1.1 --- a/Admin/CHECKLIST	Wed Mar 04 11:05:02 2009 +0100
     1.2 +++ b/Admin/CHECKLIST	Wed Mar 04 11:05:29 2009 +0100
     1.3 @@ -1,7 +1,7 @@
     1.4  Checklist for official releases
     1.5  ===============================
     1.6  
     1.7 -- test alice, mosml, polyml-5.0, polyml-4.1.3, polyml-4.1.4, polyml-4.2.0, x86-solaris, x86-cygwin;
     1.8 +- test mosml, polyml-5.0, polyml-4.1.3, polyml-4.1.4, polyml-4.2.0, x86-solaris, x86-cygwin;
     1.9  
    1.10  - test ProofGeneral;
    1.11  

     2.1 --- a/Admin/isatest/isatest-stats	Wed Mar 04 11:05:02 2009 +0100
     2.2 +++ b/Admin/isatest/isatest-stats	Wed Mar 04 11:05:29 2009 +0100
     2.3 @@ -16,6 +16,7 @@
     2.4    HOL-Algebra \
     2.5    HOL-Auth \
     2.6    HOL-Bali \
     2.7 +  HOL-Decision_Procs \
     2.8    HOL-Extraction \
     2.9    HOL-Hoare \
    2.10    HOL-HoareParallel \

     3.1 --- a/Admin/isatest/settings/sun-poly	Wed Mar 04 11:05:02 2009 +0100
     3.2 +++ b/Admin/isatest/settings/sun-poly	Wed Mar 04 11:05:29 2009 +0100
     3.3 @@ -4,7 +4,7 @@
     3.4    ML_SYSTEM="polyml-5.1"
     3.5    ML_PLATFORM="sparc-solaris"
     3.6    ML_HOME="$POLYML_HOME/$ML_PLATFORM"
     3.7 -  ML_OPTIONS="-H 1500"
     3.8 +  ML_OPTIONS="-H 800"
     3.9  
    3.10  ISABELLE_HOME_USER=/tmp/isabelle-sun-poly
    3.11  

     4.1 --- a/Admin/makedist	Wed Mar 04 11:05:02 2009 +0100
     4.2 +++ b/Admin/makedist	Wed Mar 04 11:05:29 2009 +0100
     4.3 @@ -4,7 +4,7 @@
     4.4  
     4.5  ## global settings
     4.6  
     4.7 -REPOS="https://isabelle.in.tum.de/repos/isabelle"
     4.8 +REPOS="http://isabelle.in.tum.de/repos/isabelle"
     4.9  
    4.10  DISTPREFIX=${DISTPREFIX:-~/tmp/isadist}
    4.11  
    4.12 @@ -156,7 +156,7 @@
    4.13  rm doc/codegen_process.pdf
    4.14  rm -rf doc-src
    4.15  
    4.16 -mkdir contrib
    4.17 +mkdir -p contrib
    4.18  
    4.19  cp doc/isabelle*.eps lib/logo
    4.20  

     5.1 --- a/CONTRIBUTORS	Wed Mar 04 11:05:02 2009 +0100
     5.2 +++ b/CONTRIBUTORS	Wed Mar 04 11:05:29 2009 +0100
     5.3 @@ -7,10 +7,16 @@
     5.4  Contributions to this Isabelle version
     5.5  --------------------------------------
     5.6  
     5.7 -* February 2008: Timothy Bourke, NICTA
     5.8 +* February 2009: Filip Maric, Univ. of Belgrade
     5.9 +  A Serbian theory.
    5.10 +
    5.11 +* February 2009: Jasmin Christian Blanchette, TUM
    5.12 +  Misc cleanup of HOL/refute.
    5.13 +
    5.14 +* February 2009: Timothy Bourke, NICTA
    5.15    New find_consts command.
    5.16  
    5.17 -* February 2008: Timothy Bourke, NICTA
    5.18 +* February 2009: Timothy Bourke, NICTA
    5.19    "solves" criterion for find_theorems and auto_solve option
    5.20  
    5.21  * December 2008: Clemens Ballarin, TUM
    5.22 @@ -31,6 +37,9 @@
    5.23    processes.  Additional ATP wrappers, including remote SystemOnTPTP
    5.24    services.
    5.25  
    5.26 +* September 2008: Stefan Berghofer, TUM and Marc Bezem, Univ. Bergen
    5.27 +  Prover for coherent logic.
    5.28 +
    5.29  * August 2008: Fabian Immler, TUM
    5.30    Vampire wrapper script for remote SystemOnTPTP service.
    5.31  
    5.32 @@ -46,7 +55,7 @@
    5.33    HOLCF library improvements.
    5.34  
    5.35  * 2007/2008: Stefan Berghofer, TUM
    5.36 -  HOL-Nominal package improvements.  
    5.37 +  HOL-Nominal package improvements.
    5.38  
    5.39  * March 2008: Markus Reiter, TUM
    5.40    HOL/Library/RBT: red-black trees.

