4 2049.4 2009.8 2051.1 C
-2009.9 2057.6 2009.6 2064.2 2010 2070.5 C
-2001.2 2075.4 1992 2079.1 1983.2 2084 C
-1980.3 2082.3 1977.8 2079.2 1975.2 2077.5 C
-1974.9 2079.9 1977.2 2084.6 1973.3 2085.2 C
-1964.7 2088.6 1956.8 2093.7 1948.1 2097.2 C
-1949 2097.3 1949.6 2096.9 1950.3 2096.7 C
-1958.4 2091.9 1967.1 2088.2 1975.5 2084 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-vmrs
-1948.6 2094.5 m
-1950.2 2093.7 1951.8 2092.9 1953.4 2092.1 C
-1951.8 2092.9 1950.2 2093.7 1948.6 2094.5 C
-[0 0.87 0.91 0.83] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1971.6 2082.3 m
-1971.6 2081.9 1970.7 2081.1 1970.9 2081.3 C
-1970.7 2081.6 1970.6 2081.6 1970.4 2081.3 C
-1970.8 2080.1 1968.7 2081.7 1968.3 2080.8 C
-1966.6 2080.9 1966.7 2078 1964.2 2078.2 C
-1964.8 2075 1960.1 2075.8 1960.1 2072.9 C
-1958 2072.3 1957.5 2069.3 1955.3 2069.3 C
-1953.9 2070.9 1948.8 2067.8 1950 2072 C
-1949 2074 1943.2 2070.6 1944 2074.8 C
-1942.2 2076.6 1937.6 2073.9 1938 2078.2 C
-1936.7 2078.6 1935 2078.6 1933.7 2078.2 C
-1933.5 2080 1936.8 2080.7 1937.3 2082.8 C
-1939.9 2083.5 1940.6 2086.4 1942.6 2088 C
-1945.2 2089.2 1946 2091.3 1948.4 2093.6 C
-1956 2089.5 1963.9 2086.1 1971.6 2082.3 C
-[0 0.01 1 0] vc
-f
-S
-n
-1958.2 2089.7 m
-1956.4 2090 1955.6 2091.3 1953.9 2091.9 C
-1955.6 2091.9 1956.5 2089.7 1958.2 2089.7 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1929.9 2080.4 m
-1929.5 2077.3 1929.7 2073.9 1929.6 2070.8 C
-1929.8 2074.1 1929.2 2077.8 1930.1 2080.8 C
-1935.8 2085.9 1941.4 2091.3 1946.9 2096.9 C
-1941.2 2091 1935.7 2086 1929.9 2080.4 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1930.1 2080.4 m
-1935.8 2086 1941.5 2090.7 1946.9 2096.7 C
-1941.5 2090.9 1935.7 2085.8 1930.1 2080.4 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1940.9 2087.1 m
-1941.7 2088 1944.8 2090.6 1943.6 2089.2 C
-1942.5 2089 1941.6 2087.7 1940.9 2087.1 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1972.8 2082.8 m
-1973 2075.3 1972.4 2066.9 1973.3 2059.5 C
-1972.5 2058.9 1972.8 2057.3 1973.1 2056.4 C
-1974.8 2055.2 1973.4 2055.5 1972.4 2055.4 C
-1970.1 2053.2 1967.9 2050.9 1965.6 2048.7 C
-1960.9 2049.9 1956.9 2052.7 1952.4 2054.7 C
-1949.3 2052.5 1946.3 2049.5 1943.6 2046.8 C
-1939.9 2047.7 1936.8 2050.1 1933.5 2051.8 C
-1930.9 2054.9 1933.5 2056.2 1932.3 2059.7 C
-1933.2 2059.7 1932.2 2060.5 1932.5 2060.2 C
-1933.2 2062.5 1931.6 2064.6 1932.5 2067.4 C
-1932.9 2069.7 1932.7 2072.2 1932.8 2074.6 C
-1933.6 2070.6 1932.2 2066.3 1933 2062.6 C
-1934.4 2058.2 1929.8 2053.5 1935.2 2051.1 C
-1937.7 2049.7 1940.2 2048 1942.8 2046.8 C
-1945.9 2049.2 1948.8 2052 1951.7 2054.7 C
-1952.7 2054.7 1953.6 2054.6 1954.4 2054.2 C
-1958.1 2052.5 1961.7 2049.3 1965.9 2049.2 C
-1968.2 2052.8 1975.2 2055 1972.6 2060.9 C
-1973.3 2062.4 1972.2 2065.2 1972.6 2067.6 C
-1972.7 2072.6 1972.4 2077.7 1972.8 2082.5 C
-1968.1 2084.9 1963.5 2087.5 1958.7 2089.5 C
-1963.5 2087.4 1968.2 2085 1972.8 2082.8 C
-f
-S
-n
-1935.2 2081.1 m
-1936.8 2083.4 1938.6 2084.6 1940.4 2086.6 C
-1938.8 2084.4 1936.7 2083.4 1935.2 2081.1 C
-f
-S
-n
-1983.2 2081.3 m
-1984.8 2080.5 1986.3 2079.7 1988 2078.9 C
-1986.3 2079.7 1984.8 2080.5 1983.2 2081.3 C
-f
-S
-n
-2006.2 2069.1 m
-2006.2 2068.7 2005.2 2067.9 2005.5 2068.1 C
-2005.3 2068.4 2005.2 2068.4 2005 2068.1 C
-2005.4 2066.9 2003.3 2068.5 2002.8 2067.6 C
-2001.2 2067.7 2001.2 2064.8 1998.8 2065 C
-1999.4 2061.8 1994.7 2062.6 1994.7 2059.7 C
-1992.4 2059.5 1992.4 2055.8 1990.1 2056.8 C
-1985.9 2059.5 1981.1 2061 1976.9 2063.8 C
-1977.2 2067.6 1974.9 2074.2 1978.8 2075.8 C
-1979.6 2077.8 1981.7 2078.4 1982.9 2080.4 C
-1990.6 2076.3 1998.5 2072.9 2006.2 2069.1 C
-[0 0.01 1 0] vc
-f
-S
-n
-vmrs
-1992.8 2076.5 m
-1991 2076.8 1990.2 2078.1 1988.4 2078.7 C
-1990.2 2078.7 1991 2076.5 1992.8 2076.5 C
-[0 0.87 0.91 0.83] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1975.5 2073.4 m
-1976.1 2069.7 1973.9 2064.6 1977.4 2062.4 C
-1973.9 2064.5 1976.1 2069.9 1975.5 2073.6 C
-1976 2074.8 1979.3 2077.4 1978.1 2076 C
-1977 2075.7 1975.8 2074.5 1975.5 2073.4 C
-f
-S
-n
-2007.4 2069.6 m
-2007.6 2062.1 2007 2053.7 2007.9 2046.3 C
-2007.1 2045.7 2007.3 2044.1 2007.6 2043.2 C
-2009.4 2042 2007.9 2042.3 2006.9 2042.2 C
-2002.2 2037.4 1996.7 2032.4 1992.5 2027.3 C
-1992 2027.3 1991.6 2027.3 1991.1 2027.3 C
-1991.4 2035.6 1991.4 2045.6 1991.1 2054.4 C
-1990.5 2055.5 1988.4 2056.6 1990.6 2055.4 C
-1991.6 2055.4 1991.6 2054.1 1991.6 2053.2 C
-1990.8 2044.7 1991.9 2035.4 1991.6 2027.6 C
-1991.8 2027.6 1992 2027.6 1992.3 2027.6 C
-1997 2032.8 2002.5 2037.7 2007.2 2042.9 C
-2007.3 2044.8 2006.7 2047.4 2007.6 2048.4 C
-2006.9 2055.1 2007.1 2062.5 2007.4 2069.3 C
-2002.7 2071.7 1998.1 2074.3 1993.2 2076.3 C
-1998 2074.2 2002.7 2071.8 2007.4 2069.6 C
-f
-S
-n
-2006.7 2069.1 m
-2006.3 2068.6 2005.9 2067.7 2005.7 2066.9 C
-2005.7 2059.7 2005.9 2051.4 2005.5 2045.1 C
-2004.9 2045.3 2004.7 2044.5 2004.3 2045.3 C
-2005.1 2045.3 2004.2 2045.8 2004.8 2046 C
-2004.8 2052.2 2004.8 2059.2 2004.8 2064.5 C
-2005.7 2065.7 2005.1 2065.7 2005 2066.7 C
-2003.8 2067 2002.7 2067.2 2001.9 2066.4 C
-2001.3 2064.6 1998 2063.1 1998 2061.9 C
-1996.1 2062.3 1996.6 2058.3 1994.2 2058.8 C
-1992.6 2057.7 1992.7 2054.8 1989.9 2056.6 C
-1985.6 2059.3 1980.9 2060.8 1976.7 2063.6 C
-1976 2066.9 1976 2071.2 1976.7 2074.6 C
-1977.6 2070.8 1973.1 2062.1 1980.5 2061.2 C
-1984.3 2060.3 1987.5 2058.2 1990.8 2056.4 C
-1991.7 2056.8 1992.9 2057.2 1993.5 2059.2 C
-1994.3 2058.6 1994.4 2060.6 1994.7 2059.2 C
-1995.3 2062.7 1999.2 2061.4 1998.8 2064.8 C
-2001.8 2065.4 2002.5 2068.4 2005.2 2067.4 C
-2004.9 2067.9 2006 2068 2006.4 2069.1 C
-2001.8 2071.1 1997.4 2073.9 1992.8 2075.8 C
-1997.5 2073.8 2002 2071.2 2006.7 2069.1 C
-[0 0.2 1 0] vc
-f
-S
-n
-1988.7 2056.6 m
-1985.1 2058.7 1981.1 2060.1 1977.6 2061.9 C
-1981.3 2060.5 1985.6 2058.1 1988.7 2056.6 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1977.9 2059.5 m
-1975.7 2064.5 1973.7 2054.7 1975.2 2060.9 C
-1976 2060.6 1977.6 2059.7 1977.9 2059.5 C
-f
-S
-n
-1989.6 2051.3 m
-1990.1 2042.3 1989.8 2036.6 1989.9 2028 C
-1989.8 2027 1990.8 2028.3 1990.1 2027.3 C
-1988.9 2026.7 1986.7 2026.9 1986.8 2024.7 C
-1987.4 2023 1985.9 2024.6 1985.1 2023.7 C
-1984.1 2021.4 1982.5 2020.5 1980.3 2020.6 C
-1979.9 2020.8 1979.5 2021.1 1979.3 2021.6 C
-1979.7 2025.8 1978.4 2033 1979.6 2038.1 C
-1983.7 2042.9 1968.8 2044.6 1978.8 2042.7 C
-1979.3 2042.3 1979.6 2041.9 1980 2041.5 C
-1980 2034.8 1980 2027 1980 2021.6 C
-1981.3 2020.5 1981.7 2021.5 1982.9 2021.8 C
-1983.6 2024.7 1986.1 2023.8 1986.8 2026.4 C
-1987.1 2027.7 1988.6 2027.1 1989.2 2028.3 C
-1989.1 2036.7 1989.3 2044.8 1988.9 2053.7 C
-1987.2 2054.9 1986.2 2056.8 1983.9 2057.1 C
-1986.3 2055.9 1990.9 2055 1989.6 2051.3 C
-f
-S
-n
-1971.6 2078.9 m
-1971.4 2070.5 1972.1 2062.2 1971.6 2055.9 C
-1969.9 2053.7 1967.6 2051.7 1965.6 2049.6 C
-1961.4 2050.4 1957.6 2053.6 1953.4 2055.2 C
-1949.8 2055.6 1948.2 2051.2 1945.5 2049.6 C
-1945.1 2048.8 1944.5 2047.9 1943.6 2047.5 C
-1940.1 2047.8 1937.3 2051 1934 2052.3 C
-1933.7 2052.6 1933.7 2053 1933.2 2053.2 C
-1933.7 2060.8 1933.4 2067.2 1933.5 2074.6 C
-1933.8 2068.1 1934 2060.9 1933.2 2054 C
-1935.3 2050.9 1939.3 2049.6 1942.4 2047.5 C
-1942.8 2047.5 1943.4 2047.4 1943.8 2047.7 C
-1947.1 2050.2 1950.3 2057.9 1955.3 2054.4 C
-1955.4 2054.4 1955.5 2054.3 1955.6 2054.2 C
-1955.9 2057.6 1956.1 2061.8 1955.3 2064.8 C
-1955.4 2064.3 1955.1 2063.8 1955.6 2063.6 C
-1956 2066.6 1955.3 2068.7 1958.7 2069.8 C
-1959.2 2071.7 1961.4 2071.7 1962 2074.1 C
-1964.4 2074.2 1964 2077.7 1967.3 2078.4 C
-1967 2079.7 1968.1 2079.9 1969 2080.1 C
-1971.1 2079.9 1970 2079.2 1970.4 2078 C
-1969.5 2077.2 1970.3 2075.9 1969.7 2075.1 C
-1970.1 2069.8 1970.1 2063.6 1969.7 2058.8 C
-1969.2 2058.5 1970 2058.1 1970.2 2057.8 C
-1970.4 2058.3 1971.2 2057.7 1971.4 2058.3 C
-1971.5 2065.3 1971.2 2073.6 1971.6 2081.1 C
-1974.1 2081.4 1969.8 2084.3 1972.4 2082.5 C
-1971.9 2081.4 1971.6 2080.2 1971.6 2078.9 C
-[0 0.4 1 0] vc
-f
-S
-n
-1952.4 2052 m
-1954.1 2051.3 1955.6 2050.4 1957.2 2049.6 C
-1955.6 2050.4 1954.1 2051.3 1952.4 2052 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1975.5 2039.8 m
-1975.5 2039.4 1974.5 2038.7 1974.8 2038.8 C
-1974.6 2039.1 1974.5 2039.1 1974.3 2038.8 C
-1974.6 2037.6 1972.5 2039.3 1972.1 2038.4 C
-1970.4 2038.4 1970.5 2035.5 1968 2035.7 C
-1968.6 2032.5 1964 2033.3 1964 2030.4 C
-1961.9 2029.8 1961.4 2026.8 1959.2 2026.8 C
-1957.7 2028.5 1952.6 2025.3 1953.9 2029.5 C
-1952.9 2031.5 1947 2028.2 1947.9 2032.4 C
-1946 2034.2 1941.5 2031.5 1941.9 2035.7 C
-1940.6 2036.1 1938.9 2036.1 1937.6 2035.7 C
-1937.3 2037.5 1940.7 2038.2 1941.2 2040.3 C
-1943.7 2041.1 1944.4 2043.9 1946.4 2045.6 C
-1949.1 2046.7 1949.9 2048.8 1952.2 2051.1 C
-1959.9 2047.1 1967.7 2043.6 1975.5 2039.8 C
-[0 0.01 1 0] vc
-f
-S
-n
-vmrs
-1962 2047.2 m
-1960.2 2047.5 1959.5 2048.9 1957.7 2049.4 C
-1959.5 2049.5 1960.3 2047.2 1962 2047.2 C
-[0 0.87 0.91 0.83] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-2012.4 2046.3 m
-2010.3 2051.3 2008.3 2041.5 2009.8 2047.7 C
-2010.5 2047.4 2012.2 2046.5 2012.4 2046.3 C
-f
-S
-n
-1944.8 2044.6 m
-1945.5 2045.6 1948.6 2048.1 1947.4 2046.8 C
-1946.3 2046.5 1945.5 2045.2 1944.8 2044.6 C
-f
-S
-n
-1987.2 2054.9 m
-1983.7 2057.3 1979.6 2058 1976 2060.2 C
-1974.7 2058.2 1977.2 2055.8 1974.3 2054.9 C
-1973.1 2052 1970.4 2050.2 1968 2048 C
-1968 2047.7 1968 2047.4 1968.3 2047.2 C
-1969.5 2046.1 1983 2040.8 1972.4 2044.8 C
-1971.2 2046.6 1967.9 2046 1968 2048.2 C
-1970.5 2050.7 1973.8 2052.6 1974.3 2055.6 C
-1975.1 2055 1975.7 2056.7 1975.7 2057.1 C
-1975.7 2058.2 1974.8 2059.3 1975.5 2060.4 C
-1979.3 2058.2 1983.9 2057.7 1987.2 2054.9 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1967.8 2047.5 m
-1968.5 2047 1969.1 2046.5 1969.7 2046 C
-1969.1 2046.5 1968.5 2047 1967.8 2047.5 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1976.7 2040.3 m
-1976.9 2032.8 1976.3 2024.4 1977.2 2017 C
-1976.4 2016.5 1976.6 2014.8 1976.9 2013.9 C
-1978.7 2012.7 1977.2 2013 1976.2 2012.9 C
-1971.5 2008.1 1965.9 2003.1 1961.8 1998 C
-1960.9 1998 1960.1 1998 1959.2 1998 C
-1951.5 2001.1 1944.3 2005.5 1937.1 2009.6 C
-1935 2012.9 1937 2013.6 1936.1 2017.2 C
-1937.1 2017.2 1936 2018 1936.4 2017.7 C
-1937 2020.1 1935.5 2022.1 1936.4 2024.9 C
-1936.8 2027.2 1936.5 2029.7 1936.6 2032.1 C
-1937.4 2028.2 1936 2023.8 1936.8 2020.1 C
-1938.3 2015.7 1933.6 2011 1939 2008.6 C
-1945.9 2004.5 1953.1 2000.3 1960.6 1998.3 C
-1960.9 1998.3 1961.3 1998.3 1961.6 1998.3 C
-1966.2 2003.5 1971.8 2008.4 1976.4 2013.6 C
-1976.6 2015.5 1976 2018.1 1976.9 2019.2 C
-1976.1 2025.8 1976.4 2033.2 1976.7 2040 C
-1971.9 2042.4 1967.4 2045 1962.5 2047 C
-1967.3 2044.9 1972 2042.6 1976.7 2040.3 C
-f
-S
-n
-1939 2038.6 m
-1940.6 2040.9 1942.5 2042.1 1944.3 2044.1 C
-1942.7 2041.9 1940.6 2040.9 1939 2038.6 C
-f
-S
-n
-2006.2 2065.7 m
-2006 2057.3 2006.7 2049 2006.2 2042.7 C
-2002.1 2038.4 1997.7 2033.4 1993 2030 C
-1992.9 2029.3 1992.5 2028.6 1992 2028.3 C
-1992.1 2036.6 1991.9 2046.2 1992.3 2054.9 C
-1990.8 2056.2 1989 2056.7 1987.5 2058 C
-1988.7 2057.7 1990.7 2054.4 1993 2056.4 C
-1993.4 2058.8 1996 2058.2 1996.6 2060.9 C
-1999 2061 1998.5 2064.5 2001.9 2065.2 C
-2001.5 2066.5 2002.7 2066.7 2003.6 2066.9 C
-2005.7 2066.7 2004.6 2066 2005 2064.8 C
-2004 2064 2004.8 2062.7 2004.3 2061.9 C
-2004.6 2056.6 2004.6 2050.4 2004.3 2045.6 C
-2003.7 2045.3 2004.6 2044.9 2004.8 2044.6 C
-2005 2045.1 2005.7 2044.5 2006 2045.1 C
-2006 2052.1 2005.8 2060.4 2006.2 2067.9 C
-2008.7 2068.2 2004.4 2071.1 2006.9 2069.3 C
-2006.4 2068.2 2006.2 2067 2006.2 2065.7 C
-[0 0.4 1 0] vc
-f
-S
-n
-2021.8 2041.7 m
-2018.3 2044.1 2014.1 2044.8 2010.5 2047 C
-2009.3 2045 2011.7 2042.6 2008.8 2041.7 C
-2004.3 2035.1 1997.6 2030.9 1993 2024.4 C
-1992.1 2024 1991.5 2024.3 1990.8 2024 C
-1993.2 2023.9 1995.3 2027.1 1996.8 2029 C
-2000.4 2032.6 2004.9 2036.9 2008.4 2040.8 C
-2008.2 2043.1 2011.4 2042.8 2009.8 2045.8 C
-2009.8 2046.3 2009.7 2046.9 2010 2047.2 C
-2013.8 2045 2018.5 2044.5 2021.8 2041.7 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-2001.6 2034 m
-2000.7 2033.1 1999.9 2032.3 1999 2031.4 C
-1999.9 2032.3 2000.7 2033.1 2001.6 2034 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-vmrs
-1989.4 2024.4 m
-1989.5 2025.4 1988.6 2024.3 1988.9 2024.7 C
-1990.5 2025.8 1990.7 2024.2 1992.8 2024.9 C
