1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/doc-src/Classes/IsaMakefile	Tue Mar 03 11:00:51 2009 +0100
     1.3 @@ -0,0 +1,33 @@
     1.4 +
     1.5 +## targets
     1.6 +
     1.7 +default: Thy
     1.8 +images: 
     1.9 +test: Thy
    1.10 +
    1.11 +all: images test
    1.12 +
    1.13 +
    1.14 +## global settings
    1.15 +
    1.16 +SRC = $(ISABELLE_HOME)/src
    1.17 +OUT = $(ISABELLE_OUTPUT)
    1.18 +LOG = $(OUT)/log
    1.19 +
    1.20 +USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
    1.21 +
    1.22 +
    1.23 +## Thy
    1.24 +
    1.25 +THY = $(LOG)/HOL-Thy.gz
    1.26 +
    1.27 +Thy: $(THY)
    1.28 +
    1.29 +$(THY): Thy/ROOT.ML Thy/Setup.thy Thy/Classes.thy ../antiquote_setup.ML ../more_antiquote.ML
    1.30 +	@$(USEDIR) HOL Thy
    1.31 +
    1.32 +
    1.33 +## clean
    1.34 +
    1.35 +clean:
    1.36 +	@rm -f $(THY)

     2.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     2.2 +++ b/doc-src/Classes/Makefile	Tue Mar 03 11:00:51 2009 +0100
     2.3 @@ -0,0 +1,35 @@
     2.4 +
     2.5 +## targets
     2.6 +
     2.7 +default: dvi
     2.8 +
     2.9 +
    2.10 +## dependencies
    2.11 +
    2.12 +include ../Makefile.in
    2.13 +
    2.14 +NAME = classes
    2.15 +
    2.16 +FILES = $(NAME).tex classes.tex Thy/document/Classes.tex \
    2.17 +  style.sty ../iman.sty ../extra.sty ../isar.sty \
    2.18 +  ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty \
    2.19 +  ../manual.bib ../proof.sty
    2.20 +
    2.21 +dvi: $(NAME).dvi
    2.22 +
    2.23 +$(NAME).dvi: $(FILES) isabelle_isar.eps
    2.24 +	$(LATEX) $(NAME)
    2.25 +	$(BIBTEX) $(NAME)
    2.26 +	$(LATEX) $(NAME)
    2.27 +	$(LATEX) $(NAME)
    2.28 +
    2.29 +pdf: $(NAME).pdf
    2.30 +
    2.31 +$(NAME).pdf: $(FILES) isabelle_isar.pdf
    2.32 +	$(PDFLATEX) $(NAME)
    2.33 +	$(BIBTEX) $(NAME)
    2.34 +	$(PDFLATEX) $(NAME)
    2.35 +	$(PDFLATEX) $(NAME)
    2.36 +	$(FIXBOOKMARKS) $(NAME).out
    2.37 +	$(PDFLATEX) $(NAME)
    2.38 +	$(PDFLATEX) $(NAME)

     3.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     3.2 +++ b/doc-src/Classes/Thy/Classes.thy	Tue Mar 03 11:00:51 2009 +0100
     3.3 @@ -0,0 +1,635 @@
     3.4 +theory Classes
     3.5 +imports Main Setup
     3.6 +begin
     3.7 +
     3.8 +chapter {* Haskell-style classes with Isabelle/Isar *}
     3.9 +
    3.10 +section {* Introduction *}
    3.11 +
    3.12 +text {*
    3.13 +  Type classes were introduces by Wadler and Blott \cite{wadler89how}
    3.14 +  into the Haskell language, to allow for a reasonable implementation
    3.15 +  of overloading\footnote{throughout this tutorial, we are referring
    3.16 +  to classical Haskell 1.0 type classes, not considering
    3.17 +  later additions in expressiveness}.
    3.18 +  As a canonical example, a polymorphic equality function
    3.19 +  @{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} which is overloaded on different
    3.20 +  types for @{text "\<alpha>"}, which is achieved by splitting introduction
    3.21 +  of the @{text eq} function from its overloaded definitions by means
    3.22 +  of @{text class} and @{text instance} declarations:
    3.23 +
    3.24 +  \begin{quote}
    3.25 +
    3.26 +  \noindent@{text "class eq where"}\footnote{syntax here is a kind of isabellized Haskell} \\
    3.27 +  \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
    3.28 +
    3.29 +  \medskip\noindent@{text "instance nat \<Colon> eq where"} \\
    3.30 +  \hspace*{2ex}@{text "eq 0 0 = True"} \\
    3.31 +  \hspace*{2ex}@{text "eq 0 _ = False"} \\
    3.32 +  \hspace*{2ex}@{text "eq _ 0 = False"} \\
    3.33 +  \hspace*{2ex}@{text "eq (Suc n) (Suc m) = eq n m"}
    3.34 +
    3.35 +  \medskip\noindent@{text "instance (\<alpha>\<Colon>eq, \<beta>\<Colon>eq) pair \<Colon> eq where"} \\
    3.36 +  \hspace*{2ex}@{text "eq (x1, y1) (x2, y2) = eq x1 x2 \<and> eq y1 y2"}
    3.37 +
    3.38 +  \medskip\noindent@{text "class ord extends eq where"} \\
    3.39 +  \hspace*{2ex}@{text "less_eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
    3.40 +  \hspace*{2ex}@{text "less \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
    3.41 +
    3.42 +  \end{quote}
    3.43 +
    3.44 +  \noindent Type variables are annotated with (finitely many) classes;
    3.45 +  these annotations are assertions that a particular polymorphic type
    3.46 +  provides definitions for overloaded functions.
    3.47 +
    3.48 +  Indeed, type classes not only allow for simple overloading
    3.49 +  but form a generic calculus, an instance of order-sorted
    3.50 +  algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
    3.51 +
    3.52 +  From a software engeneering point of view, type classes
    3.53 +  roughly correspond to interfaces in object-oriented languages like Java;
    3.54 +  so, it is naturally desirable that type classes do not only
    3.55 +  provide functions (class parameters) but also state specifications
    3.56 +  implementations must obey.  For example, the @{text "class eq"}
    3.57 +  above could be given the following specification, demanding that
    3.58 +  @{text "class eq"} is an equivalence relation obeying reflexivity,
    3.59 +  symmetry and transitivity:
    3.60 +
    3.61 +  \begin{quote}
    3.62 +
    3.63 +  \noindent@{text "class eq where"} \\
    3.64 +  \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
    3.65 +  @{text "satisfying"} \\
    3.66 +  \hspace*{2ex}@{text "refl: eq x x"} \\
    3.67 +  \hspace*{2ex}@{text "sym: eq x y \<longleftrightarrow> eq x y"} \\
    3.68 +  \hspace*{2ex}@{text "trans: eq x y \<and> eq y z \<longrightarrow> eq x z"}
    3.69 +
    3.70 +  \end{quote}
    3.71 +
    3.72 +  \noindent From a theoretic point of view, type classes are lightweight
    3.73 +  modules; Haskell type classes may be emulated by
    3.74 +  SML functors \cite{classes_modules}. 
    3.75 +  Isabelle/Isar offers a discipline of type classes which brings
    3.76 +  all those aspects together:
    3.77 +
    3.78 +  \begin{enumerate}
    3.79 +    \item specifying abstract parameters together with
    3.80 +       corresponding specifications,
    3.81 +    \item instantiating those abstract parameters by a particular
    3.82 +       type
    3.83 +    \item in connection with a ``less ad-hoc'' approach to overloading,
    3.84 +    \item with a direct link to the Isabelle module system
    3.85 +      (aka locales \cite{kammueller-locales}).
    3.86 +  \end{enumerate}
    3.87 +
    3.88 +  \noindent Isar type classes also directly support code generation
    3.89 +  in a Haskell like fashion.
    3.90 +
    3.91 +  This tutorial demonstrates common elements of structured specifications
    3.92 +  and abstract reasoning with type classes by the algebraic hierarchy of
    3.93 +  semigroups, monoids and groups.  Our background theory is that of
    3.94 +  Isabelle/HOL \cite{isa-tutorial}, for which some
    3.95 +  familiarity is assumed.
    3.96 +
    3.97 +  Here we merely present the look-and-feel for end users.
    3.98 +  Internally, those are mapped to more primitive Isabelle concepts.
    3.99 +  See \cite{Haftmann-Wenzel:2006:classes} for more detail.
   3.100 +*}
   3.101 +
   3.102 +section {* A simple algebra example \label{sec:example} *}
   3.103 +
   3.104 +subsection {* Class definition *}
   3.105 +
   3.106 +text {*
   3.107 +  Depending on an arbitrary type @{text "\<alpha>"}, class @{text
   3.108 +  "semigroup"} introduces a binary operator @{text "(\<otimes>)"} that is
   3.109 +  assumed to be associative:
   3.110 +*}
   3.111 +
   3.112 +class %quote semigroup =
   3.113 +  fixes mult :: "\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"    (infixl "\<otimes>" 70)
   3.114 +  assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
   3.115 +
   3.116 +text {*
   3.117 +  \noindent This @{command class} specification consists of two
   3.118 +  parts: the \qn{operational} part names the class parameter
   3.119 +  (@{element "fixes"}), the \qn{logical} part specifies properties on them
   3.120 +  (@{element "assumes"}).  The local @{element "fixes"} and
   3.121 +  @{element "assumes"} are lifted to the theory toplevel,
   3.122 +  yielding the global
   3.123 +  parameter @{term [source] "mult \<Colon> \<alpha>\<Colon>semigroup \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"} and the
   3.124 +  global theorem @{fact "semigroup.assoc:"}~@{prop [source] "\<And>x y
   3.125 +  z \<Colon> \<alpha>\<Colon>semigroup. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"}.
   3.126 +*}
   3.127 +
   3.128 +
   3.129 +subsection {* Class instantiation \label{sec:class_inst} *}
   3.130 +
   3.131 +text {*
   3.132 +  The concrete type @{typ int} is made a @{class semigroup}
   3.133 +  instance by providing a suitable definition for the class parameter
   3.134 +  @{text "(\<otimes>)"} and a proof for the specification of @{fact assoc}.
   3.135 +  This is accomplished by the @{command instantiation} target:
   3.136 +*}
   3.137 +
   3.138 +instantiation %quote int :: semigroup
   3.139 +begin
   3.140 +
   3.141 +definition %quote
   3.142 +  mult_int_def: "i \<otimes> j = i + (j\<Colon>int)"
   3.143 +
   3.144 +instance %quote proof
   3.145 +  fix i j k :: int have "(i + j) + k = i + (j + k)" by simp
   3.146 +  then show "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
   3.147 +    unfolding mult_int_def .
   3.148 +qed
   3.149 +
   3.150 +end %quote
   3.151 +
   3.152 +text {*
   3.153 +  \noindent @{command instantiation} allows to define class parameters
   3.154 +  at a particular instance using common specification tools (here,
   3.155 +  @{command definition}).  The concluding @{command instance}
   3.156 +  opens a proof that the given parameters actually conform
   3.157 +  to the class specification.  Note that the first proof step
   3.158 +  is the @{method default} method,
   3.159 +  which for such instance proofs maps to the @{method intro_classes} method.
   3.160 +  This boils down an instance judgement to the relevant primitive
   3.161 +  proof goals and should conveniently always be the first method applied
   3.162 +  in an instantiation proof.
   3.163 +
   3.164 +  From now on, the type-checker will consider @{typ int}
   3.165 +  as a @{class semigroup} automatically, i.e.\ any general results
   3.166 +  are immediately available on concrete instances.
   3.167 +
   3.168 +  \medskip Another instance of @{class semigroup} are the natural numbers:
   3.169 +*}
   3.170 +
   3.171 +instantiation %quote nat :: semigroup
   3.172 +begin
   3.173 +
   3.174 +primrec %quote mult_nat where
   3.175 +  "(0\<Colon>nat) \<otimes> n = n"
   3.176 +  | "Suc m \<otimes> n = Suc (m \<otimes> n)"
   3.177 +
   3.178 +instance %quote proof
   3.179 +  fix m n q :: nat 
   3.180 +  show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
   3.181 +    by (induct m) auto
   3.182 +qed
   3.183 +
   3.184 +end %quote
   3.185 +
   3.186 +text {*
   3.187 +  \noindent Note the occurence of the name @{text mult_nat}
   3.188 +  in the primrec declaration;  by default, the local name of
   3.189 +  a class operation @{text f} to instantiate on type constructor
   3.190 +  @{text \<kappa>} are mangled as @{text f_\<kappa>}.  In case of uncertainty,
   3.191 +  these names may be inspected using the @{command "print_context"} command
   3.192 +  or the corresponding ProofGeneral button.
   3.193 +*}
   3.194 +
   3.195 +subsection {* Lifting and parametric types *}
   3.196 +
   3.197 +text {*
   3.198 +  Overloaded definitions giving on class instantiation
   3.199 +  may include recursion over the syntactic structure of types.
   3.200 +  As a canonical example, we model product semigroups
   3.201 +  using our simple algebra:
   3.202 +*}
   3.203 +
   3.204 +instantiation %quote * :: (semigroup, semigroup) semigroup
   3.205 +begin
   3.206 +
   3.207 +definition %quote
   3.208 +  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)"
   3.209 +
   3.210 +instance %quote proof
   3.211 +  fix p\<^isub>1 p\<^isub>2 p\<^isub>3 :: "\<alpha>\<Colon>semigroup \<times> \<beta>\<Colon>semigroup"
   3.212 +  show "p\<^isub>1 \<otimes> p\<^isub>2 \<otimes> p\<^isub>3 = p\<^isub>1 \<otimes> (p\<^isub>2 \<otimes> p\<^isub>3)"
   3.213 +    unfolding mult_prod_def by (simp add: assoc)
   3.214 +qed      
   3.215 +
   3.216 +end %quote
   3.217 +
   3.218 +text {*
   3.219 +  \noindent Associativity from product semigroups is
   3.220 +  established using
   3.221 +  the definition of @{text "(\<otimes>)"} on products and the hypothetical
   3.222 +  associativity of the type components;  these hypotheses
   3.223 +  are facts due to the @{class semigroup} constraints imposed
   3.224 +  on the type components by the @{command instance} proposition.
   3.225 +  Indeed, this pattern often occurs with parametric types
   3.226 +  and type classes.
   3.227 +*}
   3.228 +
   3.229 +
   3.230 +subsection {* Subclassing *}
   3.231 +
   3.232 +text {*
   3.233 +  We define a subclass @{text monoidl} (a semigroup with a left-hand neutral)
   3.234 +  by extending @{class semigroup}
   3.235 +  with one additional parameter @{text neutral} together
   3.236 +  with its property:
   3.237 +*}
   3.238 +
   3.239 +class %quote monoidl = semigroup +
   3.240 +  fixes neutral :: "\<alpha>" ("\<one>")
   3.241 +  assumes neutl: "\<one> \<otimes> x = x"
   3.242 +
   3.243 +text {*
   3.244 +  \noindent Again, we prove some instances, by
   3.245 +  providing suitable parameter definitions and proofs for the
   3.246 +  additional specifications.  Observe that instantiations
   3.247 +  for types with the same arity may be simultaneous:
   3.248 +*}
   3.249 +
   3.250 +instantiation %quote nat and int :: monoidl
   3.251 +begin
   3.252 +
   3.253 +definition %quote
   3.254 +  neutral_nat_def: "\<one> = (0\<Colon>nat)"
   3.255 +
   3.256 +definition %quote
   3.257 +  neutral_int_def: "\<one> = (0\<Colon>int)"
   3.258 +
   3.259 +instance %quote proof
   3.260 +  fix n :: nat
   3.261 +  show "\<one> \<otimes> n = n"
   3.262 +    unfolding neutral_nat_def by simp
   3.263 +next
   3.264 +  fix k :: int
   3.265 +  show "\<one> \<otimes> k = k"
   3.266 +    unfolding neutral_int_def mult_int_def by simp
   3.267 +qed
   3.268 +
   3.269 +end %quote
   3.270 +
   3.271 +instantiation %quote * :: (monoidl, monoidl) monoidl
   3.272 +begin
   3.273 +
   3.274 +definition %quote
   3.275 +  neutral_prod_def: "\<one> = (\<one>, \<one>)"
   3.276 +
   3.277 +instance %quote proof
   3.278 +  fix p :: "\<alpha>\<Colon>monoidl \<times> \<beta>\<Colon>monoidl"
   3.279 +  show "\<one> \<otimes> p = p"
   3.280 +    unfolding neutral_prod_def mult_prod_def by (simp add: neutl)
   3.281 +qed
   3.282 +
   3.283 +end %quote
   3.284 +
   3.285 +text {*
   3.286 +  \noindent Fully-fledged monoids are modelled by another subclass
   3.287 +  which does not add new parameters but tightens the specification:
   3.288 +*}
   3.289 +
   3.290 +class %quote monoid = monoidl +
   3.291 +  assumes neutr: "x \<otimes> \<one> = x"
   3.292 +
   3.293 +instantiation %quote nat and int :: monoid 
   3.294 +begin
   3.295 +
   3.296 +instance %quote proof
   3.297 +  fix n :: nat
   3.298 +  show "n \<otimes> \<one> = n"
   3.299 +    unfolding neutral_nat_def by (induct n) simp_all
   3.300 +next
   3.301 +  fix k :: int
   3.302 +  show "k \<otimes> \<one> = k"
   3.303 +    unfolding neutral_int_def mult_int_def by simp
   3.304 +qed
   3.305 +
   3.306 +end %quote
   3.307 +
   3.308 +instantiation %quote * :: (monoid, monoid) monoid
   3.309 +begin
   3.310 +
   3.311 +instance %quote proof 
   3.312 +  fix p :: "\<alpha>\<Colon>monoid \<times> \<beta>\<Colon>monoid"
   3.313 +  show "p \<otimes> \<one> = p"
   3.314 +    unfolding neutral_prod_def mult_prod_def by (simp add: neutr)
   3.315 +qed
   3.316 +
   3.317 +end %quote
   3.318 +
   3.319 +text {*
   3.320 +  \noindent To finish our small algebra example, we add a @{text group} class
   3.321 +  with a corresponding instance:
   3.322 +*}
   3.323 +
   3.324 +class %quote group = monoidl +
   3.325 +  fixes inverse :: "\<alpha> \<Rightarrow> \<alpha>"    ("(_\<div>)" [1000] 999)
   3.326 +  assumes invl: "x\<div> \<otimes> x = \<one>"
   3.327 +
   3.328 +instantiation %quote int :: group
   3.329 +begin
   3.330 +
   3.331 +definition %quote
   3.332 +  inverse_int_def: "i\<div> = - (i\<Colon>int)"
   3.333 +
   3.334 +instance %quote proof
   3.335 +  fix i :: int
   3.336 +  have "-i + i = 0" by simp
   3.337 +  then show "i\<div> \<otimes> i = \<one>"
   3.338 +    unfolding mult_int_def neutral_int_def inverse_int_def .
   3.339 +qed
   3.340 +
   3.341 +end %quote
   3.342 +
   3.343 +
   3.344 +section {* Type classes as locales *}
   3.345 +
   3.346 +subsection {* A look behind the scene *}
   3.347 +
   3.348 +text {*
   3.349 +  The example above gives an impression how Isar type classes work
   3.350 +  in practice.  As stated in the introduction, classes also provide
   3.351 +  a link to Isar's locale system.  Indeed, the logical core of a class
   3.352 +  is nothing else than a locale:
   3.353 +*}
   3.354 +
   3.355 +class %quote idem =
   3.356 +  fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
   3.357 +  assumes idem: "f (f x) = f x"
   3.358 +
   3.359 +text {*
   3.360 +  \noindent essentially introduces the locale
   3.361 +*} setup %invisible {* Sign.add_path "foo" *}
   3.362 +
   3.363 +locale %quote idem =
   3.364 +  fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
   3.365 +  assumes idem: "f (f x) = f x"
   3.366 +
   3.367 +text {* \noindent together with corresponding constant(s): *}
   3.368 +
   3.369 +consts %quote f :: "\<alpha> \<Rightarrow> \<alpha>"
   3.370 +
   3.371 +text {*
   3.372 +  \noindent The connection to the type system is done by means
   3.373 +  of a primitive axclass
   3.374 +*} setup %invisible {* Sign.add_path "foo" *}
   3.375 +
   3.376 +axclass %quote idem < type
   3.377 +  idem: "f (f x) = f x" setup %invisible {* Sign.parent_path *}
   3.378 +
   3.379 +text {* \noindent together with a corresponding interpretation: *}
   3.380 +
   3.381 +interpretation %quote idem_class:
   3.382 +  idem "f \<Colon> (\<alpha>\<Colon>idem) \<Rightarrow> \<alpha>"
   3.383 +proof qed (rule idem)
   3.384 +
   3.385 +text {*
   3.386 +  \noindent This gives you at hand the full power of the Isabelle module system;
   3.387 +  conclusions in locale @{text idem} are implicitly propagated
   3.388 +  to class @{text idem}.
   3.389 +*} setup %invisible {* Sign.parent_path *}
   3.390 +
   3.391 +subsection {* Abstract reasoning *}
   3.392 +
   3.393 +text {*
   3.394 +  Isabelle locales enable reasoning at a general level, while results
   3.395 +  are implicitly transferred to all instances.  For example, we can
   3.396 +  now establish the @{text "left_cancel"} lemma for groups, which
   3.397 +  states that the function @{text "(x \<otimes>)"} is injective:
   3.398 +*}
   3.399 +
   3.400 +lemma %quote (in group) left_cancel: "x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"
   3.401 +proof
   3.402 +  assume "x \<otimes> y = x \<otimes> z"
   3.403 +  then have "x\<div> \<otimes> (x \<otimes> y) = x\<div> \<otimes> (x \<otimes> z)" by simp
   3.404 +  then have "(x\<div> \<otimes> x) \<otimes> y = (x\<div> \<otimes> x) \<otimes> z" using assoc by simp
   3.405 +  then show "y = z" using neutl and invl by simp
   3.406 +next
   3.407 +  assume "y = z"
   3.408 +  then show "x \<otimes> y = x \<otimes> z" by simp
   3.409 +qed
   3.410 +
   3.411 +text {*
   3.412 +  \noindent Here the \qt{@{keyword "in"} @{class group}} target specification
   3.413 +  indicates that the result is recorded within that context for later
   3.414 +  use.  This local theorem is also lifted to the global one @{fact
   3.415 +  "group.left_cancel:"} @{prop [source] "\<And>x y z \<Colon> \<alpha>\<Colon>group. x \<otimes> y = x \<otimes>
   3.416 +  z \<longleftrightarrow> y = z"}.  Since type @{text "int"} has been made an instance of
   3.417 +  @{text "group"} before, we may refer to that fact as well: @{prop
   3.418 +  [source] "\<And>x y z \<Colon> int. x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"}.
   3.419 +*}
   3.420 +
   3.421 +
   3.422 +subsection {* Derived definitions *}
   3.423 +
   3.424 +text {*
   3.425 +  Isabelle locales support a concept of local definitions
   3.426 +  in locales:
   3.427 +*}
   3.428 +
   3.429 +primrec %quote (in monoid) pow_nat :: "nat \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
   3.430 +  "pow_nat 0 x = \<one>"
   3.431 +  | "pow_nat (Suc n) x = x \<otimes> pow_nat n x"
   3.432 +
   3.433 +text {*
   3.434 +  \noindent If the locale @{text group} is also a class, this local
   3.435 +  definition is propagated onto a global definition of
   3.436 +  @{term [source] "pow_nat \<Colon> nat \<Rightarrow> \<alpha>\<Colon>monoid \<Rightarrow> \<alpha>\<Colon>monoid"}
   3.437 +  with corresponding theorems
   3.438 +
   3.439 +  @{thm pow_nat.simps [no_vars]}.
   3.440 +
   3.441 +  \noindent As you can see from this example, for local
   3.442 +  definitions you may use any specification tool
   3.443 +  which works together with locales (e.g. \cite{krauss2006}).
   3.444 +*}
   3.445 +
   3.446 +
   3.447 +subsection {* A functor analogy *}
   3.448 +
   3.449 +text {*
   3.450 +  We introduced Isar classes by analogy to type classes
   3.451 +  functional programming;  if we reconsider this in the
   3.452 +  context of what has been said about type classes and locales,
   3.453 +  we can drive this analogy further by stating that type
   3.454 +  classes essentially correspond to functors which have
   3.455 +  a canonical interpretation as type classes.
   3.456 +  Anyway, there is also the possibility of other interpretations.
   3.457 +  For example, also @{text list}s form a monoid with
   3.458 +  @{text append} and @{term "[]"} as operations, but it
   3.459 +  seems inappropriate to apply to lists
   3.460 +  the same operations as for genuinely algebraic types.
   3.461 +  In such a case, we simply can do a particular interpretation
   3.462 +  of monoids for lists:
   3.463 +*}
   3.464 +
   3.465 +interpretation %quote list_monoid!: monoid append "[]"
   3.466 +  proof qed auto
   3.467 +
   3.468 +text {*
   3.469 +  \noindent This enables us to apply facts on monoids
   3.470 +  to lists, e.g. @{thm list_monoid.neutl [no_vars]}.
   3.471 +
   3.472 +  When using this interpretation pattern, it may also
   3.473 +  be appropriate to map derived definitions accordingly:
   3.474 +*}
   3.475 +
   3.476 +primrec %quote replicate :: "nat \<Rightarrow> \<alpha> list \<Rightarrow> \<alpha> list" where
   3.477 +  "replicate 0 _ = []"
   3.478 +  | "replicate (Suc n) xs = xs @ replicate n xs"
   3.479 +
   3.480 +interpretation %quote list_monoid!: monoid append "[]" where
   3.481 +  "monoid.pow_nat append [] = replicate"
   3.482 +proof -
   3.483 +  interpret monoid append "[]" ..
   3.484 +  show "monoid.pow_nat append [] = replicate"
   3.485 +  proof
   3.486 +    fix n
   3.487 +    show "monoid.pow_nat append [] n = replicate n"
   3.488 +      by (induct n) auto
   3.489 +  qed
   3.490 +qed intro_locales
   3.491 +
   3.492 +
   3.493 +subsection {* Additional subclass relations *}
   3.494 +
   3.495 +text {*
   3.496 +  Any @{text "group"} is also a @{text "monoid"};  this
   3.497 +  can be made explicit by claiming an additional
   3.498 +  subclass relation,
   3.499 +  together with a proof of the logical difference:
   3.500 +*}
   3.501 +
   3.502 +subclass %quote (in group) monoid
   3.503 +proof
   3.504 +  fix x
   3.505 +  from invl have "x\<div> \<otimes> x = \<one>" by simp
   3.506 +  with assoc [symmetric] neutl invl have "x\<div> \<otimes> (x \<otimes> \<one>) = x\<div> \<otimes> x" by simp
   3.507 +  with left_cancel show "x \<otimes> \<one> = x" by simp
   3.508 +qed
   3.509 +
   3.510 +text {*
   3.511 +  \noindent The logical proof is carried out on the locale level.
   3.512 +  Afterwards it is propagated
   3.513 +  to the type system, making @{text group} an instance of
   3.514 +  @{text monoid} by adding an additional edge
   3.515 +  to the graph of subclass relations
   3.516 +  (cf.\ \figref{fig:subclass}).
   3.517 +
   3.518 +  \begin{figure}[htbp]
   3.519 +   \begin{center}
   3.520 +     \small
   3.521 +     \unitlength 0.6mm
   3.522 +     \begin{picture}(40,60)(0,0)
   3.523 +       \put(20,60){\makebox(0,0){@{text semigroup}}}
   3.524 +       \put(20,40){\makebox(0,0){@{text monoidl}}}
   3.525 +       \put(00,20){\makebox(0,0){@{text monoid}}}
   3.526 +       \put(40,00){\makebox(0,0){@{text group}}}
   3.527 +       \put(20,55){\vector(0,-1){10}}
   3.528 +       \put(15,35){\vector(-1,-1){10}}
   3.529 +       \put(25,35){\vector(1,-3){10}}
   3.530 +     \end{picture}
   3.531 +     \hspace{8em}
   3.532 +     \begin{picture}(40,60)(0,0)
   3.533 +       \put(20,60){\makebox(0,0){@{text semigroup}}}
   3.534 +       \put(20,40){\makebox(0,0){@{text monoidl}}}
   3.535 +       \put(00,20){\makebox(0,0){@{text monoid}}}
   3.536 +       \put(40,00){\makebox(0,0){@{text group}}}
   3.537 +       \put(20,55){\vector(0,-1){10}}
   3.538 +       \put(15,35){\vector(-1,-1){10}}
   3.539 +       \put(05,15){\vector(3,-1){30}}
   3.540 +     \end{picture}
   3.541 +     \caption{Subclass relationship of monoids and groups:
   3.542 +        before and after establishing the relationship
   3.543 +        @{text "group \<subseteq> monoid"};  transitive edges are left out.}
   3.544 +     \label{fig:subclass}
   3.545 +   \end{center}
   3.546 +  \end{figure}
   3.547 +7
   3.548 +  For illustration, a derived definition
   3.549 +  in @{text group} which uses @{text pow_nat}:
   3.550 +*}
   3.551 +
   3.552 +definition %quote (in group) pow_int :: "int \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
   3.553 +  "pow_int k x = (if k >= 0
   3.554 +    then pow_nat (nat k) x
   3.555 +    else (pow_nat (nat (- k)) x)\<div>)"
   3.556 +
   3.557 +text {*
   3.558 +  \noindent yields the global definition of
   3.559 +  @{term [source] "pow_int \<Colon> int \<Rightarrow> \<alpha>\<Colon>group \<Rightarrow> \<alpha>\<Colon>group"}
   3.560 +  with the corresponding theorem @{thm pow_int_def [no_vars]}.
   3.561 +*}
   3.562 +
   3.563 +subsection {* A note on syntax *}
   3.564 +
   3.565 +text {*
   3.566 +  As a commodity, class context syntax allows to refer
   3.567 +  to local class operations and their global counterparts
   3.568 +  uniformly;  type inference resolves ambiguities.  For example:
   3.569 +*}
   3.570 +
   3.571 +context %quote semigroup
   3.572 +begin
   3.573 +
   3.574 +term %quote "x \<otimes> y" -- {* example 1 *}
   3.575 +term %quote "(x\<Colon>nat) \<otimes> y" -- {* example 2 *}
   3.576 +
   3.577 +end  %quote
   3.578 +
   3.579 +term %quote "x \<otimes> y" -- {* example 3 *}
   3.580 +
   3.581 +text {*
   3.582 +  \noindent Here in example 1, the term refers to the local class operation
   3.583 +  @{text "mult [\<alpha>]"}, whereas in example 2 the type constraint
   3.584 +  enforces the global class operation @{text "mult [nat]"}.
   3.585 +  In the global context in example 3, the reference is
   3.586 +  to the polymorphic global class operation @{text "mult [?\<alpha> \<Colon> semigroup]"}.
   3.587 +*}
   3.588 +
   3.589 +section {* Further issues *}
   3.590 +
   3.591 +subsection {* Type classes and code generation *}
   3.592 +
   3.593 +text {*
   3.594 +  Turning back to the first motivation for type classes,
   3.595 +  namely overloading, it is obvious that overloading
   3.596 +  stemming from @{command class} statements and
   3.597 +  @{command instantiation}
   3.598 +  targets naturally maps to Haskell type classes.
   3.599 +  The code generator framework \cite{isabelle-codegen} 
   3.600 +  takes this into account.  Concerning target languages
   3.601 +  lacking type classes (e.g.~SML), type classes
   3.602 +  are implemented by explicit dictionary construction.
   3.603 +  As example, let's go back to the power function:
   3.604 +*}
   3.605 +
   3.606 +definition %quote example :: int where
   3.607 +  "example = pow_int 10 (-2)"
   3.608 +
   3.609 +text {*
   3.610 +  \noindent This maps to Haskell as:
   3.611 +*}
   3.612 +
   3.613 +text %quote {*@{code_stmts example (Haskell)}*}
   3.614 +
   3.615 +text {*
   3.616 +  \noindent The whole code in SML with explicit dictionary passing:
   3.617 +*}
   3.618 +
   3.619 +text %quote {*@{code_stmts example (SML)}*}
   3.620 +
   3.621 +subsection {* Inspecting the type class universe *}
   3.622 +
   3.623 +text {*
   3.624 +  To facilitate orientation in complex subclass structures,
   3.625 +  two diagnostics commands are provided:
   3.626 +
   3.627 +  \begin{description}
   3.628 +
   3.629 +    \item[@{command "print_classes"}] print a list of all classes
   3.630 +      together with associated operations etc.
   3.631 +
   3.632 +    \item[@{command "class_deps"}] visualizes the subclass relation
   3.633 +      between all classes as a Hasse diagram.
   3.634 +
   3.635 +  \end{description}
   3.636 +*}
   3.637 +
   3.638 +end