-1993.8 2025.9 1995 2027.1 1995.9 2028 C
-1994.3 2026 1991.9 2023.4 1989.4 2024.4 C
-[0 0.87 0.91 0.83] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1984.8 2019.9 m
-1984.6 2018.6 1986.3 2017.2 1987.7 2016.8 C
-1987.2 2017.5 1982.9 2017.9 1984.4 2020.6 C
-1984.1 2019.9 1984.9 2020 1984.8 2019.9 C
-f
-S
-n
-1981.7 2017 m
-1979.6 2022 1977.6 2012.3 1979.1 2018.4 C
-1979.8 2018.1 1981.5 2017.2 1981.7 2017 C
-f
-S
-n
-1884.3 2019.2 m
-1884.7 2010.5 1884.5 2000.6 1884.5 1991.8 C
-1886.6 1989.3 1889.9 1988.9 1892.4 1987 C
-1890.8 1988.7 1886 1989.1 1884.3 1992.3 C
-1884.7 2001 1884.5 2011.3 1884.5 2019.9 C
-1891 2025.1 1895.7 2031.5 1902 2036.9 C
-1896.1 2031 1890 2024.9 1884.3 2019.2 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1884 2019.4 m
-1884.5 2010.6 1884.2 2000.4 1884.3 1991.8 C
-1884.8 1990.4 1887.8 1989 1884.8 1990.8 C
-1884.3 1991.3 1884.3 1992 1884 1992.5 C
-1884.5 2001.2 1884.2 2011.1 1884.3 2019.9 C
-1887.9 2023.1 1891.1 2026.4 1894.4 2030 C
-1891.7 2026.1 1887.1 2022.9 1884 2019.4 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1885 2011.7 m
-1885 2006.9 1885 2001.9 1885 1997.1 C
-1885 2001.9 1885 2006.9 1885 2011.7 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1975.5 2036.4 m
-1975.2 2028 1976 2019.7 1975.5 2013.4 C
-1971.1 2008.5 1965.6 2003.6 1961.6 1999 C
-1958.8 1998 1956 2000 1953.6 2001.2 C
-1948.2 2004.7 1941.9 2006.5 1937.1 2010.8 C
-1937.5 2018.3 1937.3 2024.7 1937.3 2032.1 C
-1937.6 2025.6 1937.9 2018.4 1937.1 2011.5 C
-1937.3 2011 1937.6 2010.5 1937.8 2010 C
-1944.6 2005.7 1951.9 2002.3 1959.2 1999 C
-1960.1 1998.5 1960.1 1999.8 1960.4 2000.4 C
-1959.7 2006.9 1959.7 2014.2 1959.4 2021.1 C
-1959 2021.1 1959.2 2021.9 1959.2 2022.3 C
-1959.2 2021.9 1959 2021.3 1959.4 2021.1 C
-1959.8 2024.1 1959.2 2026.2 1962.5 2027.3 C
-1963 2029.2 1965.3 2029.2 1965.9 2031.6 C
-1968.3 2031.8 1967.8 2035.2 1971.2 2036 C
-1970.8 2037.2 1971.9 2037.5 1972.8 2037.6 C
-1974.9 2037.4 1973.9 2036.7 1974.3 2035.5 C
-1973.3 2034.7 1974.1 2033.4 1973.6 2032.6 C
-1973.9 2027.3 1973.9 2021.1 1973.6 2016.3 C
-1973 2016 1973.9 2015.6 1974 2015.3 C
-1974.3 2015.9 1975 2015.3 1975.2 2015.8 C
-1975.3 2022.8 1975.1 2031.2 1975.5 2038.6 C
-1977.9 2039 1973.7 2041.8 1976.2 2040 C
-1975.7 2039 1975.5 2037.8 1975.5 2036.4 C
-[0 0.4 1 0] vc
-f
-S
-n
-1991.1 2012.4 m
-1987.5 2014.8 1983.4 2015.6 1979.8 2017.7 C
-1978.5 2015.7 1981 2013.3 1978.1 2012.4 C
-1973.6 2005.8 1966.8 2001.6 1962.3 1995.2 C
-1961.4 1994.7 1960.8 1995 1960.1 1994.7 C
-1962.5 1994.6 1964.6 1997.8 1966.1 1999.7 C
-1969.7 2003.3 1974.2 2007.6 1977.6 2011.5 C
-1977.5 2013.8 1980.6 2013.5 1979.1 2016.5 C
-1979.1 2017 1979 2017.6 1979.3 2018 C
-1983.1 2015.7 1987.8 2015.2 1991.1 2012.4 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1970.9 2004.8 m
-1970 2003.9 1969.2 2003 1968.3 2002.1 C
-1969.2 2003 1970 2003.9 1970.9 2004.8 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1887.9 1994.9 m
-1888.5 1992.3 1891.4 1992.2 1893.2 1990.8 C
-1898.4 1987.5 1904 1984.8 1909.5 1982.2 C
-1909.7 1982.7 1910.3 1982.1 1910.4 1982.7 C
-1909.5 1990.5 1910.1 1996.4 1910 2004.5 C
-1909.1 2003.4 1909.7 2005.8 1909.5 2006.4 C
-1910.4 2006 1909.7 2008 1910.2 2007.9 C
-1911.3 2010.6 1912.5 2012.6 1915.7 2013.4 C
-1915.8 2013.7 1915.5 2014.4 1916 2014.4 C
-1916.3 2015 1915.4 2016 1915.2 2016 C
-1916.1 2015.5 1916.5 2014.5 1916 2013.6 C
-1913.4 2013.3 1913.1 2010.5 1910.9 2009.8 C
-1910.7 2008.8 1910.4 2007.9 1910.2 2006.9 C
-1911.1 1998.8 1909.4 1990.7 1910.7 1982.4 C
-1910 1982.1 1908.9 1982.1 1908.3 1982.4 C
-1901.9 1986.1 1895 1988.7 1888.8 1993 C
-1888 1993.4 1888.4 1994.3 1887.6 1994.7 C
-1888.1 2001.3 1887.8 2008.6 1887.9 2015.1 C
-1887.3 2017.5 1887.9 2015.4 1888.4 2014.4 C
-1887.8 2008 1888.4 2001.3 1887.9 1994.9 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-vmrs
-1887.9 2018.4 m
-1887.5 2016.9 1888.5 2016 1888.8 2014.8 C
-1890.1 2014.8 1891.1 2016.6 1892.4 2015.3 C
-1892.4 2014.4 1893.8 2012.9 1894.4 2012.4 C
-1895.9 2012.4 1896.6 2013.9 1897.7 2012.7 C
-1898.4 2011.7 1898.6 2010.4 1899.6 2009.8 C
-1901.7 2009.9 1902.9 2010.4 1904 2009.1 C
-1904.3 2007.4 1904 2007.6 1904.9 2007.2 C
-1906.2 2007 1907.6 2006.5 1908.8 2006.7 C
-1910.6 2008.2 1909.8 2011.5 1912.6 2012 C
-1912.4 2013 1913.8 2012.7 1914 2013.2 C
-1911.5 2011.1 1909.1 2007.9 1909.2 2004.3 C
-1909.5 2003.5 1909.9 2004.9 1909.7 2004.3 C
-1909.9 1996.2 1909.3 1990.5 1910.2 1982.7 C
-1909.5 1982.6 1909.5 1982.6 1908.8 1982.7 C
-1903.1 1985.7 1897 1987.9 1891.7 1992 C
-1890.5 1993 1888.2 1992.9 1888.1 1994.9 C
-1888.7 2001.4 1888.1 2008.4 1888.6 2014.8 C
-1888.3 2016 1887.2 2016.9 1887.6 2018.4 C
-1892.3 2023.9 1897.6 2027.9 1902.3 2033.3 C
-1898 2028.2 1892.1 2023.8 1887.9 2018.4 C
-[0.4 0.4 0 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1910.9 1995.2 m
-1910.4 1999.8 1911 2003.3 1910.9 2008.1 C
-1910.9 2003.8 1910.9 1999.2 1910.9 1995.2 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1911.2 2004.3 m
-1911.2 2001.9 1911.2 1999.7 1911.2 1997.3 C
-1911.2 1999.7 1911.2 2001.9 1911.2 2004.3 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1958.7 1995.2 m
-1959 1995.6 1956.2 1995 1956.5 1996.8 C
-1955.8 1997.6 1954.2 1998.5 1953.6 1997.3 C
-1953.6 1990.8 1954.9 1989.6 1953.4 1983.9 C
-1953.4 1983.3 1953.3 1982.1 1954.4 1982 C
-1955.5 1982.6 1956.5 1981.3 1957.5 1981 C
-1956.3 1981.8 1954.7 1982.6 1953.9 1981.5 C
-1951.4 1983 1954.7 1988.8 1952.9 1990.6 C
-1953.8 1990.6 1953.2 1992.7 1953.4 1993.7 C
-1953.8 1994.5 1952.3 1996.1 1953.2 1997.8 C
-1956.3 1999.4 1957.5 1994 1959.9 1995.6 C
-1962 1994.4 1963.7 1997.7 1965.2 1998.8 C
-1963.5 1996.7 1961.2 1994.1 1958.7 1995.2 C
-f
-S
-n
-1945 2000.7 m
-1945.4 1998.7 1945.4 1997.9 1945 1995.9 C
-1944.5 1995.3 1944.2 1992.6 1945.7 1993.2 C
-1946 1992.2 1948.7 1992.5 1948.4 1990.6 C
-1947.5 1990.3 1948.1 1988.7 1947.9 1988.2 C
-1948.9 1987.8 1950.5 1986.8 1950.5 1984.6 C
-1951.5 1980.9 1946.7 1983 1947.2 1979.8 C
-1944.5 1979.9 1945.2 1976.6 1943.1 1976.7 C
-1941.8 1975.7 1942.1 1972.7 1939.2 1973.8 C
-1938.2 1974.6 1939.3 1971.6 1938.3 1970.9 C
-1938.8 1969.2 1933.4 1970.3 1937.3 1970 C
-1939.4 1971.2 1937.2 1973 1937.6 1974.3 C
-1937.2 1976.3 1937.1 1981.2 1937.8 1984.1 C
-1938.8 1982.3 1937.9 1976.6 1938.5 1973.1 C
-1938.9 1975 1938.5 1976.4 1939.7 1977.2 C
-1939.5 1983.5 1938.9 1991.3 1940.2 1997.3 C
-1939.4 1999.1 1938.6 1997.1 1937.8 1997.1 C
-1937.4 1996.7 1937.6 1996.1 1937.6 1995.6 C
-1936.5 1998.5 1940.1 1998.4 1940.9 2000.7 C
-1942.1 2000.4 1943.2 2001.3 1943.1 2002.4 C
-1943.6 2003.1 1941.1 2004.6 1942.8 2003.8 C
-1943.9 2002.5 1942.6 2000.6 1945 2000.7 C
-[0.65 0.65 0 0.42] vc
-f
-S
-n
-1914.5 2006.4 m
-1914.1 2004.9 1915.2 2004 1915.5 2002.8 C
-1916.7 2002.8 1917.8 2004.6 1919.1 2003.3 C
-1919 2002.4 1920.4 2000.9 1921 2000.4 C
-1922.5 2000.4 1923.2 2001.9 1924.4 2000.7 C
-1925 1999.7 1925.3 1998.4 1926.3 1997.8 C
-1928.4 1997.9 1929.5 1998.4 1930.6 1997.1 C
-1930.9 1995.4 1930.7 1995.6 1931.6 1995.2 C
-1932.8 1995 1934.3 1994.5 1935.4 1994.7 C
-1936.1 1995.8 1936.9 1996.2 1936.6 1997.8 C
-1938.9 1999.4 1939.7 2001.3 1942.4 2002.4 C
-1942.4 2002.5 1942.2 2003 1942.6 2002.8 C
-1942.9 2000.4 1939.2 2001.8 1939.2 1999.7 C
-1936.2 1998.6 1937 1995.3 1935.9 1993.5 C
-1937.1 1986.5 1935.2 1977.9 1937.6 1971.2 C
-1937.6 1970.3 1936.6 1971 1936.4 1970.4 C
-1930.2 1973.4 1924 1976 1918.4 1980 C
-1917.2 1981 1914.9 1980.9 1914.8 1982.9 C
-1915.3 1989.4 1914.7 1996.4 1915.2 2002.8 C
-1914.9 2004 1913.9 2004.9 1914.3 2006.4 C
-1919 2011.9 1924.2 2015.9 1928.9 2021.3 C
-1924.6 2016.2 1918.7 2011.8 1914.5 2006.4 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1914.5 1982.9 m
-1915.1 1980.3 1918 1980.2 1919.8 1978.8 C
-1925 1975.5 1930.6 1972.8 1936.1 1970.2 C
-1939.4 1970.6 1936.1 1974.2 1936.6 1976.4 C
-1936.5 1981.9 1936.8 1987.5 1936.4 1992.8 C
-1935.9 1992.8 1936.2 1993.5 1936.1 1994 C
-1937.1 1993.6 1936.2 1995.9 1936.8 1995.9 C
-1937 1998 1939.5 1999.7 1940.4 2000.7 C
-1940.1 1998.6 1935 1997.2 1937.6 1993.7 C
-1938.3 1985.7 1935.9 1976.8 1937.8 1970.7 C
-1936.9 1969.8 1935.4 1970.3 1934.4 1970.7 C
-1928.3 1974.4 1921.4 1976.7 1915.5 1981 C
-1914.6 1981.4 1915.1 1982.3 1914.3 1982.7 C
-1914.7 1989.3 1914.5 1996.6 1914.5 2003.1 C
-1913.9 2005.5 1914.5 2003.4 1915 2002.4 C
-1914.5 1996 1915.1 1989.3 1914.5 1982.9 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1939.2 1994.9 m
-1939.3 1995 1939.4 1995.1 1939.5 1995.2 C
-1939.1 1989 1939.3 1981.6 1939 1976.7 C
-1938.6 1976.3 1938.6 1974.6 1938.5 1973.3 C
-1938.7 1976.1 1938.1 1979.4 1939 1981.7 C
-1937.3 1986 1937.7 1991.6 1938 1996.4 C
-1937.3 1994.3 1939.6 1996.2 1939.2 1994.9 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1938.3 1988.4 m
-1938.5 1990.5 1937.9 1994.1 1938.8 1994.7 C
-1937.9 1992.6 1939 1990.6 1938.3 1988.4 C
-[0 0.87 0.91 0.83] vc
-f
-S
-n
-1938.8 1985.8 m
-1938.5 1985.9 1938.4 1985.7 1938.3 1985.6 C
-1938.4 1986.2 1938 1989.5 1938.8 1987.2 C
-1938.8 1986.8 1938.8 1986.3 1938.8 1985.8 C
-f
-S
-n
-vmrs
-1972.8 2062.1 m
-1971.9 2061 1972.5 2059.4 1972.4 2058 C
-1972.2 2063.8 1971.9 2073.7 1972.4 2081.3 C
-1972.5 2074.9 1971.9 2067.9 1972.8 2062.1 C
-[0 1 1 0.36] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1940.2 2071.7 m
-1941.3 2072 1943.1 2072.3 1944 2071.5 C
-1943.6 2069.9 1945.2 2069.1 1946 2068.8 C
-1950 2071.1 1948.7 2065.9 1951.7 2066.2 C
-1953.5 2063.9 1956.9 2069.4 1955.6 2063.8 C
-1955.5 2064.2 1955.7 2064.8 1955.3 2065 C
-1954.3 2063.7 1956.2 2063.6 1955.6 2062.1 C
-1954.5 2060 1958.3 2050.3 1952.2 2055.6 C
-1949.1 2053.8 1946 2051 1943.8 2048 C
-1940.3 2048 1937.5 2051.3 1934.2 2052.5 C
-1933.1 2054.6 1934.4 2057.3 1934 2060 C
-1934 2065.1 1934 2069.7 1934 2074.6 C
-1934.4 2069 1934.1 2061.5 1934.2 2054.9 C
-1934.6 2054.5 1935.3 2054.7 1935.9 2054.7 C
-1937 2055.3 1935.9 2056.1 1935.9 2056.8 C
-1936.5 2063 1935.6 2070.5 1935.9 2074.6 C
-1936.7 2074.4 1937.3 2075.2 1938 2074.6 C
-1937.9 2073.6 1939.1 2072.1 1940.2 2071.7 C
-[0 0.2 1 0] vc
-f
-S
-n
-1933.2 2074.1 m
-1933.2 2071.5 1933.2 2069 1933.2 2066.4 C
-1933.2 2069 1933.2 2071.5 1933.2 2074.1 C
-[0 1 1 0.36] vc
-f
-S
-n
-2007.4 2048.9 m
-2006.5 2047.8 2007.1 2046.2 2006.9 2044.8 C
-2006.7 2050.6 2006.5 2060.5 2006.9 2068.1 C
-2007.1 2061.7 2006.5 2054.7 2007.4 2048.9 C
-f
-S
-n
-1927.2 2062.4 m
-1925.8 2060.1 1928.1 2058.2 1927 2056.4 C
-1927.3 2055.5 1926.5 2053.5 1926.8 2051.8 C
-1926.8 2052.8 1926 2052.5 1925.3 2052.5 C
-1924.1 2052.8 1925 2050.5 1924.4 2050.1 C
-1925.3 2050.2 1925.4 2048.8 1926.3 2049.4 C
-1926.5 2052.3 1928.4 2047.2 1928.4 2051.1 C
-1928.9 2050.5 1929 2051.4 1928.9 2051.8 C
-1928.9 2052 1928.9 2052.3 1928.9 2052.5 C
-1929.4 2051.4 1928.9 2049 1930.1 2048.2 C
-1928.9 2047.1 1930.5 2047.1 1930.4 2046.5 C
-1931.9 2046.2 1933.1 2046.1 1934.7 2046.5 C
-1934.6 2046.9 1935.2 2047.9 1934.4 2048.4 C
-1936.9 2048.1 1933.6 2043.8 1935.9 2043.9 C
-1935.7 2043.9 1934.8 2041.3 1933.2 2041.7 C
-1932.5 2041.6 1932.4 2039.6 1932.3 2041 C
-1930.8 2042.6 1929 2040.6 1927.7 2042 C
-1927.5 2041.4 1927.1 2040.9 1927.2 2040.3 C
-1927.8 2040.6 1927.4 2039.1 1928.2 2038.6 C
-1929.4 2038 1930.5 2038.8 1931.3 2037.9 C
-1931.7 2039 1932.5 2038.6 1931.8 2037.6 C
-1930.9 2037 1928.7 2037.8 1928.2 2037.9 C
-1926.7 2037.8 1928 2039 1927 2038.8 C
-1927.4 2040.4 1925.6 2040.8 1925.1 2041 C
-1924.3 2040.4 1923.2 2040.5 1922.2 2040.5 C
-1921.4 2041.7 1921 2043.9 1919.3 2043.9 C
-1918.8 2043.4 1917.2 2043.3 1916.4 2043.4 C
-1915.9 2044.4 1915.7 2046 1914.3 2046.5 C
-1913.1 2046.6 1912 2044.5 1911.4 2046.3 C
-1912.8 2046.5 1913.8 2047.4 1915.7 2047 C
-1916.9 2047.7 1915.6 2048.8 1916 2049.4 C
-1915.4 2049.3 1913.9 2050.3 1913.3 2051.1 C
-1913.9 2054.1 1916 2050.2 1916.7 2053 C
-1916.9 2053.8 1915.5 2054.1 1916.7 2054.4 C
-1917 2054.7 1920.2 2054.3 1919.3 2056.6 C
-1918.8 2056.1 1920.2 2058.6 1920.3 2057.6 C
-1921.2 2057.9 1922.1 2057.5 1922.4 2059 C
-1922.3 2059.1 1922.2 2059.3 1922 2059.2 C
-1922.1 2059.7 1922.4 2060.3 1922.9 2060.7 C
-1923.2 2060.1 1923.8 2060.4 1924.6 2060.7 C
-1925.9 2062.6 1923.2 2062 1925.6 2063.6 C
-1926.1 2063.1 1927.3 2062.5 1927.2 2062.4 C
-[0.21 0.21 0 0] vc
-f
-S
-n
-1933.2 2063.3 m
-1933.2 2060.7 1933.2 2058.2 1933.2 2055.6 C
-1933.2 2058.2 1933.2 2060.7 1933.2 2063.3 C
-[0 1 1 0.36] vc
-f
-S
-n
-1965.2 2049.2 m
-1967.1 2050.1 1969.9 2053.7 1972.1 2056.4 C
-1970.5 2054 1967.6 2051.3 1965.2 2049.2 C
-f
-S
-n
-1991.8 2034.8 m
-1991.7 2041.5 1992 2048.5 1991.6 2055.2 C
-1990.5 2056.4 1991.9 2054.9 1991.8 2054.4 C
-1991.8 2047.9 1991.8 2041.3 1991.8 2034.8 C
-f
-S
-n
-1988.9 2053.2 m
-1988.9 2044.3 1988.9 2036.6 1988.9 2028.3 C
-1985.7 2028.2 1987.2 2023.5 1983.9 2024.2 C
-1983.9 2022.4 1982 2021.6 1981 2021.3 C
-1980.6 2021.1 1980.6 2021.7 1980.3 2021.6 C
-1980.3 2027 1980.3 2034.8 1980.3 2041.5 C
-1979.3 2043.2 1977.6 2043 1976.2 2043.6 C
-1977.1 2043.8 1978.5 2043.2 1978.8 2044.1 C
-1978.5 2045.3 1979.9 2045.3 1980.3 2045.8 C
-1980.5 2046.8 1980.7 2046.2 1981.5 2046.5 C
-1982.4 2047.1 1982 2048.6 1982.7 2049.4 C
-1984.2 2049.6 1984.6 2052.2 1986.8 2051.6 C
-1987.1 2048.6 1985.1 2042.7 1986.5 2040.5 C
-1986.3 2036.7 1986.9 2031.7 1986 2029.2 C
-1986.3 2027.1 1986.9 2028.6 1987.7 2027.6 C
-1987.7 2028.3 1988.7 2028 1988.7 2028.8 C
-1988.1 2033 1988.7 2037.5 1988.2 2041.7 C
-1987.8 2041.4 1988 2040.8 1988 2040.3 C
-1988 2041 1988 2042.4 1988 2042.4 C
-1988 2042.4 1988.1 2042.3 1988.2 2042.2 C
-1989.3 2046 1988 2050.2 1988.4 2054 C
-1987.8 2054.4 1987.1 2054.7 1986.5 2055.4 C
-1987.4 2054.4 1988.4 2054.6 1988.9 2053.2 C
-[0 1 1 0.23] vc
-f
-S
-n
-1950.8 2054.4 m
-1949.7 2053.4 1948.7 2052.3 1947.6 2051.3 C
-1948.7 2052.3 1949.7 2053.4 1950.8 2054.4 C
-[0 1 1 0.36] vc
-f
-S
-n
-vmrs
-2006.7 2043.2 m
-2004.5 2040.8 2002.4 2038.4 2000.2 2036 C
-2002.4 2038.4 2004.5 2040.8 2006.7 2043.2 C
-[0 1 1 0.36] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1976.7 2019.6 m
-1975.8 2018.6 1976.4 2016.9 1976.2 2015.6 C
-1976 2021.3 1975.8 2031.2 1976.2 2038.8 C
-1976.4 2032.4 1975.8 2025.5 1976.7 2019.6 C
-f
-S
-n
-1988.4 2053.5 m
-1988.6 2049.2 1988.1 2042.8 1988 2040 C
-1988.4 2040.4 1988.1 2041 1988.2 2041.5 C
-1988.3 2037.2 1988 2032.7 1988.4 2028.5 C
-1987.6 2027.1 1987.2 2028.6 1986.8 2028 C
-1985.9 2028.5 1986.5 2029.7 1986.3 2030.4 C
-1986.9 2029.8 1986.6 2031 1987 2031.2 C
-1987.4 2039.6 1985 2043 1987.2 2050.4 C
-1987.2 2051.6 1985.9 2052.3 1984.6 2051.3 C
-1981.9 2049.7 1982.9 2047 1980.3 2046.5 C
-1980.3 2045.2 1978.1 2046.2 1978.6 2043.9 C
-1975.6 2043.3 1979.3 2045.6 1979.6 2046.5 C
-1980.8 2046.6 1981.5 2048.5 1982.2 2049.9 C
-1983.7 2050.8 1984.8 2052.8 1986.5 2053 C
-1986.7 2053.5 1987.5 2054.1 1987 2054.7 C
-1987.4 2053.9 1988.3 2054.3 1988.4 2053.5 C
-[0 1 1 0.23] vc
-f
-S
-n
-1988 2038.1 m
-1988 2036.7 1988 2035.4 1988 2034 C
-1988 2035.4 1988 2036.7 1988 2038.1 C
-[0 1 1 0.36] vc
-f
-S
-n
-1999.7 2035.7 m
-1997.6 2033.5 1995.4 2031.2 1993.2 2029 C
-1995.4 2031.2 1997.6 2033.5 1999.7 2035.7 C
-f
-S
-n
-1944 2029.2 m
-1945.2 2029.5 1946.9 2029.8 1947.9 2029 C
-1947.4 2027.4 1949 2026.7 1949.8 2026.4 C
-1953.9 2028.6 1952.6 2023.4 1955.6 2023.7 C
-1957.4 2021.4 1960.7 2027 1959.4 2021.3 C
-1959.3 2021.7 1959.6 2022.3 1959.2 2022.5 C
-1958.1 2021.2 1960.1 2021.1 1959.4 2019.6 C
-1959.1 2012.7 1959.9 2005.1 1959.6 1999.2 C
-1955.3 2000.1 1951.3 2003.1 1947.2 2005 C
-1943.9 2006 1941.2 2008.7 1938 2010 C
-1936.9 2012.1 1938.2 2014.8 1937.8 2017.5 C
-1937.8 2022.6 1937.8 2027.3 1937.8 2032.1 C
-1938.2 2026.5 1938 2019 1938 2012.4 C
-1938.5 2012 1939.2 2012.3 1939.7 2012.2 C
-1940.8 2012.8 1939.7 2013.6 1939.7 2014.4 C
-1940.4 2020.5 1939.4 2028 1939.7 2032.1 C
-1940.6 2031.9 1941.2 2032.7 1941.9 2032.1 C
-1941.7 2031.2 1943 2029.7 1944 2029.2 C
-[0 0.2 1 0] vc
-f
-S
-n
-1937.1 2031.6 m
-1937.1 2029.1 1937.1 2026.5 1937.1 2024 C
-1937.1 2026.5 1937.1 2029.1 1937.1 2031.6 C
-[0 1 1 0.36] vc
-f
-S
-n
-1991.8 2028 m
-1992.5 2027.8 1993.2 2029.9 1994 2030.2 C
-1992.9 2029.6 1993.1 2028.1 1991.8 2028 C
-[0 1 1 0.23] vc
-f
-S
-n
-1991.8 2027.8 m
-1992.4 2027.6 1992.6 2028.3 1993 2028.5 C
-1992.6 2028.2 1992.2 2027.6 1991.6 2027.8 C
-1991.6 2028.5 1991.6 2029.1 1991.6 2029.7 C
-1991.6 2029.1 1991.4 2028.3 1991.8 2027.8 C
-[0 1 1 0.36] vc
-f
-S
-n
-1985.8 2025.4 m
-1985.3 2025.2 1984.8 2024.7 1984.1 2024.9 C
-1983.3 2025.3 1983.6 2027.3 1983.9 2027.6 C
-1985 2028 1986.9 2026.9 1985.8 2025.4 C
-[0 1 1 0.23] vc
-f
-S
-n
-vmrs
-1993.5 2024.4 m
-1992.4 2023.7 1991.3 2022.9 1990.1 2023.2 C
-1990.7 2023.7 1989.8 2023.8 1989.4 2023.7 C
-1989.1 2023.7 1988.6 2023.9 1988.4 2023.5 C
-1988.5 2023.2 1988.3 2022.7 1988.7 2022.5 C
-1989 2022.6 1988.9 2023 1988.9 2023.2 C
-1989.1 2022.8 1990.4 2022.3 1990.6 2021.3 C
-1990.4 2021.8 1990 2021.3 1990.1 2021.1 C
-1990.1 2020.9 1990.1 2020.1 1990.1 2020.6 C
-1989.9 2021.1 1989.5 2020.6 1989.6 2020.4 C
-1989.6 2019.8 1988.7 2019.6 1988.2 2019.2 C
-1987.5 2018.7 1987.7 2020.2 1987 2019.4 C
-1987.5 2020.4 1986 2021.1 1987.5 2021.8 C
-1986.8 2023.1 1986.6 2021.1 1986 2021.1 C
-1986.1 2020.1 1985.9 2019 1986.3 2018.2 C
-1986.7 2018.4 1986.5 2019 1986.5 2019.4 C
-1986.5 2018.7 1986.4 2017.8 1987.2 2017.7 C
-1986.5 2017.2 1985.5 2019.3 1985.3 2020.4 C
-1986.2 2022 1987.3 2023.5 1989.2 2024.2 C
-1990.8 2024.3 1991.6 2022.9 1993.2 2024.4 C
-1993.8 2025.4 1995 2026.6 1995.9 2027.1 C
-1995 2026.5 1994.1 2025.5 1993.5 2024.4 C
-[0 1 1 0.36] vc
-f
-0.4 w
-2 J
-2 M
-[0 0.5 0.5 0.2] vc
-S
-n
-2023 2040.3 m
-2023.2 2036 2022.7 2029.6 2022.5 2026.8 C
-2022.9 2027.2 2022.7 2027.8 2022.8 2028.3 C
-2022.8 2024 2022.6 2019.5 2023 2015.3 C
-2022.2 2013.9 2021.7 2015.4 2021.3 2014.8 C
-2020.4 2015.3 2021 2016.5 2020.8 2017.2 C
-2021.4 2016.6 2021.1 2017.8 2021.6 2018 C
-2022 2026.4 2019.6 2029.8 2021.8 2037.2 C
-2021.7 2038.4 2020.5 2039.1 2019.2 2038.1 C
-2016.5 2036.5 2017.5 2033.8 2014.8 2033.3 C
-2014.9 2032 2012.6 2033 2013.2 2030.7 C
-2011.9 2030.8 2011.2 2030.1 2010.8 2029.2 C
-2010.8 2029.1 2010.8 2028.2 2010.8 2028.8 C
-2010 2028.8 2010.4 2026.5 2008.6 2027.3 C
-2007.9 2026.6 2007.3 2025.9 2007.9 2027.1 C
-2009.7 2028 2010 2030.1 2012.2 2030.9 C
-2012.9 2032.1 2013.7 2033.6 2015.1 2033.6 C
-2015.7 2035.1 2016.9 2036.7 2018.4 2038.4 C
-2019.8 2039.3 2022 2039.4 2021.6 2041.5 C
-2021.9 2040.7 2022.9 2041.1 2023 2040.3 C
-[0 1 1 0.23] vc
-f
-S
-n
-2022.5 2024.9 m
-2022.5 2023.5 2022.5 2022.2 2022.5 2020.8 C
-2022.5 2022.2 2022.5 2023.5 2022.5 2024.9 C
-[0 1 1 0.36] vc
-f
-S
-n
-1983.2 2022.8 m
-1982.4 2022.5 1982.1 2021.6 1981.2 2022.3 C
-1981.1 2022.9 1980.5 2024 1981 2024.2 C
-1981.8 2024.6 1982.9 2024.4 1983.2 2022.8 C
-[0 1 1 0.23] vc
-f
-S
-n
-1931.1 2019.9 m
-1929.6 2017.7 1932 2015.7 1930.8 2013.9 C
-1931.1 2013 1930.3 2011 1930.6 2009.3 C
-1930.6 2010.3 1929.8 2010 1929.2 2010 C
-1928 2010.3 1928.8 2008.1 1928.2 2007.6 C
-1929.1 2007.8 1929.3 2006.3 1930.1 2006.9 C
-1930.3 2009.8 1932.2 2004.8 1932.3 2008.6 C
-1932.7 2008 1932.8 2009 1932.8 2009.3 C
-1932.8 2009.6 1932.8 2009.8 1932.8 2010 C
-1933.2 2009 1932.7 2006.6 1934 2005.7 C
-1932.7 2004.6 1934.3 2004.6 1934.2 2004 C
-1935.8 2003.7 1937 2003.6 1938.5 2004 C
-1938.5 2004.5 1939.1 2005.4 1938.3 2006 C
-1940.7 2005.7 1937.4 2001.3 1939.7 2001.4 C
-1939.5 2001.4 1938.6 1998.8 1937.1 1999.2 C
-1936.3 1999.1 1936.2 1997.1 1936.1 1998.5 C
-1934.7 2000.1 1932.9 1998.2 1931.6 1999.5 C
-1931.3 1998.9 1930.9 1998.5 1931.1 1997.8 C
-1931.6 1998.2 1931.3 1996.6 1932 1996.1 C
-1933.2 1995.5 1934.3 1996.4 1935.2 1995.4 C
-1935.5 1996.5 1936.3 1996.1 1935.6 1995.2 C
-1934.7 1994.5 1932.5 1995.3 1932 1995.4 C
-1930.5 1995.3 1931.9 1996.5 1930.8 1996.4 C
-1931.2 1997.9 1929.5 1998.3 1928.9 1998.5 C
-1928.1 1997.9 1927.1 1998 1926 1998 C
-1925.3 1999.2 1924.8 2001.4 1923.2 2001.4 C
-1922.6 2000.9 1921 2000.9 1920.3 2000.9 C
-1919.7 2001.9 1919.6 2003.5 1918.1 2004 C
-1916.9 2004.1 1915.8 2002 1915.2 2003.8 C
-1916.7 2004 1917.6 2004.9 1919.6 2004.5 C
-1920.7 2005.2 1919.4 2006.3 1919.8 2006.9 C
-1919.2 2006.9 1917.7 2007.8 1917.2 2008.6 C
-1917.8 2011.6 1919.8 2007.8 1920.5 2010.5 C
-1920.8 2011.3 1919.3 2011.6 1920.5 2012 C
-1920.8 2012.3 1924 2011.8 1923.2 2014.1 C
-1922.6 2013.6 1924.1 2016.1 1924.1 2015.1 C
-1925.1 2015.4 1925.9 2015 1926.3 2016.5 C
-1926.2 2016.6 1926 2016.8 1925.8 2016.8 C
-1925.9 2017.2 1926.2 2017.8 1926.8 2018.2 C
-1927.1 2017.6 1927.7 2018 1928.4 2018.2 C
-1929.7 2020.1 1927.1 2019.5 1929.4 2021.1 C
-1929.9 2020.7 1931.1 2020 1931.1 2019.9 C
-[0.21 0.21 0 0] vc
-f
-S
-n
-1937.1 2020.8 m
-1937.1 2018.3 1937.1 2015.7 1937.1 2013.2 C
-1937.1 2015.7 1937.1 2018.3 1937.1 2020.8 C
-[0 1 1 0.36] vc
-f
-S
-n
-2020.4 2012.2 m
-2019.8 2012 2019.3 2011.5 2018.7 2011.7 C
-2017.9 2012.1 2018.1 2014.1 2018.4 2014.4 C
-2019.6 2014.8 2021.4 2013.7 2020.4 2012.2 C
-[0 1 1 0.23] vc
-f
-S
-n
-1976 2013.9 m
-1973.8 2011.5 1971.6 2009.1 1969.5 2006.7 C
-1971.6 2009.1 1973.8 2011.5 1976 2013.9 C
-[0 1 1 0.36] vc
-f
-S
-n
-1995.4 2012.7 m
-1996.1 2010.3 1993.8 2006.2 1997.3 2005.7 C
-1998.9 2005.4 2000 2003.7 2001.4 2003.1 C
-2003.9 2003.1 2005.3 2001.3 2006.9 1999.7 C
-2004.5 2003.5 2000 2002.2 1997.6 2005.7 C
-1996.5 2005.9 1994.8 2006.1 1995.2 2007.6 C
-1995.7 2009.4 1995.2 2011.6 1994.7 2012.9 C
-1992 2015.8 1987.8 2015.7 1985.3 2018.7 C
-1988.3 2016.3 1992.3 2015.3 1995.4 2012.7 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1995.6 2012.4 m
-1995.6 2011.2 1995.6 2010 1995.6 2008.8 C
-1995.6 2010 1995.6 2011.2 1995.6 2012.4 C
-[0 1 1 0.36] vc
-f
-S
-n
-vmrs
-2017.7 2009.6 m
-2016.9 2009.3 2016.7 2008.4 2015.8 2009.1 C
-2014.2 2010.6 2016 2010.6 2016.5 2011.5 C
-2017.2 2010.9 2018.1 2010.8 2017.7 2009.6 C
-[0 1 1 0.23] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-2014.4 2006.4 m
-2013.5 2006.8 2012.1 2005.6 2012 2006.7 C
-2013 2007.3 2011.9 2009.2 2012.9 2008.4 C
-2014.2 2008.3 2014.6 2007.8 2014.4 2006.4 C
-f
-S
-n
-1969 2006.4 m
-1966.5 2003.8 1964 2001.2 1961.6 1998.5 C
-1964 2001.2 1966.5 2003.8 1969 2006.4 C
-[0 1 1 0.36] vc
-f
-S
-n
-2012 2005.2 m
-2012.2 2004.2 2011.4 2003.3 2010.3 2003.3 C
-2009 2003.6 2010 2004.7 2009.6 2004.8 C
-2009.3 2005.7 2011.4 2006.7 2012 2005.2 C
-[0 1 1 0.23] vc
-f
-S
-n
-1962.8 1995.2 m
-1961.7 1994.4 1960.6 1993.7 1959.4 1994 C
-1959.5 1994.9 1957.5 1994.1 1956.8 1994.7 C
-1955.9 1995.5 1956.7 1997 1955.1 1997.3 C
-1956.9 1996.7 1956.8 1994 1959.2 1994.7 C
-1961.1 1991 1968.9 2003.2 1962.8 1995.2 C
-[0 1 1 0.36] vc
-f
-S
-n
-1954.6 1995.6 m
-1955.9 1994.7 1955.1 1989.8 1955.3 1988 C
-1954.5 1988.3 1954.9 1986.6 1954.4 1986 C
-1955.7 1989.2 1953.9 1991.1 1954.8 1994.2 C
-1954.5 1995.9 1953.5 1995.3 1953.9 1997.3 C
-1955.3 1998.3 1953.2 1995.5 1954.6 1995.6 C
-f
-S
-n
-1992.3 2011 m
-1992.5 2006.7 1992 2000.3 1991.8 1997.6 C
-1992.2 1997.9 1992 1998.5 1992 1999 C
-1992.1 1994.7 1991.9 1990.2 1992.3 1986 C
-1991.4 1984.6 1991 1986.1 1990.6 1985.6 C
-1989.7 1986 1990.3 1987.2 1990.1 1988 C
-1990.7 1987.4 1990.4 1988.5 1990.8 1988.7 C
-1991.3 1997.1 1988.9 2000.6 1991.1 2007.9 C
-1991 2009.1 1989.8 2009.9 1988.4 2008.8 C
-1985.7 2007.2 1986.8 2004.5 1984.1 2004 C
-1984.2 2002.7 1981.9 2003.7 1982.4 2001.4 C
-1981.2 2001.5 1980.5 2000.8 1980 2000 C
-1980 1999.8 1980 1998.9 1980 1999.5 C
-1979.3 1999.5 1979.7 1997.2 1977.9 1998 C
-1977.2 1997.3 1976.6 1996.7 1977.2 1997.8 C
-1979 1998.7 1979.3 2000.8 1981.5 2001.6 C
-1982.2 2002.8 1983 2004.3 1984.4 2004.3 C
-1985 2005.8 1986.2 2007.5 1987.7 2009.1 C
-1989 2010 1991.3 2010.2 1990.8 2012.2 C
-1991.2 2011.4 1992.2 2011.8 1992.3 2011 C
-[0 1 1 0.23] vc
-f
-S
-n
-1991.8 1995.6 m
-1991.8 1994.3 1991.8 1992.9 1991.8 1991.6 C
-1991.8 1992.9 1991.8 1994.3 1991.8 1995.6 C
-[0 1 1 0.36] vc
-f
-S
-n
-1959.2 1994.2 m
-1958.8 1993.3 1960.7 1993.9 1961.1 1993.7 C
-1961.5 1993.9 1961.2 1994.4 1961.8 1994.2 C
-1960.9 1994 1960.8 1992.9 1959.9 1992.5 C
-1959.6 1993.5 1958.3 1993.5 1958.2 1994.2 C
-1958.1 1994.1 1958 1994 1958 1994 C
-1957.2 1994.9 1958 1993.4 1956.8 1993 C
-1955.6 1992.5 1956 1991 1956.3 1989.9 C
-1956.5 1989.8 1956.6 1990 1956.8 1990.1 C
-1957.1 1989 1956 1989.1 1955.8 1988.2 C
-1955.1 1990.4 1956.2 1995 1954.8 1995.9 C
-1954.1 1995.5 1954.5 1996.5 1954.4 1997.1 C
-1955 1996.8 1954.8 1997.4 1955.6 1996.8 C
-1956 1996 1956.3 1993.2 1958.7 1994.2 C
-1958.9 1994.2 1959.7 1994.2 1959.2 1994.2 C
-[0 1 1 0.23] vc
-f
-S
-n
-1958.2 1994 m
-1958.4 1993.5 1959.7 1993.1 1959.9 1992 C
-1959.7 1992.5 1959.3 1992 1959.4 1991.8 C
-1959.4 1991.6 1959.4 1990.8 1959.4 1991.3 C
-1959.2 1991.8 1958.8 1991.3 1958.9 1991.1 C
-1958.9 1990.5 1958 1990.3 1957.5 1989.9 C
-1956.8 1989.5 1956.9 1991 1956.3 1990.1 C
-1956.7 1991 1955.4 1992.1 1956.5 1992.3 C
-1956.8 1993.5 1958.3 1992.9 1957.2 1994 C
-1957.8 1994.3 1958.1 1992.4 1958.2 1994 C