     4.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     4.2 +++ b/doc-src/Classes/Thy/ROOT.ML	Tue Mar 03 11:00:51 2009 +0100
     4.3 @@ -0,0 +1,6 @@
     4.4 +
     4.5 +(* $Id$ *)
     4.6 +
     4.7 +no_document use_thy "Setup";
     4.8 +
     4.9 +use_thy "Classes";

     5.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     5.2 +++ b/doc-src/Classes/Thy/Setup.thy	Tue Mar 03 11:00:51 2009 +0100
     5.3 @@ -0,0 +1,34 @@
     5.4 +theory Setup
     5.5 +imports Main Code_Integer
     5.6 +uses
     5.7 +  "../../../antiquote_setup"
     5.8 +  "../../../more_antiquote"
     5.9 +begin
    5.10 +
    5.11 +ML {* Code_Target.code_width := 74 *}
    5.12 +
    5.13 +syntax
    5.14 +  "_alpha" :: "type"  ("\<alpha>")
    5.15 +  "_alpha_ofsort" :: "sort \<Rightarrow> type"  ("\<alpha>()\<Colon>_" [0] 1000)
    5.16 +  "_beta" :: "type"  ("\<beta>")
    5.17 +  "_beta_ofsort" :: "sort \<Rightarrow> type"  ("\<beta>()\<Colon>_" [0] 1000)
    5.18 +
    5.19 +parse_ast_translation {*
    5.20 +  let
    5.21 +    fun alpha_ast_tr [] = Syntax.Variable "'a"
    5.22 +      | alpha_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
    5.23 +    fun alpha_ofsort_ast_tr [ast] =
    5.24 +      Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'a", ast]
    5.25 +      | alpha_ofsort_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
    5.26 +    fun beta_ast_tr [] = Syntax.Variable "'b"
    5.27 +      | beta_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
    5.28 +    fun beta_ofsort_ast_tr [ast] =
    5.29 +      Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'b", ast]
    5.30 +      | beta_ofsort_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
    5.31 +  in [
    5.32 +    ("_alpha", alpha_ast_tr), ("_alpha_ofsort", alpha_ofsort_ast_tr),
    5.33 +    ("_beta", beta_ast_tr), ("_beta_ofsort", beta_ofsort_ast_tr)
    5.34 +  ] end
    5.35 +*}
    5.36 +
    5.37 +end
    5.38 \ No newline at end of file

     6.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     6.2 +++ b/doc-src/Classes/Thy/document/Classes.tex	Tue Mar 03 11:00:51 2009 +0100
     6.3 @@ -0,0 +1,1346 @@
     6.4 +%
     6.5 +\begin{isabellebody}%
     6.6 +\def\isabellecontext{Classes}%
     6.7 +%
     6.8 +\isadelimtheory
     6.9 +%
    6.10 +\endisadelimtheory
    6.11 +%
    6.12 +\isatagtheory
    6.13 +\isacommand{theory}\isamarkupfalse%
    6.14 +\ Classes\isanewline
    6.15 +\isakeyword{imports}\ Main\ Setup\isanewline
    6.16 +\isakeyword{begin}%
    6.17 +\endisatagtheory
    6.18 +{\isafoldtheory}%
    6.19 +%
    6.20 +\isadelimtheory
    6.21 +%
    6.22 +\endisadelimtheory
    6.23 +%
    6.24 +\isamarkupchapter{Haskell-style classes with Isabelle/Isar%
    6.25 +}
    6.26 +\isamarkuptrue%
    6.27 +%
    6.28 +\isamarkupsection{Introduction%
    6.29 +}
    6.30 +\isamarkuptrue%
    6.31 +%
    6.32 +\begin{isamarkuptext}%
    6.33 +Type classes were introduces by Wadler and Blott \cite{wadler89how}
    6.34 +  into the Haskell language, to allow for a reasonable implementation
    6.35 +  of overloading\footnote{throughout this tutorial, we are referring
    6.36 +  to classical Haskell 1.0 type classes, not considering
    6.37 +  later additions in expressiveness}.
    6.38 +  As a canonical example, a polymorphic equality function
    6.39 +  \isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} which is overloaded on different
    6.40 +  types for \isa{{\isasymalpha}}, which is achieved by splitting introduction
    6.41 +  of the \isa{eq} function from its overloaded definitions by means
    6.42 +  of \isa{class} and \isa{instance} declarations:
    6.43 +
    6.44 +  \begin{quote}
    6.45 +
    6.46 +  \noindent\isa{class\ eq\ where}\footnote{syntax here is a kind of isabellized Haskell} \\
    6.47 +  \hspace*{2ex}\isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool}
    6.48 +
    6.49 +  \medskip\noindent\isa{instance\ nat\ {\isasymColon}\ eq\ where} \\
    6.50 +  \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isadigit{0}}\ {\isacharequal}\ True} \\
    6.51 +  \hspace*{2ex}\isa{eq\ {\isadigit{0}}\ {\isacharunderscore}\ {\isacharequal}\ False} \\
    6.52 +  \hspace*{2ex}\isa{eq\ {\isacharunderscore}\ {\isadigit{0}}\ {\isacharequal}\ False} \\
    6.53 +  \hspace*{2ex}\isa{eq\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharparenleft}Suc\ m{\isacharparenright}\ {\isacharequal}\ eq\ n\ m}
    6.54 +
    6.55 +  \medskip\noindent\isa{instance\ {\isacharparenleft}{\isasymalpha}{\isasymColon}eq{\isacharcomma}\ {\isasymbeta}{\isasymColon}eq{\isacharparenright}\ pair\ {\isasymColon}\ eq\ where} \\
    6.56 +  \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}}}
    6.57 +
    6.58 +  \medskip\noindent\isa{class\ ord\ extends\ eq\ where} \\
    6.59 +  \hspace*{2ex}\isa{less{\isacharunderscore}eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} \\
    6.60 +  \hspace*{2ex}\isa{less\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool}
    6.61 +
    6.62 +  \end{quote}
    6.63 +
    6.64 +  \noindent Type variables are annotated with (finitely many) classes;
    6.65 +  these annotations are assertions that a particular polymorphic type
    6.66 +  provides definitions for overloaded functions.
    6.67 +
    6.68 +  Indeed, type classes not only allow for simple overloading
    6.69 +  but form a generic calculus, an instance of order-sorted
    6.70 +  algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
    6.71 +
    6.72 +  From a software engeneering point of view, type classes
    6.73 +  roughly correspond to interfaces in object-oriented languages like Java;
    6.74 +  so, it is naturally desirable that type classes do not only
    6.75 +  provide functions (class parameters) but also state specifications
    6.76 +  implementations must obey.  For example, the \isa{class\ eq}
    6.77 +  above could be given the following specification, demanding that
    6.78 +  \isa{class\ eq} is an equivalence relation obeying reflexivity,
    6.79 +  symmetry and transitivity:
    6.80 +
    6.81 +  \begin{quote}
    6.82 +
    6.83 +  \noindent\isa{class\ eq\ where} \\
    6.84 +  \hspace*{2ex}\isa{eq\ {\isasymColon}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ bool} \\
    6.85 +  \isa{satisfying} \\
    6.86 +  \hspace*{2ex}\isa{refl{\isacharcolon}\ eq\ x\ x} \\
    6.87 +  \hspace*{2ex}\isa{sym{\isacharcolon}\ eq\ x\ y\ {\isasymlongleftrightarrow}\ eq\ x\ y} \\
    6.88 +  \hspace*{2ex}\isa{trans{\isacharcolon}\ eq\ x\ y\ {\isasymand}\ eq\ y\ z\ {\isasymlongrightarrow}\ eq\ x\ z}
    6.89 +
    6.90 +  \end{quote}
    6.91 +
    6.92 +  \noindent From a theoretic point of view, type classes are lightweight
    6.93 +  modules; Haskell type classes may be emulated by
    6.94 +  SML functors \cite{classes_modules}. 
    6.95 +  Isabelle/Isar offers a discipline of type classes which brings
    6.96 +  all those aspects together:
    6.97 +
    6.98 +  \begin{enumerate}
    6.99 +    \item specifying abstract parameters together with
   6.100 +       corresponding specifications,
   6.101 +    \item instantiating those abstract parameters by a particular
   6.102 +       type
   6.103 +    \item in connection with a ``less ad-hoc'' approach to overloading,
   6.104 +    \item with a direct link to the Isabelle module system
   6.105 +      (aka locales \cite{kammueller-locales}).
   6.106 +  \end{enumerate}
   6.107 +
   6.108 +  \noindent Isar type classes also directly support code generation
   6.109 +  in a Haskell like fashion.
   6.110 +
   6.111 +  This tutorial demonstrates common elements of structured specifications
   6.112 +  and abstract reasoning with type classes by the algebraic hierarchy of
   6.113 +  semigroups, monoids and groups.  Our background theory is that of
   6.114 +  Isabelle/HOL \cite{isa-tutorial}, for which some
   6.115 +  familiarity is assumed.
   6.116 +
   6.117 +  Here we merely present the look-and-feel for end users.
   6.118 +  Internally, those are mapped to more primitive Isabelle concepts.
   6.119 +  See \cite{Haftmann-Wenzel:2006:classes} for more detail.%
   6.120 +\end{isamarkuptext}%
   6.121 +\isamarkuptrue%
   6.122 +%
   6.123 +\isamarkupsection{A simple algebra example \label{sec:example}%
   6.124 +}
   6.125 +\isamarkuptrue%
   6.126 +%
   6.127 +\isamarkupsubsection{Class definition%
   6.128 +}
   6.129 +\isamarkuptrue%
   6.130 +%
   6.131 +\begin{isamarkuptext}%
   6.132 +Depending on an arbitrary type \isa{{\isasymalpha}}, class \isa{semigroup} introduces a binary operator \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} that is
   6.133 +  assumed to be associative:%
   6.134 +\end{isamarkuptext}%
   6.135 +\isamarkuptrue%
   6.136 +%
   6.137 +\isadelimquote
   6.138 +%
   6.139 +\endisadelimquote
   6.140 +%
   6.141 +\isatagquote
   6.142 +\isacommand{class}\isamarkupfalse%
   6.143 +\ semigroup\ {\isacharequal}\isanewline
   6.144 +\ \ \isakeyword{fixes}\ mult\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymotimes}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
   6.145 +\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}%
   6.146 +\endisatagquote
   6.147 +{\isafoldquote}%
   6.148 +%
   6.149 +\isadelimquote
   6.150 +%
   6.151 +\endisadelimquote
   6.152 +%
   6.153 +\begin{isamarkuptext}%
   6.154 +\noindent This \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} specification consists of two
   6.155 +  parts: the \qn{operational} part names the class parameter
   6.156 +  (\hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}}), the \qn{logical} part specifies properties on them
   6.157 +  (\hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}}).  The local \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}} and
   6.158 +  \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}} are lifted to the theory toplevel,
   6.159 +  yielding the global
   6.160 +  parameter \isa{{\isachardoublequote}mult\ {\isasymColon}\ {\isasymalpha}{\isasymColon}semigroup\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequote}} and the
   6.161 +  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}}.%
   6.162 +\end{isamarkuptext}%
   6.163 +\isamarkuptrue%
   6.164 +%
   6.165 +\isamarkupsubsection{Class instantiation \label{sec:class_inst}%
   6.166 +}
   6.167 +\isamarkuptrue%
   6.168 +%
   6.169 +\begin{isamarkuptext}%
   6.170 +The concrete type \isa{int} is made a \isa{semigroup}
   6.171 +  instance by providing a suitable definition for the class parameter
   6.172 +  \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} and a proof for the specification of \hyperlink{fact.assoc}{\mbox{\isa{assoc}}}.
   6.173 +  This is accomplished by the \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} target:%
   6.174 +\end{isamarkuptext}%
   6.175 +\isamarkuptrue%
   6.176 +%
   6.177 +\isadelimquote
   6.178 +%
   6.179 +\endisadelimquote
   6.180 +%
   6.181 +\isatagquote
   6.182 +\isacommand{instantiation}\isamarkupfalse%
   6.183 +\ int\ {\isacharcolon}{\isacharcolon}\ semigroup\isanewline
   6.184 +\isakeyword{begin}\isanewline
   6.185 +\isanewline
   6.186 +\isacommand{definition}\isamarkupfalse%
   6.187 +\isanewline
   6.188 +\ \ mult{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymotimes}\ j\ {\isacharequal}\ i\ {\isacharplus}\ {\isacharparenleft}j{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.189 +\isanewline
   6.190 +\isacommand{instance}\isamarkupfalse%
   6.191 +\ \isacommand{proof}\isamarkupfalse%
   6.192 +\isanewline
   6.193 +\ \ \isacommand{fix}\isamarkupfalse%
   6.194 +\ i\ j\ k\ {\isacharcolon}{\isacharcolon}\ int\ \isacommand{have}\isamarkupfalse%
   6.195 +\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isacharplus}\ j{\isacharparenright}\ {\isacharplus}\ k\ {\isacharequal}\ i\ {\isacharplus}\ {\isacharparenleft}j\ {\isacharplus}\ k{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
   6.196 +\ simp\isanewline
   6.197 +\ \ \isacommand{then}\isamarkupfalse%
   6.198 +\ \isacommand{show}\isamarkupfalse%
   6.199 +\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isasymotimes}\ j{\isacharparenright}\ {\isasymotimes}\ k\ {\isacharequal}\ i\ {\isasymotimes}\ {\isacharparenleft}j\ {\isasymotimes}\ k{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.200 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.201 +\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse%
   6.202 +\isanewline
   6.203 +\isacommand{qed}\isamarkupfalse%
   6.204 +\isanewline
   6.205 +\isanewline
   6.206 +\isacommand{end}\isamarkupfalse%
   6.207 +%
   6.208 +\endisatagquote
   6.209 +{\isafoldquote}%
   6.210 +%
   6.211 +\isadelimquote
   6.212 +%
   6.213 +\endisadelimquote
   6.214 +%
   6.215 +\begin{isamarkuptext}%
   6.216 +\noindent \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} allows to define class parameters
   6.217 +  at a particular instance using common specification tools (here,
   6.218 +  \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}).  The concluding \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}
   6.219 +  opens a proof that the given parameters actually conform
   6.220 +  to the class specification.  Note that the first proof step
   6.221 +  is the \hyperlink{method.default}{\mbox{\isa{default}}} method,
   6.222 +  which for such instance proofs maps to the \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}} method.
   6.223 +  This boils down an instance judgement to the relevant primitive
   6.224 +  proof goals and should conveniently always be the first method applied
   6.225 +  in an instantiation proof.
   6.226 +
   6.227 +  From now on, the type-checker will consider \isa{int}
   6.228 +  as a \isa{semigroup} automatically, i.e.\ any general results
   6.229 +  are immediately available on concrete instances.
   6.230 +
   6.231 +  \medskip Another instance of \isa{semigroup} are the natural numbers:%
   6.232 +\end{isamarkuptext}%
   6.233 +\isamarkuptrue%
   6.234 +%
   6.235 +\isadelimquote
   6.236 +%
   6.237 +\endisadelimquote
   6.238 +%
   6.239 +\isatagquote
   6.240 +\isacommand{instantiation}\isamarkupfalse%
   6.241 +\ nat\ {\isacharcolon}{\isacharcolon}\ semigroup\isanewline
   6.242 +\isakeyword{begin}\isanewline
   6.243 +\isanewline
   6.244 +\isacommand{primrec}\isamarkupfalse%
   6.245 +\ mult{\isacharunderscore}nat\ \isakeyword{where}\isanewline
   6.246 +\ \ {\isachardoublequoteopen}{\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
   6.247 +\ \ {\isacharbar}\ {\isachardoublequoteopen}Suc\ m\ {\isasymotimes}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}m\ {\isasymotimes}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.248 +\isanewline
   6.249 +\isacommand{instance}\isamarkupfalse%
   6.250 +\ \isacommand{proof}\isamarkupfalse%
   6.251 +\isanewline
   6.252 +\ \ \isacommand{fix}\isamarkupfalse%
   6.253 +\ m\ n\ q\ {\isacharcolon}{\isacharcolon}\ nat\ \isanewline
   6.254 +\ \ \isacommand{show}\isamarkupfalse%
   6.255 +\ {\isachardoublequoteopen}m\ {\isasymotimes}\ n\ {\isasymotimes}\ q\ {\isacharequal}\ m\ {\isasymotimes}\ {\isacharparenleft}n\ {\isasymotimes}\ q{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.256 +\ \ \ \ \isacommand{by}\isamarkupfalse%
   6.257 +\ {\isacharparenleft}induct\ m{\isacharparenright}\ auto\isanewline
   6.258 +\isacommand{qed}\isamarkupfalse%
   6.259 +\isanewline
   6.260 +\isanewline
   6.261 +\isacommand{end}\isamarkupfalse%
   6.262 +%
   6.263 +\endisatagquote
   6.264 +{\isafoldquote}%
   6.265 +%
   6.266 +\isadelimquote
   6.267 +%
   6.268 +\endisadelimquote
   6.269 +%
   6.270 +\begin{isamarkuptext}%
   6.271 +\noindent Note the occurence of the name \isa{mult{\isacharunderscore}nat}
   6.272 +  in the primrec declaration;  by default, the local name of
   6.273 +  a class operation \isa{f} to instantiate on type constructor
   6.274 +  \isa{{\isasymkappa}} are mangled as \isa{f{\isacharunderscore}{\isasymkappa}}.  In case of uncertainty,
   6.275 +  these names may be inspected using the \hyperlink{command.print-context}{\mbox{\isa{\isacommand{print{\isacharunderscore}context}}}} command
   6.276 +  or the corresponding ProofGeneral button.%
   6.277 +\end{isamarkuptext}%
   6.278 +\isamarkuptrue%
   6.279 +%
   6.280 +\isamarkupsubsection{Lifting and parametric types%
   6.281 +}
   6.282 +\isamarkuptrue%
   6.283 +%
   6.284 +\begin{isamarkuptext}%
   6.285 +Overloaded definitions giving on class instantiation
   6.286 +  may include recursion over the syntactic structure of types.
   6.287 +  As a canonical example, we model product semigroups
   6.288 +  using our simple algebra:%
   6.289 +\end{isamarkuptext}%
   6.290 +\isamarkuptrue%
   6.291 +%
   6.292 +\isadelimquote
   6.293 +%
   6.294 +\endisadelimquote
   6.295 +%
   6.296 +\isatagquote
   6.297 +\isacommand{instantiation}\isamarkupfalse%
   6.298 +\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
   6.299 +\isakeyword{begin}\isanewline
   6.300 +\isanewline
   6.301 +\isacommand{definition}\isamarkupfalse%
   6.302 +\isanewline
   6.303 +\ \ 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
   6.304 +\isanewline
   6.305 +\isacommand{instance}\isamarkupfalse%
   6.306 +\ \isacommand{proof}\isamarkupfalse%
   6.307 +\isanewline
   6.308 +\ \ \isacommand{fix}\isamarkupfalse%
   6.309 +\ p\isactrlisub {\isadigit{1}}\ p\isactrlisub {\isadigit{2}}\ p\isactrlisub {\isadigit{3}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}semigroup\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}semigroup{\isachardoublequoteclose}\isanewline
   6.310 +\ \ \isacommand{show}\isamarkupfalse%
   6.311 +\ {\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
   6.312 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.313 +\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
   6.314 +\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
   6.315 +\isacommand{qed}\isamarkupfalse%
   6.316 +\ \ \ \ \ \ \isanewline
   6.317 +\isanewline
   6.318 +\isacommand{end}\isamarkupfalse%
   6.319 +%
   6.320 +\endisatagquote
   6.321 +{\isafoldquote}%
   6.322 +%
   6.323 +\isadelimquote
   6.324 +%
   6.325 +\endisadelimquote
   6.326 +%
   6.327 +\begin{isamarkuptext}%
   6.328 +\noindent Associativity from product semigroups is
   6.329 +  established using
   6.330 +  the definition of \isa{{\isacharparenleft}{\isasymotimes}{\isacharparenright}} on products and the hypothetical
   6.331 +  associativity of the type components;  these hypotheses
   6.332 +  are facts due to the \isa{semigroup} constraints imposed
   6.333 +  on the type components by the \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} proposition.
   6.334 +  Indeed, this pattern often occurs with parametric types
   6.335 +  and type classes.%
   6.336 +\end{isamarkuptext}%
   6.337 +\isamarkuptrue%
   6.338 +%
   6.339 +\isamarkupsubsection{Subclassing%
   6.340 +}
   6.341 +\isamarkuptrue%
   6.342 +%
   6.343 +\begin{isamarkuptext}%
   6.344 +We define a subclass \isa{monoidl} (a semigroup with a left-hand neutral)
   6.345 +  by extending \isa{semigroup}
   6.346 +  with one additional parameter \isa{neutral} together
   6.347 +  with its property:%
   6.348 +\end{isamarkuptext}%
   6.349 +\isamarkuptrue%
   6.350 +%
   6.351 +\isadelimquote
   6.352 +%
   6.353 +\endisadelimquote
   6.354 +%
   6.355 +\isatagquote
   6.356 +\isacommand{class}\isamarkupfalse%
   6.357 +\ monoidl\ {\isacharequal}\ semigroup\ {\isacharplus}\isanewline
   6.358 +\ \ \isakeyword{fixes}\ neutral\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isachardoublequoteclose}\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}{\isacharparenright}\isanewline
   6.359 +\ \ \isakeyword{assumes}\ neutl{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}%
   6.360 +\endisatagquote
   6.361 +{\isafoldquote}%
   6.362 +%
   6.363 +\isadelimquote
   6.364 +%
   6.365 +\endisadelimquote
   6.366 +%
   6.367 +\begin{isamarkuptext}%
   6.368 +\noindent Again, we prove some instances, by
   6.369 +  providing suitable parameter definitions and proofs for the
   6.370 +  additional specifications.  Observe that instantiations
   6.371 +  for types with the same arity may be simultaneous:%
   6.372 +\end{isamarkuptext}%
   6.373 +\isamarkuptrue%
   6.374 +%
   6.375 +\isadelimquote
   6.376 +%
   6.377 +\endisadelimquote
   6.378 +%
   6.379 +\isatagquote
   6.380 +\isacommand{instantiation}\isamarkupfalse%
   6.381 +\ nat\ \isakeyword{and}\ int\ {\isacharcolon}{\isacharcolon}\ monoidl\isanewline
   6.382 +\isakeyword{begin}\isanewline
   6.383 +\isanewline
   6.384 +\isacommand{definition}\isamarkupfalse%
   6.385 +\isanewline
   6.386 +\ \ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.387 +\isanewline
   6.388 +\isacommand{definition}\isamarkupfalse%
   6.389 +\isanewline
   6.390 +\ \ neutral{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.391 +\isanewline
   6.392 +\isacommand{instance}\isamarkupfalse%
   6.393 +\ \isacommand{proof}\isamarkupfalse%
   6.394 +\isanewline
   6.395 +\ \ \isacommand{fix}\isamarkupfalse%
   6.396 +\ n\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
   6.397 +\ \ \isacommand{show}\isamarkupfalse%
   6.398 +\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
   6.399 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.400 +\ neutral{\isacharunderscore}nat{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
   6.401 +\ simp\isanewline
   6.402 +\isacommand{next}\isamarkupfalse%
   6.403 +\isanewline
   6.404 +\ \ \isacommand{fix}\isamarkupfalse%
   6.405 +\ k\ {\isacharcolon}{\isacharcolon}\ int\isanewline
   6.406 +\ \ \isacommand{show}\isamarkupfalse%
   6.407 +\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ k\ {\isacharequal}\ k{\isachardoublequoteclose}\isanewline
   6.408 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.409 +\ neutral{\isacharunderscore}int{\isacharunderscore}def\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
   6.410 +\ simp\isanewline
   6.411 +\isacommand{qed}\isamarkupfalse%
   6.412 +\isanewline
   6.413 +\isanewline
   6.414 +\isacommand{end}\isamarkupfalse%
   6.415 +\isanewline
   6.416 +\isanewline
   6.417 +\isacommand{instantiation}\isamarkupfalse%
   6.418 +\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}monoidl{\isacharcomma}\ monoidl{\isacharparenright}\ monoidl\isanewline
   6.419 +\isakeyword{begin}\isanewline
   6.420 +\isanewline
   6.421 +\isacommand{definition}\isamarkupfalse%
   6.422 +\isanewline
   6.423 +\ \ neutral{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ {\isacharparenleft}{\isasymone}{\isacharcomma}\ {\isasymone}{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.424 +\isanewline
   6.425 +\isacommand{instance}\isamarkupfalse%
   6.426 +\ \isacommand{proof}\isamarkupfalse%
   6.427 +\isanewline
   6.428 +\ \ \isacommand{fix}\isamarkupfalse%
   6.429 +\ p\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}monoidl\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}monoidl{\isachardoublequoteclose}\isanewline
   6.430 +\ \ \isacommand{show}\isamarkupfalse%
   6.431 +\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
   6.432 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.433 +\ neutral{\isacharunderscore}prod{\isacharunderscore}def\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
   6.434 +\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutl{\isacharparenright}\isanewline
   6.435 +\isacommand{qed}\isamarkupfalse%
   6.436 +\isanewline
   6.437 +\isanewline
   6.438 +\isacommand{end}\isamarkupfalse%
   6.439 +%
   6.440 +\endisatagquote
   6.441 +{\isafoldquote}%
   6.442 +%
   6.443 +\isadelimquote
   6.444 +%
   6.445 +\endisadelimquote
   6.446 +%
   6.447 +\begin{isamarkuptext}%
   6.448 +\noindent Fully-fledged monoids are modelled by another subclass
   6.449 +  which does not add new parameters but tightens the specification:%
   6.450 +\end{isamarkuptext}%
   6.451 +\isamarkuptrue%
   6.452 +%
   6.453 +\isadelimquote
   6.454 +%
   6.455 +\endisadelimquote
   6.456 +%
   6.457 +\isatagquote
   6.458 +\isacommand{class}\isamarkupfalse%
   6.459 +\ monoid\ {\isacharequal}\ monoidl\ {\isacharplus}\isanewline
   6.460 +\ \ \isakeyword{assumes}\ neutr{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
   6.461 +\isanewline
   6.462 +\isacommand{instantiation}\isamarkupfalse%
   6.463 +\ nat\ \isakeyword{and}\ int\ {\isacharcolon}{\isacharcolon}\ monoid\ \isanewline
   6.464 +\isakeyword{begin}\isanewline
   6.465 +\isanewline
   6.466 +\isacommand{instance}\isamarkupfalse%
   6.467 +\ \isacommand{proof}\isamarkupfalse%
   6.468 +\isanewline
   6.469 +\ \ \isacommand{fix}\isamarkupfalse%
   6.470 +\ n\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
   6.471 +\ \ \isacommand{show}\isamarkupfalse%
   6.472 +\ {\isachardoublequoteopen}n\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
   6.473 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.474 +\ neutral{\isacharunderscore}nat{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
   6.475 +\ {\isacharparenleft}induct\ n{\isacharparenright}\ simp{\isacharunderscore}all\isanewline
   6.476 +\isacommand{next}\isamarkupfalse%
   6.477 +\isanewline
   6.478 +\ \ \isacommand{fix}\isamarkupfalse%
   6.479 +\ k\ {\isacharcolon}{\isacharcolon}\ int\isanewline
   6.480 +\ \ \isacommand{show}\isamarkupfalse%
   6.481 +\ {\isachardoublequoteopen}k\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ k{\isachardoublequoteclose}\isanewline
   6.482 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.483 +\ neutral{\isacharunderscore}int{\isacharunderscore}def\ mult{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
   6.484 +\ simp\isanewline
   6.485 +\isacommand{qed}\isamarkupfalse%
   6.486 +\isanewline
   6.487 +\isanewline
   6.488 +\isacommand{end}\isamarkupfalse%
   6.489 +\isanewline
   6.490 +\isanewline
   6.491 +\isacommand{instantiation}\isamarkupfalse%
   6.492 +\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}monoid{\isacharcomma}\ monoid{\isacharparenright}\ monoid\isanewline
   6.493 +\isakeyword{begin}\isanewline
   6.494 +\isanewline
   6.495 +\isacommand{instance}\isamarkupfalse%
   6.496 +\ \isacommand{proof}\isamarkupfalse%
   6.497 +\ \isanewline
   6.498 +\ \ \isacommand{fix}\isamarkupfalse%
   6.499 +\ p\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}{\isasymColon}monoid\ {\isasymtimes}\ {\isasymbeta}{\isasymColon}monoid{\isachardoublequoteclose}\isanewline
   6.500 +\ \ \isacommand{show}\isamarkupfalse%
   6.501 +\ {\isachardoublequoteopen}p\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
   6.502 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.503 +\ neutral{\isacharunderscore}prod{\isacharunderscore}def\ mult{\isacharunderscore}prod{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
   6.504 +\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutr{\isacharparenright}\isanewline
   6.505 +\isacommand{qed}\isamarkupfalse%
   6.506 +\isanewline
   6.507 +\isanewline
   6.508 +\isacommand{end}\isamarkupfalse%
   6.509 +%
   6.510 +\endisatagquote
   6.511 +{\isafoldquote}%
   6.512 +%
   6.513 +\isadelimquote
   6.514 +%
   6.515 +\endisadelimquote
   6.516 +%
   6.517 +\begin{isamarkuptext}%
   6.518 +\noindent To finish our small algebra example, we add a \isa{group} class
   6.519 +  with a corresponding instance:%
   6.520 +\end{isamarkuptext}%
   6.521 +\isamarkuptrue%
   6.522 +%
   6.523 +\isadelimquote
   6.524 +%
   6.525 +\endisadelimquote
   6.526 +%
   6.527 +\isatagquote
   6.528 +\isacommand{class}\isamarkupfalse%
   6.529 +\ group\ {\isacharequal}\ monoidl\ {\isacharplus}\isanewline
   6.530 +\ \ \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
   6.531 +\ \ \isakeyword{assumes}\ invl{\isacharcolon}\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
   6.532 +\isanewline
   6.533 +\isacommand{instantiation}\isamarkupfalse%
   6.534 +\ int\ {\isacharcolon}{\isacharcolon}\ group\isanewline
   6.535 +\isakeyword{begin}\isanewline
   6.536 +\isanewline
   6.537 +\isacommand{definition}\isamarkupfalse%
   6.538 +\isanewline
   6.539 +\ \ inverse{\isacharunderscore}int{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}i{\isasymdiv}\ {\isacharequal}\ {\isacharminus}\ {\isacharparenleft}i{\isasymColon}int{\isacharparenright}{\isachardoublequoteclose}\isanewline
   6.540 +\isanewline
   6.541 +\isacommand{instance}\isamarkupfalse%
   6.542 +\ \isacommand{proof}\isamarkupfalse%
   6.543 +\isanewline
   6.544 +\ \ \isacommand{fix}\isamarkupfalse%
   6.545 +\ i\ {\isacharcolon}{\isacharcolon}\ int\isanewline
   6.546 +\ \ \isacommand{have}\isamarkupfalse%
   6.547 +\ {\isachardoublequoteopen}{\isacharminus}i\ {\isacharplus}\ i\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
   6.548 +\ simp\isanewline
   6.549 +\ \ \isacommand{then}\isamarkupfalse%
   6.550 +\ \isacommand{show}\isamarkupfalse%
   6.551 +\ {\isachardoublequoteopen}i{\isasymdiv}\ {\isasymotimes}\ i\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
   6.552 +\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
   6.553 +\ mult{\isacharunderscore}int{\isacharunderscore}def\ neutral{\isacharunderscore}int{\isacharunderscore}def\ inverse{\isacharunderscore}int{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse%
   6.554 +\isanewline
   6.555 +\isacommand{qed}\isamarkupfalse%
   6.556 +\isanewline
   6.557 +\isanewline
   6.558 +\isacommand{end}\isamarkupfalse%
   6.559 +%
   6.560 +\endisatagquote
   6.561 +{\isafoldquote}%
   6.562 +%
   6.563 +\isadelimquote
   6.564 +%
   6.565 +\endisadelimquote
   6.566 +%
   6.567 +\isamarkupsection{Type classes as locales%
   6.568 +}
   6.569 +\isamarkuptrue%
   6.570 +%
   6.571 +\isamarkupsubsection{A look behind the scene%
   6.572 +}
   6.573 +\isamarkuptrue%
   6.574 +%
   6.575 +\begin{isamarkuptext}%
   6.576 +The example above gives an impression how Isar type classes work
   6.577 +  in practice.  As stated in the introduction, classes also provide
   6.578 +  a link to Isar's locale system.  Indeed, the logical core of a class
   6.579 +  is nothing else than a locale:%
   6.580 +\end{isamarkuptext}%
   6.581 +\isamarkuptrue%
   6.582 +%
   6.583 +\isadelimquote
   6.584 +%
   6.585 +\endisadelimquote
   6.586 +%
   6.587 +\isatagquote
   6.588 +\isacommand{class}\isamarkupfalse%
   6.589 +\ idem\ {\isacharequal}\isanewline
   6.590 +\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
   6.591 +\ \ \isakeyword{assumes}\ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
   6.592 +\endisatagquote
   6.593 +{\isafoldquote}%
   6.594 +%
   6.595 +\isadelimquote
   6.596 +%
   6.597 +\endisadelimquote
   6.598 +%
   6.599 +\begin{isamarkuptext}%
   6.600 +\noindent essentially introduces the locale%
   6.601 +\end{isamarkuptext}%
   6.602 +\isamarkuptrue%
   6.603 +%
   6.604 +\isadeliminvisible
   6.605 +\ %
   6.606 +\endisadeliminvisible
   6.607 +%
   6.608 +\isataginvisible
   6.609 +\isacommand{setup}\isamarkupfalse%
   6.610 +\ {\isacharverbatimopen}\ Sign{\isachardot}add{\isacharunderscore}path\ {\isachardoublequote}foo{\isachardoublequote}\ {\isacharverbatimclose}%
   6.611 +\endisataginvisible
   6.612 +{\isafoldinvisible}%
   6.613 +%
   6.614 +\isadeliminvisible
   6.615 +%
   6.616 +\endisadeliminvisible
   6.617 +\isanewline
   6.618 +%
   6.619 +\isadelimquote
   6.620 +\isanewline
   6.621 +%
   6.622 +\endisadelimquote
   6.623 +%
   6.624 +\isatagquote
   6.625 +\isacommand{locale}\isamarkupfalse%
   6.626 +\ idem\ {\isacharequal}\isanewline
   6.627 +\ \ \isakeyword{fixes}\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
   6.628 +\ \ \isakeyword{assumes}\ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
   6.629 +\endisatagquote
   6.630 +{\isafoldquote}%
   6.631 +%
   6.632 +\isadelimquote
   6.633 +%
   6.634 +\endisadelimquote
   6.635 +%
   6.636 +\begin{isamarkuptext}%
   6.637 +\noindent together with corresponding constant(s):%
   6.638 +\end{isamarkuptext}%
   6.639 +\isamarkuptrue%
   6.640 +%
   6.641 +\isadelimquote
   6.642 +%
   6.643 +\endisadelimquote
   6.644 +%
   6.645 +\isatagquote
   6.646 +\isacommand{consts}\isamarkupfalse%
   6.647 +\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}%
   6.648 +\endisatagquote
   6.649 +{\isafoldquote}%
   6.650 +%
   6.651 +\isadelimquote
   6.652 +%
   6.653 +\endisadelimquote
   6.654 +%
   6.655 +\begin{isamarkuptext}%
   6.656 +\noindent The connection to the type system is done by means
   6.657 +  of a primitive axclass%
   6.658 +\end{isamarkuptext}%
   6.659 +\isamarkuptrue%
   6.660 +%
   6.661 +\isadeliminvisible
   6.662 +\ %
   6.663 +\endisadeliminvisible
   6.664 +%
   6.665 +\isataginvisible
   6.666 +\isacommand{setup}\isamarkupfalse%
   6.667 +\ {\isacharverbatimopen}\ Sign{\isachardot}add{\isacharunderscore}path\ {\isachardoublequote}foo{\isachardoublequote}\ {\isacharverbatimclose}%
   6.668 +\endisataginvisible
   6.669 +{\isafoldinvisible}%
   6.670 +%
   6.671 +\isadeliminvisible
   6.672 +%
   6.673 +\endisadeliminvisible
   6.674 +\isanewline
   6.675 +%
   6.676 +\isadelimquote
   6.677 +\isanewline
   6.678 +%
   6.679 +\endisadelimquote
   6.680 +%
   6.681 +\isatagquote
   6.682 +\isacommand{axclass}\isamarkupfalse%
   6.683 +\ idem\ {\isacharless}\ type\isanewline
   6.684 +\ \ idem{\isacharcolon}\ {\isachardoublequoteopen}f\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharequal}\ f\ x{\isachardoublequoteclose}%
   6.685 +\endisatagquote
   6.686 +{\isafoldquote}%
   6.687 +%
   6.688 +\isadelimquote
   6.689 +%
   6.690 +\endisadelimquote
   6.691 +%
   6.692 +\isadeliminvisible
   6.693 +\ %
   6.694 +\endisadeliminvisible
   6.695 +%
   6.696 +\isataginvisible
   6.697 +\isacommand{setup}\isamarkupfalse%
   6.698 +\ {\isacharverbatimopen}\ Sign{\isachardot}parent{\isacharunderscore}path\ {\isacharverbatimclose}%
   6.699 +\endisataginvisible
   6.700 +{\isafoldinvisible}%
   6.701 +%
   6.702 +\isadeliminvisible
   6.703 +%
   6.704 +\endisadeliminvisible
   6.705 +%
   6.706 +\begin{isamarkuptext}%
   6.707 +\noindent together with a corresponding interpretation:%
   6.708 +\end{isamarkuptext}%
   6.709 +\isamarkuptrue%
   6.710 +%
   6.711 +\isadelimquote
   6.712 +%
   6.713 +\endisadelimquote
   6.714 +%
   6.715 +\isatagquote
   6.716 +\isacommand{interpretation}\isamarkupfalse%
   6.717 +\ idem{\isacharunderscore}class{\isacharcolon}\isanewline
   6.718 +\ \ idem\ {\isachardoublequoteopen}f\ {\isasymColon}\ {\isacharparenleft}{\isasymalpha}{\isasymColon}idem{\isacharparenright}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\isanewline
   6.719 +\isacommand{proof}\isamarkupfalse%
   6.720 +\ \isacommand{qed}\isamarkupfalse%
   6.721 +\ {\isacharparenleft}rule\ idem{\isacharparenright}%
   6.722 +\endisatagquote
   6.723 +{\isafoldquote}%
   6.724 +%
   6.725 +\isadelimquote
   6.726 +%
   6.727 +\endisadelimquote
   6.728 +%
   6.729 +\begin{isamarkuptext}%
   6.730 +\noindent This gives you at hand the full power of the Isabelle module system;
   6.731 +  conclusions in locale \isa{idem} are implicitly propagated
   6.732 +  to class \isa{idem}.%
   6.733 +\end{isamarkuptext}%
   6.734 +\isamarkuptrue%
   6.735 +%
   6.736 +\isadeliminvisible
   6.737 +\ %
   6.738 +\endisadeliminvisible
   6.739 +%
   6.740 +\isataginvisible
   6.741 +\isacommand{setup}\isamarkupfalse%
   6.742 +\ {\isacharverbatimopen}\ Sign{\isachardot}parent{\isacharunderscore}path\ {\isacharverbatimclose}%
   6.743 +\endisataginvisible
   6.744 +{\isafoldinvisible}%
   6.745 +%
   6.746 +\isadeliminvisible
   6.747 +%
   6.748 +\endisadeliminvisible
   6.749 +%
   6.750 +\isamarkupsubsection{Abstract reasoning%
   6.751 +}
   6.752 +\isamarkuptrue%
   6.753 +%
   6.754 +\begin{isamarkuptext}%
   6.755 +Isabelle locales enable reasoning at a general level, while results
   6.756 +  are implicitly transferred to all instances.  For example, we can
   6.757 +  now establish the \isa{left{\isacharunderscore}cancel} lemma for groups, which
   6.758 +  states that the function \isa{{\isacharparenleft}x\ {\isasymotimes}{\isacharparenright}} is injective:%
   6.759 +\end{isamarkuptext}%
   6.760 +\isamarkuptrue%
   6.761 +%
   6.762 +\isadelimquote
   6.763 +%
   6.764 +\endisadelimquote
   6.765 +%
   6.766 +\isatagquote
   6.767 +\isacommand{lemma}\isamarkupfalse%
   6.768 +\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ left{\isacharunderscore}cancel{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z\ {\isasymlongleftrightarrow}\ y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
   6.769 +\isacommand{proof}\isamarkupfalse%
   6.770 +\isanewline
   6.771 +\ \ \isacommand{assume}\isamarkupfalse%
   6.772 +\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z{\isachardoublequoteclose}\isanewline
   6.773 +\ \ \isacommand{then}\isamarkupfalse%
   6.774 +\ \isacommand{have}\isamarkupfalse%
   6.775 +\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isacharequal}\ x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
   6.776 +\ simp\isanewline
   6.777 +\ \ \isacommand{then}\isamarkupfalse%
   6.778 +\ \isacommand{have}\isamarkupfalse%
   6.779 +\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasymdiv}\ {\isasymotimes}\ x{\isacharparenright}\ {\isasymotimes}\ y\ {\isacharequal}\ {\isacharparenleft}x{\isasymdiv}\ {\isasymotimes}\ x{\isacharparenright}\ {\isasymotimes}\ z{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
   6.780 +\ assoc\ \isacommand{by}\isamarkupfalse%
   6.781 +\ simp\isanewline
   6.782 +\ \ \isacommand{then}\isamarkupfalse%
   6.783 +\ \isacommand{show}\isamarkupfalse%
   6.784 +\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
   6.785 +\ neutl\ \isakeyword{and}\ invl\ \isacommand{by}\isamarkupfalse%
   6.786 +\ simp\isanewline
   6.787 +\isacommand{next}\isamarkupfalse%
   6.788 +\isanewline
   6.789 +\ \ \isacommand{assume}\isamarkupfalse%
   6.790 +\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
   6.791 +\ \ \isacommand{then}\isamarkupfalse%
   6.792 +\ \isacommand{show}\isamarkupfalse%
   6.793 +\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y\ {\isacharequal}\ x\ {\isasymotimes}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
   6.794 +\ simp\isanewline
   6.795 +\isacommand{qed}\isamarkupfalse%
   6.796 +%
   6.797 +\endisatagquote
   6.798 +{\isafoldquote}%
   6.799 +%
   6.800 +\isadelimquote
   6.801 +%
   6.802 +\endisadelimquote
   6.803 +%
   6.804 +\begin{isamarkuptext}%
   6.805 +\noindent Here the \qt{\hyperlink{keyword.in}{\mbox{\isa{\isakeyword{in}}}} \isa{group}} target specification
   6.806 +  indicates that the result is recorded within that context for later
   6.807 +  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
   6.808 +  \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}}.%
   6.809 +\end{isamarkuptext}%
   6.810 +\isamarkuptrue%
   6.811 +%
   6.812 +\isamarkupsubsection{Derived definitions%
   6.813 +}
   6.814 +\isamarkuptrue%
   6.815 +%
   6.816 +\begin{isamarkuptext}%
   6.817 +Isabelle locales support a concept of local definitions
   6.818 +  in locales:%
   6.819 +\end{isamarkuptext}%
   6.820 +\isamarkuptrue%
   6.821 +%
   6.822 +\isadelimquote
   6.823 +%
   6.824 +\endisadelimquote
   6.825 +%
   6.826 +\isatagquote
   6.827 +\isacommand{primrec}\isamarkupfalse%
   6.828 +\ {\isacharparenleft}\isakeyword{in}\ monoid{\isacharparenright}\ pow{\isacharunderscore}nat\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
   6.829 +\ \ {\isachardoublequoteopen}pow{\isacharunderscore}nat\ {\isadigit{0}}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
   6.830 +\ \ {\isacharbar}\ {\isachardoublequoteopen}pow{\isacharunderscore}nat\ {\isacharparenleft}Suc\ n{\isacharparenright}\ x\ {\isacharequal}\ x\ {\isasymotimes}\ pow{\isacharunderscore}nat\ n\ x{\isachardoublequoteclose}%
   6.831 +\endisatagquote
   6.832 +{\isafoldquote}%
   6.833 +%
   6.834 +\isadelimquote
   6.835 +%
   6.836 +\endisadelimquote
   6.837 +%
   6.838 +\begin{isamarkuptext}%
   6.839 +\noindent If the locale \isa{group} is also a class, this local
   6.840 +  definition is propagated onto a global definition of
   6.841 +  \isa{{\isachardoublequote}pow{\isacharunderscore}nat\ {\isasymColon}\ nat\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}monoid\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}monoid{\isachardoublequote}}
   6.842 +  with corresponding theorems
   6.843 +
   6.844 +  \isa{pow{\isacharunderscore}nat\ {\isadigit{0}}\ x\ {\isacharequal}\ {\isasymone}\isasep\isanewline%
   6.845 +pow{\isacharunderscore}nat\ {\isacharparenleft}Suc\ n{\isacharparenright}\ x\ {\isacharequal}\ x\ {\isasymotimes}\ pow{\isacharunderscore}nat\ n\ x}.
   6.846 +
   6.847 +  \noindent As you can see from this example, for local
   6.848 +  definitions you may use any specification tool
   6.849 +  which works together with locales (e.g. \cite{krauss2006}).%
   6.850 +\end{isamarkuptext}%
   6.851 +\isamarkuptrue%
   6.852 +%
   6.853 +\isamarkupsubsection{A functor analogy%
   6.854 +}
   6.855 +\isamarkuptrue%
   6.856 +%
   6.857 +\begin{isamarkuptext}%
   6.858 +We introduced Isar classes by analogy to type classes
   6.859 +  functional programming;  if we reconsider this in the
   6.860 +  context of what has been said about type classes and locales,
   6.861 +  we can drive this analogy further by stating that type
   6.862 +  classes essentially correspond to functors which have
   6.863 +  a canonical interpretation as type classes.
   6.864 +  Anyway, there is also the possibility of other interpretations.
   6.865 +  For example, also \isa{list}s form a monoid with
   6.866 +  \isa{append} and \isa{{\isacharbrackleft}{\isacharbrackright}} as operations, but it
   6.867 +  seems inappropriate to apply to lists
   6.868 +  the same operations as for genuinely algebraic types.
   6.869 +  In such a case, we simply can do a particular interpretation
   6.870 +  of monoids for lists:%
   6.871 +\end{isamarkuptext}%
   6.872 +\isamarkuptrue%
   6.873 +%
   6.874 +\isadelimquote
   6.875 +%
   6.876 +\endisadelimquote
   6.877 +%
   6.878 +\isatagquote
   6.879 +\isacommand{interpretation}\isamarkupfalse%
   6.880 +\ list{\isacharunderscore}monoid{\isacharbang}{\isacharcolon}\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
   6.881 +\ \ \isacommand{proof}\isamarkupfalse%
   6.882 +\ \isacommand{qed}\isamarkupfalse%
   6.883 +\ auto%
   6.884 +\endisatagquote
   6.885 +{\isafoldquote}%
   6.886 +%
   6.887 +\isadelimquote
   6.888 +%
   6.889 +\endisadelimquote
   6.890 +%
   6.891 +\begin{isamarkuptext}%
   6.892 +\noindent This enables us to apply facts on monoids
   6.893 +  to lists, e.g. \isa{{\isacharbrackleft}{\isacharbrackright}\ {\isacharat}\ x\ {\isacharequal}\ x}.
   6.894 +
   6.895 +  When using this interpretation pattern, it may also
   6.896 +  be appropriate to map derived definitions accordingly:%
   6.897 +\end{isamarkuptext}%
   6.898 +\isamarkuptrue%
   6.899 +%
   6.900 +\isadelimquote
   6.901 +%
   6.902 +\endisadelimquote
   6.903 +%
   6.904 +\isatagquote
   6.905 +\isacommand{primrec}\isamarkupfalse%
   6.906 +\ replicate\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isasymalpha}\ list\ {\isasymRightarrow}\ {\isasymalpha}\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
   6.907 +\ \ {\isachardoublequoteopen}replicate\ {\isadigit{0}}\ {\isacharunderscore}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
   6.908 +\ \ {\isacharbar}\ {\isachardoublequoteopen}replicate\ {\isacharparenleft}Suc\ n{\isacharparenright}\ xs\ {\isacharequal}\ xs\ {\isacharat}\ replicate\ n\ xs{\isachardoublequoteclose}\isanewline
   6.909 +\isanewline
   6.910 +\isacommand{interpretation}\isamarkupfalse%
   6.911 +\ list{\isacharunderscore}monoid{\isacharbang}{\isacharcolon}\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
   6.912 +\ \ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ replicate{\isachardoublequoteclose}\isanewline
   6.913 +\isacommand{proof}\isamarkupfalse%
   6.914 +\ {\isacharminus}\isanewline
   6.915 +\ \ \isacommand{interpret}\isamarkupfalse%
   6.916 +\ monoid\ append\ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
   6.917 +\isanewline
   6.918 +\ \ \isacommand{show}\isamarkupfalse%
   6.919 +\ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ replicate{\isachardoublequoteclose}\isanewline
   6.920 +\ \ \isacommand{proof}\isamarkupfalse%
   6.921 +\isanewline
   6.922 +\ \ \ \ \isacommand{fix}\isamarkupfalse%
   6.923 +\ n\isanewline
   6.924 +\ \ \ \ \isacommand{show}\isamarkupfalse%
   6.925 +\ {\isachardoublequoteopen}monoid{\isachardot}pow{\isacharunderscore}nat\ append\ {\isacharbrackleft}{\isacharbrackright}\ n\ {\isacharequal}\ replicate\ n{\isachardoublequoteclose}\isanewline
   6.926 +\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
   6.927 +\ {\isacharparenleft}induct\ n{\isacharparenright}\ auto\isanewline
   6.928 +\ \ \isacommand{qed}\isamarkupfalse%
   6.929 +\isanewline
   6.930 +\isacommand{qed}\isamarkupfalse%
   6.931 +\ intro{\isacharunderscore}locales%
   6.932 +\endisatagquote
   6.933 +{\isafoldquote}%
   6.934 +%
   6.935 +\isadelimquote
   6.936 +%
   6.937 +\endisadelimquote
   6.938 +%
   6.939 +\isamarkupsubsection{Additional subclass relations%
   6.940 +}
   6.941 +\isamarkuptrue%
   6.942 +%
   6.943 +\begin{isamarkuptext}%
   6.944 +Any \isa{group} is also a \isa{monoid};  this
   6.945 +  can be made explicit by claiming an additional
   6.946 +  subclass relation,
   6.947 +  together with a proof of the logical difference:%
   6.948 +\end{isamarkuptext}%
   6.949 +\isamarkuptrue%
   6.950 +%
   6.951 +\isadelimquote
   6.952 +%
   6.953 +\endisadelimquote
   6.954 +%
   6.955 +\isatagquote
   6.956 +\isacommand{subclass}\isamarkupfalse%
   6.957 +\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ monoid\isanewline
   6.958 +\isacommand{proof}\isamarkupfalse%
   6.959 +\isanewline
   6.960 +\ \ \isacommand{fix}\isamarkupfalse%
   6.961 +\ x\isanewline
   6.962 +\ \ \isacommand{from}\isamarkupfalse%
   6.963 +\ invl\ \isacommand{have}\isamarkupfalse%
   6.964 +\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
   6.965 +\ simp\isanewline
   6.966 +\ \ \isacommand{with}\isamarkupfalse%
   6.967 +\ assoc\ {\isacharbrackleft}symmetric{\isacharbrackright}\ neutl\ invl\ \isacommand{have}\isamarkupfalse%
   6.968 +\ {\isachardoublequoteopen}x{\isasymdiv}\ {\isasymotimes}\ {\isacharparenleft}x\ {\isasymotimes}\ {\isasymone}{\isacharparenright}\ {\isacharequal}\ x{\isasymdiv}\ {\isasymotimes}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
   6.969 +\ simp\isanewline
   6.970 +\ \ \isacommand{with}\isamarkupfalse%
   6.971 +\ left{\isacharunderscore}cancel\ \isacommand{show}\isamarkupfalse%
   6.972 +\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
   6.973 +\ simp\isanewline
   6.974 +\isacommand{qed}\isamarkupfalse%
   6.975 +%
   6.976 +\endisatagquote
   6.977 +{\isafoldquote}%
   6.978 +%
   6.979 +\isadelimquote
   6.980 +%
   6.981 +\endisadelimquote
   6.982 +%
   6.983 +\begin{isamarkuptext}%
   6.984 +\noindent The logical proof is carried out on the locale level.
   6.985 +  Afterwards it is propagated
   6.986 +  to the type system, making \isa{group} an instance of
   6.987 +  \isa{monoid} by adding an additional edge
   6.988 +  to the graph of subclass relations
   6.989 +  (cf.\ \figref{fig:subclass}).
   6.990 +
   6.991 +  \begin{figure}[htbp]
   6.992 +   \begin{center}
   6.993 +     \small
   6.994 +     \unitlength 0.6mm
   6.995 +     \begin{picture}(40,60)(0,0)
   6.996 +       \put(20,60){\makebox(0,0){\isa{semigroup}}}
   6.997 +       \put(20,40){\makebox(0,0){\isa{monoidl}}}
   6.998 +       \put(00,20){\makebox(0,0){\isa{monoid}}}
   6.999 +       \put(40,00){\makebox(0,0){\isa{group}}}
  6.1000 +       \put(20,55){\vector(0,-1){10}}
  6.1001 +       \put(15,35){\vector(-1,-1){10}}
  6.1002 +       \put(25,35){\vector(1,-3){10}}
  6.1003 +     \end{picture}
  6.1004 +     \hspace{8em}
  6.1005 +     \begin{picture}(40,60)(0,0)
  6.1006 +       \put(20,60){\makebox(0,0){\isa{semigroup}}}
  6.1007 +       \put(20,40){\makebox(0,0){\isa{monoidl}}}
  6.1008 +       \put(00,20){\makebox(0,0){\isa{monoid}}}
  6.1009 +       \put(40,00){\makebox(0,0){\isa{group}}}
  6.1010 +       \put(20,55){\vector(0,-1){10}}
  6.1011 +       \put(15,35){\vector(-1,-1){10}}
  6.1012 +       \put(05,15){\vector(3,-1){30}}
  6.1013 +     \end{picture}
  6.1014 +     \caption{Subclass relationship of monoids and groups:
  6.1015 +        before and after establishing the relationship
  6.1016 +        \isa{group\ {\isasymsubseteq}\ monoid};  transitive edges left out.}
  6.1017 +     \label{fig:subclass}
  6.1018 +   \end{center}
  6.1019 +  \end{figure}
  6.1020 +7
  6.1021 +  For illustration, a derived definition
  6.1022 +  in \isa{group} which uses \isa{pow{\isacharunderscore}nat}:%
  6.1023 +\end{isamarkuptext}%
  6.1024 +\isamarkuptrue%
  6.1025 +%
  6.1026 +\isadelimquote
  6.1027 +%
  6.1028 +\endisadelimquote
  6.1029 +%
  6.1030 +\isatagquote
  6.1031 +\isacommand{definition}\isamarkupfalse%
  6.1032 +\ {\isacharparenleft}\isakeyword{in}\ group{\isacharparenright}\ pow{\isacharunderscore}int\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}int\ {\isasymRightarrow}\ {\isasymalpha}\ {\isasymRightarrow}\ {\isasymalpha}{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
  6.1033 +\ \ {\isachardoublequoteopen}pow{\isacharunderscore}int\ k\ x\ {\isacharequal}\ {\isacharparenleft}if\ k\ {\isachargreater}{\isacharequal}\ {\isadigit{0}}\isanewline
  6.1034 +\ \ \ \ then\ pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ k{\isacharparenright}\ x\isanewline
  6.1035 +\ \ \ \ else\ {\isacharparenleft}pow{\isacharunderscore}nat\ {\isacharparenleft}nat\ {\isacharparenleft}{\isacharminus}\ k{\isacharparenright}{\isacharparenright}\ x{\isacharparenright}{\isasymdiv}{\isacharparenright}{\isachardoublequoteclose}%
  6.1036 +\endisatagquote
  6.1037 +{\isafoldquote}%
  6.1038 +%
  6.1039 +\isadelimquote
  6.1040 +%
  6.1041 +\endisadelimquote
  6.1042 +%
  6.1043 +\begin{isamarkuptext}%
  6.1044 +\noindent yields the global definition of
  6.1045 +  \isa{{\isachardoublequote}pow{\isacharunderscore}int\ {\isasymColon}\ int\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}group\ {\isasymRightarrow}\ {\isasymalpha}{\isasymColon}group{\isachardoublequote}}
  6.1046 +  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}}.%
  6.1047 +\end{isamarkuptext}%
  6.1048 +\isamarkuptrue%
  6.1049 +%
  6.1050 +\isamarkupsubsection{A note on syntax%
  6.1051 +}
  6.1052 +\isamarkuptrue%
  6.1053 +%
  6.1054 +\begin{isamarkuptext}%
  6.1055 +As a commodity, class context syntax allows to refer
  6.1056 +  to local class operations and their global counterparts
  6.1057 +  uniformly;  type inference resolves ambiguities.  For example:%
  6.1058 +\end{isamarkuptext}%
  6.1059 +\isamarkuptrue%
  6.1060 +%
  6.1061 +\isadelimquote
  6.1062 +%
  6.1063 +\endisadelimquote
  6.1064 +%
  6.1065 +\isatagquote
  6.1066 +\isacommand{context}\isamarkupfalse%
  6.1067 +\ semigroup\isanewline
  6.1068 +\isakeyword{begin}\isanewline
  6.1069 +\isanewline
  6.1070 +\isacommand{term}\isamarkupfalse%
  6.1071 +\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
  6.1072 +\isamarkupcmt{example 1%
  6.1073 +}
  6.1074 +\isanewline
  6.1075 +\isacommand{term}\isamarkupfalse%
  6.1076 +\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasymColon}nat{\isacharparenright}\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
  6.1077 +\isamarkupcmt{example 2%
  6.1078 +}
  6.1079 +\isanewline
  6.1080 +\isanewline
  6.1081 +\isacommand{end}\isamarkupfalse%
  6.1082 +\isanewline
  6.1083 +\isanewline
  6.1084 +\isacommand{term}\isamarkupfalse%
  6.1085 +\ {\isachardoublequoteopen}x\ {\isasymotimes}\ y{\isachardoublequoteclose}\ %
  6.1086 +\isamarkupcmt{example 3%
  6.1087 +}
  6.1088 +%
  6.1089 +\endisatagquote
  6.1090 +{\isafoldquote}%
  6.1091 +%
  6.1092 +\isadelimquote
  6.1093 +%
  6.1094 +\endisadelimquote
  6.1095 +%
  6.1096 +\begin{isamarkuptext}%
  6.1097 +\noindent Here in example 1, the term refers to the local class operation
  6.1098 +  \isa{mult\ {\isacharbrackleft}{\isasymalpha}{\isacharbrackright}}, whereas in example 2 the type constraint
  6.1099 +  enforces the global class operation \isa{mult\ {\isacharbrackleft}nat{\isacharbrackright}}.
  6.1100 +  In the global context in example 3, the reference is
  6.1101 +  to the polymorphic global class operation \isa{mult\ {\isacharbrackleft}{\isacharquery}{\isasymalpha}\ {\isasymColon}\ semigroup{\isacharbrackright}}.%
  6.1102 +\end{isamarkuptext}%
  6.1103 +\isamarkuptrue%
  6.1104 +%
  6.1105 +\isamarkupsection{Further issues%
  6.1106 +}
  6.1107 +\isamarkuptrue%
  6.1108 +%
  6.1109 +\isamarkupsubsection{Type classes and code generation%
  6.1110 +}
  6.1111 +\isamarkuptrue%
  6.1112 +%
  6.1113 +\begin{isamarkuptext}%
  6.1114 +Turning back to the first motivation for type classes,
  6.1115 +  namely overloading, it is obvious that overloading
  6.1116 +  stemming from \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} statements and
  6.1117 +  \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}}
  6.1118 +  targets naturally maps to Haskell type classes.
  6.1119 +  The code generator framework \cite{isabelle-codegen} 
  6.1120 +  takes this into account.  Concerning target languages
  6.1121 +  lacking type classes (e.g.~SML), type classes
  6.1122 +  are implemented by explicit dictionary construction.
  6.1123 +  As example, let's go back to the power function:%
  6.1124 +\end{isamarkuptext}%
  6.1125 +\isamarkuptrue%
  6.1126 +%
  6.1127 +\isadelimquote
  6.1128 +%
  6.1129 +\endisadelimquote
  6.1130 +%
  6.1131 +\isatagquote
  6.1132 +\isacommand{definition}\isamarkupfalse%
  6.1133 +\ example\ {\isacharcolon}{\isacharcolon}\ int\ \isakeyword{where}\isanewline
  6.1134 +\ \ {\isachardoublequoteopen}example\ {\isacharequal}\ pow{\isacharunderscore}int\ {\isadigit{1}}{\isadigit{0}}\ {\isacharparenleft}{\isacharminus}{\isadigit{2}}{\isacharparenright}{\isachardoublequoteclose}%
  6.1135 +\endisatagquote
  6.1136 +{\isafoldquote}%
  6.1137 +%
  6.1138 +\isadelimquote
  6.1139 +%
  6.1140 +\endisadelimquote
  6.1141 +%
  6.1142 +\begin{isamarkuptext}%
  6.1143 +\noindent This maps to Haskell as:%
  6.1144 +\end{isamarkuptext}%
  6.1145 +\isamarkuptrue%
  6.1146 +%
  6.1147 +\isadelimquote
  6.1148 +%
  6.1149 +\endisadelimquote
  6.1150 +%
  6.1151 +\isatagquote
  6.1152 +%
  6.1153 +\begin{isamarkuptext}%
  6.1154 +\isatypewriter%
  6.1155 +\noindent%
  6.1156 +\hspace*{0pt}module Example where {\char123}\\
  6.1157 +\hspace*{0pt}\\
  6.1158 +\hspace*{0pt}\\
  6.1159 +\hspace*{0pt}data Nat = Zero{\char95}nat | Suc Nat;\\
  6.1160 +\hspace*{0pt}\\
  6.1161 +\hspace*{0pt}nat{\char95}aux ::~Integer -> Nat -> Nat;\\
  6.1162 +\hspace*{0pt}nat{\char95}aux i n = (if i <= 0 then n else nat{\char95}aux (i - 1) (Suc n));\\
  6.1163 +\hspace*{0pt}\\
  6.1164 +\hspace*{0pt}nat ::~Integer -> Nat;\\
  6.1165 +\hspace*{0pt}nat i = nat{\char95}aux i Zero{\char95}nat;\\
  6.1166 +\hspace*{0pt}\\
  6.1167 +\hspace*{0pt}class Semigroup a where {\char123}\\
  6.1168 +\hspace*{0pt} ~mult ::~a -> a -> a;\\
  6.1169 +\hspace*{0pt}{\char125};\\
  6.1170 +\hspace*{0pt}\\
  6.1171 +\hspace*{0pt}class (Semigroup a) => Monoidl a where {\char123}\\
  6.1172 +\hspace*{0pt} ~neutral ::~a;\\
  6.1173 +\hspace*{0pt}{\char125};\\
  6.1174 +\hspace*{0pt}\\
  6.1175 +\hspace*{0pt}class (Monoidl a) => Monoid a where {\char123}\\
  6.1176 +\hspace*{0pt}{\char125};\\
  6.1177 +\hspace*{0pt}\\
  6.1178 +\hspace*{0pt}class (Monoid a) => Group a where {\char123}\\
  6.1179 +\hspace*{0pt} ~inverse ::~a -> a;\\
  6.1180 +\hspace*{0pt}{\char125};\\
  6.1181 +\hspace*{0pt}\\
  6.1182 +\hspace*{0pt}inverse{\char95}int ::~Integer -> Integer;\\
  6.1183 +\hspace*{0pt}inverse{\char95}int i = negate i;\\
  6.1184 +\hspace*{0pt}\\
  6.1185 +\hspace*{0pt}neutral{\char95}int ::~Integer;\\
  6.1186 +\hspace*{0pt}neutral{\char95}int = 0;\\
  6.1187 +\hspace*{0pt}\\
  6.1188 +\hspace*{0pt}mult{\char95}int ::~Integer -> Integer -> Integer;\\
  6.1189 +\hspace*{0pt}mult{\char95}int i j = i + j;\\
  6.1190 +\hspace*{0pt}\\
  6.1191 +\hspace*{0pt}instance Semigroup Integer where {\char123}\\
  6.1192 +\hspace*{0pt} ~mult = mult{\char95}int;\\
  6.1193 +\hspace*{0pt}{\char125};\\
  6.1194 +\hspace*{0pt}\\
  6.1195 +\hspace*{0pt}instance Monoidl Integer where {\char123}\\
  6.1196 +\hspace*{0pt} ~neutral = neutral{\char95}int;\\
  6.1197 +\hspace*{0pt}{\char125};\\
  6.1198 +\hspace*{0pt}\\
  6.1199 +\hspace*{0pt}instance Monoid Integer where {\char123}\\
  6.1200 +\hspace*{0pt}{\char125};\\
  6.1201 +\hspace*{0pt}\\
  6.1202 +\hspace*{0pt}instance Group Integer where {\char123}\\
  6.1203 +\hspace*{0pt} ~inverse = inverse{\char95}int;\\
  6.1204 +\hspace*{0pt}{\char125};\\
  6.1205 +\hspace*{0pt}\\
  6.1206 +\hspace*{0pt}pow{\char95}nat ::~forall a.~(Monoid a) => Nat -> a -> a;\\
  6.1207 +\hspace*{0pt}pow{\char95}nat Zero{\char95}nat x = neutral;\\
  6.1208 +\hspace*{0pt}pow{\char95}nat (Suc n) x = mult x (pow{\char95}nat n x);\\
  6.1209 +\hspace*{0pt}\\
  6.1210 +\hspace*{0pt}pow{\char95}int ::~forall a.~(Group a) => Integer -> a -> a;\\
  6.1211 +\hspace*{0pt}pow{\char95}int k x =\\
  6.1212 +\hspace*{0pt} ~(if 0 <= k then pow{\char95}nat (nat k) x\\
  6.1213 +\hspace*{0pt} ~~~else inverse (pow{\char95}nat (nat (negate k)) x));\\
  6.1214 +\hspace*{0pt}\\
  6.1215 +\hspace*{0pt}example ::~Integer;\\
  6.1216 +\hspace*{0pt}example = pow{\char95}int 10 (-2);\\
  6.1217 +\hspace*{0pt}\\
  6.1218 +\hspace*{0pt}{\char125}%
  6.1219 +\end{isamarkuptext}%
  6.1220 +\isamarkuptrue%
  6.1221 +%
  6.1222 +\endisatagquote
  6.1223 +{\isafoldquote}%
  6.1224 +%
  6.1225 +\isadelimquote
  6.1226 +%
  6.1227 +\endisadelimquote
  6.1228 +%
  6.1229 +\begin{isamarkuptext}%
  6.1230 +\noindent The whole code in SML with explicit dictionary passing:%
  6.1231 +\end{isamarkuptext}%
  6.1232 +\isamarkuptrue%
  6.1233 +%
  6.1234 +\isadelimquote
  6.1235 +%
  6.1236 +\endisadelimquote
  6.1237 +%
  6.1238 +\isatagquote
  6.1239 +%
  6.1240 +\begin{isamarkuptext}%
  6.1241 +\isatypewriter%
  6.1242 +\noindent%
  6.1243 +\hspace*{0pt}structure Example = \\
  6.1244 +\hspace*{0pt}struct\\
  6.1245 +\hspace*{0pt}\\
  6.1246 +\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
  6.1247 +\hspace*{0pt}\\
  6.1248 +\hspace*{0pt}fun nat{\char95}aux i n =\\
  6.1249 +\hspace*{0pt} ~(if IntInf.<= (i,~(0 :~IntInf.int)) then n\\
  6.1250 +\hspace*{0pt} ~~~else nat{\char95}aux (IntInf.- (i,~(1 :~IntInf.int))) (Suc n));\\
  6.1251 +\hspace*{0pt}\\
  6.1252 +\hspace*{0pt}fun nat i = nat{\char95}aux i Zero{\char95}nat;\\
  6.1253 +\hspace*{0pt}\\
  6.1254 +\hspace*{0pt}type 'a semigroup = {\char123}mult :~'a -> 'a -> 'a{\char125};\\
  6.1255 +\hspace*{0pt}fun mult (A{\char95}:'a semigroup) = {\char35}mult A{\char95};\\
  6.1256 +\hspace*{0pt}\\
  6.1257 +\hspace*{0pt}type 'a monoidl =\\
  6.1258 +\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}semigroup{\char95}monoidl :~'a semigroup,~neutral :~'a{\char125};\\
  6.1259 +\hspace*{0pt}fun semigroup{\char95}monoidl (A{\char95}:'a monoidl) = {\char35}Classes{\char95}{\char95}semigroup{\char95}monoidl A{\char95};\\
  6.1260 +\hspace*{0pt}fun neutral (A{\char95}:'a monoidl) = {\char35}neutral A{\char95};\\
  6.1261 +\hspace*{0pt}\\
  6.1262 +\hspace*{0pt}type 'a monoid = {\char123}Classes{\char95}{\char95}monoidl{\char95}monoid :~'a monoidl{\char125};\\
  6.1263 +\hspace*{0pt}fun monoidl{\char95}monoid (A{\char95}:'a monoid) = {\char35}Classes{\char95}{\char95}monoidl{\char95}monoid A{\char95};\\
  6.1264 +\hspace*{0pt}\\
  6.1265 +\hspace*{0pt}type 'a group = {\char123}Classes{\char95}{\char95}monoid{\char95}group :~'a monoid,~inverse :~'a -> 'a{\char125};\\
  6.1266 +\hspace*{0pt}fun monoid{\char95}group (A{\char95}:'a group) = {\char35}Classes{\char95}{\char95}monoid{\char95}group A{\char95};\\
  6.1267 +\hspace*{0pt}fun inverse (A{\char95}:'a group) = {\char35}inverse A{\char95};\\
  6.1268 +\hspace*{0pt}\\
  6.1269 +\hspace*{0pt}fun inverse{\char95}int i = IntInf.{\char126}~i;\\
  6.1270 +\hspace*{0pt}\\
  6.1271 +\hspace*{0pt}val neutral{\char95}int :~IntInf.int = (0 :~IntInf.int)\\
  6.1272 +\hspace*{0pt}\\
  6.1273 +\hspace*{0pt}fun mult{\char95}int i j = IntInf.+ (i,~j);\\
  6.1274 +\hspace*{0pt}\\
  6.1275 +\hspace*{0pt}val semigroup{\char95}int = {\char123}mult = mult{\char95}int{\char125}~:~IntInf.int semigroup;\\
  6.1276 +\hspace*{0pt}\\
  6.1277 +\hspace*{0pt}val monoidl{\char95}int =\\
  6.1278 +\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}semigroup{\char95}monoidl = semigroup{\char95}int,~neutral = neutral{\char95}int{\char125}~:\\
  6.1279 +\hspace*{0pt} ~IntInf.int monoidl;\\
  6.1280 +\hspace*{0pt}\\
  6.1281 +\hspace*{0pt}val monoid{\char95}int = {\char123}Classes{\char95}{\char95}monoidl{\char95}monoid = monoidl{\char95}int{\char125}~:\\
  6.1282 +\hspace*{0pt} ~IntInf.int monoid;\\
  6.1283 +\hspace*{0pt}\\
  6.1284 +\hspace*{0pt}val group{\char95}int =\\
  6.1285 +\hspace*{0pt} ~{\char123}Classes{\char95}{\char95}monoid{\char95}group = monoid{\char95}int,~inverse = inverse{\char95}int{\char125}~:\\
  6.1286 +\hspace*{0pt} ~IntInf.int group;\\
  6.1287 +\hspace*{0pt}\\
  6.1288 +\hspace*{0pt}fun pow{\char95}nat A{\char95}~Zero{\char95}nat x = neutral (monoidl{\char95}monoid A{\char95})\\
  6.1289 +\hspace*{0pt} ~| pow{\char95}nat A{\char95}~(Suc n) x =\\
  6.1290 +\hspace*{0pt} ~~~mult ((semigroup{\char95}monoidl o monoidl{\char95}monoid) A{\char95}) x (pow{\char95}nat A{\char95}~n x);\\
  6.1291 +\hspace*{0pt}\\
  6.1292 +\hspace*{0pt}fun pow{\char95}int A{\char95}~k x =\\
  6.1293 +\hspace*{0pt} ~(if IntInf.<= ((0 :~IntInf.int),~k)\\
  6.1294 +\hspace*{0pt} ~~~then pow{\char95}nat (monoid{\char95}group A{\char95}) (nat k) x\\
  6.1295 +\hspace*{0pt} ~~~else inverse A{\char95}~(pow{\char95}nat (monoid{\char95}group A{\char95}) (nat (IntInf.{\char126}~k)) x));\\
  6.1296 +\hspace*{0pt}\\
  6.1297 +\hspace*{0pt}val example :~IntInf.int =\\
  6.1298 +\hspace*{0pt} ~pow{\char95}int group{\char95}int (10 :~IntInf.int) ({\char126}2 :~IntInf.int)\\
  6.1299 +\hspace*{0pt}\\
  6.1300 +\hspace*{0pt}end;~(*struct Example*)%
  6.1301 +\end{isamarkuptext}%
  6.1302 +\isamarkuptrue%
  6.1303 +%
  6.1304 +\endisatagquote
  6.1305 +{\isafoldquote}%
  6.1306 +%
  6.1307 +\isadelimquote
  6.1308 +%
  6.1309 +\endisadelimquote
  6.1310 +%
  6.1311 +\isamarkupsubsection{Inspecting the type class universe%
  6.1312 +}
  6.1313 +\isamarkuptrue%
  6.1314 +%
  6.1315 +\begin{isamarkuptext}%
  6.1316 +To facilitate orientation in complex subclass structures,
  6.1317 +  two diagnostics commands are provided:
  6.1318 +
  6.1319 +  \begin{description}
  6.1320 +
  6.1321 +    \item[\hyperlink{command.print-classes}{\mbox{\isa{\isacommand{print{\isacharunderscore}classes}}}}] print a list of all classes
  6.1322 +      together with associated operations etc.
  6.1323 +
  6.1324 +    \item[\hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isacharunderscore}deps}}}}] visualizes the subclass relation
  6.1325 +      between all classes as a Hasse diagram.
  6.1326 +
  6.1327 +  \end{description}%
  6.1328 +\end{isamarkuptext}%
  6.1329 +\isamarkuptrue%
  6.1330 +%
  6.1331 +\isadelimtheory
  6.1332 +%
  6.1333 +\endisadelimtheory
  6.1334 +%
  6.1335 +\isatagtheory
  6.1336 +\isacommand{end}\isamarkupfalse%
  6.1337 +%
  6.1338 +\endisatagtheory
  6.1339 +{\isafoldtheory}%
  6.1340 +%
  6.1341 +\isadelimtheory
  6.1342 +%
  6.1343 +\endisadelimtheory
  6.1344 +\isanewline
  6.1345 +\end{isabellebody}%
  6.1346 +%%% Local Variables:
  6.1347 +%%% mode: latex
  6.1348 +%%% TeX-master: "root"
  6.1349 +%%% End:

     7.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     7.2 +++ b/doc-src/Classes/classes.tex	Tue Mar 03 11:00:51 2009 +0100
     7.3 @@ -0,0 +1,50 @@
     7.4 +
     7.5 +\documentclass[12pt,a4paper,fleqn]{report}
     7.6 +\usepackage{latexsym,graphicx}
     7.7 +\usepackage[refpage]{nomencl}
     7.8 +\usepackage{../iman,../extra,../isar,../proof}
     7.9 +\usepackage{../isabelle,../isabellesym}
    7.10 +\usepackage{style}
    7.11 +\usepackage{../pdfsetup}
    7.12 +
    7.13 +
    7.14 +\hyphenation{Isabelle}
    7.15 +\hyphenation{Isar}
    7.16 +\isadroptag{theory}
    7.17 +
    7.18 +\title{\includegraphics[scale=0.5]{isabelle_isar}
    7.19 +  \\[4ex] Haskell-style type classes with Isabelle/Isar}
    7.20 +\author{\emph{Florian Haftmann}}
    7.21 +
    7.22 +\begin{document}
    7.23 +
    7.24 +\maketitle
    7.25 +
    7.26 +\begin{abstract}
    7.27 +  This tutorial introduces the look-and-feel of Isar type classes
    7.28 +  to the end-user; Isar type classes are a convenient mechanism
    7.29 +  for organizing specifications, overcoming some drawbacks
    7.30 +  of raw axiomatic type classes. Essentially, they combine
    7.31 +  an operational aspect (in the manner of Haskell) with
    7.32 +  a logical aspect, both managed uniformly.
    7.33 +\end{abstract}
    7.34 +
    7.35 +\thispagestyle{empty}\clearpage
    7.36 +
    7.37 +\pagenumbering{roman}
    7.38 +\clearfirst
    7.39 +
    7.40 +\input{Thy/document/Classes.tex}
    7.41 +
    7.42 +\begingroup
    7.43 +\bibliographystyle{plain} \small\raggedright\frenchspacing
    7.44 +\bibliography{../manual}
    7.45 +\endgroup
    7.46 +
    7.47 +\end{document}
    7.48 +
    7.49 +
    7.50 +%%% Local Variables: 
    7.51 +%%% mode: latex
    7.52 +%%% TeX-master: t
    7.53 +%%% End: 