-[0 0.5 0.5 0.2] vc
-f
-S
-n
-vmrs
-1954.4 1982.7 m
-1956.1 1982.7 1954.1 1982.5 1953.9 1982.9 C
-1953.9 1983.7 1953.7 1984.7 1954.1 1985.3 C
-1954.4 1984.2 1953.6 1983.6 1954.4 1982.7 C
-[0 1 1 0.36] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1989.6 1982.9 m
-1989.1 1982.7 1988.6 1982.3 1988 1982.4 C
-1987.2 1982.8 1987.4 1984.8 1987.7 1985.1 C
-1988.9 1985.6 1990.7 1984.4 1989.6 1982.9 C
-[0 1 1 0.23] vc
-f
-S
-n
-1987 1980.3 m
-1986.2 1980 1986 1979.1 1985.1 1979.8 C
-1983.5 1981.4 1985.3 1981.4 1985.8 1982.2 C
-1986.5 1981.7 1987.4 1981.5 1987 1980.3 C
-f
-S
-n
-1983.6 1977.2 m
-1982.7 1977.5 1981.4 1976.3 1981.2 1977.4 C
-1982.3 1978 1981.2 1979.9 1982.2 1979.1 C
-1983.5 1979 1983.9 1978.5 1983.6 1977.2 C
-f
-S
-n
-1981.2 1976 m
-1981.5 1974.9 1980.6 1974 1979.6 1974 C
-1978.3 1974.3 1979.3 1975.4 1978.8 1975.5 C
-1978.6 1976.4 1980.7 1977.4 1981.2 1976 C
-f
-S
-n
-1972.1 2082.3 m
-1971.8 2081.8 1971.3 2080.9 1971.2 2080.1 C
-1971.1 2072.9 1971.3 2064.6 1970.9 2058.3 C
-1970.3 2058.5 1970.1 2057.7 1969.7 2058.5 C
-1970.6 2058.5 1969.7 2059 1970.2 2059.2 C
-1970.2 2065.4 1970.2 2072.4 1970.2 2077.7 C
-1971.1 2078.9 1970.6 2078.9 1970.4 2079.9 C
-1969.2 2080.2 1968.2 2080.4 1967.3 2079.6 C
-1966.8 2077.8 1963.4 2076.3 1963.5 2075.1 C
-1961.5 2075.5 1962 2071.5 1959.6 2072 C
-1959.2 2070 1956.5 2069.3 1955.8 2067.6 C
-1956 2068.4 1955.3 2069.7 1956.5 2069.8 C
-1958.6 2068.9 1958.1 2073.5 1960.1 2072.4 C
-1960.7 2075.9 1964.7 2074.6 1964.2 2078 C
-1967.2 2078.6 1967.9 2081.6 1970.7 2080.6 C
-1970.3 2081.1 1971.5 2081.2 1971.9 2082.3 C
-1967.2 2084.3 1962.9 2087.1 1958.2 2089 C
-1962.9 2087 1967.4 2084.4 1972.1 2082.3 C
-[0 0.2 1 0] vc
-f
-S
-n
-1971.9 2080.1 m
-1971.9 2075.1 1971.9 2070 1971.9 2065 C
-1971.9 2070 1971.9 2075.1 1971.9 2080.1 C
-[0 1 1 0.23] vc
-f
-S
-n
-2010.8 2050.6 m
-2013.2 2049 2010.5 2050.1 2010.5 2051.3 C
-2010.5 2057.7 2010.5 2064.1 2010.5 2070.5 C
-2008.7 2072.4 2006 2073.3 2003.6 2074.4 C
-2016.4 2073.7 2008 2058.4 2010.8 2050.6 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-2006.4 2066.9 m
-2006.4 2061.9 2006.4 2056.8 2006.4 2051.8 C
-2006.4 2056.8 2006.4 2061.9 2006.4 2066.9 C
-[0 1 1 0.23] vc
-f
-S
-n
-1971.9 2060.7 m
-1972.2 2060.3 1971.4 2068.2 1972.4 2061.9 C
-1971.8 2061.6 1972.4 2060.9 1971.9 2060.7 C
-f
-S
-n
-vmrs
-1986.5 2055.2 m
-1987.5 2054.3 1986.3 2053.4 1986 2052.8 C
-1983.8 2052.7 1983.6 2050.1 1981.7 2049.6 C
-1981.2 2048.7 1980.8 2047 1980.3 2046.8 C
-1978.5 2047 1978 2044.6 1976.7 2043.9 C
-1974 2044.4 1972 2046.6 1969.2 2047 C
-1969 2047.2 1968.8 2047.5 1968.5 2047.7 C
-1970.6 2049.6 1973.1 2051.3 1974.3 2054.2 C
-1975.7 2054.5 1977 2055.2 1976.4 2057.1 C
-1976.7 2058 1975.5 2058.5 1976 2059.5 C
-1979.2 2058 1983 2056.6 1986.5 2055.2 C
-[0 0.5 0.5 0.2] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1970.2 2054.2 m
-1971.5 2055.3 1972.5 2056.8 1972.1 2058.3 C
-1972.8 2056.5 1971.6 2055.6 1970.2 2054.2 C
-[0 1 1 0.23] vc
-f
-S
-n
-1992 2052.5 m
-1992 2053.4 1992.2 2054.4 1991.8 2055.2 C
-1992.2 2054.4 1992 2053.4 1992 2052.5 C
-f
-S
-n
-1957.2 2053 m
-1958.1 2052.6 1959 2052.2 1959.9 2051.8 C
-1959 2052.2 1958.1 2052.6 1957.2 2053 C
-f
-S
-n
-2006.4 2047.5 m
-2006.8 2047.1 2006 2055 2006.9 2048.7 C
-2006.4 2048.4 2007 2047.7 2006.4 2047.5 C
-f
-S
-n
-2004.8 2041 m
-2006.1 2042.1 2007.1 2043.6 2006.7 2045.1 C
-2007.3 2043.3 2006.2 2042.4 2004.8 2041 C
-f
-S
-n
-1976 2039.8 m
-1975.6 2039.3 1975.2 2038.4 1975 2037.6 C
-1974.9 2030.4 1975.2 2022.1 1974.8 2015.8 C
-1974.2 2016 1974 2015.3 1973.6 2016 C
-1974.4 2016 1973.5 2016.5 1974 2016.8 C
-1974 2022.9 1974 2030 1974 2035.2 C
-1974.9 2036.4 1974.4 2036.4 1974.3 2037.4 C
-1973.1 2037.7 1972 2037.9 1971.2 2037.2 C
-1970.6 2035.3 1967.3 2033.9 1967.3 2032.6 C
-1965.3 2033 1965.9 2029.1 1963.5 2029.5 C
-1963 2027.6 1960.4 2026.8 1959.6 2025.2 C
-1959.8 2025.9 1959.2 2027.2 1960.4 2027.3 C
-1962.5 2026.4 1961.9 2031 1964 2030 C
-1964.6 2033.4 1968.5 2032.1 1968 2035.5 C
-1971 2036.1 1971.8 2039.1 1974.5 2038.1 C
-1974.2 2038.7 1975.3 2038.7 1975.7 2039.8 C
-1971 2041.8 1966.7 2044.6 1962 2046.5 C
-1966.8 2044.5 1971.3 2041.9 1976 2039.8 C
-[0 0.2 1 0] vc
-f
-S
-n
-1975.7 2037.6 m
-1975.7 2032.6 1975.7 2027.6 1975.7 2022.5 C
-1975.7 2027.6 1975.7 2032.6 1975.7 2037.6 C
-[0 1 1 0.23] vc
-f
-S
-n
-1992 2035.5 m
-1992 2034.2 1992 2032.9 1992 2031.6 C
-1992 2032.9 1992 2034.2 1992 2035.5 C
-f
-S
-n
-2015.3 2036 m
-2015.4 2034.1 2013.3 2034 2012.9 2033.3 C
-2011.5 2031 2009.3 2029.4 2007.4 2028 C
-2006.9 2027.1 2006.6 2023.8 2005 2024.9 C
-2004 2024.9 2002.9 2024.9 2001.9 2024.9 C
-2001.4 2026.5 2001 2028.4 2003.8 2028.3 C
-2006.6 2030.4 2008.9 2033.7 2011.2 2036.2 C
-2011.8 2036.4 2012.9 2035.8 2012.9 2036.7 C
-2013 2035.5 2015.3 2037.4 2015.3 2036 C
-[0 0 0 0] vc
-f
-S
-n
-vmrs
-2009.1 2030.4 m
-2009.1 2029 2007.5 2029.4 2006.9 2028.3 C
-2007.2 2027.1 2006.5 2025.5 2005.7 2024.7 C
-2004.6 2025.1 2003.1 2024.9 2001.9 2024.9 C
-2001.8 2026.2 2000.9 2027 2002.4 2028 C
-2004.5 2027.3 2004.9 2029.4 2006.9 2029 C
-2007 2030.2 2007.6 2030.7 2008.4 2031.4 C
-2008.8 2031.5 2009.1 2031.1 2009.1 2030.4 C
-[0 0 0 0.18] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-2003.8 2029.5 m
-2003 2029.4 2001.9 2029.1 2002.4 2030.4 C
-2003.1 2031.3 2005.2 2030.3 2003.8 2029.5 C
-[0 1 1 0.23] vc
-f
-S
-n
-1999.2 2025.2 m
-1999.1 2025.6 1998 2025.7 1998.8 2026.6 C
-2000.9 2028.5 1999.5 2023.4 1999.2 2025.2 C
-f
-S
-n
-2007.6 2024.2 m
-2007.6 2022.9 2008.4 2024.2 2007.6 2022.8 C
-2007.6 2017.5 2007.8 2009.1 2007.4 2003.8 C
-2007.9 2003.7 2008.7 2002.8 2009.1 2002.1 C
-2009.6 2000.8 2008.3 2000.8 2007.9 2000.2 C
-2004.9 2000 2008.9 2001.3 2007.2 2002.1 C
-2006.7 2007.7 2007 2015.1 2006.9 2021.1 C
-2006.7 2022.1 2005.4 2022.8 2006.2 2023.5 C
-2006.6 2023.1 2008 2025.9 2007.6 2024.2 C
-f
-S
-n
-1989.9 2023.5 m
-1989.5 2022.6 1991.4 2023.2 1991.8 2023 C
-1992.2 2023.2 1991.9 2023.7 1992.5 2023.5 C
-1991.6 2023.2 1991.6 2022.2 1990.6 2021.8 C
-1990.4 2022.8 1989 2022.8 1988.9 2023.5 C
-1988.5 2023 1988.7 2022.6 1988.7 2023.5 C
-1989.1 2023.5 1990.2 2023.5 1989.9 2023.5 C
-f
-[0 0.5 0.5 0.2] vc
-S
-n
-2003.3 2023.5 m
-2003.1 2023.3 2003.1 2023.2 2003.3 2023 C
-2003.7 2023.1 2003.9 2022.9 2003.8 2022.5 C
-2003.4 2022.2 2001.2 2022.3 2002.4 2023 C
-2002.6 2022.9 2002.7 2023.1 2002.8 2023.2 C
-2000.7 2023.7 2003.9 2023.4 2003.3 2023.5 C
-[0 1 1 0.23] vc
-f
-S
-n
-1986.8 2019.4 m
-1987.8 2019.8 1987.5 2018.6 1987.2 2018 C
-1986.2 2017.8 1987.3 2020.5 1986.3 2019.2 C
-1986.3 2017.7 1986.3 2020.6 1986.3 2021.3 C
-1988.5 2023.1 1985.6 2020.3 1986.8 2019.4 C
-f
-S
-n
-1975.7 2018.2 m
-1976.1 2017.8 1975.2 2025.7 1976.2 2019.4 C
-1975.7 2019.2 1976.3 2018.4 1975.7 2018.2 C
-f
-S
-n
-1974 2011.7 m
-1975.4 2012.8 1976.4 2014.3 1976 2015.8 C
-1976.6 2014 1975.5 2013.1 1974 2011.7 C
-f
-S
-n
-1984.6 2006.7 m
-1984.7 2004.8 1982.6 2004.8 1982.2 2004 C
-1980.8 2001.7 1978.6 2000.1 1976.7 1998.8 C
-1976.1 1997.8 1975.8 1994.5 1974.3 1995.6 C
-1973.3 1995.6 1972.2 1995.6 1971.2 1995.6 C
-1970.7 1997.2 1970.3 1999.1 1973.1 1999 C
-1975.8 2001.2 1978.2 2004.4 1980.5 2006.9 C
-1981.1 2007.1 1982.1 2006.5 1982.2 2007.4 C
-1982.3 2006.2 1984.5 2008.1 1984.6 2006.7 C
-[0 0 0 0] vc
-f
-S
-n
-vmrs
-1978.4 2001.2 m
-1978.4 1999.7 1976.8 2000.1 1976.2 1999 C
-1976.5 1997.8 1975.8 1996.2 1975 1995.4 C
-1973.9 1995.8 1972.4 1995.6 1971.2 1995.6 C
-1971 1997 1970.2 1997.7 1971.6 1998.8 C
-1973.8 1998 1974.2 2000.1 1976.2 1999.7 C
-1976.3 2000.9 1976.9 2001.4 1977.6 2002.1 C
-1978.1 2002.2 1978.4 2001.8 1978.4 2001.2 C
-[0 0 0 0.18] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1973.1 2000.2 m
-1972.3 2000.1 1971.2 1999.8 1971.6 2001.2 C
-1972.4 2002 1974.5 2001 1973.1 2000.2 C
-[0 1 1 0.23] vc
-f
-S
-n
-1960.8 1998.5 m
-1961.6 1998.2 1962.6 2000.3 1963.2 2000.9 C
-1962.3 2000.1 1962.2 1998.7 1960.8 1998.5 C
-f
-S
-n
-1968.5 1995.9 m
-1968.4 1996.4 1967.3 1996.4 1968 1997.3 C
-1970.1 1999.2 1968.8 1994.1 1968.5 1995.9 C
-f
-S
-n
-1976.9 1994.9 m
-1976.9 1993.7 1977.6 1994.9 1976.9 1993.5 C
-1976.9 1988.2 1977.1 1979.8 1976.7 1974.5 C
-1977.2 1974.5 1978 1973.5 1978.4 1972.8 C
-1978.8 1971.5 1977.6 1971.5 1977.2 1970.9 C
-1974.2 1970.7 1978.2 1972 1976.4 1972.8 C
-1976 1978.4 1976.3 1985.8 1976.2 1991.8 C
-1976 1992.8 1974.6 1993.5 1975.5 1994.2 C
-1975.9 1993.8 1977.3 1996.6 1976.9 1994.9 C
-f
-S
-n
-1972.6 1994.2 m
-1972.4 1994 1972.4 1993.9 1972.6 1993.7 C
-1973 1993.8 1973.1 1993.7 1973.1 1993.2 C
-1972.7 1992.9 1970.5 1993.1 1971.6 1993.7 C
-1971.9 1993.7 1972 1993.8 1972.1 1994 C
-1970 1994.4 1973.1 1994.1 1972.6 1994.2 C
-f
-S
-n
-1948.1 2093.8 m
-1947 2092.7 1945.9 2091.6 1944.8 2090.4 C
-1945.9 2091.6 1947 2092.7 1948.1 2093.8 C
-[0 0.4 1 0] vc
-f
-S
-n
-1953.4 2091.4 m
-1954.8 2090.7 1956.3 2090 1957.7 2089.2 C
-1956.3 2090 1954.8 2090.7 1953.4 2091.4 C
-[0 0.2 1 0] vc
-f
-S
-n
-1954.1 2091.4 m
-1956.6 2089.6 1957.2 2089.6 1954.1 2091.4 C
-[0 0.4 1 0] vc
-f
-S
-n
-1962.3 2087.3 m
-1963.7 2086.6 1965.2 2085.9 1966.6 2085.2 C
-1965.2 2085.9 1963.7 2086.6 1962.3 2087.3 C
-f
-S
-n
-vmrs
-1967.1 2084.9 m
-1968.3 2084.4 1969.7 2083.8 1970.9 2083.2 C
-1969.7 2083.8 1968.3 2084.4 1967.1 2084.9 C
-[0 0.4 1 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1982.7 2080.6 m
-1981.5 2079.5 1980.5 2078.4 1979.3 2077.2 C
-1980.5 2078.4 1981.5 2079.5 1982.7 2080.6 C
-f
-S
-n
-1988 2078.2 m
-1989.4 2077.5 1990.8 2076.8 1992.3 2076 C
-1990.8 2076.8 1989.4 2077.5 1988 2078.2 C
-[0 0.2 1 0] vc
-f
-S
-n
-1988.7 2078.2 m
-1991.1 2076.4 1991.8 2076.4 1988.7 2078.2 C
-[0 0.4 1 0] vc
-f
-S
-n
-1976.2 2063.8 m
-1978.6 2062.2 1976 2063.3 1976 2064.5 C
-1976.1 2067.8 1975.5 2071.4 1976.4 2074.4 C
-1975.7 2071.1 1975.9 2067.2 1976.2 2063.8 C
-f
-S
-n
-1996.8 2074.1 m
-1998.3 2073.4 1999.7 2072.7 2001.2 2072 C
-1999.7 2072.7 1998.3 2073.4 1996.8 2074.1 C
-f
-S
-n
-2001.6 2071.7 m
-2002.9 2071.2 2004.2 2070.6 2005.5 2070 C
-2004.2 2070.6 2002.9 2071.2 2001.6 2071.7 C
-f
-S
-n
-1981.5 2060.7 m
-1980.2 2061.2 1978.9 2061.5 1977.9 2062.6 C
-1978.9 2061.5 1980.2 2061.2 1981.5 2060.7 C
-f
-S
-n
-1982 2060.4 m
-1982.7 2060.1 1983.6 2059.8 1984.4 2059.5 C
-1983.6 2059.8 1982.7 2060.1 1982 2060.4 C
-f
-S
-n
-1952 2051.3 m
-1950.8 2050.2 1949.7 2049.1 1948.6 2048 C
-1949.7 2049.1 1950.8 2050.2 1952 2051.3 C
-f
-S
-n
-vmrs
-1977.4 2047.7 m
-1975.8 2047.8 1974.8 2046.1 1974.5 2045.3 C
-1974.9 2044.4 1976 2044.5 1976.7 2044.8 C
-1977.9 2045 1977 2048.4 1979.3 2047.5 C
-1979.9 2047.5 1980.8 2048.6 1979.8 2049.2 C
-1978.2 2050.4 1980.8 2049.5 1980.3 2049.4 C
-1981.4 2049.8 1980.3 2048.4 1980.3 2048 C
-1979.8 2047.5 1979 2046.6 1978.4 2046.5 C
-1977.3 2045.9 1977.2 2043.3 1975.2 2044.6 C
-1974.7 2045.3 1973.6 2045 1973.3 2045.8 C
-1975 2046.3 1975.8 2049.8 1978.1 2049.4 C
-1978.4 2050.9 1978.7 2048.5 1977.9 2049.2 C
-1977.7 2048.7 1977.2 2047.8 1977.4 2047.7 C
-[0 0.5 0.5 0.2] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1957.2 2048.9 m
-1958.7 2048.2 1960.1 2047.5 1961.6 2046.8 C
-1960.1 2047.5 1958.7 2048.2 1957.2 2048.9 C
-[0 0.2 1 0] vc
-f
-S
-n
-1958 2048.9 m
-1960.4 2047.1 1961.1 2047.1 1958 2048.9 C
-[0 0.4 1 0] vc
-f
-S
-n
-1966.1 2044.8 m
-1967.6 2044.1 1969 2043.4 1970.4 2042.7 C
-1969 2043.4 1967.6 2044.1 1966.1 2044.8 C
-f
-S
-n
-1970.9 2042.4 m
-1972.2 2041.9 1973.5 2041.3 1974.8 2040.8 C
-1973.5 2041.3 1972.2 2041.9 1970.9 2042.4 C
-f
-S
-n
-2012 2034.5 m
-2010.4 2034.6 2009.3 2032.9 2009.1 2032.1 C
-2009.4 2031 2010.3 2031.3 2011.2 2031.6 C
-2012.5 2031.8 2011.6 2035.2 2013.9 2034.3 C
-2014.4 2034.3 2015.4 2035.4 2014.4 2036 C
-2012.7 2037.2 2015.3 2036.3 2014.8 2036.2 C
-2015.9 2036.6 2014.8 2035.2 2014.8 2034.8 C
-2014.4 2034.3 2013.6 2033.4 2012.9 2033.3 C
-2011.5 2031 2009.3 2029.4 2007.4 2028 C
-2007.5 2026.5 2007.3 2027.9 2007.2 2028.3 C
-2007.9 2028.8 2008.7 2029.1 2009.3 2030 C
-2009.6 2030.7 2009 2031.9 2008.4 2031.6 C
-2006.7 2031 2007.7 2028 2005 2028.8 C
-2004.8 2028.6 2004.3 2028.2 2003.8 2028.3 C
-2006.6 2030.4 2008.9 2033.7 2011.2 2036.2 C
-2011.8 2036.4 2012.9 2035.8 2012.9 2036.7 C
-2012.7 2036.1 2011.8 2035 2012 2034.5 C
-[0 0.5 0.5 0.2] vc
-f
-S
-n
-1981.2 2005.2 m
-1979.7 2005.3 1978.6 2003.6 1978.4 2002.8 C
-1978.7 2001.8 1979.6 2002.1 1980.5 2002.4 C
-1981.8 2002.5 1980.9 2005.9 1983.2 2005 C
-1983.7 2005.1 1984.7 2006.1 1983.6 2006.7 C
-1982 2007.9 1984.6 2007 1984.1 2006.9 C
-1985.2 2007.3 1984.1 2006 1984.1 2005.5 C
-1983.6 2005 1982.9 2004.1 1982.2 2004 C
-1980.8 2001.7 1978.6 2000.1 1976.7 1998.8 C
-1976.7 1997.2 1976.6 1998.6 1976.4 1999 C
-1977.2 1999.5 1978 1999.8 1978.6 2000.7 C
-1978.8 2001.5 1978.3 2002.7 1977.6 2002.4 C
-1976 2001.8 1977 1998.7 1974.3 1999.5 C
-1974.1 1999.3 1973.6 1998.9 1973.1 1999 C
-1975.8 2001.2 1978.2 2004.4 1980.5 2006.9 C
-1981.1 2007.1 1982.1 2006.5 1982.2 2007.4 C
-1982 2006.8 1981.1 2005.7 1981.2 2005.2 C
-f
-S
-n
-1966.8 1976.4 m
-1969.4 1973 1974.4 1974.6 1976.2 1970.4 C
-1972.7 1974 1968 1975.1 1964 1977.4 C
-1960.9 1979.9 1957.1 1981.8 1953.9 1982.7 C
-1958.4 1981.1 1962.6 1978.8 1966.8 1976.4 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1948.4 2093.8 m
-1949.8 2093.1 1951.2 2092.5 1952.7 2091.9 C
-1951.2 2092.5 1949.8 2093.1 1948.4 2093.8 C
-[0 0.2 1 0] vc
-f
-S
-n
-1948.1 2093.6 m
-1947.3 2092.8 1946.5 2091.9 1945.7 2091.2 C
-1946.5 2091.9 1947.3 2092.8 1948.1 2093.6 C
-f
-S
-n
-vmrs
-1942.1 2087.8 m
-1943.5 2088.4 1944.3 2089.5 1945.2 2090.7 C
-1944.8 2089.3 1943.3 2088.3 1942.1 2087.8 C
-[0 0.2 1 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1933.5 2078.4 m
-1933.5 2078 1933.2 2079 1933.7 2079.4 C
-1935 2080.4 1936.2 2081.3 1937.1 2082.8 C
-1936.7 2080.7 1933.7 2080.7 1933.5 2078.4 C
-f
-S
-n
-1982.9 2080.6 m
-1984.4 2079.9 1985.8 2079.3 1987.2 2078.7 C
-1985.8 2079.3 1984.4 2079.9 1982.9 2080.6 C
-f
-S
-n
-1982.7 2080.4 m
-1981.9 2079.6 1981.1 2078.7 1980.3 2078 C
-1981.1 2078.7 1981.9 2079.6 1982.7 2080.4 C
-f
-S
-n
-1977.4 2075.1 m
-1977.9 2075.3 1979.1 2076.4 1979.8 2077.5 C
-1979 2076.8 1978.7 2075.1 1977.4 2075.1 C
-f
-S
-n
-1952.2 2051.3 m
-1953.6 2050.7 1955.1 2050.1 1956.5 2049.4 C
-1955.1 2050.1 1953.6 2050.7 1952.2 2051.3 C
-f
-S
-n
-1952 2051.1 m
-1951.2 2050.3 1950.3 2049.5 1949.6 2048.7 C
-1950.3 2049.5 1951.2 2050.3 1952 2051.1 C
-f
-S
-n
-1946 2045.3 m
-1947.3 2045.9 1948.1 2047 1949.1 2048.2 C
-1948.6 2046.8 1947.1 2045.8 1946 2045.3 C
-f
-S
-n
-1937.3 2036 m
-1937.4 2035.5 1937 2036.5 1937.6 2036.9 C
-1938.8 2037.9 1940.1 2038.8 1940.9 2040.3 C
-1940.6 2038.2 1937.6 2038.2 1937.3 2036 C
-f
-S
-n
-1935.2 2073.2 m
-1936.4 2069.9 1935.8 2061.8 1935.6 2056.4 C
-1935.8 2055.9 1936.3 2055.7 1936.1 2055.2 C
-1935.7 2054.7 1935 2055 1934.4 2054.9 C
-1934.4 2061.5 1934.4 2068.7 1934.4 2074.6 C
-1935.7 2075.1 1936 2073.7 1935.2 2073.2 C
-[0 0.01 1 0] vc
-f
-S
-n
-vmrs
-1939 2030.7 m
-1940.3 2027.4 1939.7 2019.3 1939.5 2013.9 C
-1939.7 2013.5 1940.1 2013.2 1940 2012.7 C
-1939.5 2012.3 1938.8 2012.5 1938.3 2012.4 C
-1938.3 2019 1938.3 2026.2 1938.3 2032.1 C
-1939.5 2032.7 1939.8 2031.2 1939 2030.7 C
-[0 0.01 1 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1975.2 2077.2 m
-1975.3 2077.3 1975.4 2077.4 1975.5 2077.5 C
-1974.7 2073.2 1974.9 2067.5 1975.2 2063.6 C
-1975.4 2064 1974.6 2063.9 1974.8 2064.3 C
-1974.9 2069.9 1974.3 2076.5 1975.2 2081.1 C
-1974.9 2079.9 1974.9 2078.4 1975.2 2077.2 C
-[0.92 0.92 0 0.67] vc
-f
-S
-n
-1930.8 2067.4 m
-1931.5 2070.1 1929.6 2072.1 1930.6 2074.6 C
-1931 2072.6 1930.8 2069.8 1930.8 2067.4 C
-f
-S
-n
-2010 2050.1 m
-2009.8 2050.5 2009.5 2050.9 2009.3 2051.1 C
-2009.5 2056.7 2008.9 2063.3 2009.8 2067.9 C
-2009.5 2062.1 2009.3 2054.7 2010 2050.1 C
-f
-S
-n
-1930.1 2060.9 m
-1929.3 2057.1 1930.7 2054.8 1929.9 2051.3 C
-1930.2 2050.2 1931.1 2049.6 1931.8 2049.2 C
-1931.4 2049.6 1930.4 2049.5 1930.1 2050.1 C
-1928.4 2054.8 1933.4 2063.5 1925.3 2064.3 C
-1927.2 2063.9 1928.5 2062.1 1930.1 2060.9 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1929.6 2061.2 m
-1929.6 2057.6 1929.6 2054.1 1929.6 2050.6 C
-1930 2049.9 1930.5 2049.4 1931.1 2049.2 C
-1930 2048.6 1930.5 2050.2 1929.4 2049.6 C
-1928 2054.4 1932.8 2063 1925.3 2064 C
-1926.9 2063.3 1928.3 2062.4 1929.6 2061.2 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1930.8 2061.6 m
-1930.5 2058 1931.6 2054 1930.8 2051.3 C
-1930.3 2054.5 1930.9 2058.5 1930.4 2061.9 C
-1930.5 2061.2 1931 2062.2 1930.8 2061.6 C
-[0.92 0.92 0 0.67] vc
-f
-S
-n
-1941.2 2045.1 m
-1939.7 2042.6 1937.3 2041.2 1935.4 2039.3 C
-1934.2 2040 1933.7 2036.4 1934 2039.3 C
-1934.9 2040.1 1936.1 2039.9 1936.8 2040.8 C
-1935.3 2044.2 1942.3 2041.7 1939.5 2046 C
-1937.1 2048.5 1940.5 2045.6 1941.2 2045.1 C
-f
-S
-n
-1910 2045.8 m
-1910 2039.4 1910 2033 1910 2026.6 C
-1910 2033 1910 2039.4 1910 2045.8 C
-f
-S
-n
-1978.8 2022.3 m
-1979.1 2021.7 1979.4 2020.4 1978.6 2021.6 C
-1978.6 2026.9 1978.6 2033 1978.6 2037.6 C
-1979.2 2037 1979.1 2038.2 1979.1 2038.6 C
-1978.7 2033.6 1978.9 2026.8 1978.8 2022.3 C
-f
-S
-n
-vmrs
-2026.1 2041.2 m
-2026.1 2034.8 2026.1 2028.3 2026.1 2021.8 C
-2026.1 2028.5 2026.3 2035.4 2025.9 2042 C
-2024.4 2042.9 2022.9 2044.1 2021.3 2044.8 C
-2023.1 2044 2025.1 2042.8 2026.1 2041.2 C
-[0.07 0.06 0 0.58] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-2026.4 2021.8 m
-2026.3 2028.5 2026.5 2035.4 2026.1 2042 C
-2025.6 2042.8 2024.7 2042.7 2024.2 2043.4 C
-2024.7 2042.7 2025.5 2042.7 2026.1 2042.2 C
-2026.5 2035.5 2026.3 2027.9 2026.4 2021.8 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-2025.6 2038.4 m
-2025.6 2033 2025.6 2027.6 2025.6 2022.3 C
-2025.6 2027.6 2025.6 2033 2025.6 2038.4 C
-[0.92 0.92 0 0.67] vc
-f
-S
-n
-1934 2023.5 m
-1934 2024.7 1933.8 2026 1934.2 2027.1 C
-1934 2025.5 1934.7 2024.6 1934 2023.5 C
-f
-S
-n
-1928.2 2023.5 m
-1928 2024.6 1927.4 2023.1 1926.8 2023.2 C
-1926.2 2021 1921.4 2019.3 1923.2 2018 C
-1922.7 2016.5 1923.2 2019.3 1922.2 2018.2 C
-1924.4 2020.4 1926.2 2023.3 1928.9 2024.9 C
-1927.9 2024.2 1929.8 2023.5 1928.2 2023.5 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1934 2019.2 m
-1932 2019.6 1930.8 2022.6 1928.7 2021.8 C
-1924.5 2016.5 1918.2 2011.8 1914 2006.7 C
-1914 2005.7 1914 2004.6 1914 2003.6 C
-1913.6 2004.3 1913.9 2005.8 1913.8 2006.9 C
-1919 2012.4 1924.1 2016.5 1929.2 2022.3 C
-1931 2021.7 1932.2 2019.8 1934 2019.2 C
-f
-S
-n
-1928.7 2024.9 m
-1926.3 2022.7 1924.1 2020.4 1921.7 2018.2 C
-1924.1 2020.4 1926.3 2022.7 1928.7 2024.9 C
-[0.65 0.65 0 0.42] vc
-f
-S
-n
-1914.3 2006.7 m
-1918.7 2011.8 1924.5 2016.4 1928.9 2021.6 C
-1924.2 2016.1 1919 2012.1 1914.3 2006.7 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1924.8 2020.8 m
-1921.2 2016.9 1925.6 2022.5 1926 2021.1 C
-1924.2 2021 1926.7 2019.6 1924.8 2020.8 C
-[0.92 0.92 0 0.67] vc
-f
-S
-n
-1934 2018.4 m
-1933.2 2014.7 1934.5 2012.3 1933.7 2008.8 C
-1934 2007.8 1935 2007.2 1935.6 2006.7 C
-1935.3 2007.1 1934.3 2007 1934 2007.6 C
-1932.2 2012.3 1937.2 2021 1929.2 2021.8 C
-1931.1 2021.4 1932.3 2019.6 1934 2018.4 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-vmrs
-1933.5 2018.7 m
-1933.5 2015.1 1933.5 2011.7 1933.5 2008.1 C
-1933.8 2007.4 1934.3 2006.9 1934.9 2006.7 C
-1933.8 2006.1 1934.3 2007.7 1933.2 2007.2 C
-1931.9 2012 1936.7 2020.5 1929.2 2021.6 C
-1930.7 2020.8 1932.2 2019.9 1933.5 2018.7 C
-[0.4 0.4 0 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1934.7 2019.2 m
-1934.3 2015.6 1935.4 2011.5 1934.7 2008.8 C
-1934.1 2012 1934.7 2016 1934.2 2019.4 C
-1934.4 2018.7 1934.8 2019.8 1934.7 2019.2 C
-[0.92 0.92 0 0.67] vc
-f
-S
-n
-1917.6 2013.6 m
-1917.8 2011.1 1916.8 2014.2 1917.2 2012.2 C
-1916.3 2012.9 1914.8 2011.8 1914.3 2010.8 C
-1914.2 2010.5 1914.4 2010.4 1914.5 2010.3 C
-1913.9 2008.8 1913.9 2011.9 1914.3 2012 C
-1916.3 2012 1917.6 2013.6 1916.7 2015.6 C
-1913.7 2017.4 1919.6 2014.8 1917.6 2013.6 C
-f
-S
-n
-1887.2 2015.3 m
-1887.2 2008.9 1887.2 2002.5 1887.2 1996.1 C
-1887.2 2002.5 1887.2 2008.9 1887.2 2015.3 C
-f
-S
-n
-1916.7 2014.4 m
-1917 2012.1 1913 2013 1913.8 2010.8 C
-1912.1 2009.8 1910.9 2009.4 1910.7 2007.9 C
-1910.4 2010.6 1913.4 2010.4 1914 2012.4 C
-1914.9 2012.8 1916.6 2012.9 1916.4 2014.4 C
-1916.9 2015.1 1914.5 2016.6 1916.2 2015.8 C
-1916.4 2015.3 1916.7 2015 1916.7 2014.4 C
-[0.65 0.65 0 0.42] vc
-f
-S
-n
-1914 2009.3 m
-1912.8 2010.9 1909.6 2005.3 1911.9 2009.8 C
-1912.3 2009.6 1913.6 2010.2 1914 2009.3 C
-[0.92 0.92 0 0.67] vc
-f
-S
-n
-1951.2 1998.8 m
-1949 1996.4 1951.5 1994 1950.3 1991.8 C
-1949.1 1989.1 1954 1982.7 1948.8 1981.2 C
-1949.2 1981.5 1951 1982.4 1950.8 1983.6 C
-1951.9 1988.6 1947.1 1986.5 1948.1 1990.4 C
-1948.5 1990.3 1948.7 1990.7 1948.6 1991.1 C
-1949 1992.5 1947.3 1991.9 1948.1 1992.5 C
-1947.1 1992.7 1945.7 1993.5 1945.2 1994.7 C
-1944.5 1996.8 1947.7 2000.5 1943.8 2001.4 C
-1943.4 2002 1943.7 2004 1942.4 2004.5 C
-1945.2 2002.2 1948.9 2000.9 1951.2 1998.8 C
-f
-S
-n
-1994.9 1993 m
-1995.1 1996.5 1994.5 2000.3 1995.4 2003.6 C
-1994.5 2000.3 1995.1 1996.5 1994.9 1993 C
-f
-S
-n
-1913.8 2003.3 m
-1913.8 1996.9 1913.8 1990.5 1913.8 1984.1 C
-1913.8 1990.5 1913.8 1996.9 1913.8 2003.3 C
-f
-S
-n
-1941.9 1998 m
-1940.5 1997.3 1940.7 1999.4 1940.7 2000 C
-1942.8 2001.3 1942.6 1998.8 1941.9 1998 C
-[0 0 0 0] vc
-f
-S
-n
-vmrs
-1942.1 1999.2 m
-1942.2 1998.9 1941.8 1998.8 1941.6 1998.5 C
-1940.4 1998 1940.7 1999.7 1940.7 2000 C
-1941.6 2000.3 1942.6 2000.4 1942.1 1999.2 C
-[0.92 0.92 0 0.67] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1940 1997.1 m
-1939.8 1996 1939.7 1995.9 1939.2 1995.2 C
-1939.1 1995.3 1938.5 1997.9 1937.8 1996.4 C
-1938 1997.3 1939.4 1998.6 1940 1997.1 C
-f
-S
-n
-1911.2 1995.9 m
-1911.2 1991.6 1911.3 1987.2 1911.4 1982.9 C
-1911.3 1987.2 1911.2 1991.6 1911.2 1995.9 C
-f
-S
-n
-1947.2 1979.1 m
-1945.1 1978.8 1944.6 1975.7 1942.4 1975 C
-1940.5 1972.6 1942.2 1973.7 1942.4 1975.7 C
-1945.8 1975.5 1944.2 1979.8 1947.6 1979.6 C
-1948.3 1982.3 1948.5 1980 1947.2 1979.1 C
-f
-S
-n
-1939.5 1973.3 m
-1940.1 1972.6 1939.8 1974.2 1940.2 1973.1 C
-1939.1 1972.8 1938.8 1968.5 1935.9 1969.7 C
-1937.4 1969.2 1938.5 1970.6 1939 1971.4 C
-1939.2 1972.7 1938.6 1973.9 1939.5 1973.3 C
-f
-S
-n
-1975.2 2073.2 m
-1975.2 2070.2 1975.2 2067.2 1975.2 2064.3 C
-1975.2 2067.2 1975.2 2070.2 1975.2 2073.2 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1929.9 2065.7 m
-1928.1 2065.6 1926 2068.8 1924.1 2066.9 C
-1918.1 2060.9 1912.9 2055.7 1907.1 2049.9 C
-1906.7 2047.1 1906.9 2043.9 1906.8 2041 C
-1906.8 2043.9 1906.8 2046.8 1906.8 2049.6 C
-1913.2 2055.5 1918.7 2061.9 1925.1 2067.6 C
-1927.1 2067.9 1928.6 2064.4 1930.1 2066.2 C
-1929.7 2070.3 1929.9 2074.7 1929.9 2078.9 C
-1929.6 2074.4 1930.5 2070.1 1929.9 2065.7 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1930.1 2061.6 m
-1928.1 2062.1 1927 2065.1 1924.8 2064.3 C
-1920.7 2058.9 1914.4 2054.3 1910.2 2049.2 C
-1910.2 2048.1 1910.2 2047.1 1910.2 2046 C
-1909.8 2046.8 1910 2048.3 1910 2049.4 C
-1915.1 2054.9 1920.3 2059 1925.3 2064.8 C
-1927.1 2064.2 1928.4 2062.3 1930.1 2061.6 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1932 2049.9 m
-1932.3 2050.3 1932 2050.4 1932.8 2050.4 C
-1932 2050.4 1932.2 2049.2 1931.3 2049.6 C
-1931.4 2050.5 1930.3 2050.4 1930.4 2051.3 C
-1931.1 2051.1 1930.7 2049.4 1932 2049.9 C
-f
-S
-n
-1938.3 2046 m
-1936.3 2046.8 1935.2 2047.2 1934.2 2048.9 C
-1935.3 2047.7 1936.8 2046.2 1938.3 2046 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-vmrs
-1938.3 2047 m
-1937.9 2046.9 1936.6 2047.1 1936.1 2048 C
-1936.5 2047.5 1937.3 2046.7 1938.3 2047 C
-[0.18 0.18 0 0.78] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1910.2 2043.2 m
-1910.1 2037.5 1910 2031.8 1910 2026.1 C
-1910 2031.8 1910.1 2037.5 1910.2 2043.2 C
-f
-S
-n
-1933.5 2032.1 m
-1933.7 2035.2 1932.8 2035.8 1933.7 2038.6 C
-1933.3 2036.6 1934.6 2018 1933.5 2032.1 C
-f
-S
-n
-1907.3 2021.8 m
-1906.6 2025.9 1909.4 2032.6 1903.2 2034 C
-1902.8 2034.1 1902.4 2033.9 1902 2033.8 C
-1897.9 2028.5 1891.6 2023.8 1887.4 2018.7 C
-1887.4 2017.7 1887.4 2016.6 1887.4 2015.6 C
-1887 2016.3 1887.2 2017.8 1887.2 2018.9 C
-1892.3 2024.4 1897.5 2028.5 1902.5 2034.3 C
-1904.3 2033.6 1905.7 2032 1907.3 2030.9 C
-1907.3 2027.9 1907.3 2024.9 1907.3 2021.8 C
-f
-S
-n
-1933.7 2023.2 m
-1932 2021.7 1931.1 2024.9 1929.4 2024.9 C
-1931.2 2024.7 1932.4 2021.5 1933.7 2023.2 C
-f
-S
-n
-1989.2 2024.4 m
-1987.4 2023.7 1985.8 2022.2 1985.1 2020.4 C
-1984.6 2020.1 1986 2018.9 1985.1 2019.2 C
-1985.6 2020.8 1984.1 2019.4 1984.6 2021.1 C
-1986.3 2022.3 1988.1 2025.3 1989.2 2024.4 C
-f
-S
-n
-1904.4 2031.9 m
-1903 2029.7 1905.3 2027.7 1904.2 2025.9 C
-1904.5 2025 1903.7 2023 1904 2021.3 C
-1904 2022.3 1903.2 2022 1902.5 2022 C
-1901.3 2022.3 1902.2 2020.1 1901.6 2019.6 C
-1902.5 2019.8 1902.6 2018.3 1903.5 2018.9 C