     8.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     8.2 +++ b/doc-src/Classes/style.sty	Tue Mar 03 11:00:51 2009 +0100
     8.3 @@ -0,0 +1,43 @@
     8.4 +
     8.5 +%% toc
     8.6 +\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
     8.7 +\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
     8.8 +
     8.9 +%% references
    8.10 +\newcommand{\secref}[1]{\S\ref{#1}}
    8.11 +\newcommand{\figref}[1]{figure~\ref{#1}}
    8.12 +
    8.13 +%% logical markup
    8.14 +\newcommand{\strong}[1]{{\bfseries {#1}}}
    8.15 +\newcommand{\qn}[1]{\emph{#1}}
    8.16 +
    8.17 +%% typographic conventions
    8.18 +\newcommand{\qt}[1]{``{#1}''}
    8.19 +
    8.20 +%% verbatim text
    8.21 +\newcommand{\isatypewriter}{\fontsize{9pt}{0pt}\tt\renewcommand{\baselinestretch}{1}\setlength{\baselineskip}{9pt}}
    8.22 +
    8.23 +%% quoted segments
    8.24 +\makeatletter
    8.25 +\isakeeptag{quote}
    8.26 +\newenvironment{quotesegment}{\begin{quote}\isa@parindent\parindent\parindent0pt\isa@parskip\parskip\parskip0pt}{\end{quote}}
    8.27 +\renewcommand{\isatagquote}{\begin{quotesegment}}
    8.28 +\renewcommand{\endisatagquote}{\end{quotesegment}}
    8.29 +\makeatother
    8.30 +
    8.31 +%% presentation
    8.32 +\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
    8.33 +
    8.34 +\pagestyle{headings}
    8.35 +\binperiod
    8.36 +\underscoreoff
    8.37 +
    8.38 +\renewcommand{\isadigit}[1]{\isamath{#1}}
    8.39 +
    8.40 +\isabellestyle{it}
    8.41 +
    8.42 +
    8.43 +%%% Local Variables: 
    8.44 +%%% mode: latex
    8.45 +%%% TeX-master: "implementation"
    8.46 +%%% End: 

     9.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     9.2 +++ b/doc-src/Codegen/IsaMakefile	Tue Mar 03 11:00:51 2009 +0100
     9.3 @@ -0,0 +1,33 @@
     9.4 +
     9.5 +## targets
     9.6 +
     9.7 +default: Thy
     9.8 +images: 
     9.9 +test: Thy
    9.10 +
    9.11 +all: images test
    9.12 +
    9.13 +
    9.14 +## global settings
    9.15 +
    9.16 +SRC = $(ISABELLE_HOME)/src
    9.17 +OUT = $(ISABELLE_OUTPUT)
    9.18 +LOG = $(OUT)/log
    9.19 +
    9.20 +USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
    9.21 +
    9.22 +
    9.23 +## Thy
    9.24 +
    9.25 +THY = $(LOG)/HOL-Thy.gz
    9.26 +
    9.27 +Thy: $(THY)
    9.28 +
    9.29 +$(THY): Thy/ROOT.ML Thy/*.thy ../antiquote_setup.ML ../more_antiquote.ML
    9.30 +	@$(USEDIR) HOL Thy
    9.31 +
    9.32 +
    9.33 +## clean
    9.34 +
    9.35 +clean:
    9.36 +	@rm -f $(THY)

    10.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    10.2 +++ b/doc-src/Codegen/Makefile	Tue Mar 03 11:00:51 2009 +0100
    10.3 @@ -0,0 +1,35 @@
    10.4 +
    10.5 +## targets
    10.6 +
    10.7 +default: dvi
    10.8 +
    10.9 +
   10.10 +## dependencies
   10.11 +
   10.12 +include ../Makefile.in
   10.13 +
   10.14 +NAME = codegen
   10.15 +
   10.16 +FILES = $(NAME).tex Thy/document/*.tex \
   10.17 +  style.sty ../iman.sty ../extra.sty ../isar.sty \
   10.18 +  ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty \
   10.19 +  ../manual.bib ../proof.sty
   10.20 +
   10.21 +dvi: $(NAME).dvi
   10.22 +
   10.23 +$(NAME).dvi: $(FILES) isabelle_isar.eps codegen_process.ps
   10.24 +	$(LATEX) $(NAME)
   10.25 +	$(BIBTEX) $(NAME)
   10.26 +	$(LATEX) $(NAME)
   10.27 +	$(LATEX) $(NAME)
   10.28 +
   10.29 +pdf: $(NAME).pdf
   10.30 +
   10.31 +$(NAME).pdf: $(FILES) isabelle_isar.pdf codegen_process.pdf
   10.32 +	$(PDFLATEX) $(NAME)
   10.33 +	$(BIBTEX) $(NAME)
   10.34 +	$(PDFLATEX) $(NAME)
   10.35 +	$(PDFLATEX) $(NAME)
   10.36 +	$(FIXBOOKMARKS) $(NAME).out
   10.37 +	$(PDFLATEX) $(NAME)
   10.38 +	$(PDFLATEX) $(NAME)