-1903.7 2021.8 1905.6 2016.8 1905.6 2020.6 C
-1905.9 2020 1906.3 2020.8 1906.1 2021.1 C
-1905.8 2022.7 1906.7 2020.4 1906.4 2019.9 C
-1906.4 2018.5 1908.2 2017.8 1906.8 2016.5 C
-1906.9 2015.7 1907.7 2017.1 1907.1 2016.3 C
-1908.5 2015.8 1910.3 2015.1 1911.6 2016 C
-1912.2 2016.2 1911.9 2018 1911.6 2018 C
-1914.5 2017.1 1910.4 2013.6 1913.3 2013.4 C
-1912.4 2011.3 1910.5 2011.8 1909.5 2010 C
-1910 2010.5 1909 2010.8 1908.8 2011.2 C
-1907.5 2009.9 1906.1 2011.7 1904.9 2011.5 C
-1904.7 2010.9 1904.3 2010.5 1904.4 2009.8 C
-1905 2010.2 1904.6 2008.6 1905.4 2008.1 C
-1906.6 2007.5 1907.7 2008.4 1908.5 2007.4 C
-1908.9 2008.5 1909.7 2008.1 1909 2007.2 C
-1908.1 2006.5 1905.9 2007.3 1905.4 2007.4 C
-1903.9 2007.3 1905.2 2008.5 1904.2 2008.4 C
-1904.6 2009.9 1902.8 2010.3 1902.3 2010.5 C
-1901.5 2009.9 1900.4 2010 1899.4 2010 C
-1898.6 2011.2 1898.2 2013.4 1896.5 2013.4 C
-1896 2012.9 1894.4 2012.9 1893.6 2012.9 C
-1893.1 2013.9 1892.9 2015.5 1891.5 2016 C
-1890.3 2016.1 1889.2 2014 1888.6 2015.8 C
-1890 2016 1891 2016.9 1892.9 2016.5 C
-1894.1 2017.2 1892.8 2018.3 1893.2 2018.9 C
-1892.6 2018.9 1891.1 2019.8 1890.5 2020.6 C
-1891.1 2023.6 1893.2 2019.8 1893.9 2022.5 C
-1894.1 2023.3 1892.7 2023.6 1893.9 2024 C
-1894.2 2024.3 1897.4 2023.8 1896.5 2026.1 C
-1896 2025.6 1897.4 2028.1 1897.5 2027.1 C
-1898.4 2027.4 1899.3 2027 1899.6 2028.5 C
-1899.5 2028.6 1899.4 2028.8 1899.2 2028.8 C
-1899.3 2029.2 1899.6 2029.8 1900.1 2030.2 C
-1900.4 2029.6 1901 2030 1901.8 2030.2 C
-1903.1 2032.1 1900.4 2031.5 1902.8 2033.1 C
-1903.3 2032.7 1904.5 2032 1904.4 2031.9 C
-[0.21 0.21 0 0] vc
-f
-S
-n
-1909.2 2019.4 m
-1908.8 2020.3 1910.2 2019.8 1909.2 2019.2 C
-1908.3 2019.3 1907.6 2020.2 1907.6 2021.3 C
-1908.5 2021 1907.6 2019 1909.2 2019.4 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1915.5 2015.6 m
-1913.5 2016.3 1912.4 2016.8 1911.4 2018.4 C
-1912.5 2017.2 1914 2015.7 1915.5 2015.6 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1915.5 2016.5 m
-1915.1 2016.4 1913.8 2016.6 1913.3 2017.5 C
-1913.7 2017 1914.5 2016.2 1915.5 2016.5 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-vmrs
-1887.4 2012.7 m
-1887.3 2007 1887.2 2001.3 1887.2 1995.6 C
-1887.2 2001.3 1887.3 2007 1887.4 2012.7 C
-[0.18 0.18 0 0.78] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1935.9 2007.4 m
-1936.2 2007.8 1935.8 2007.9 1936.6 2007.9 C
-1935.9 2007.9 1936.1 2006.7 1935.2 2007.2 C
-1935.2 2008.1 1934.1 2007.9 1934.2 2008.8 C
-1935 2008.7 1934.6 2006.9 1935.9 2007.4 C
-f
-S
-n
-1942.1 2003.6 m
-1940.1 2004.3 1939.1 2004.8 1938 2006.4 C
-1939.1 2005.2 1940.6 2003.7 1942.1 2003.6 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1942.1 2004.5 m
-1941.8 2004.4 1940.4 2004.6 1940 2005.5 C
-1940.4 2005 1941.2 2004.2 1942.1 2004.5 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-1914 2000.7 m
-1914 1995 1913.9 1989.3 1913.8 1983.6 C
-1913.9 1989.3 1914 1995 1914 2000.7 C
-f
-S
-n
-1941.6 1998.3 m
-1943.4 2001.9 1942.4 1996 1940.9 1998.3 C
-1941.2 1998.3 1941.4 1998.3 1941.6 1998.3 C
-f
-S
-n
-1954.8 1989.9 m
-1953.9 1989.6 1954.7 1991.6 1953.9 1991.1 C
-1954.5 1993.1 1953.6 1998 1954.6 1993.2 C
-1954 1992.2 1954.7 1990.7 1954.8 1989.9 C
-f
-S
-n
-1947.6 1992.5 m
-1946.2 1993.5 1944.9 1993 1944.8 1994.7 C
-1945.5 1994 1947 1992.2 1947.6 1992.5 C
-f
-S
-n
-1910.7 1982.2 m
-1910.3 1981.8 1909.7 1982 1909.2 1982 C
-1909.7 1982 1910.3 1981.9 1910.7 1982.2 C
-1911 1987.1 1910 1992.6 1910.7 1997.3 C
-1910.7 1992.3 1910.7 1987.2 1910.7 1982.2 C
-[0.65 0.65 0 0.42] vc
-f
-S
-n
-1910.9 1992.8 m
-1910.9 1991.3 1910.9 1989.7 1910.9 1988.2 C
-1910.9 1989.7 1910.9 1991.3 1910.9 1992.8 C
-[0.18 0.18 0 0.78] vc
-f
-S
-n
-vmrs
-1953.6 1983.6 m
-1954.1 1985.3 1953.2 1988.6 1954.8 1989.4 C
-1954.1 1987.9 1954.4 1985.4 1953.6 1983.6 C
-[0.18 0.18 0 0.78] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1910.7 1982 m
-1911.6 1982.9 1911 1984.4 1911.2 1985.6 C
-1911 1984.4 1911.6 1982.9 1910.7 1982 C
-f
-S
-n
-1947.2 1979.6 m
-1947.5 1980.6 1948.3 1980.6 1947.4 1979.6 C
-1946.2 1979.4 1945.7 1978.8 1947.2 1979.6 C
-f
-S
-n
-1930.4 2061.4 m
-1930.4 2058 1930.4 2053.5 1930.4 2051.1 C
-1930.7 2054.6 1929.8 2057.4 1930.1 2061.2 C
-1929.5 2061.9 1929.7 2061.2 1930.4 2061.4 C
-[0.65 0.65 0 0.42] vc
-f
-S
-n
-1939.5 2044.8 m
-1940 2041.5 1935.2 2044.3 1936.4 2040.8 C
-1934.9 2040.9 1934.1 2039.7 1933.5 2038.6 C
-1933.3 2035.4 1933.2 2040 1934 2040.3 C
-1936.2 2040.6 1936.3 2043.6 1938.5 2043.4 C
-1939.7 2044.2 1939.4 2045.6 1938.3 2046.5 C
-1939.1 2046.6 1939.6 2045.6 1939.5 2044.8 C
-f
-S
-n
-1910.4 2045.3 m
-1910.4 2039.5 1910.4 2033.6 1910.4 2027.8 C
-1910.4 2033.6 1910.4 2039.5 1910.4 2045.3 C
-f
-S
-n
-1906.8 2030.9 m
-1907.6 2026.8 1905 2020.8 1909 2018.7 C
-1906.5 2018.9 1906.8 2022.4 1906.8 2024.7 C
-1906.4 2028.2 1907.9 2032 1903 2033.8 C
-1902.2 2034 1903.8 2033.4 1904.2 2033.1 C
-1905.1 2032.4 1905.9 2031.5 1906.8 2030.9 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1907.1 2030.7 m
-1907.1 2028.8 1907.1 2027 1907.1 2025.2 C
-1907.1 2027 1907.1 2028.8 1907.1 2030.7 C
-[0.65 0.65 0 0.42] vc
-f
-S
-n
-1932 2023.2 m
-1932.2 2023.6 1931.7 2023.7 1931.6 2024 C
-1932 2023.7 1932.3 2022.8 1933 2023 C
-1933.9 2024.3 1933.3 2026.2 1933.5 2027.8 C
-1933.5 2026.4 1934.9 2022.2 1932 2023.2 C
-f
-S
-n
-2026.1 2021.6 m
-2026.1 2020.8 2026.1 2019.9 2026.1 2019.2 C
-2026.1 2019.9 2026.1 2020.8 2026.1 2021.6 C
-f
-S
-n
-vmrs
-1934.2 2018.9 m
-1934.2 2015.5 1934.2 2011 1934.2 2008.6 C
-1934.5 2012.1 1933.7 2014.9 1934 2018.7 C
-1933.4 2019.5 1933.5 2018.7 1934.2 2018.9 C
-[0.65 0.65 0 0.42] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1887.6 2014.8 m
-1887.6 2009 1887.6 2003.1 1887.6 1997.3 C
-1887.6 2003.1 1887.6 2009 1887.6 2014.8 C
-f
-S
-n
-1914.3 2002.8 m
-1914.3 1997 1914.3 1991.1 1914.3 1985.3 C
-1914.3 1991.1 1914.3 1997 1914.3 2002.8 C
-f
-S
-n
-1995.4 1992.3 m
-1995.4 1991.5 1995.4 1990.7 1995.4 1989.9 C
-1995.4 1990.7 1995.4 1991.5 1995.4 1992.3 C
-f
-S
-n
-1896 1988.4 m
-1896.9 1988 1897.8 1987.7 1898.7 1987.2 C
-1897.8 1987.7 1896.9 1988 1896 1988.4 C
-f
-S
-n
-1899.4 1986.8 m
-1900.4 1986.3 1901.3 1985.8 1902.3 1985.3 C
-1901.3 1985.8 1900.4 1986.3 1899.4 1986.8 C
-f
-S
-n
-1902.8 1985.1 m
-1905.2 1984 1905.2 1984 1902.8 1985.1 C
-f
-S
-n
-1949.1 1983.4 m
-1950.2 1984.4 1947.8 1984.6 1949.3 1985.1 C
-1949.5 1984.4 1949.6 1984.1 1949.1 1983.4 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1906.1 1983.4 m
-1908.6 1982 1908.6 1982 1906.1 1983.4 C
-[0.65 0.65 0 0.42] vc
-f
-S
-n
-1922.7 1976.4 m
-1923.6 1976 1924.4 1975.7 1925.3 1975.2 C
-1924.4 1975.7 1923.6 1976 1922.7 1976.4 C
-f
-S
-n
-vmrs
-1926 1974.8 m
-1927 1974.3 1928 1973.8 1928.9 1973.3 C
-1928 1973.8 1927 1974.3 1926 1974.8 C
-[0.65 0.65 0 0.42] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1929.4 1973.1 m
-1931.9 1972 1931.9 1972 1929.4 1973.1 C
-f
-S
-n
-1932.8 1971.4 m
-1935.3 1970 1935.3 1970 1932.8 1971.4 C
-f
-S
-n
-1949.6 2097.2 m
-1951.1 2096.4 1952.6 2095.5 1954.1 2094.8 C
-1952.6 2095.5 1951.1 2096.4 1949.6 2097.2 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1955.1 2094.3 m
-1956.7 2093.5 1958.3 2092.7 1959.9 2091.9 C
-1958.3 2092.7 1956.7 2093.5 1955.1 2094.3 C
-f
-S
-n
-1960.4 2091.6 m
-1961.3 2091.2 1962.1 2090.9 1963 2090.4 C
-1962.1 2090.9 1961.3 2091.2 1960.4 2091.6 C
-f
-S
-n
-1963.5 2090.2 m
-1964.4 2089.7 1965.2 2089.2 1966.1 2088.8 C
-1965.2 2089.2 1964.4 2089.7 1963.5 2090.2 C
-f
-S
-n
-1966.6 2088.5 m
-1969.5 2087.1 1972.4 2085.8 1975.2 2084.4 C
-1972.4 2085.8 1969.5 2087.1 1966.6 2088.5 C
-f
-S
-n
-1965.2 2086.1 m
-1965.9 2085.7 1966.8 2085.3 1967.6 2084.9 C
-1966.8 2085.3 1965.9 2085.7 1965.2 2086.1 C
-f
-S
-n
-1968.3 2084.7 m
-1969.2 2084.3 1970 2083.9 1970.9 2083.5 C
-1970 2083.9 1969.2 2084.3 1968.3 2084.7 C
-f
-S
-n
-vmrs
-1984.1 2084 m
-1985.6 2083.2 1987.2 2082.3 1988.7 2081.6 C
-1987.2 2082.3 1985.6 2083.2 1984.1 2084 C
-[0.07 0.06 0 0.58] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1976 2078.7 m
-1978.1 2080.1 1980 2082 1982 2083.7 C
-1980 2081.9 1977.9 2080.3 1976 2078.2 C
-1975.5 2079.9 1975.8 2081.9 1975.7 2083.7 C
-1975.8 2082 1975.5 2080.2 1976 2078.7 C
-f
-S
-n
-1989.6 2081.1 m
-1991.3 2080.3 1992.8 2079.5 1994.4 2078.7 C
-1992.8 2079.5 1991.3 2080.3 1989.6 2081.1 C
-f
-S
-n
-1933.2 2074.6 m
-1932.4 2076.2 1932.8 2077.5 1933 2078.7 C
-1933 2077.6 1932.9 2074.8 1933.2 2074.6 C
-f
-S
-n
-1994.9 2078.4 m
-1995.8 2078 1996.7 2077.7 1997.6 2077.2 C
-1996.7 2077.7 1995.8 2078 1994.9 2078.4 C
-f
-S
-n
-1998 2077 m
-1998.9 2076.5 1999.8 2076 2000.7 2075.6 C
-1999.8 2076 1998.9 2076.5 1998 2077 C
-f
-S
-n
-2001.2 2075.3 m
-2004 2073.9 2006.9 2072.6 2009.8 2071.2 C
-2006.9 2072.6 2004 2073.9 2001.2 2075.3 C
-f
-S
-n
-1980.5 2060.7 m
-1979.9 2060.7 1976.7 2062.8 1975.7 2064.5 C
-1975.7 2067.5 1975.7 2070.5 1975.7 2073.4 C
-1976.3 2068.7 1973.9 2061.6 1980.5 2060.7 C
-f
-S
-n
-1999.7 2072.9 m
-2000.5 2072.5 2001.3 2072.1 2002.1 2071.7 C
-2001.3 2072.1 2000.5 2072.5 1999.7 2072.9 C
-f
-S
-n
-2002.8 2071.5 m
-2003.7 2071.1 2004.6 2070.7 2005.5 2070.3 C
-2004.6 2070.7 2003.7 2071.1 2002.8 2071.5 C
-f
-S
-n
-vmrs
-2015.1 2047.5 m
-2014.4 2047.5 2011.2 2049.6 2010.3 2051.3 C
-2010.3 2057.7 2010.3 2064.1 2010.3 2070.5 C
-2010.3 2063.9 2010.1 2057.1 2010.5 2050.6 C
-2012 2049.3 2013.5 2048.3 2015.1 2047.5 C
-[0.07 0.06 0 0.58] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1910.4 2049.2 m
-1914.8 2054.3 1920.7 2058.9 1925.1 2064 C
-1920.4 2058.6 1915.1 2054.6 1910.4 2049.2 C
-f
-S
-n
-1988.2 2057.3 m
-1989.1 2056.8 1989.9 2056.2 1990.8 2055.6 C
-1989.9 2056.2 1989.1 2056.8 1988.2 2057.3 C
-f
-S
-n
-1991.6 2051.3 m
-1991.6 2046.3 1991.6 2041.2 1991.6 2036.2 C
-1991.6 2041.2 1991.6 2046.3 1991.6 2051.3 C
-f
-S
-n
-1935.6 2047.5 m
-1932.9 2051.7 1939.7 2043.8 1935.6 2047.5 C
-f
-S
-n
-1938.8 2043.9 m
-1938.1 2043.3 1938.2 2043.7 1937.3 2043.4 C
-1938.7 2043 1938.2 2044.9 1939 2045.3 C
-1938.2 2045.3 1938.7 2046.6 1937.8 2046.5 C
-1939.1 2046.2 1939.1 2044.5 1938.8 2043.9 C
-f
-S
-n
-1972.4 2045.6 m
-1973.4 2045 1974.5 2044.4 1975.5 2043.9 C
-1974.5 2044.4 1973.4 2045 1972.4 2045.6 C
-f
-S
-n
-1969 2043.6 m
-1969.8 2043.2 1970.6 2042.9 1971.4 2042.4 C
-1970.6 2042.9 1969.8 2043.2 1969 2043.6 C
-f
-S
-n
-1972.1 2042.2 m
-1973 2041.8 1973.9 2041.4 1974.8 2041 C
-1973.9 2041.4 1973 2041.8 1972.1 2042.2 C
-f
-S
-n
-1906.6 2035 m
-1905 2034.7 1904.8 2036.6 1903.5 2036.9 C
-1904.9 2037 1905.8 2033.4 1907.1 2035.7 C
-1907.1 2037.2 1907.1 2038.6 1907.1 2040 C
-1906.9 2038.4 1907.5 2036.4 1906.6 2035 C
-f
-S
-n
-vmrs
-1937.1 2032.1 m
-1936.2 2033.7 1936.6 2035 1936.8 2036.2 C
-1936.8 2035.1 1936.8 2032.4 1937.1 2032.1 C
-[0.07 0.06 0 0.58] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1887.6 2018.7 m
-1892 2023.8 1897.9 2028.4 1902.3 2033.6 C
-1897.6 2028.1 1892.3 2024.1 1887.6 2018.7 C
-f
-S
-n
-1999.7 2031.4 m
-1998.7 2030.3 1997.6 2029.2 1996.6 2028 C
-1997.6 2029.2 1998.7 2030.3 1999.7 2031.4 C
-f
-S
-n
-1912.8 2017 m
-1910.6 2021.1 1913.6 2015.3 1914.5 2016 C
-1914 2016.3 1913.4 2016.7 1912.8 2017 C
-f
-S
-n
-1939.5 2005 m
-1936.7 2009.2 1943.6 2001.3 1939.5 2005 C
-f
-S
-n
-1942.6 2001.4 m
-1941.9 2000.8 1942 2001.2 1941.2 2000.9 C
-1942.5 2000.6 1942.1 2002.4 1942.8 2002.8 C
-1942 2002.8 1942.5 2004.1 1941.6 2004 C
-1943 2003.7 1942.9 2002.1 1942.6 2001.4 C
-f
-S
-n
-2006.2 2000.7 m
-2005.4 2001.5 2004 2002.8 2004 2002.8 C
-2004.5 2002.4 2005.5 2001.4 2006.2 2000.7 C
-f
-S
-n
-1998.5 2001.6 m
-1997.7 2002 1996.8 2002.4 1995.9 2002.6 C
-1995.5 1999.3 1995.7 1995.7 1995.6 1992.3 C
-1995.6 1995.7 1995.6 1999.2 1995.6 2002.6 C
-1996.6 2002.4 1997.7 2002.2 1998.5 2001.6 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1996.1 2002.8 m
-1995.9 2002.8 1995.8 2002.8 1995.6 2002.8 C
-1995.2 1999.5 1995.5 1995.9 1995.4 1992.5 C
-1995.4 1995.9 1995.4 1999.4 1995.4 2002.8 C
-1996.4 2003.1 1998.2 2001.6 1996.1 2002.8 C
-[0.07 0.06 0 0.58] vc
-f
-S
-n
-1969 2002.1 m
-1968 2001 1966.9 1999.9 1965.9 1998.8 C
-1966.9 1999.9 1968 2001 1969 2002.1 C
-f
-S
-n
-vmrs
-2000 2001.2 m
-2002.1 2000 2004.1 1998.9 2006.2 1997.8 C
-2004.1 1998.9 2002.1 2000 2000 2001.2 C
-[0.07 0.06 0 0.58] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1895.8 1984.8 m
-1898.3 1983.6 1900.8 1982.3 1903.2 1981 C
-1900.8 1982.3 1898.3 1983.6 1895.8 1984.8 C
-f
-S
-n
-1905.2 1980.3 m
-1906.4 1979.9 1907.6 1979.5 1908.8 1979.1 C
-1907.6 1979.5 1906.4 1979.9 1905.2 1980.3 C
-f
-S
-n
-1964.7 1977.4 m
-1963.8 1977.5 1962.5 1980.2 1960.8 1980 C
-1962.5 1980.2 1963.3 1978 1964.7 1977.4 C
-f
-S
-n
-1952 1979.6 m
-1955.2 1979.2 1955.2 1979.2 1952 1979.6 C
-f
-S
-n
-1937.8 1966.4 m
-1941.2 1969.5 1946.1 1976.4 1951.5 1979.3 C
-1946.1 1976.7 1942.8 1970.4 1937.8 1966.4 C
-f
-S
-n
-1911.9 1978.6 m
-1914.3 1977.4 1916.7 1976.2 1919.1 1975 C
-1916.7 1976.2 1914.3 1977.4 1911.9 1978.6 C
-f
-S
-n
-1975.5 1971.4 m
-1974.6 1972.2 1973.3 1973.6 1973.3 1973.6 C
-1973.7 1973.1 1974.8 1972.1 1975.5 1971.4 C
-f
-S
-n
-1922.4 1972.8 m
-1924.9 1971.6 1927.4 1970.3 1929.9 1969 C
-1927.4 1970.3 1924.9 1971.6 1922.4 1972.8 C
-f
-S
-n
-1969.2 1971.9 m
-1971.1 1970.9 1972.9 1969.8 1974.8 1968.8 C
-1972.9 1969.8 1971.1 1970.9 1969.2 1971.9 C
-f
-S
-n
-vmrs
-1931.8 1968.3 m
-1933 1967.9 1934.2 1967.5 1935.4 1967.1 C
-1934.2 1967.5 1933 1967.9 1931.8 1968.3 C
-[0.07 0.06 0 0.58] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1940.7 2072.4 m
-1941.5 2072.4 1942.3 2072.3 1943.1 2072.2 C
-1942.3 2072.3 1941.5 2072.4 1940.7 2072.4 C
-[0 0 0 0.18] vc
-f
-S
-n
-1948.6 2069.3 m
-1947 2069.5 1945.7 2068.9 1944.8 2069.8 C
-1945.9 2068.5 1948.4 2070.2 1948.6 2069.3 C
-f
-S
-n
-1954.6 2066.4 m
-1954.7 2067.9 1955.6 2067.3 1955.6 2068.8 C
-1955.4 2067.8 1956 2066.6 1954.6 2066.4 C
-f
-S
-n
-1929.2 2061.2 m
-1927.8 2062.1 1926.3 2064.1 1924.8 2063.3 C
-1926.3 2064.6 1928 2062 1929.2 2061.2 C
-f
-S
-n
-1924.4 2067.4 m
-1918.5 2061.6 1912.7 2055.9 1906.8 2050.1 C
-1912.7 2055.9 1918.5 2061.6 1924.4 2067.4 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1924.6 2062.8 m
-1923.9 2062.1 1923.2 2061.2 1922.4 2060.4 C
-1923.2 2061.2 1923.9 2062.1 1924.6 2062.8 C
-[0 0 0 0.18] vc
-f
-S
-n
-1919.3 2057.3 m
-1917.5 2055.6 1915.7 2053.8 1913.8 2052 C
-1915.7 2053.8 1917.5 2055.6 1919.3 2057.3 C
-f
-S
-n
-1929.2 2055.2 m
-1929.2 2054.2 1929.2 2053.2 1929.2 2052.3 C
-1929.2 2053.2 1929.2 2054.2 1929.2 2055.2 C
-f
-S
-n
-1926.3 2049.6 m
-1925.4 2049 1925.4 2050.5 1924.4 2050.4 C
-1925.3 2051.3 1924.5 2051.9 1925.6 2052.5 C
-1926.9 2052.6 1926 2050.6 1926.3 2049.6 C
-f
-S
-n
-vmrs
-1911.2 2046.8 m
-1910.1 2048.9 1911.9 2050.1 1913.1 2051.3 C
-1912.1 2049.9 1910.6 2048.8 1911.2 2046.8 C
-[0 0 0 0.18] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1934 2048.7 m
-1932.6 2048.7 1930.1 2047.7 1929.6 2049.4 C
-1930.9 2048.6 1933.3 2049 1934 2048.7 C
-f
-S
-n
-1980 2048.4 m
-1979.5 2046.8 1976.3 2047.9 1977.2 2045.6 C
-1976.8 2045.1 1976.1 2044.7 1975.2 2044.8 C
-1973.7 2046 1976.3 2046.4 1976.7 2047.5 C
-1977.8 2047.2 1978.2 2050 1979.6 2049.2 C
-1980 2049 1979.6 2048.6 1980 2048.4 C
-f
-S
-n
-1938.3 2045.6 m
-1938.2 2044.4 1936.8 2043.8 1935.9 2043.4 C
-1936.4 2044.4 1939.1 2044.3 1937.6 2045.8 C
-1937 2046.1 1935.9 2046.1 1935.9 2046.8 C
-1936.7 2046.3 1937.8 2046.2 1938.3 2045.6 C
-f
-S
-n
-1932.5 2040 m
-1932.8 2038.1 1932 2038.9 1932.3 2040.3 C
-1933.1 2040.3 1932.7 2041.7 1933.7 2041.5 C
-1933.1 2041 1932.9 2040.5 1932.5 2040 C
-f
-S
-n
-2014.6 2035.2 m
-2014.1 2033.6 2010.9 2034.7 2011.7 2032.4 C
-2011.3 2031.9 2009.4 2030.7 2009.3 2032.1 C
-2009.5 2033.7 2012.9 2033.8 2012.4 2035.7 C
-2013 2036.4 2014.2 2036.5 2014.6 2035.2 C
-f
-S
-n
-1906.4 2030.7 m
-1905 2031.6 1903.5 2033.6 1902 2032.8 C
-1903.4 2034 1905.6 2031.4 1906.4 2030.7 C
-f
-S
-n
-1901.8 2037.2 m
-1899.5 2034.8 1897.2 2032.5 1894.8 2030.2 C
-1897.2 2032.5 1899.5 2034.8 1901.8 2037.2 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1901.8 2032.4 m
-1901.1 2031.6 1900.4 2030.7 1899.6 2030 C
-1900.4 2030.7 1901.1 2031.6 1901.8 2032.4 C
-[0 0 0 0.18] vc
-f
-S
-n
-1944.5 2030 m
-1945.3 2029.9 1946.1 2029.8 1946.9 2029.7 C
-1946.1 2029.8 1945.3 2029.9 1944.5 2030 C
-f
-S
-n
-vmrs
-1997.8 2027.8 m
-1997.7 2027.9 1997.6 2028.1 1997.3 2028 C
-1997.4 2029.1 1998.5 2029.5 1999.2 2030 C
-2000.1 2029.5 1998.9 2028 1997.8 2027.8 C
-[0 0 0 0.18] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1906.4 2029.2 m
-1906.4 2026.6 1906.4 2024 1906.4 2021.3 C
-1906.4 2024 1906.4 2026.6 1906.4 2029.2 C
-f
-S
-n
-2006.2 2025.9 m
-2006 2025.9 2005.8 2025.8 2005.7 2025.6 C
-2005.7 2025.5 2005.7 2025.3 2005.7 2025.2 C
-2004.6 2025.8 2002.7 2024.7 2001.9 2026.1 C
-2001.9 2027.9 2007.8 2029.2 2006.2 2025.9 C
-[0 0 0 0] vc
-f
-S
-n
-1952.4 2026.8 m
-1950.9 2027 1949.6 2026.4 1948.6 2027.3 C
-1949.7 2026.1 1952.2 2027.7 1952.4 2026.8 C
-[0 0 0 0.18] vc
-f
-S
-n
-1896.5 2026.8 m
-1894.7 2025.1 1892.9 2023.3 1891 2021.6 C
-1892.9 2023.3 1894.7 2025.1 1896.5 2026.8 C
-f
-S
-n
-1958.4 2024 m
-1958.5 2025.5 1959.4 2024.8 1959.4 2026.4 C
-1959.3 2025.3 1959.8 2024.1 1958.4 2024 C
-f
-S
-n
-1903.5 2019.2 m
-1902.6 2018.6 1902.6 2020 1901.6 2019.9 C
-1902.5 2020.8 1901.7 2021.4 1902.8 2022 C
-1904.1 2022.2 1903.2 2020.1 1903.5 2019.2 C
-f
-S
-n
-1933 2018.7 m
-1931.7 2019.6 1930.1 2021.6 1928.7 2020.8 C
-1930.1 2022.1 1931.8 2019.5 1933 2018.7 C
-f
-S
-n
-1888.4 2016.3 m
-1887.3 2018.4 1889.1 2019.6 1890.3 2020.8 C
-1889.3 2019.5 1887.8 2018.3 1888.4 2016.3 C
-f
-S
-n
-1928.4 2020.4 m
-1927.7 2019.6 1927 2018.7 1926.3 2018 C
-1927 2018.7 1927.7 2019.6 1928.4 2020.4 C
-f
-S
-n
-vmrs
-1911.2 2018.2 m
-1909.8 2018.3 1907.3 2017.2 1906.8 2018.9 C
-1908.1 2018.1 1910.5 2018.6 1911.2 2018.2 C
-[0 0 0 0.18] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1915.5 2015.1 m
-1915.4 2013.9 1914 2013.3 1913.1 2012.9 C
-1913.6 2013.9 1916.3 2013.8 1914.8 2015.3 C
-1914.2 2015.6 1913.1 2015.6 1913.1 2016.3 C
-1913.9 2015.9 1915 2015.7 1915.5 2015.1 C
-f
-S
-n
-1923.2 2014.8 m
-1921.3 2013.1 1919.5 2011.3 1917.6 2009.6 C
-1919.5 2011.3 1921.3 2013.1 1923.2 2014.8 C
-f
-S
-n
-1933 2012.7 m
-1933 2011.7 1933 2010.8 1933 2009.8 C
-1933 2010.8 1933 2011.7 1933 2012.7 C
-f
-S
-n
-1909.7 2008.1 m
-1908.9 2009.2 1910.1 2009.9 1910.4 2011 C
-1911.1 2010.7 1908.9 2009.7 1909.7 2008.1 C
-f
-S
-n
-1930.1 2007.2 m
-1929.2 2006.6 1929.2 2008 1928.2 2007.9 C
-1929.1 2008.8 1928.4 2009.4 1929.4 2010 C
-1930.7 2010.2 1929.9 2008.1 1930.1 2007.2 C
-f
-S
-n
-1915 2004.3 m
-1914 2006.4 1915.7 2007.6 1916.9 2008.8 C
-1915.9 2007.5 1914.4 2006.3 1915 2004.3 C
-f
-S
-n
-1937.8 2006.2 m
-1936.4 2006.3 1934 2005.2 1933.5 2006.9 C
-1934.7 2006.1 1937.1 2006.6 1937.8 2006.2 C
-f
-S
-n
-1983.9 2006 m
-1983.3 2004.3 1980.2 2005.4 1981 2003.1 C
-1980.6 2002.7 1978.7 2001.5 1978.6 2002.8 C
-1978.8 2004.4 1982.1 2004.5 1981.7 2006.4 C
-1982.3 2007.2 1983.5 2007.2 1983.9 2006 C
-f
-S
-n
-1942.1 2003.1 m
-1942 2001.9 1940.6 2001.3 1939.7 2000.9 C
-1940.2 2001.9 1943 2001.8 1941.4 2003.3 C
-1940.9 2003.6 1939.7 2003.6 1939.7 2004.3 C
-1940.5 2003.9 1941.6 2003.7 1942.1 2003.1 C
-f
-S
-n
-vmrs
-1967.1 1998.5 m
-1967 1998.6 1966.8 1998.8 1966.6 1998.8 C
-1966.7 1999.8 1967.8 2000.2 1968.5 2000.7 C
-1969.4 2000.2 1968.2 1998.8 1967.1 1998.5 C
-[0 0 0 0.18] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1936.4 1997.6 m
-1936.7 1995.6 1935.8 1996.4 1936.1 1997.8 C
-1936.9 1997.9 1936.5 1999.2 1937.6 1999 C
-1937 1998.5 1936.8 1998 1936.4 1997.6 C
-f
-S
-n
-1975.5 1996.6 m
-1975.2 1996.7 1975.1 1996.5 1975 1996.4 C
-1975 1996.2 1975 1996.1 1975 1995.9 C
-1973.9 1996.5 1972 1995.5 1971.2 1996.8 C
-1971.2 1998.6 1977 1999.9 1975.5 1996.6 C
-[0 0 0 0] vc
-f
-S
-n
-1949.3 2097.4 m
-1950.3 2096.9 1951.2 2096.4 1952.2 2096 C
-1951.2 2096.4 1950.3 2096.9 1949.3 2097.4 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1960.8 2091.6 m
-1961.7 2091.2 1962.6 2090.9 1963.5 2090.4 C
-1962.6 2090.9 1961.7 2091.2 1960.8 2091.6 C
-f
-S
-n
-1964.4 2090 m
-1965.7 2089.2 1967 2088.5 1968.3 2087.8 C
-1967 2088.5 1965.7 2089.2 1964.4 2090 C
-f
-S
-n
-1976 2083.7 m
-1976.3 2082.3 1975.2 2079.1 1976.9 2079.4 C
-1978.8 2080.7 1980.3 2082.9 1982.2 2084.2 C
-1980.6 2083.1 1978.2 2080.2 1976 2078.9 C
-1975.6 2081.2 1977 2084.9 1973.8 2085.4 C
-1972.2 2086.1 1970.7 2087 1969 2087.6 C
-1971.4 2086.5 1974.1 2085.6 1976 2083.7 C
-f
-S
-n
-1983.9 2084.2 m
-1984.8 2083.7 1985.8 2083.2 1986.8 2082.8 C
-1985.8 2083.2 1984.8 2083.7 1983.9 2084.2 C
-f
-S
-n
-1995.4 2078.4 m
-1996.3 2078 1997.1 2077.7 1998 2077.2 C
-1997.1 2077.7 1996.3 2078 1995.4 2078.4 C
-f
-S
-n
-1999 2076.8 m
-2000.3 2076 2001.6 2075.3 2002.8 2074.6 C
-2001.6 2075.3 2000.3 2076 1999 2076.8 C
-f
-S
-n
-vmrs
-1929.6 2065.7 m
-1930.1 2065.6 1929.8 2068.6 1929.9 2070 C
-1929.8 2068.6 1930.1 2067 1929.6 2065.7 C
-[0.4 0.4 0 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1906.6 2049.4 m
-1906.6 2046.7 1906.6 2043.9 1906.6 2041.2 C
-1906.6 2043.9 1906.6 2046.7 1906.6 2049.4 C
-f
-S
-n
-2016 2047.5 m
-2014.8 2048 2013.5 2048.3 2012.4 2049.4 C
-2013.5 2048.3 2014.8 2048 2016 2047.5 C
-f
-S
-n
-2016.5 2047.2 m
-2017.3 2046.9 2018.1 2046.6 2018.9 2046.3 C
-2018.1 2046.6 2017.3 2046.9 2016.5 2047.2 C
-f
-S
-n
-1912.4 2028.5 m
-1911.8 2032.4 1912.4 2037.2 1911.9 2041.2 C
-1911.5 2037.2 1911.7 2032.9 1911.6 2028.8 C
-1911.6 2033.5 1911.6 2038.9 1911.6 2042.9 C
-1912.5 2042.2 1911.6 2043.9 1912.6 2043.6 C
-1912.9 2039.3 1913.1 2033.3 1912.4 2028.5 C
-[0.21 0.21 0 0] vc
-f
-S
-n
-1906.8 2040.8 m
-1906.8 2039 1906.8 2037.2 1906.8 2035.5 C
-1906.8 2037.2 1906.8 2039 1906.8 2040.8 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1905.9 2035.2 m
-1904.9 2036.4 1903.7 2037.2 1902.3 2037.4 C
-1903.7 2037.2 1904.9 2036.4 1905.9 2035.2 C
-f
-S
-n
-1906.1 2031.2 m
-1907 2031.1 1906.4 2028 1906.6 2030.7 C
-1905.5 2032.1 1904 2032.8 1902.5 2033.6 C
-1903.9 2033.2 1905 2032.1 1906.1 2031.2 C
-f
-S
-n
-1908.3 2018.7 m
-1905.2 2018.6 1907.1 2023.2 1906.6 2025.4 C
-1906.8 2023 1905.9 2019.5 1908.3 2018.7 C
-f
-S
-n
-1889.6 1998 m
-1889 2001.9 1889.6 2006.7 1889.1 2010.8 C
-1888.7 2006.7 1888.9 2002.4 1888.8 1998.3 C
-1888.8 2003 1888.8 2008.4 1888.8 2012.4 C
-1889.7 2011.7 1888.8 2013.4 1889.8 2013.2 C
-1890.1 2008.8 1890.3 2002.8 1889.6 1998 C
-[0.21 0.21 0 0] vc
-f
-S
-n
-vmrs
-1999 2001.4 m
-2001 2000.3 2003 1999.2 2005 1998 C
-2003 1999.2 2001 2000.3 1999 2001.4 C
-[0.4 0.4 0 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1916.2 1986 m
-1915.7 1989.9 1916.3 1994.7 1915.7 1998.8 C
-1915.3 1994.7 1915.5 1990.4 1915.5 1986.3 C
-1915.5 1991 1915.5 1996.4 1915.5 2000.4 C
-1916.3 1999.7 1915.5 2001.4 1916.4 2001.2 C
-1916.7 1996.8 1917 1990.8 1916.2 1986 C
-[0.21 0.21 0 0] vc
-f
-S
-n
-1886.9 1989.6 m
-1887.8 1989.2 1888.7 1988.9 1889.6 1988.4 C
-1888.7 1988.9 1887.8 1989.2 1886.9 1989.6 C
-[0.4 0.4 0 0] vc
-f
-S
-n
-1892.4 1986.8 m
-1895.1 1985.1 1897.9 1983.6 1900.6 1982 C
-1897.9 1983.6 1895.1 1985.1 1892.4 1986.8 C
-f
-S
-n
-1907.3 1979.3 m
-1908.5 1978.9 1909.7 1978.5 1910.9 1978.1 C
-1909.7 1978.5 1908.5 1978.9 1907.3 1979.3 C
-f
-S
-n
-1938.5 1966.6 m
-1942.6 1970.1 1945.9 1976.4 1951.7 1979.1 C
-1946.2 1976.1 1943.1 1970.9 1938.5 1966.6 C
-f
-S
-n
-1955.1 1978.6 m
-1955.9 1978.2 1956.7 1977.8 1957.5 1977.4 C
-1956.7 1977.8 1955.9 1978.2 1955.1 1978.6 C
-f
-S
-n
-1913.6 1977.6 m
-1914.5 1977.2 1915.3 1976.9 1916.2 1976.4 C
-1915.3 1976.9 1914.5 1977.2 1913.6 1977.6 C
-f
-S
-n
-1919.1 1974.8 m
-1921.8 1973.1 1924.5 1971.6 1927.2 1970 C
-1924.5 1971.6 1921.8 1973.1 1919.1 1974.8 C
-f
-S
-n
-1963.5 1974.5 m
-1964.5 1974 1965.6 1973.4 1966.6 1972.8 C
-1965.6 1973.4 1964.5 1974 1963.5 1974.5 C
-f
-S
-n
-vmrs
-1967.8 1972.4 m
-1970 1971.2 1972.1 1970 1974.3 1968.8 C
-1972.1 1970 1970 1971.2 1967.8 1972.4 C
-[0.4 0.4 0 0] vc
-f
-0.4 w
-2 J
-2 M
-S
-n
-1934 1967.3 m
-1935.2 1966.9 1936.4 1966.5 1937.6 1966.1 C
-1936.4 1966.5 1935.2 1966.9 1934 1967.3 C
-f
-S
-n
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Binary file doc-src/IsarAdvanced/Functions/isabelle_isar.pdf has changed
--- a/doc-src/IsarAdvanced/Functions/mathpartir.sty Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,421 +0,0 @@
-% Mathpartir --- Math Paragraph for Typesetting Inference Rules
-%
-% Copyright (C) 2001, 2002, 2003, 2004, 2005 Didier Rémy
-%
-% Author : Didier Remy
-% Version : 1.2.0
-% Bug Reports : to author
-% Web Site : http://pauillac.inria.fr/~remy/latex/
-%
-% Mathpartir is free software; you can redistribute it and/or modify
-% it under the terms of the GNU General Public License as published by
-% the Free Software Foundation; either version 2, or (at your option)
-% any later version.