    11.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    11.2 +++ b/doc-src/Codegen/Thy/Adaption.thy	Tue Mar 03 11:00:51 2009 +0100
    11.3 @@ -0,0 +1,361 @@
    11.4 +theory Adaption
    11.5 +imports Setup
    11.6 +begin
    11.7 +
    11.8 +setup %invisible {* Code_Target.extend_target ("\<SML>", ("SML", K I)) *}
    11.9 +
   11.10 +section {* Adaption to target languages \label{sec:adaption} *}
   11.11 +
   11.12 +subsection {* Adapting code generation *}
   11.13 +
   11.14 +text {*
   11.15 +  The aspects of code generation introduced so far have two aspects
   11.16 +  in common:
   11.17 +
   11.18 +  \begin{itemize}
   11.19 +    \item They act uniformly, without reference to a specific
   11.20 +       target language.
   11.21 +    \item They are \emph{safe} in the sense that as long as you trust
   11.22 +       the code generator meta theory and implementation, you cannot
   11.23 +       produce programs that yield results which are not derivable
   11.24 +       in the logic.
   11.25 +  \end{itemize}
   11.26 +
   11.27 +  \noindent In this section we will introduce means to \emph{adapt} the serialiser
   11.28 +  to a specific target language, i.e.~to print program fragments
   11.29 +  in a way which accommodates \qt{already existing} ingredients of
   11.30 +  a target language environment, for three reasons:
   11.31 +
   11.32 +  \begin{itemize}
   11.33 +    \item improving readability and aesthetics of generated code
   11.34 +    \item gaining efficiency
   11.35 +    \item interface with language parts which have no direct counterpart
   11.36 +      in @{text "HOL"} (say, imperative data structures)
   11.37 +  \end{itemize}
   11.38 +
   11.39 +  \noindent Generally, you should avoid using those features yourself
   11.40 +  \emph{at any cost}:
   11.41 +
   11.42 +  \begin{itemize}
   11.43 +    \item The safe configuration methods act uniformly on every target language,
   11.44 +      whereas for adaption you have to treat each target language separate.
   11.45 +    \item Application is extremely tedious since there is no abstraction
   11.46 +      which would allow for a static check, making it easy to produce garbage.
   11.47 +    \item More or less subtle errors can be introduced unconsciously.
   11.48 +  \end{itemize}
   11.49 +
   11.50 +  \noindent However, even if you ought refrain from setting up adaption
   11.51 +  yourself, already the @{text "HOL"} comes with some reasonable default
   11.52 +  adaptions (say, using target language list syntax).  There also some
   11.53 +  common adaption cases which you can setup by importing particular
   11.54 +  library theories.  In order to understand these, we provide some clues here;
   11.55 +  these however are not supposed to replace a careful study of the sources.
   11.56 +*}
   11.57 +
   11.58 +subsection {* The adaption principle *}
   11.59 +
   11.60 +text {*
   11.61 +  The following figure illustrates what \qt{adaption} is conceptually
   11.62 +  supposed to be:
   11.63 +
   11.64 +  \begin{figure}[here]
   11.65 +    \begin{tikzpicture}[scale = 0.5]
   11.66 +      \tikzstyle water=[color = blue, thick]
   11.67 +      \tikzstyle ice=[color = black, very thick, cap = round, join = round, fill = white]
   11.68 +      \tikzstyle process=[color = green, semithick, ->]
   11.69 +      \tikzstyle adaption=[color = red, semithick, ->]
   11.70 +      \tikzstyle target=[color = black]
   11.71 +      \foreach \x in {0, ..., 24}
   11.72 +        \draw[style=water] (\x, 0.25) sin + (0.25, 0.25) cos + (0.25, -0.25) sin
   11.73 +          + (0.25, -0.25) cos + (0.25, 0.25);
   11.74 +      \draw[style=ice] (1, 0) --
   11.75 +        (3, 6) node[above, fill=white] {logic} -- (5, 0) -- cycle;
   11.76 +      \draw[style=ice] (9, 0) --
   11.77 +        (11, 6) node[above, fill=white] {intermediate language} -- (13, 0) -- cycle;
   11.78 +      \draw[style=ice] (15, -6) --
   11.79 +        (19, 6) node[above, fill=white] {target language} -- (23, -6) -- cycle;
   11.80 +      \draw[style=process]
   11.81 +        (3.5, 3) .. controls (7, 5) .. node[fill=white] {translation} (10.5, 3);
   11.82 +      \draw[style=process]
   11.83 +        (11.5, 3) .. controls (15, 5) .. node[fill=white] (serialisation) {serialisation} (18.5, 3);
   11.84 +      \node (adaption) at (11, -2) [style=adaption] {adaption};
   11.85 +      \node at (19, 3) [rotate=90] {generated};
   11.86 +      \node at (19.5, -5) {language};
   11.87 +      \node at (19.5, -3) {library};
   11.88 +      \node (includes) at (19.5, -1) {includes};
   11.89 +      \node (reserved) at (16.5, -3) [rotate=72] {reserved}; % proper 71.57
   11.90 +      \draw[style=process]
   11.91 +        (includes) -- (serialisation);
   11.92 +      \draw[style=process]
   11.93 +        (reserved) -- (serialisation);
   11.94 +      \draw[style=adaption]
   11.95 +        (adaption) -- (serialisation);
   11.96 +      \draw[style=adaption]
   11.97 +        (adaption) -- (includes);
   11.98 +      \draw[style=adaption]
   11.99 +        (adaption) -- (reserved);
  11.100 +    \end{tikzpicture}
  11.101 +    \caption{The adaption principle}
  11.102 +    \label{fig:adaption}
  11.103 +  \end{figure}
  11.104 +
  11.105 +  \noindent In the tame view, code generation acts as broker between
  11.106 +  @{text logic}, @{text "intermediate language"} and
  11.107 +  @{text "target language"} by means of @{text translation} and
  11.108 +  @{text serialisation};  for the latter, the serialiser has to observe
  11.109 +  the structure of the @{text language} itself plus some @{text reserved}
  11.110 +  keywords which have to be avoided for generated code.
  11.111 +  However, if you consider @{text adaption} mechanisms, the code generated
  11.112 +  by the serializer is just the tip of the iceberg:
  11.113 +
  11.114 +  \begin{itemize}
  11.115 +    \item @{text serialisation} can be \emph{parametrised} such that
  11.116 +      logical entities are mapped to target-specific ones
  11.117 +      (e.g. target-specific list syntax,
  11.118 +        see also \secref{sec:adaption_mechanisms})
  11.119 +    \item Such parametrisations can involve references to a
  11.120 +      target-specific standard @{text library} (e.g. using
  11.121 +      the @{text Haskell} @{verbatim Maybe} type instead
  11.122 +      of the @{text HOL} @{type "option"} type);
  11.123 +      if such are used, the corresponding identifiers
  11.124 +      (in our example, @{verbatim Maybe}, @{verbatim Nothing}
  11.125 +      and @{verbatim Just}) also have to be considered @{text reserved}.
  11.126 +    \item Even more, the user can enrich the library of the
  11.127 +      target-language by providing code snippets
  11.128 +      (\qt{@{text "includes"}}) which are prepended to
  11.129 +      any generated code (see \secref{sec:include});  this typically
  11.130 +      also involves further @{text reserved} identifiers.
  11.131 +  \end{itemize}
  11.132 +
  11.133 +  \noindent As figure \ref{fig:adaption} illustrates, all these adaption mechanisms
  11.134 +  have to act consistently;  it is at the discretion of the user
  11.135 +  to take care for this.
  11.136 +*}
  11.137 +
  11.138 +subsection {* Common adaption patterns *}
  11.139 +
  11.140 +text {*
  11.141 +  The @{theory HOL} @{theory Main} theory already provides a code
  11.142 +  generator setup
  11.143 +  which should be suitable for most applications.  Common extensions
  11.144 +  and modifications are available by certain theories of the @{text HOL}
  11.145 +  library; beside being useful in applications, they may serve
  11.146 +  as a tutorial for customising the code generator setup (see below
  11.147 +  \secref{sec:adaption_mechanisms}).
  11.148 +
  11.149 +  \begin{description}
  11.150 +
  11.151 +    \item[@{theory "Code_Integer"}] represents @{text HOL} integers by big
  11.152 +       integer literals in target languages.
  11.153 +    \item[@{theory "Code_Char"}] represents @{text HOL} characters by 
  11.154 +       character literals in target languages.
  11.155 +    \item[@{theory "Code_Char_chr"}] like @{text "Code_Char"},
  11.156 +       but also offers treatment of character codes; includes
  11.157 +       @{theory "Code_Char"}.
  11.158 +    \item[@{theory "Efficient_Nat"}] \label{eff_nat} implements natural numbers by integers,
  11.159 +       which in general will result in higher efficiency; pattern
  11.160 +       matching with @{term "0\<Colon>nat"} / @{const "Suc"}
  11.161 +       is eliminated;  includes @{theory "Code_Integer"}
  11.162 +       and @{theory "Code_Index"}.
  11.163 +    \item[@{theory "Code_Index"}] provides an additional datatype
  11.164 +       @{typ index} which is mapped to target-language built-in integers.
  11.165 +       Useful for code setups which involve e.g. indexing of
  11.166 +       target-language arrays.
  11.167 +    \item[@{theory "Code_Message"}] provides an additional datatype
  11.168 +       @{typ message_string} which is isomorphic to strings;
  11.169 +       @{typ message_string}s are mapped to target-language strings.
  11.170 +       Useful for code setups which involve e.g. printing (error) messages.
  11.171 +
  11.172 +  \end{description}
  11.173 +
  11.174 +  \begin{warn}
  11.175 +    When importing any of these theories, they should form the last
  11.176 +    items in an import list.  Since these theories adapt the
  11.177 +    code generator setup in a non-conservative fashion,
  11.178 +    strange effects may occur otherwise.
  11.179 +  \end{warn}
  11.180 +*}
  11.181 +
  11.182 +
  11.183 +subsection {* Parametrising serialisation \label{sec:adaption_mechanisms} *}
  11.184 +
  11.185 +text {*
  11.186 +  Consider the following function and its corresponding
  11.187 +  SML code:
  11.188 +*}
  11.189 +
  11.190 +primrec %quote in_interval :: "nat \<times> nat \<Rightarrow> nat \<Rightarrow> bool" where
  11.191 +  "in_interval (k, l) n \<longleftrightarrow> k \<le> n \<and> n \<le> l"
  11.192 +(*<*)
  11.193 +code_type %invisible bool
  11.194 +  (SML)
  11.195 +code_const %invisible True and False and "op \<and>" and Not
  11.196 +  (SML and and and)
  11.197 +(*>*)
  11.198 +text %quote {*@{code_stmts in_interval (SML)}*}
  11.199 +
  11.200 +text {*
  11.201 +  \noindent Though this is correct code, it is a little bit unsatisfactory:
  11.202 +  boolean values and operators are materialised as distinguished
  11.203 +  entities with have nothing to do with the SML-built-in notion
  11.204 +  of \qt{bool}.  This results in less readable code;
  11.205 +  additionally, eager evaluation may cause programs to
  11.206 +  loop or break which would perfectly terminate when
  11.207 +  the existing SML @{verbatim "bool"} would be used.  To map
  11.208 +  the HOL @{typ bool} on SML @{verbatim "bool"}, we may use
  11.209 +  \qn{custom serialisations}:
  11.210 +*}
  11.211 +
  11.212 +code_type %quotett bool
  11.213 +  (SML "bool")
  11.214 +code_const %quotett True and False and "op \<and>"
  11.215 +  (SML "true" and "false" and "_ andalso _")
  11.216 +
  11.217 +text {*
  11.218 +  \noindent The @{command code_type} command takes a type constructor
  11.219 +  as arguments together with a list of custom serialisations.
  11.220 +  Each custom serialisation starts with a target language
  11.221 +  identifier followed by an expression, which during
  11.222 +  code serialisation is inserted whenever the type constructor
  11.223 +  would occur.  For constants, @{command code_const} implements
  11.224 +  the corresponding mechanism.  Each ``@{verbatim "_"}'' in
  11.225 +  a serialisation expression is treated as a placeholder
  11.226 +  for the type constructor's (the constant's) arguments.
  11.227 +*}
  11.228 +
  11.229 +text %quote {*@{code_stmts in_interval (SML)}*}
  11.230 +
  11.231 +text {*
  11.232 +  \noindent This still is not perfect: the parentheses
  11.233 +  around the \qt{andalso} expression are superfluous.
  11.234 +  Though the serialiser
  11.235 +  by no means attempts to imitate the rich Isabelle syntax
  11.236 +  framework, it provides some common idioms, notably
  11.237 +  associative infixes with precedences which may be used here:
  11.238 +*}
  11.239 +
  11.240 +code_const %quotett "op \<and>"
  11.241 +  (SML infixl 1 "andalso")
  11.242 +
  11.243 +text %quote {*@{code_stmts in_interval (SML)}*}
  11.244 +
  11.245 +text {*
  11.246 +  \noindent The attentive reader may ask how we assert that no generated
  11.247 +  code will accidentally overwrite.  For this reason the serialiser has
  11.248 +  an internal table of identifiers which have to be avoided to be used
  11.249 +  for new declarations.  Initially, this table typically contains the
  11.250 +  keywords of the target language.  It can be extended manually, thus avoiding
  11.251 +  accidental overwrites, using the @{command "code_reserved"} command:
  11.252 +*}
  11.253 +
  11.254 +code_reserved %quote "\<SML>" bool true false andalso
  11.255 +
  11.256 +text {*
  11.257 +  \noindent Next, we try to map HOL pairs to SML pairs, using the
  11.258 +  infix ``@{verbatim "*"}'' type constructor and parentheses:
  11.259 +*}
  11.260 +(*<*)
  11.261 +code_type %invisible *
  11.262 +  (SML)
  11.263 +code_const %invisible Pair
  11.264 +  (SML)
  11.265 +(*>*)
  11.266 +code_type %quotett *
  11.267 +  (SML infix 2 "*")
  11.268 +code_const %quotett Pair
  11.269 +  (SML "!((_),/ (_))")
  11.270 +
  11.271 +text {*
  11.272 +  \noindent The initial bang ``@{verbatim "!"}'' tells the serialiser
  11.273 +  never to put
  11.274 +  parentheses around the whole expression (they are already present),
  11.275 +  while the parentheses around argument place holders
  11.276 +  tell not to put parentheses around the arguments.
  11.277 +  The slash ``@{verbatim "/"}'' (followed by arbitrary white space)
  11.278 +  inserts a space which may be used as a break if necessary
  11.279 +  during pretty printing.
  11.280 +
  11.281 +  These examples give a glimpse what mechanisms
  11.282 +  custom serialisations provide; however their usage
  11.283 +  requires careful thinking in order not to introduce
  11.284 +  inconsistencies -- or, in other words:
  11.285 +  custom serialisations are completely axiomatic.
  11.286 +
  11.287 +  A further noteworthy details is that any special
  11.288 +  character in a custom serialisation may be quoted
  11.289 +  using ``@{verbatim "'"}''; thus, in
  11.290 +  ``@{verbatim "fn '_ => _"}'' the first
  11.291 +  ``@{verbatim "_"}'' is a proper underscore while the
  11.292 +  second ``@{verbatim "_"}'' is a placeholder.
  11.293 +*}
  11.294 +
  11.295 +
  11.296 +subsection {* @{text Haskell} serialisation *}
  11.297 +
  11.298 +text {*
  11.299 +  For convenience, the default
  11.300 +  @{text HOL} setup for @{text Haskell} maps the @{class eq} class to
  11.301 +  its counterpart in @{text Haskell}, giving custom serialisations
  11.302 +  for the class @{class eq} (by command @{command code_class}) and its operation
  11.303 +  @{const HOL.eq}
  11.304 +*}
  11.305 +
  11.306 +code_class %quotett eq
  11.307 +  (Haskell "Eq")
  11.308 +
  11.309 +code_const %quotett "op ="
  11.310 +  (Haskell infixl 4 "==")
  11.311 +
  11.312 +text {*
  11.313 +  \noindent A problem now occurs whenever a type which
  11.314 +  is an instance of @{class eq} in @{text HOL} is mapped
  11.315 +  on a @{text Haskell}-built-in type which is also an instance
  11.316 +  of @{text Haskell} @{text Eq}:
  11.317 +*}
  11.318 +
  11.319 +typedecl %quote bar
  11.320 +
  11.321 +instantiation %quote bar :: eq
  11.322 +begin
  11.323 +
  11.324 +definition %quote "eq_class.eq (x\<Colon>bar) y \<longleftrightarrow> x = y"
  11.325 +
  11.326 +instance %quote by default (simp add: eq_bar_def)
  11.327 +
  11.328 +end %quote
  11.329 +
  11.330 +code_type %quotett bar
  11.331 +  (Haskell "Integer")
  11.332 +
  11.333 +text {*
  11.334 +  \noindent The code generator would produce
  11.335 +  an additional instance, which of course is rejected by the @{text Haskell}
  11.336 +  compiler.
  11.337 +  To suppress this additional instance, use
  11.338 +  @{text "code_instance"}:
  11.339 +*}
  11.340 +
  11.341 +code_instance %quotett bar :: eq
  11.342 +  (Haskell -)
  11.343 +
  11.344 +
  11.345 +subsection {* Enhancing the target language context \label{sec:include} *}
  11.346 +
  11.347 +text {*
  11.348 +  In rare cases it is necessary to \emph{enrich} the context of a
  11.349 +  target language;  this is accomplished using the @{command "code_include"}
  11.350 +  command:
  11.351 +*}
  11.352 +
  11.353 +code_include %quotett Haskell "Errno"
  11.354 +{*errno i = error ("Error number: " ++ show i)*}
  11.355 +
  11.356 +code_reserved %quotett Haskell Errno
  11.357 +
  11.358 +text {*
  11.359 +  \noindent Such named @{text include}s are then prepended to every generated code.
  11.360 +  Inspect such code in order to find out how @{command "code_include"} behaves
  11.361 +  with respect to a particular target language.
  11.362 +*}
  11.363 +
  11.364 +end

    12.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    12.2 +++ b/doc-src/Codegen/Thy/Codegen.thy	Tue Mar 03 11:00:51 2009 +0100
    12.3 @@ -0,0 +1,2 @@
    12.4 +
    12.5 +end

    13.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    13.2 +++ b/doc-src/Codegen/Thy/Further.thy	Tue Mar 03 11:00:51 2009 +0100
    13.3 @@ -0,0 +1,113 @@
    13.4 +theory Further
    13.5 +imports Setup
    13.6 +begin
    13.7 +
    13.8 +section {* Further issues \label{sec:further} *}
    13.9 +
   13.10 +subsection {* Further reading *}
   13.11 +
   13.12 +text {*
   13.13 +  Do dive deeper into the issue of code generation, you should visit
   13.14 +  the Isabelle/Isar Reference Manual \cite{isabelle-isar-ref} which
   13.15 +  contains exhaustive syntax diagrams.
   13.16 +*}
   13.17 +
   13.18 +subsection {* Modules *}
   13.19 +
   13.20 +text {*
   13.21 +  When invoking the @{command export_code} command it is possible to leave
   13.22 +  out the @{keyword "module_name"} part;  then code is distributed over
   13.23 +  different modules, where the module name space roughly is induced
   13.24 +  by the @{text Isabelle} theory name space.
   13.25 +
   13.26 +  Then sometimes the awkward situation occurs that dependencies between
   13.27 +  definitions introduce cyclic dependencies between modules, which in the
   13.28 +  @{text Haskell} world leaves you to the mercy of the @{text Haskell} implementation
   13.29 +  you are using,  while for @{text SML}/@{text OCaml} code generation is not possible.
   13.30 +
   13.31 +  A solution is to declare module names explicitly.
   13.32 +  Let use assume the three cyclically dependent
   13.33 +  modules are named \emph{A}, \emph{B} and \emph{C}.
   13.34 +  Then, by stating
   13.35 +*}
   13.36 +
   13.37 +code_modulename %quote SML
   13.38 +  A ABC
   13.39 +  B ABC
   13.40 +  C ABC
   13.41 +
   13.42 +text {*
   13.43 +  we explicitly map all those modules on \emph{ABC},
   13.44 +  resulting in an ad-hoc merge of this three modules
   13.45 +  at serialisation time.
   13.46 +*}
   13.47 +
   13.48 +subsection {* Evaluation oracle *}
   13.49 +
   13.50 +text {*
   13.51 +  Code generation may also be used to \emph{evaluate} expressions
   13.52 +  (using @{text SML} as target language of course).
   13.53 +  For instance, the @{command value} allows to reduce an expression to a
   13.54 +  normal form with respect to the underlying code equations:
   13.55 +*}
   13.56 +
   13.57 +value %quote "42 / (12 :: rat)"
   13.58 +
   13.59 +text {*
   13.60 +  \noindent will display @{term "7 / (2 :: rat)"}.
   13.61 +
   13.62 +  The @{method eval} method tries to reduce a goal by code generation to @{term True}
   13.63 +  and solves it in that case, but fails otherwise:
   13.64 +*}
   13.65 +
   13.66 +lemma %quote "42 / (12 :: rat) = 7 / 2"
   13.67 +  by %quote eval
   13.68 +
   13.69 +text {*
   13.70 +  \noindent The soundness of the @{method eval} method depends crucially 
   13.71 +  on the correctness of the code generator;  this is one of the reasons
   13.72 +  why you should not use adaption (see \secref{sec:adaption}) frivolously.
   13.73 +*}
   13.74 +
   13.75 +subsection {* Code antiquotation *}
   13.76 +
   13.77 +text {*
   13.78 +  In scenarios involving techniques like reflection it is quite common
   13.79 +  that code generated from a theory forms the basis for implementing
   13.80 +  a proof procedure in @{text SML}.  To facilitate interfacing of generated code
   13.81 +  with system code, the code generator provides a @{text code} antiquotation:
   13.82 +*}
   13.83 +
   13.84 +datatype %quote form = T | F | And form form | Or form form
   13.85 +
   13.86 +ML %quote {*
   13.87 +  fun eval_form @{code T} = true
   13.88 +    | eval_form @{code F} = false
   13.89 +    | eval_form (@{code And} (p, q)) =
   13.90 +        eval_form p andalso eval_form q
   13.91 +    | eval_form (@{code Or} (p, q)) =
   13.92 +        eval_form p orelse eval_form q;
   13.93 +*}
   13.94 +
   13.95 +text {*
   13.96 +  \noindent @{text code} takes as argument the name of a constant;  after the
   13.97 +  whole @{text SML} is read, the necessary code is generated transparently
   13.98 +  and the corresponding constant names are inserted.  This technique also
   13.99 +  allows to use pattern matching on constructors stemming from compiled
  13.100 +  @{text datatypes}.
  13.101 +
  13.102 +  For a less simplistic example, theory @{theory Ferrack} is
  13.103 +  a good reference.
  13.104 +*}
  13.105 +
  13.106 +subsection {* Imperative data structures *}
  13.107 +
  13.108 +text {*
  13.109 +  If you consider imperative data structures as inevitable for a specific
  13.110 +  application, you should consider
  13.111 +  \emph{Imperative Functional Programming with Isabelle/HOL}
  13.112 +  (\cite{bulwahn-et-al:2008:imperative});
  13.113 +  the framework described there is available in theory @{theory Imperative_HOL}.
  13.114 +*}
  13.115 +
  13.116 +end

    14.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    14.2 +++ b/doc-src/Codegen/Thy/Introduction.thy	Tue Mar 03 11:00:51 2009 +0100
    14.3 @@ -0,0 +1,206 @@
    14.4 +theory Introduction
    14.5 +imports Setup
    14.6 +begin
    14.7 +
    14.8 +chapter {* Code generation from @{text "Isabelle/HOL"} theories *}
    14.9 +
   14.10 +section {* Introduction and Overview *}
   14.11 +
   14.12 +text {*
   14.13 +  This tutorial introduces a generic code generator for the
   14.14 +  @{text Isabelle} system.
   14.15 +  Generic in the sense that the
   14.16 +  \qn{target language} for which code shall ultimately be
   14.17 +  generated is not fixed but may be an arbitrary state-of-the-art
   14.18 +  functional programming language (currently, the implementation
   14.19 +  supports @{text SML} \cite{SML}, @{text OCaml} \cite{OCaml} and @{text Haskell}
   14.20 +  \cite{haskell-revised-report}).
   14.21 +
   14.22 +  Conceptually the code generator framework is part
   14.23 +  of Isabelle's @{theory Pure} meta logic framework; the logic
   14.24 +  @{theory HOL} which is an extension of @{theory Pure}
   14.25 +  already comes with a reasonable framework setup and thus provides
   14.26 +  a good working horse for raising code-generation-driven
   14.27 +  applications.  So, we assume some familiarity and experience
   14.28 +  with the ingredients of the @{theory HOL} distribution theories.
   14.29 +  (see also \cite{isa-tutorial}).
   14.30 +
   14.31 +  The code generator aims to be usable with no further ado
   14.32 +  in most cases while allowing for detailed customisation.
   14.33 +  This manifests in the structure of this tutorial: after a short
   14.34 +  conceptual introduction with an example (\secref{sec:intro}),
   14.35 +  we discuss the generic customisation facilities (\secref{sec:program}).
   14.36 +  A further section (\secref{sec:adaption}) is dedicated to the matter of
   14.37 +  \qn{adaption} to specific target language environments.  After some
   14.38 +  further issues (\secref{sec:further}) we conclude with an overview
   14.39 +  of some ML programming interfaces (\secref{sec:ml}).
   14.40 +
   14.41 +  \begin{warn}
   14.42 +    Ultimately, the code generator which this tutorial deals with
   14.43 +    is supposed to replace the existing code generator
   14.44 +    by Stefan Berghofer \cite{Berghofer-Nipkow:2002}.
   14.45 +    So, for the moment, there are two distinct code generators
   14.46 +    in Isabelle.  In case of ambiguity, we will refer to the framework
   14.47 +    described here as @{text "generic code generator"}, to the
   14.48 +    other as @{text "SML code generator"}.
   14.49 +    Also note that while the framework itself is
   14.50 +    object-logic independent, only @{theory HOL} provides a reasonable
   14.51 +    framework setup.    
   14.52 +  \end{warn}
   14.53 +
   14.54 +*}
   14.55 +
   14.56 +subsection {* Code generation via shallow embedding \label{sec:intro} *}
   14.57 +
   14.58 +text {*
   14.59 +  The key concept for understanding @{text Isabelle}'s code generation is
   14.60 +  \emph{shallow embedding}, i.e.~logical entities like constants, types and
   14.61 +  classes are identified with corresponding concepts in the target language.
   14.62 +
   14.63 +  Inside @{theory HOL}, the @{command datatype} and
   14.64 +  @{command definition}/@{command primrec}/@{command fun} declarations form
   14.65 +  the core of a functional programming language.  The default code generator setup
   14.66 +  allows to turn those into functional programs immediately.
   14.67 +  This means that \qt{naive} code generation can proceed without further ado.
   14.68 +  For example, here a simple \qt{implementation} of amortised queues:
   14.69 +*}
   14.70 +
   14.71 +datatype %quote 'a queue = AQueue "'a list" "'a list"
   14.72 +
   14.73 +definition %quote empty :: "'a queue" where
   14.74 +  "empty = AQueue [] []"
   14.75 +
   14.76 +primrec %quote enqueue :: "'a \<Rightarrow> 'a queue \<Rightarrow> 'a queue" where
   14.77 +  "enqueue x (AQueue xs ys) = AQueue (x # xs) ys"
   14.78 +
   14.79 +fun %quote dequeue :: "'a queue \<Rightarrow> 'a option \<times> 'a queue" where
   14.80 +    "dequeue (AQueue [] []) = (None, AQueue [] [])"
   14.81 +  | "dequeue (AQueue xs (y # ys)) = (Some y, AQueue xs ys)"
   14.82 +  | "dequeue (AQueue xs []) =
   14.83 +      (case rev xs of y # ys \<Rightarrow> (Some y, AQueue [] ys))"
   14.84 +
   14.85 +text {* \noindent Then we can generate code e.g.~for @{text SML} as follows: *}
   14.86 +
   14.87 +export_code %quote empty dequeue enqueue in SML
   14.88 +  module_name Example file "examples/example.ML"
   14.89 +
   14.90 +text {* \noindent resulting in the following code: *}
   14.91 +
   14.92 +text %quote {*@{code_stmts empty enqueue dequeue (SML)}*}
   14.93 +
   14.94 +text {*
   14.95 +  \noindent The @{command export_code} command takes a space-separated list of
   14.96 +  constants for which code shall be generated;  anything else needed for those
   14.97 +  is added implicitly.  Then follows a target language identifier
   14.98 +  (@{text SML}, @{text OCaml} or @{text Haskell}) and a freely chosen module name.
   14.99 +  A file name denotes the destination to store the generated code.  Note that
  14.100 +  the semantics of the destination depends on the target language:  for
  14.101 +  @{text SML} and @{text OCaml} it denotes a \emph{file}, for @{text Haskell}
  14.102 +  it denotes a \emph{directory} where a file named as the module name
  14.103 +  (with extension @{text ".hs"}) is written:
  14.104 +*}
  14.105 +
  14.106 +export_code %quote empty dequeue enqueue in Haskell
  14.107 +  module_name Example file "examples/"
  14.108 +
  14.109 +text {*
  14.110 +  \noindent This is how the corresponding code in @{text Haskell} looks like:
  14.111 +*}
  14.112 +
  14.113 +text %quote {*@{code_stmts empty enqueue dequeue (Haskell)}*}
  14.114 +
  14.115 +text {*
  14.116 +  \noindent This demonstrates the basic usage of the @{command export_code} command;
  14.117 +  for more details see \secref{sec:further}.
  14.118 +*}
  14.119 +
  14.120 +subsection {* Code generator architecture \label{sec:concept} *}
  14.121 +
  14.122 +text {*
  14.123 +  What you have seen so far should be already enough in a lot of cases.  If you
  14.124 +  are content with this, you can quit reading here.  Anyway, in order to customise
  14.125 +  and adapt the code generator, it is inevitable to gain some understanding
  14.126 +  how it works.
  14.127 +
  14.128 +  \begin{figure}[h]
  14.129 +    \begin{tikzpicture}[x = 4.2cm, y = 1cm]
  14.130 +      \tikzstyle entity=[rounded corners, draw, thick, color = black, fill = white];
  14.131 +      \tikzstyle process=[ellipse, draw, thick, color = green, fill = white];
  14.132 +      \tikzstyle process_arrow=[->, semithick, color = green];
  14.133 +      \node (HOL) at (0, 4) [style=entity] {@{text "Isabelle/HOL"} theory};
  14.134 +      \node (eqn) at (2, 2) [style=entity] {code equations};
  14.135 +      \node (iml) at (2, 0) [style=entity] {intermediate language};
  14.136 +      \node (seri) at (1, 0) [style=process] {serialisation};
  14.137 +      \node (SML) at (0, 3) [style=entity] {@{text SML}};
  14.138 +      \node (OCaml) at (0, 2) [style=entity] {@{text OCaml}};
  14.139 +      \node (further) at (0, 1) [style=entity] {@{text "\<dots>"}};
  14.140 +      \node (Haskell) at (0, 0) [style=entity] {@{text Haskell}};
  14.141 +      \draw [style=process_arrow] (HOL) .. controls (2, 4) ..
  14.142 +        node [style=process, near start] {selection}
  14.143 +        node [style=process, near end] {preprocessing}
  14.144 +        (eqn);
  14.145 +      \draw [style=process_arrow] (eqn) -- node (transl) [style=process] {translation} (iml);
  14.146 +      \draw [style=process_arrow] (iml) -- (seri);
  14.147 +      \draw [style=process_arrow] (seri) -- (SML);
  14.148 +      \draw [style=process_arrow] (seri) -- (OCaml);
  14.149 +      \draw [style=process_arrow, dashed] (seri) -- (further);
  14.150 +      \draw [style=process_arrow] (seri) -- (Haskell);
  14.151 +    \end{tikzpicture}
  14.152 +    \caption{Code generator architecture}
  14.153 +    \label{fig:arch}
  14.154 +  \end{figure}
  14.155 +
  14.156 +  The code generator employs a notion of executability
  14.157 +  for three foundational executable ingredients known
  14.158 +  from functional programming:
  14.159 +  \emph{code equations}, \emph{datatypes}, and
  14.160 +  \emph{type classes}.  A code equation as a first approximation
  14.161 +  is a theorem of the form @{text "f t\<^isub>1 t\<^isub>2 \<dots> t\<^isub>n \<equiv> t"}
  14.162 +  (an equation headed by a constant @{text f} with arguments
  14.163 +  @{text "t\<^isub>1 t\<^isub>2 \<dots> t\<^isub>n"} and right hand side @{text t}).
  14.164 +  Code generation aims to turn code equations
  14.165 +  into a functional program.  This is achieved by three major
  14.166 +  components which operate sequentially, i.e. the result of one is
  14.167 +  the input
  14.168 +  of the next in the chain,  see diagram \ref{fig:arch}:
  14.169 +
  14.170 +  \begin{itemize}
  14.171 +
  14.172 +    \item Out of the vast collection of theorems proven in a
  14.173 +      \qn{theory}, a reasonable subset modelling
  14.174 +      code equations is \qn{selected}.
  14.175 +
  14.176 +    \item On those selected theorems, certain
  14.177 +      transformations are carried out
  14.178 +      (\qn{preprocessing}).  Their purpose is to turn theorems
  14.179 +      representing non- or badly executable
  14.180 +      specifications into equivalent but executable counterparts.
  14.181 +      The result is a structured collection of \qn{code theorems}.
  14.182 +
  14.183 +    \item Before the selected code equations are continued with,
  14.184 +      they can be \qn{preprocessed}, i.e. subjected to theorem
  14.185 +      transformations.  This \qn{preprocessor} is an interface which
  14.186 +      allows to apply
  14.187 +      the full expressiveness of ML-based theorem transformations
  14.188 +      to code generation;  motivating examples are shown below, see
  14.189 +      \secref{sec:preproc}.
  14.190 +      The result of the preprocessing step is a structured collection
  14.191 +      of code equations.
  14.192 +
  14.193 +    \item These code equations are \qn{translated} to a program
  14.194 +      in an abstract intermediate language.  Think of it as a kind
  14.195 +      of \qt{Mini-Haskell} with four \qn{statements}: @{text data}
  14.196 +      (for datatypes), @{text fun} (stemming from code equations),
  14.197 +      also @{text class} and @{text inst} (for type classes).
  14.198 +
  14.199 +    \item Finally, the abstract program is \qn{serialised} into concrete
  14.200 +      source code of a target language.
  14.201 +
  14.202 +  \end{itemize}
  14.203 +
  14.204 +  \noindent From these steps, only the two last are carried out outside the logic;  by
  14.205 +  keeping this layer as thin as possible, the amount of code to trust is
  14.206 +  kept to a minimum.
  14.207 +*}
  14.208 +
  14.209 +end

    15.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    15.2 +++ b/doc-src/Codegen/Thy/ML.thy	Tue Mar 03 11:00:51 2009 +0100
    15.3 @@ -0,0 +1,177 @@
    15.4 +theory "ML"
    15.5 +imports Setup
    15.6 +begin
    15.7 +
    15.8 +section {* ML system interfaces \label{sec:ml} *}
    15.9 +
   15.10 +text {*
   15.11 +  Since the code generator framework not only aims to provide
   15.12 +  a nice Isar interface but also to form a base for
   15.13 +  code-generation-based applications, here a short
   15.14 +  description of the most important ML interfaces.
   15.15 +*}
   15.16 +
   15.17 +subsection {* Executable theory content: @{text Code} *}
   15.18 +
   15.19 +text {*
   15.20 +  This Pure module implements the core notions of
   15.21 +  executable content of a theory.
   15.22 +*}
   15.23 +
   15.24 +subsubsection {* Managing executable content *}
   15.25 +
   15.26 +text %mlref {*
   15.27 +  \begin{mldecls}
   15.28 +  @{index_ML Code.add_eqn: "thm -> theory -> theory"} \\
   15.29 +  @{index_ML Code.del_eqn: "thm -> theory -> theory"} \\
   15.30 +  @{index_ML Code.add_eqnl: "string * (thm * bool) list lazy -> theory -> theory"} \\
   15.31 +  @{index_ML Code.map_pre: "(simpset -> simpset) -> theory -> theory"} \\
   15.32 +  @{index_ML Code.map_post: "(simpset -> simpset) -> theory -> theory"} \\
   15.33 +  @{index_ML Code.add_functrans: "string * (theory -> (thm * bool) list -> (thm * bool) list option)
   15.34 +    -> theory -> theory"} \\
   15.35 +  @{index_ML Code.del_functrans: "string -> theory -> theory"} \\
   15.36 +  @{index_ML Code.add_datatype: "(string * typ) list -> theory -> theory"} \\
   15.37 +  @{index_ML Code.get_datatype: "theory -> string
   15.38 +    -> (string * sort) list * (string * typ list) list"} \\
   15.39 +  @{index_ML Code.get_datatype_of_constr: "theory -> string -> string option"}
   15.40 +  \end{mldecls}
   15.41 +
   15.42 +  \begin{description}
   15.43 +
   15.44 +  \item @{ML Code.add_eqn}~@{text "thm"}~@{text "thy"} adds function
   15.45 +     theorem @{text "thm"} to executable content.
   15.46 +
   15.47 +  \item @{ML Code.del_eqn}~@{text "thm"}~@{text "thy"} removes function
   15.48 +     theorem @{text "thm"} from executable content, if present.
   15.49 +
   15.50 +  \item @{ML Code.add_eqnl}~@{text "(const, lthms)"}~@{text "thy"} adds
   15.51 +     suspended code equations @{text lthms} for constant
   15.52 +     @{text const} to executable content.
   15.53 +
   15.54 +  \item @{ML Code.map_pre}~@{text "f"}~@{text "thy"} changes
   15.55 +     the preprocessor simpset.
   15.56 +
   15.57 +  \item @{ML Code.add_functrans}~@{text "(name, f)"}~@{text "thy"} adds
   15.58 +     function transformer @{text f} (named @{text name}) to executable content;
   15.59 +     @{text f} is a transformer of the code equations belonging
   15.60 +     to a certain function definition, depending on the
   15.61 +     current theory context.  Returning @{text NONE} indicates that no
   15.62 +     transformation took place;  otherwise, the whole process will be iterated
   15.63 +     with the new code equations.
   15.64 +
   15.65 +  \item @{ML Code.del_functrans}~@{text "name"}~@{text "thy"} removes
   15.66 +     function transformer named @{text name} from executable content.
   15.67 +
   15.68 +  \item @{ML Code.add_datatype}~@{text cs}~@{text thy} adds
   15.69 +     a datatype to executable content, with generation
   15.70 +     set @{text cs}.
   15.71 +
   15.72 +  \item @{ML Code.get_datatype_of_constr}~@{text "thy"}~@{text "const"}
   15.73 +     returns type constructor corresponding to
   15.74 +     constructor @{text const}; returns @{text NONE}
   15.75 +     if @{text const} is no constructor.
   15.76 +
   15.77 +  \end{description}
   15.78 +*}
   15.79 +
   15.80 +subsection {* Auxiliary *}
   15.81 +
   15.82 +text %mlref {*
   15.83 +  \begin{mldecls}
   15.84 +  @{index_ML Code_Unit.read_const: "theory -> string -> string"} \\
   15.85 +  @{index_ML Code_Unit.head_eqn: "theory -> thm -> string * ((string * sort) list * typ)"} \\
   15.86 +  @{index_ML Code_Unit.rewrite_eqn: "simpset -> thm -> thm"} \\
   15.87 +  \end{mldecls}
   15.88 +
   15.89 +  \begin{description}
   15.90 +
   15.91 +  \item @{ML Code_Unit.read_const}~@{text thy}~@{text s}
   15.92 +     reads a constant as a concrete term expression @{text s}.
   15.93 +
   15.94 +  \item @{ML Code_Unit.head_eqn}~@{text thy}~@{text thm}
   15.95 +     extracts the constant and its type from a code equation @{text thm}.
   15.96 +
   15.97 +  \item @{ML Code_Unit.rewrite_eqn}~@{text ss}~@{text thm}
   15.98 +     rewrites a code equation @{text thm} with a simpset @{text ss};
   15.99 +     only arguments and right hand side are rewritten,
  15.100 +     not the head of the code equation.
  15.101 +
  15.102 +  \end{description}
  15.103 +
  15.104 +*}
  15.105 +
  15.106 +subsection {* Implementing code generator applications *}
  15.107 +
  15.108 +text {*
  15.109 +  Implementing code generator applications on top
  15.110 +  of the framework set out so far usually not only
  15.111 +  involves using those primitive interfaces
  15.112 +  but also storing code-dependent data and various
  15.113 +  other things.
  15.114 +*}
  15.115 +
  15.116 +subsubsection {* Data depending on the theory's executable content *}
  15.117 +
  15.118 +text {*
  15.119 +  Due to incrementality of code generation, changes in the
  15.120 +  theory's executable content have to be propagated in a
  15.121 +  certain fashion.  Additionally, such changes may occur
  15.122 +  not only during theory extension but also during theory
  15.123 +  merge, which is a little bit nasty from an implementation
  15.124 +  point of view.  The framework provides a solution
  15.125 +  to this technical challenge by providing a functorial
  15.126 +  data slot @{ML_functor CodeDataFun}; on instantiation
  15.127 +  of this functor, the following types and operations
  15.128 +  are required:
  15.129 +
  15.130 +  \medskip
  15.131 +  \begin{tabular}{l}
  15.132 +  @{text "type T"} \\
  15.133 +  @{text "val empty: T"} \\
  15.134 +  @{text "val purge: theory \<rightarrow> string list option \<rightarrow> T \<rightarrow> T"}
  15.135 +  \end{tabular}
  15.136 +
  15.137 +  \begin{description}
  15.138 +
  15.139 +  \item @{text T} the type of data to store.
  15.140 +
  15.141 +  \item @{text empty} initial (empty) data.
  15.142 +
  15.143 +  \item @{text purge}~@{text thy}~@{text consts} propagates changes in executable content;
  15.144 +    @{text consts} indicates the kind
  15.145 +    of change: @{ML NONE} stands for a fundamental change
  15.146 +    which invalidates any existing code, @{text "SOME consts"}
  15.147 +    hints that executable content for constants @{text consts}
  15.148 +    has changed.
  15.149 +
  15.150 +  \end{description}
  15.151 +
  15.152 +  \noindent An instance of @{ML_functor CodeDataFun} provides the following
  15.153 +  interface:
  15.154 +
  15.155 +  \medskip
  15.156 +  \begin{tabular}{l}
  15.157 +  @{text "get: theory \<rightarrow> T"} \\
  15.158 +  @{text "change: theory \<rightarrow> (T \<rightarrow> T) \<rightarrow> T"} \\
  15.159 +  @{text "change_yield: theory \<rightarrow> (T \<rightarrow> 'a * T) \<rightarrow> 'a * T"}
  15.160 +  \end{tabular}
  15.161 +
  15.162 +  \begin{description}
  15.163 +
  15.164 +  \item @{text get} retrieval of the current data.
  15.165 +
  15.166 +  \item @{text change} update of current data (cached!)
  15.167 +    by giving a continuation.
  15.168 +
  15.169 +  \item @{text change_yield} update with side result.
  15.170 +
  15.171 +  \end{description}
  15.172 +*}
  15.173 +
  15.174 +text {*
  15.175 +  \bigskip
  15.176 +
  15.177 +  \emph{Happy proving, happy hacking!}
  15.178 +*}
  15.179 +
  15.180 +end