-%
-% Mathpartir is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-% GNU General Public License for more details
-% (http://pauillac.inria.fr/~remy/license/GPL).
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% File mathpartir.sty (LaTeX macros)
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{mathpartir}
- [2005/12/20 version 1.2.0 Math Paragraph for Typesetting Inference Rules]
-
-%%
-
-%% Identification
-%% Preliminary declarations
-
-\RequirePackage {keyval}
-
-%% Options
-%% More declarations
-
-%% PART I: Typesetting maths in paragraphe mode
-
-\newdimen \mpr@tmpdim
-
-% To ensure hevea \hva compatibility, \hva should expands to nothing
-% in mathpar or in inferrule
-\let \mpr@hva \empty
-
-%% normal paragraph parametters, should rather be taken dynamically
-\def \mpr@savepar {%
- \edef \MathparNormalpar
- {\noexpand \lineskiplimit \the\lineskiplimit
- \noexpand \lineskip \the\lineskip}%
- }
-
-\def \mpr@rulelineskip {\lineskiplimit=0.3em\lineskip=0.2em plus 0.1em}
-\def \mpr@lesslineskip {\lineskiplimit=0.6em\lineskip=0.5em plus 0.2em}
-\def \mpr@lineskip {\lineskiplimit=1.2em\lineskip=1.2em plus 0.2em}
-\let \MathparLineskip \mpr@lineskip
-\def \mpr@paroptions {\MathparLineskip}
-\let \mpr@prebindings \relax
-
-\newskip \mpr@andskip \mpr@andskip 2em plus 0.5fil minus 0.5em
-
-\def \mpr@goodbreakand
- {\hskip -\mpr@andskip \penalty -1000\hskip \mpr@andskip}
-\def \mpr@and {\hskip \mpr@andskip}
-\def \mpr@andcr {\penalty 50\mpr@and}
-\def \mpr@cr {\penalty -10000\mpr@and}
-\def \mpr@eqno #1{\mpr@andcr #1\hskip 0em plus -1fil \penalty 10}
-
-\def \mpr@bindings {%
- \let \and \mpr@andcr
- \let \par \mpr@andcr
- \let \\\mpr@cr
- \let \eqno \mpr@eqno
- \let \hva \mpr@hva
- }
-\let \MathparBindings \mpr@bindings
-
-% \@ifundefined {ignorespacesafterend}
-% {\def \ignorespacesafterend {\aftergroup \ignorespaces}
-
-\newenvironment{mathpar}[1][]
- {$$\mpr@savepar \parskip 0em \hsize \linewidth \centering
- \vbox \bgroup \mpr@prebindings \mpr@paroptions #1\ifmmode $\else
- \noindent $\displaystyle\fi
- \MathparBindings}
- {\unskip \ifmmode $\fi\egroup $$\ignorespacesafterend}
-
-% \def \math@mathpar #1{\setbox0 \hbox {$\displaystyle #1$}\ifnum
-% \wd0 < \hsize $$\box0$$\else \bmathpar #1\emathpar \fi}
-
-%%% HOV BOXES
-
-\def \mathvbox@ #1{\hbox \bgroup \mpr@normallineskip
- \vbox \bgroup \tabskip 0em \let \\ \cr
- \halign \bgroup \hfil $##$\hfil\cr #1\crcr \egroup \egroup
- \egroup}
-
-\def \mathhvbox@ #1{\setbox0 \hbox {\let \\\qquad $#1$}\ifnum \wd0 < \hsize
- \box0\else \mathvbox {#1}\fi}
-
-
-%% Part II -- operations on lists
-
-\newtoks \mpr@lista
-\newtoks \mpr@listb
-
-\long \def\mpr@cons #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
-{#2}\edef #2{\the \mpr@lista \the \mpr@listb}}
-
-\long \def\mpr@snoc #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
-{#2}\edef #2{\the \mpr@listb\the\mpr@lista}}
-
-\long \def \mpr@concat#1=#2\mpr@to#3{\mpr@lista \expandafter {#2}\mpr@listb
-\expandafter {#3}\edef #1{\the \mpr@listb\the\mpr@lista}}
-
-\def \mpr@head #1\mpr@to #2{\expandafter \mpr@head@ #1\mpr@head@ #1#2}
-\long \def \mpr@head@ #1#2\mpr@head@ #3#4{\def #4{#1}\def#3{#2}}
-
-\def \mpr@flatten #1\mpr@to #2{\expandafter \mpr@flatten@ #1\mpr@flatten@ #1#2}
-\long \def \mpr@flatten@ \\#1\\#2\mpr@flatten@ #3#4{\def #4{#1}\def #3{\\#2}}
-
-\def \mpr@makelist #1\mpr@to #2{\def \mpr@all {#1}%
- \mpr@lista {\\}\mpr@listb \expandafter {\mpr@all}\edef \mpr@all {\the
- \mpr@lista \the \mpr@listb \the \mpr@lista}\let #2\empty
- \def \mpr@stripof ##1##2\mpr@stripend{\def \mpr@stripped{##2}}\loop
- \mpr@flatten \mpr@all \mpr@to \mpr@one
- \expandafter \mpr@snoc \mpr@one \mpr@to #2\expandafter \mpr@stripof
- \mpr@all \mpr@stripend
- \ifx \mpr@stripped \empty \let \mpr@isempty 0\else \let \mpr@isempty 1\fi
- \ifx 1\mpr@isempty
- \repeat
-}
-
-\def \mpr@rev #1\mpr@to #2{\let \mpr@tmp \empty
- \def \\##1{\mpr@cons ##1\mpr@to \mpr@tmp}#1\let #2\mpr@tmp}
-
-%% Part III -- Type inference rules
-
-\newif \if@premisse
-\newbox \mpr@hlist
-\newbox \mpr@vlist
-\newif \ifmpr@center \mpr@centertrue
-\def \mpr@htovlist {%
- \setbox \mpr@hlist
- \hbox {\strut
- \ifmpr@center \hskip -0.5\wd\mpr@hlist\fi
- \unhbox \mpr@hlist}%
- \setbox \mpr@vlist
- \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
- \else \unvbox \mpr@vlist \box \mpr@hlist
- \fi}%
-}
-% OLD version
-% \def \mpr@htovlist {%
-% \setbox \mpr@hlist
-% \hbox {\strut \hskip -0.5\wd\mpr@hlist \unhbox \mpr@hlist}%
-% \setbox \mpr@vlist
-% \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
-% \else \unvbox \mpr@vlist \box \mpr@hlist
-% \fi}%
-% }
-
-\def \mpr@item #1{$\displaystyle #1$}
-\def \mpr@sep{2em}
-\def \mpr@blank { }
-\def \mpr@hovbox #1#2{\hbox
- \bgroup
- \ifx #1T\@premissetrue
- \else \ifx #1B\@premissefalse
- \else
- \PackageError{mathpartir}
- {Premisse orientation should either be T or B}
- {Fatal error in Package}%
- \fi \fi
- \def \@test {#2}\ifx \@test \mpr@blank\else
- \setbox \mpr@hlist \hbox {}%
- \setbox \mpr@vlist \vbox {}%
- \if@premisse \let \snoc \mpr@cons \else \let \snoc \mpr@snoc \fi
- \let \@hvlist \empty \let \@rev \empty
- \mpr@tmpdim 0em
- \expandafter \mpr@makelist #2\mpr@to \mpr@flat
- \if@premisse \mpr@rev \mpr@flat \mpr@to \@rev \else \let \@rev \mpr@flat \fi
- \def \\##1{%
- \def \@test {##1}\ifx \@test \empty
- \mpr@htovlist
- \mpr@tmpdim 0em %%% last bug fix not extensively checked
- \else
- \setbox0 \hbox{\mpr@item {##1}}\relax
- \advance \mpr@tmpdim by \wd0
- %\mpr@tmpdim 1.02\mpr@tmpdim
- \ifnum \mpr@tmpdim < \hsize
- \ifnum \wd\mpr@hlist > 0
- \if@premisse
- \setbox \mpr@hlist
- \hbox {\unhbox0 \hskip \mpr@sep \unhbox \mpr@hlist}%
- \else
- \setbox \mpr@hlist
- \hbox {\unhbox \mpr@hlist \hskip \mpr@sep \unhbox0}%
- \fi
- \else
- \setbox \mpr@hlist \hbox {\unhbox0}%
- \fi
- \else
- \ifnum \wd \mpr@hlist > 0
- \mpr@htovlist
- \mpr@tmpdim \wd0
- \fi
- \setbox \mpr@hlist \hbox {\unhbox0}%
- \fi
- \advance \mpr@tmpdim by \mpr@sep
- \fi
- }%
- \@rev
- \mpr@htovlist
- \ifmpr@center \hskip \wd\mpr@vlist\fi \box \mpr@vlist
- \fi
- \egroup
-}
-
-%%% INFERENCE RULES
-
-\@ifundefined{@@over}{%
- \let\@@over\over % fallback if amsmath is not loaded
- \let\@@overwithdelims\overwithdelims
- \let\@@atop\atop \let\@@atopwithdelims\atopwithdelims
- \let\@@above\above \let\@@abovewithdelims\abovewithdelims
- }{}
-
-%% The default
-
-\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
- $\displaystyle {#1\mpr@over #2}$}}
-\let \mpr@fraction \mpr@@fraction
-
-%% A generic solution to arrow
-
-\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
- \def \mpr@tail{#1}%
- \def \mpr@body{#2}%
- \def \mpr@head{#3}%
- \setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
- \setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
- \setbox3=\hbox{$\mkern -3mu \mpr@body\mkern -3mu$}%
- \dimen0=\dp1\advance\dimen0 by \ht3\relax\dp1\dimen0\relax
- \dimen0=\ht2\advance\dimen0 by \dp3\relax\ht2\dimen0\relax
- \setbox0=\hbox {$\box1 \@@atop \box2$}%
- \dimen0=\wd0\box0
- \box0 \hskip -\dimen0\relax
- \hbox to \dimen0 {$%
- \mathrel{\mpr@tail}\joinrel
- \xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
- $}}}
-
-%% Old stuff should be removed in next version
-\def \mpr@@reduce #1#2{\hbox
- {$\lower 0.01pt \mpr@@fraction {#1}{#2}\mkern -15mu\rightarrow$}}
-\def \mpr@@rewrite #1#2#3{\hbox
- {$\lower 0.01pt \mpr@@fraction {#2}{#3}\mkern -8mu#1$}}
-\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
-
-\def \mpr@empty {}
-\def \mpr@inferrule
- {\bgroup
- \ifnum \linewidth<\hsize \hsize \linewidth\fi
- \mpr@rulelineskip
- \let \and \qquad
- \let \hva \mpr@hva
- \let \@rulename \mpr@empty
- \let \@rule@options \mpr@empty
- \let \mpr@over \@@over
- \mpr@inferrule@}
-\newcommand {\mpr@inferrule@}[3][]
- {\everymath={\displaystyle}%
- \def \@test {#2}\ifx \empty \@test
- \setbox0 \hbox {$\vcenter {\mpr@hovbox{B}{#3}}$}%
- \else
- \def \@test {#3}\ifx \empty \@test
- \setbox0 \hbox {$\vcenter {\mpr@hovbox{T}{#2}}$}%
- \else
- \setbox0 \mpr@fraction {\mpr@hovbox{T}{#2}}{\mpr@hovbox{B}{#3}}%
- \fi \fi
- \def \@test {#1}\ifx \@test\empty \box0
- \else \vbox
-%%% Suggestion de Francois pour les etiquettes longues
-%%% {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
- {\hbox {\RefTirName {#1}}\box0}\fi
- \egroup}
-
-\def \mpr@vdotfil #1{\vbox to #1{\leaders \hbox{$\cdot$} \vfil}}
-
-% They are two forms
-% \inferrule [label]{[premisses}{conclusions}
-% or
-% \inferrule* [options]{[premisses}{conclusions}
-%
-% Premisses and conclusions are lists of elements separated by \\
-% Each \\ produces a break, attempting horizontal breaks if possible,
-% and vertical breaks if needed.
-%
-% An empty element obtained by \\\\ produces a vertical break in all cases.
-%
-% The former rule is aligned on the fraction bar.
-% The optional label appears on top of the rule
-% The second form to be used in a derivation tree is aligned on the last
-% line of its conclusion
-%
-% The second form can be parameterized, using the key=val interface. The
-% folloiwng keys are recognized:
-%
-% width set the width of the rule to val
-% narrower set the width of the rule to val\hsize
-% before execute val at the beginning/left
-% lab put a label [Val] on top of the rule
-% lskip add negative skip on the right
-% left put a left label [Val]
-% Left put a left label [Val], ignoring its width
-% right put a right label [Val]
-% Right put a right label [Val], ignoring its width
-% leftskip skip negative space on the left-hand side
-% rightskip skip negative space on the right-hand side
-% vdots lift the rule by val and fill vertical space with dots
-% after execute val at the end/right
-%
-% Note that most options must come in this order to avoid strange
-% typesetting (in particular leftskip must preceed left and Left and
-% rightskip must follow Right or right; vdots must come last
-% or be only followed by rightskip.
-%
-
-%% Keys that make sence in all kinds of rules
-\def \mprset #1{\setkeys{mprset}{#1}}
-\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
-\define@key {mprset}{center}[]{\mpr@centertrue}
-\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-
-\newbox \mpr@right
-\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
-\define@key {mpr}{center}[]{\mpr@centertrue}
-\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{width}{\hsize #1}
-\define@key {mpr}{sep}{\def\mpr@sep{#1}}
-\define@key {mpr}{before}{#1}
-\define@key {mpr}{lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
-\define@key {mpr}{Lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
-\define@key {mpr}{narrower}{\hsize #1\hsize}
-\define@key {mpr}{leftskip}{\hskip -#1}
-\define@key {mpr}{reduce}[]{\let \mpr@fraction \mpr@@reduce}
-\define@key {mpr}{rightskip}
- {\setbox \mpr@right \hbox {\unhbox \mpr@right \hskip -#1}}
-\define@key {mpr}{LEFT}{\setbox0 \hbox {$#1$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{Left}{\llap{$\TirName {#1}\;$}}
-\define@key {mpr}{right}
- {\setbox0 \hbox {$\;\TirName {#1}$}\relax \advance \hsize by -\wd0
- \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
-\define@key {mpr}{RIGHT}
- {\setbox0 \hbox {$#1$}\relax \advance \hsize by -\wd0
- \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
-\define@key {mpr}{Right}
- {\setbox \mpr@right \hbox {\unhbox \mpr@right \rlap {$\;\TirName {#1}$}}}
-\define@key {mpr}{vdots}{\def \mpr@vdots {\@@atop \mpr@vdotfil{#1}}}
-\define@key {mpr}{after}{\edef \mpr@after {\mpr@after #1}}
-
-\newdimen \rule@dimen
-\newcommand \mpr@inferstar@ [3][]{\setbox0
- \hbox {\let \mpr@rulename \mpr@empty \let \mpr@vdots \relax
- \setbox \mpr@right \hbox{}%
- $\setkeys{mpr}{#1}%
- \ifx \mpr@rulename \mpr@empty \mpr@inferrule {#2}{#3}\else
- \mpr@inferrule [{\mpr@rulename}]{#2}{#3}\fi
- \box \mpr@right \mpr@vdots$}
- \setbox1 \hbox {\strut}
- \rule@dimen \dp0 \advance \rule@dimen by -\dp1
- \raise \rule@dimen \box0}
-
-\def \mpr@infer {\@ifnextchar *{\mpr@inferstar}{\mpr@inferrule}}
-\newcommand \mpr@err@skipargs[3][]{}
-\def \mpr@inferstar*{\ifmmode
- \let \@do \mpr@inferstar@
- \else
- \let \@do \mpr@err@skipargs
- \PackageError {mathpartir}
- {\string\inferrule* can only be used in math mode}{}%
- \fi \@do}
-
-
-%%% Exports
-
-% Envirnonment mathpar
-
-\let \inferrule \mpr@infer
-
-% make a short name \infer is not already defined
-\@ifundefined {infer}{\let \infer \mpr@infer}{}
-
-\def \TirNameStyle #1{\small \textsc{#1}}
-\def \tir@name #1{\hbox {\small \TirNameStyle{#1}}}
-\let \TirName \tir@name
-\let \DefTirName \TirName
-\let \RefTirName \TirName
-
-%%% Other Exports
-
-% \let \listcons \mpr@cons
-% \let \listsnoc \mpr@snoc
-% \let \listhead \mpr@head
-% \let \listmake \mpr@makelist
-
-
-
-
-\endinput
--- a/doc-src/IsarAdvanced/Functions/style.sty Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,46 +0,0 @@
-%% toc
-\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
-\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
-
-%% references
-\newcommand{\secref}[1]{\S\ref{#1}}
-\newcommand{\chref}[1]{chapter~\ref{#1}}
-\newcommand{\figref}[1]{figure~\ref{#1}}
-
-%% math
-\newcommand{\text}[1]{\mbox{#1}}
-\newcommand{\isasymvartheta}{\isamath{\theta}}
-\newcommand{\isactrlvec}[1]{\emph{$\overline{#1}$}}
-
-\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
-
-\pagestyle{headings}
-\sloppy
-\binperiod
-\underscoreon
-
-\renewcommand{\isadigit}[1]{\isamath{#1}}
-
-\newenvironment{mldecls}{\par\noindent\begingroup\footnotesize\def\isanewline{\\}\begin{tabular}{l}}{\end{tabular}\smallskip\endgroup}
-
-\isafoldtag{FIXME}
-\isakeeptag{mlref}
-\renewcommand{\isatagmlref}{\subsection*{\makebox[0pt][r]{\fbox{\ML}~~}Reference}\begingroup\def\isastyletext{\rm}\small}
-\renewcommand{\endisatagmlref}{\endgroup}
-
-\newcommand{\isasymGUESS}{\isakeyword{guess}}
-\newcommand{\isasymOBTAIN}{\isakeyword{obtain}}
-\newcommand{\isasymTHEORY}{\isakeyword{theory}}
-\newcommand{\isasymUSES}{\isakeyword{uses}}
-\newcommand{\isasymEND}{\isakeyword{end}}
-\newcommand{\isasymCONSTS}{\isakeyword{consts}}
-\newcommand{\isasymDEFS}{\isakeyword{defs}}
-\newcommand{\isasymTHEOREM}{\isakeyword{theorem}}
-\newcommand{\isasymDEFINITION}{\isakeyword{definition}}
-
-\isabellestyle{it}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "implementation"
-%%% End:
--- a/doc-src/IsarAdvanced/Makefile.in Mon Mar 02 16:58:39 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,13 +0,0 @@
-# $Id$
-#
-
-include ../../Makefile.in
-
-SEDINDEX = ../../sedindex
-FIXBOOKMARKS = perl -pi ../../fixbookmarks.pl
-
-isabelle_isar.eps:
- test -r isabelle_isar.eps || ln -s ../../gfx/isabelle_isar.eps .
-
-isabelle_isar.pdf:
- test -r isabelle_isar.pdf || ln -s ../../gfx/isabelle_isar.pdf .