    16.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    16.2 +++ b/doc-src/Codegen/Thy/Program.thy	Tue Mar 03 11:00:51 2009 +0100
    16.3 @@ -0,0 +1,534 @@
    16.4 +theory Program
    16.5 +imports Introduction
    16.6 +begin
    16.7 +
    16.8 +section {* Turning Theories into Programs \label{sec:program} *}
    16.9 +
   16.10 +subsection {* The @{text "Isabelle/HOL"} default setup *}
   16.11 +
   16.12 +text {*
   16.13 +  We have already seen how by default equations stemming from
   16.14 +  @{command definition}/@{command primrec}/@{command fun}
   16.15 +  statements are used for code generation.  This default behaviour
   16.16 +  can be changed, e.g. by providing different code equations.
   16.17 +  All kinds of customisation shown in this section is \emph{safe}
   16.18 +  in the sense that the user does not have to worry about
   16.19 +  correctness -- all programs generatable that way are partially
   16.20 +  correct.
   16.21 +*}
   16.22 +
   16.23 +subsection {* Selecting code equations *}
   16.24 +
   16.25 +text {*
   16.26 +  Coming back to our introductory example, we
   16.27 +  could provide an alternative code equations for @{const dequeue}
   16.28 +  explicitly:
   16.29 +*}
   16.30 +
   16.31 +lemma %quote [code]:
   16.32 +  "dequeue (AQueue xs []) =
   16.33 +     (if xs = [] then (None, AQueue [] [])
   16.34 +       else dequeue (AQueue [] (rev xs)))"
   16.35 +  "dequeue (AQueue xs (y # ys)) =
   16.36 +     (Some y, AQueue xs ys)"
   16.37 +  by (cases xs, simp_all) (cases "rev xs", simp_all)
   16.38 +
   16.39 +text {*
   16.40 +  \noindent The annotation @{text "[code]"} is an @{text Isar}
   16.41 +  @{text attribute} which states that the given theorems should be
   16.42 +  considered as code equations for a @{text fun} statement --
   16.43 +  the corresponding constant is determined syntactically.  The resulting code:
   16.44 +*}
   16.45 +
   16.46 +text %quote {*@{code_stmts dequeue (consts) dequeue (Haskell)}*}
   16.47 +
   16.48 +text {*
   16.49 +  \noindent You may note that the equality test @{term "xs = []"} has been
   16.50 +  replaced by the predicate @{term "null xs"}.  This is due to the default
   16.51 +  setup in the \qn{preprocessor} to be discussed further below (\secref{sec:preproc}).
   16.52 +
   16.53 +  Changing the default constructor set of datatypes is also
   16.54 +  possible.  See \secref{sec:datatypes} for an example.
   16.55 +
   16.56 +  As told in \secref{sec:concept}, code generation is based
   16.57 +  on a structured collection of code theorems.
   16.58 +  For explorative purpose, this collection
   16.59 +  may be inspected using the @{command code_thms} command:
   16.60 +*}
   16.61 +
   16.62 +code_thms %quote dequeue
   16.63 +
   16.64 +text {*
   16.65 +  \noindent prints a table with \emph{all} code equations
   16.66 +  for @{const dequeue}, including
   16.67 +  \emph{all} code equations those equations depend
   16.68 +  on recursively.
   16.69 +  
   16.70 +  Similarly, the @{command code_deps} command shows a graph
   16.71 +  visualising dependencies between code equations.
   16.72 +*}
   16.73 +
   16.74 +subsection {* @{text class} and @{text instantiation} *}
   16.75 +
   16.76 +text {*
   16.77 +  Concerning type classes and code generation, let us examine an example
   16.78 +  from abstract algebra:
   16.79 +*}
   16.80 +
   16.81 +class %quote semigroup =
   16.82 +  fixes mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<otimes>" 70)
   16.83 +  assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
   16.84 +
   16.85 +class %quote monoid = semigroup +
   16.86 +  fixes neutral :: 'a ("\<one>")
   16.87 +  assumes neutl: "\<one> \<otimes> x = x"
   16.88 +    and neutr: "x \<otimes> \<one> = x"
   16.89 +
   16.90 +instantiation %quote nat :: monoid
   16.91 +begin
   16.92 +
   16.93 +primrec %quote mult_nat where
   16.94 +    "0 \<otimes> n = (0\<Colon>nat)"
   16.95 +  | "Suc m \<otimes> n = n + m \<otimes> n"
   16.96 +
   16.97 +definition %quote neutral_nat where
   16.98 +  "\<one> = Suc 0"
   16.99 +
  16.100 +lemma %quote add_mult_distrib:
  16.101 +  fixes n m q :: nat
  16.102 +  shows "(n + m) \<otimes> q = n \<otimes> q + m \<otimes> q"
  16.103 +  by (induct n) simp_all
  16.104 +
  16.105 +instance %quote proof
  16.106 +  fix m n q :: nat
  16.107 +  show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
  16.108 +    by (induct m) (simp_all add: add_mult_distrib)
  16.109 +  show "\<one> \<otimes> n = n"
  16.110 +    by (simp add: neutral_nat_def)
  16.111 +  show "m \<otimes> \<one> = m"
  16.112 +    by (induct m) (simp_all add: neutral_nat_def)
  16.113 +qed
  16.114 +
  16.115 +end %quote
  16.116 +
  16.117 +text {*
  16.118 +  \noindent We define the natural operation of the natural numbers
  16.119 +  on monoids:
  16.120 +*}
  16.121 +
  16.122 +primrec %quote (in monoid) pow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" where
  16.123 +    "pow 0 a = \<one>"
  16.124 +  | "pow (Suc n) a = a \<otimes> pow n a"
  16.125 +
  16.126 +text {*
  16.127 +  \noindent This we use to define the discrete exponentiation function:
  16.128 +*}
  16.129 +
  16.130 +definition %quote bexp :: "nat \<Rightarrow> nat" where
  16.131 +  "bexp n = pow n (Suc (Suc 0))"
  16.132 +
  16.133 +text {*
  16.134 +  \noindent The corresponding code:
  16.135 +*}
  16.136 +
  16.137 +text %quote {*@{code_stmts bexp (Haskell)}*}
  16.138 +
  16.139 +text {*
  16.140 +  \noindent This is a convenient place to show how explicit dictionary construction
  16.141 +  manifests in generated code (here, the same example in @{text SML}):
  16.142 +*}
  16.143 +
  16.144 +text %quote {*@{code_stmts bexp (SML)}*}
  16.145 +
  16.146 +text {*
  16.147 +  \noindent Note the parameters with trailing underscore (@{verbatim "A_"})
  16.148 +    which are the dictionary parameters.
  16.149 +*}
  16.150 +
  16.151 +subsection {* The preprocessor \label{sec:preproc} *}
  16.152 +
  16.153 +text {*
  16.154 +  Before selected function theorems are turned into abstract
  16.155 +  code, a chain of definitional transformation steps is carried
  16.156 +  out: \emph{preprocessing}.  In essence, the preprocessor
  16.157 +  consists of two components: a \emph{simpset} and \emph{function transformers}.
  16.158 +
  16.159 +  The \emph{simpset} allows to employ the full generality of the Isabelle
  16.160 +  simplifier.  Due to the interpretation of theorems
  16.161 +  as code equations, rewrites are applied to the right
  16.162 +  hand side and the arguments of the left hand side of an
  16.163 +  equation, but never to the constant heading the left hand side.
  16.164 +  An important special case are \emph{inline theorems} which may be
  16.165 +  declared and undeclared using the
  16.166 +  \emph{code inline} or \emph{code inline del} attribute respectively.
  16.167 +
  16.168 +  Some common applications:
  16.169 +*}
  16.170 +
  16.171 +text_raw {*
  16.172 +  \begin{itemize}
  16.173 +*}
  16.174 +
  16.175 +text {*
  16.176 +     \item replacing non-executable constructs by executable ones:
  16.177 +*}     
  16.178 +
  16.179 +lemma %quote [code inline]:
  16.180 +  "x \<in> set xs \<longleftrightarrow> x mem xs" by (induct xs) simp_all
  16.181 +
  16.182 +text {*
  16.183 +     \item eliminating superfluous constants:
  16.184 +*}
  16.185 +
  16.186 +lemma %quote [code inline]:
  16.187 +  "1 = Suc 0" by simp
  16.188 +
  16.189 +text {*
  16.190 +     \item replacing executable but inconvenient constructs:
  16.191 +*}
  16.192 +
  16.193 +lemma %quote [code inline]:
  16.194 +  "xs = [] \<longleftrightarrow> List.null xs" by (induct xs) simp_all
  16.195 +
  16.196 +text_raw {*
  16.197 +  \end{itemize}
  16.198 +*}
  16.199 +
  16.200 +text {*
  16.201 +  \noindent \emph{Function transformers} provide a very general interface,
  16.202 +  transforming a list of function theorems to another
  16.203 +  list of function theorems, provided that neither the heading
  16.204 +  constant nor its type change.  The @{term "0\<Colon>nat"} / @{const Suc}
  16.205 +  pattern elimination implemented in
  16.206 +  theory @{text Efficient_Nat} (see \secref{eff_nat}) uses this
  16.207 +  interface.
  16.208 +
  16.209 +  \noindent The current setup of the preprocessor may be inspected using
  16.210 +  the @{command print_codesetup} command.
  16.211 +  @{command code_thms} provides a convenient
  16.212 +  mechanism to inspect the impact of a preprocessor setup
  16.213 +  on code equations.
  16.214 +
  16.215 +  \begin{warn}
  16.216 +    The attribute \emph{code unfold}
  16.217 +    associated with the @{text "SML code generator"} also applies to
  16.218 +    the @{text "generic code generator"}:
  16.219 +    \emph{code unfold} implies \emph{code inline}.
  16.220 +  \end{warn}
  16.221 +*}
  16.222 +
  16.223 +subsection {* Datatypes \label{sec:datatypes} *}
  16.224 +
  16.225 +text {*
  16.226 +  Conceptually, any datatype is spanned by a set of
  16.227 +  \emph{constructors} of type @{text "\<tau> = \<dots> \<Rightarrow> \<kappa> \<alpha>\<^isub>1 \<dots> \<alpha>\<^isub>n"} where @{text
  16.228 +  "{\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>n}"} is exactly the set of \emph{all} type variables in
  16.229 +  @{text "\<tau>"}.  The HOL datatype package by default registers any new
  16.230 +  datatype in the table of datatypes, which may be inspected using the
  16.231 +  @{command print_codesetup} command.
  16.232 +
  16.233 +  In some cases, it is appropriate to alter or extend this table.  As
  16.234 +  an example, we will develop an alternative representation of the
  16.235 +  queue example given in \secref{sec:intro}.  The amortised
  16.236 +  representation is convenient for generating code but exposes its
  16.237 +  \qt{implementation} details, which may be cumbersome when proving
  16.238 +  theorems about it.  Therefore, here a simple, straightforward
  16.239 +  representation of queues:
  16.240 +*}
  16.241 +
  16.242 +datatype %quote 'a queue = Queue "'a list"
  16.243 +
  16.244 +definition %quote empty :: "'a queue" where
  16.245 +  "empty = Queue []"
  16.246 +
  16.247 +primrec %quote enqueue :: "'a \<Rightarrow> 'a queue \<Rightarrow> 'a queue" where
  16.248 +  "enqueue x (Queue xs) = Queue (xs @ [x])"
  16.249 +
  16.250 +fun %quote dequeue :: "'a queue \<Rightarrow> 'a option \<times> 'a queue" where
  16.251 +    "dequeue (Queue []) = (None, Queue [])"
  16.252 +  | "dequeue (Queue (x # xs)) = (Some x, Queue xs)"
  16.253 +
  16.254 +text {*
  16.255 +  \noindent This we can use directly for proving;  for executing,
  16.256 +  we provide an alternative characterisation:
  16.257 +*}
  16.258 +
  16.259 +definition %quote AQueue :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a queue" where
  16.260 +  "AQueue xs ys = Queue (ys @ rev xs)"
  16.261 +
  16.262 +code_datatype %quote AQueue
  16.263 +
  16.264 +text {*
  16.265 +  \noindent Here we define a \qt{constructor} @{const "AQueue"} which
  16.266 +  is defined in terms of @{text "Queue"} and interprets its arguments
  16.267 +  according to what the \emph{content} of an amortised queue is supposed
  16.268 +  to be.  Equipped with this, we are able to prove the following equations
  16.269 +  for our primitive queue operations which \qt{implement} the simple
  16.270 +  queues in an amortised fashion:
  16.271 +*}
  16.272 +
  16.273 +lemma %quote empty_AQueue [code]:
  16.274 +  "empty = AQueue [] []"
  16.275 +  unfolding AQueue_def empty_def by simp
  16.276 +
  16.277 +lemma %quote enqueue_AQueue [code]:
  16.278 +  "enqueue x (AQueue xs ys) = AQueue (x # xs) ys"
  16.279 +  unfolding AQueue_def by simp
  16.280 +
  16.281 +lemma %quote dequeue_AQueue [code]:
  16.282 +  "dequeue (AQueue xs []) =
  16.283 +    (if xs = [] then (None, AQueue [] [])
  16.284 +    else dequeue (AQueue [] (rev xs)))"
  16.285 +  "dequeue (AQueue xs (y # ys)) = (Some y, AQueue xs ys)"
  16.286 +  unfolding AQueue_def by simp_all
  16.287 +
  16.288 +text {*
  16.289 +  \noindent For completeness, we provide a substitute for the
  16.290 +  @{text case} combinator on queues:
  16.291 +*}
  16.292 +
  16.293 +definition %quote
  16.294 +  aqueue_case_def: "aqueue_case = queue_case"
  16.295 +
  16.296 +lemma %quote aqueue_case [code, code inline]:
  16.297 +  "queue_case = aqueue_case"
  16.298 +  unfolding aqueue_case_def ..
  16.299 +
  16.300 +lemma %quote case_AQueue [code]:
  16.301 +  "aqueue_case f (AQueue xs ys) = f (ys @ rev xs)"
  16.302 +  unfolding aqueue_case_def AQueue_def by simp
  16.303 +
  16.304 +text {*
  16.305 +  \noindent The resulting code looks as expected:
  16.306 +*}
  16.307 +
  16.308 +text %quote {*@{code_stmts empty enqueue dequeue (SML)}*}
  16.309 +
  16.310 +text {*
  16.311 +  \noindent From this example, it can be glimpsed that using own
  16.312 +  constructor sets is a little delicate since it changes the set of
  16.313 +  valid patterns for values of that type.  Without going into much
  16.314 +  detail, here some practical hints:
  16.315 +
  16.316 +  \begin{itemize}
  16.317 +
  16.318 +    \item When changing the constructor set for datatypes, take care
  16.319 +      to provide an alternative for the @{text case} combinator
  16.320 +      (e.g.~by replacing it using the preprocessor).
  16.321 +
  16.322 +    \item Values in the target language need not to be normalised --
  16.323 +      different values in the target language may represent the same
  16.324 +      value in the logic.
  16.325 +
  16.326 +    \item Usually, a good methodology to deal with the subtleties of
  16.327 +      pattern matching is to see the type as an abstract type: provide
  16.328 +      a set of operations which operate on the concrete representation
  16.329 +      of the type, and derive further operations by combinations of
  16.330 +      these primitive ones, without relying on a particular
  16.331 +      representation.
  16.332 +
  16.333 +  \end{itemize}
  16.334 +*}
  16.335 +
  16.336 +
  16.337 +subsection {* Equality and wellsortedness *}
  16.338 +
  16.339 +text {*
  16.340 +  Surely you have already noticed how equality is treated
  16.341 +  by the code generator:
  16.342 +*}
  16.343 +
  16.344 +primrec %quote collect_duplicates :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
  16.345 +  "collect_duplicates xs ys [] = xs"
  16.346 +  | "collect_duplicates xs ys (z#zs) = (if z \<in> set xs
  16.347 +      then if z \<in> set ys
  16.348 +        then collect_duplicates xs ys zs
  16.349 +        else collect_duplicates xs (z#ys) zs
  16.350 +      else collect_duplicates (z#xs) (z#ys) zs)"
  16.351 +
  16.352 +text {*
  16.353 +  \noindent The membership test during preprocessing is rewritten,
  16.354 +  resulting in @{const List.member}, which itself
  16.355 +  performs an explicit equality check.
  16.356 +*}
  16.357 +
  16.358 +text %quote {*@{code_stmts collect_duplicates (SML)}*}
  16.359 +
  16.360 +text {*
  16.361 +  \noindent Obviously, polymorphic equality is implemented the Haskell
  16.362 +  way using a type class.  How is this achieved?  HOL introduces
  16.363 +  an explicit class @{class eq} with a corresponding operation
  16.364 +  @{const eq_class.eq} such that @{thm eq [no_vars]}.
  16.365 +  The preprocessing framework does the rest by propagating the
  16.366 +  @{class eq} constraints through all dependent code equations.
  16.367 +  For datatypes, instances of @{class eq} are implicitly derived
  16.368 +  when possible.  For other types, you may instantiate @{text eq}
  16.369 +  manually like any other type class.
  16.370 +
  16.371 +  Though this @{text eq} class is designed to get rarely in
  16.372 +  the way, a subtlety
  16.373 +  enters the stage when definitions of overloaded constants
  16.374 +  are dependent on operational equality.  For example, let
  16.375 +  us define a lexicographic ordering on tuples
  16.376 +  (also see theory @{theory Product_ord}):
  16.377 +*}
  16.378 +
  16.379 +instantiation %quote "*" :: (order, order) order
  16.380 +begin
  16.381 +
  16.382 +definition %quote [code del]:
  16.383 +  "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
  16.384 +
  16.385 +definition %quote [code del]:
  16.386 +  "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
  16.387 +
  16.388 +instance %quote proof
  16.389 +qed (auto simp: less_eq_prod_def less_prod_def intro: order_less_trans)
  16.390 +
  16.391 +end %quote
  16.392 +
  16.393 +lemma %quote order_prod [code]:
  16.394 +  "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
  16.395 +     x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
  16.396 +  "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
  16.397 +     x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
  16.398 +  by (simp_all add: less_prod_def less_eq_prod_def)
  16.399 +
  16.400 +text {*
  16.401 +  \noindent Then code generation will fail.  Why?  The definition
  16.402 +  of @{term "op \<le>"} depends on equality on both arguments,
  16.403 +  which are polymorphic and impose an additional @{class eq}
  16.404 +  class constraint, which the preprocessor does not propagate
  16.405 +  (for technical reasons).
  16.406 +
  16.407 +  The solution is to add @{class eq} explicitly to the first sort arguments in the
  16.408 +  code theorems:
  16.409 +*}
  16.410 +
  16.411 +lemma %quote order_prod_code [code]:
  16.412 +  "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
  16.413 +     x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
  16.414 +  "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
  16.415 +     x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
  16.416 +  by (simp_all add: less_prod_def less_eq_prod_def)
  16.417 +
  16.418 +text {*
  16.419 +  \noindent Then code generation succeeds:
  16.420 +*}
  16.421 +
  16.422 +text %quote {*@{code_stmts "op \<le> \<Colon> _ \<times> _ \<Rightarrow> _ \<times> _ \<Rightarrow> bool" (SML)}*}
  16.423 +
  16.424 +text {*
  16.425 +  In some cases, the automatically derived code equations
  16.426 +  for equality on a particular type may not be appropriate.
  16.427 +  As example, watch the following datatype representing
  16.428 +  monomorphic parametric types (where type constructors
  16.429 +  are referred to by natural numbers):
  16.430 +*}
  16.431 +
  16.432 +datatype %quote monotype = Mono nat "monotype list"
  16.433 +(*<*)
  16.434 +lemma monotype_eq:
  16.435 +  "eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<equiv> 
  16.436 +     eq_class.eq tyco1 tyco2 \<and> eq_class.eq typargs1 typargs2" by (simp add: eq)
  16.437 +(*>*)
  16.438 +
  16.439 +text {*
  16.440 +  \noindent Then code generation for SML would fail with a message
  16.441 +  that the generated code contains illegal mutual dependencies:
  16.442 +  the theorem @{thm monotype_eq [no_vars]} already requires the
  16.443 +  instance @{text "monotype \<Colon> eq"}, which itself requires
  16.444 +  @{thm monotype_eq [no_vars]};  Haskell has no problem with mutually
  16.445 +  recursive @{text instance} and @{text function} definitions,
  16.446 +  but the SML serialiser does not support this.
  16.447 +
  16.448 +  In such cases, you have to provide your own equality equations
  16.449 +  involving auxiliary constants.  In our case,
  16.450 +  @{const [show_types] list_all2} can do the job:
  16.451 +*}
  16.452 +
  16.453 +lemma %quote monotype_eq_list_all2 [code]:
  16.454 +  "eq_class.eq (Mono tyco1 typargs1) (Mono tyco2 typargs2) \<longleftrightarrow>
  16.455 +     eq_class.eq tyco1 tyco2 \<and> list_all2 eq_class.eq typargs1 typargs2"
  16.456 +  by (simp add: eq list_all2_eq [symmetric])
  16.457 +
  16.458 +text {*
  16.459 +  \noindent does not depend on instance @{text "monotype \<Colon> eq"}:
  16.460 +*}
  16.461 +
  16.462 +text %quote {*@{code_stmts "eq_class.eq :: monotype \<Rightarrow> monotype \<Rightarrow> bool" (SML)}*}
  16.463 +
  16.464 +
  16.465 +subsection {* Explicit partiality *}
  16.466 +
  16.467 +text {*
  16.468 +  Partiality usually enters the game by partial patterns, as
  16.469 +  in the following example, again for amortised queues:
  16.470 +*}
  16.471 +
  16.472 +definition %quote strict_dequeue :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
  16.473 +  "strict_dequeue q = (case dequeue q
  16.474 +    of (Some x, q') \<Rightarrow> (x, q'))"
  16.475 +
  16.476 +lemma %quote strict_dequeue_AQueue [code]:
  16.477 +  "strict_dequeue (AQueue xs (y # ys)) = (y, AQueue xs ys)"
  16.478 +  "strict_dequeue (AQueue xs []) =
  16.479 +    (case rev xs of y # ys \<Rightarrow> (y, AQueue [] ys))"
  16.480 +  by (simp_all add: strict_dequeue_def dequeue_AQueue split: list.splits)
  16.481 +
  16.482 +text {*
  16.483 +  \noindent In the corresponding code, there is no equation
  16.484 +  for the pattern @{term "AQueue [] []"}:
  16.485 +*}
  16.486 +
  16.487 +text %quote {*@{code_stmts strict_dequeue (consts) strict_dequeue (Haskell)}*}
  16.488 +
  16.489 +text {*
  16.490 +  \noindent In some cases it is desirable to have this
  16.491 +  pseudo-\qt{partiality} more explicitly, e.g.~as follows:
  16.492 +*}
  16.493 +
  16.494 +axiomatization %quote empty_queue :: 'a
  16.495 +
  16.496 +definition %quote strict_dequeue' :: "'a queue \<Rightarrow> 'a \<times> 'a queue" where
  16.497 +  "strict_dequeue' q = (case dequeue q of (Some x, q') \<Rightarrow> (x, q') | _ \<Rightarrow> empty_queue)"
  16.498 +
  16.499 +lemma %quote strict_dequeue'_AQueue [code]:
  16.500 +  "strict_dequeue' (AQueue xs []) = (if xs = [] then empty_queue
  16.501 +     else strict_dequeue' (AQueue [] (rev xs)))"
  16.502 +  "strict_dequeue' (AQueue xs (y # ys)) =
  16.503 +     (y, AQueue xs ys)"
  16.504 +  by (simp_all add: strict_dequeue'_def dequeue_AQueue split: list.splits)
  16.505 +
  16.506 +text {*
  16.507 +  Observe that on the right hand side of the definition of @{const
  16.508 +  "strict_dequeue'"} the constant @{const empty_queue} occurs
  16.509 +  which is unspecified.
  16.510 +
  16.511 +  Normally, if constants without any code equations occur in a
  16.512 +  program, the code generator complains (since in most cases this is
  16.513 +  not what the user expects).  But such constants can also be thought
  16.514 +  of as function definitions with no equations which always fail,
  16.515 +  since there is never a successful pattern match on the left hand
  16.516 +  side.  In order to categorise a constant into that category
  16.517 +  explicitly, use @{command "code_abort"}:
  16.518 +*}
  16.519 +
  16.520 +code_abort %quote empty_queue
  16.521 +
  16.522 +text {*
  16.523 +  \noindent Then the code generator will just insert an error or
  16.524 +  exception at the appropriate position:
  16.525 +*}
  16.526 +
  16.527 +text %quote {*@{code_stmts strict_dequeue' (consts) empty_queue strict_dequeue' (Haskell)}*}
  16.528 +
  16.529 +text {*
  16.530 +  \noindent This feature however is rarely needed in practice.
  16.531 +  Note also that the @{text HOL} default setup already declares
  16.532 +  @{const undefined} as @{command "code_abort"}, which is most
  16.533 +  likely to be used in such situations.
  16.534 +*}
  16.535 +
  16.536 +end
  16.537 + 
  16.538 \ No newline at end of file