     6.1 --- a/NEWS	Wed Mar 04 11:05:02 2009 +0100
     6.2 +++ b/NEWS	Wed Mar 04 11:05:29 2009 +0100
     6.3 @@ -6,6 +6,10 @@
     6.4  
     6.5  *** General ***
     6.6  
     6.7 +* The main reference manuals (isar-ref, implementation, system) have
     6.8 +been updated and extended.  Formally checked references as hyperlinks
     6.9 +are now available in uniform manner.
    6.10 +
    6.11  * Simplified main Isabelle executables, with less surprises on
    6.12  case-insensitive file-systems (such as Mac OS).
    6.13  
    6.14 @@ -47,9 +51,6 @@
    6.15  regular 4-core machine, if the initial heap space is made reasonably
    6.16  large (cf. Poly/ML option -H).  [Poly/ML 5.2.1 or later]
    6.17  
    6.18 -* The Isabelle System Manual (system) has been updated, with formally
    6.19 -checked references as hyperlinks.
    6.20 -
    6.21  * Generalized Isar history, with support for linear undo, direct state
    6.22  addressing etc.
    6.23  
    6.24 @@ -63,6 +64,8 @@
    6.25  * There is a new syntactic category "float_const" for signed decimal
    6.26  fractions (e.g. 123.45 or -123.45).
    6.27  
    6.28 +* New prover for coherent logic (see src/Tools/coherent.ML).
    6.29 +
    6.30  
    6.31  *** Pure ***
    6.32  
    6.33 @@ -111,30 +114,32 @@
    6.34    unify_trace_bound = 50 (formerly 25)
    6.35    unify_search_bound = 60 (formerly 30)
    6.36  
    6.37 -* Different bookkeeping for code equations:
    6.38 -  a) On theory merge, the last set of code equations for a particular constant
    6.39 -     is taken (in accordance with the policy applied by other parts of the
    6.40 -     code generator framework).
    6.41 -  b) Code equations stemming from explicit declarations (e.g. code attribute)
    6.42 -     gain priority over default code equations stemming from definition, primrec,
    6.43 -     fun etc.
    6.44 -  INCOMPATIBILITY.
    6.45 -
    6.46 -* Global versions of theorems stemming from classes do not carry
    6.47 -a parameter prefix any longer.  INCOMPATIBILITY.
    6.48 +* Different bookkeeping for code equations (INCOMPATIBILITY):
    6.49 +
    6.50 +  a) On theory merge, the last set of code equations for a particular
    6.51 +     constant is taken (in accordance with the policy applied by other
    6.52 +     parts of the code generator framework).
    6.53 +
    6.54 +  b) Code equations stemming from explicit declarations (e.g. code
    6.55 +     attribute) gain priority over default code equations stemming
    6.56 +     from definition, primrec, fun etc.
    6.57 +
    6.58 +* Global versions of theorems stemming from classes do not carry a
    6.59 +parameter prefix any longer.  INCOMPATIBILITY.
    6.60  
    6.61  * Dropped locale element "includes".  This is a major INCOMPATIBILITY.
    6.62  In existing theorem specifications replace the includes element by the
    6.63 -respective context elements of the included locale, omitting those that
    6.64 -are already present in the theorem specification.  Multiple assume
    6.65 -elements of a locale should be replaced by a single one involving the
    6.66 -locale predicate.  In the proof body, declarations (most notably
    6.67 -theorems) may be regained by interpreting the respective locales in the
    6.68 -proof context as required (command "interpret").
    6.69 +respective context elements of the included locale, omitting those
    6.70 +that are already present in the theorem specification.  Multiple
    6.71 +assume elements of a locale should be replaced by a single one
    6.72 +involving the locale predicate.  In the proof body, declarations (most
    6.73 +notably theorems) may be regained by interpreting the respective
    6.74 +locales in the proof context as required (command "interpret").
    6.75 +
    6.76  If using "includes" in replacement of a target solely because the
    6.77  parameter types in the theorem are not as general as in the target,
    6.78 -consider declaring a new locale with additional type constraints on the
    6.79 -parameters (context element "constrains").
    6.80 +consider declaring a new locale with additional type constraints on
    6.81 +the parameters (context element "constrains").
    6.82  
    6.83  * Dropped "locale (open)".  INCOMPATIBILITY.
    6.84  
    6.85 @@ -145,9 +150,9 @@
    6.86  * Interpretation commands no longer accept interpretation attributes.
    6.87  INCOMPATBILITY.
    6.88  
    6.89 -* Complete re-implementation of locales.  INCOMPATIBILITY.
    6.90 -The most important changes are listed below.  See documentation
    6.91 -(forthcoming) and tutorial (also forthcoming) for details.
    6.92 +* Complete re-implementation of locales.  INCOMPATIBILITY.  The most
    6.93 +important changes are listed below.  See documentation (forthcoming)
    6.94 +and tutorial (also forthcoming) for details.
    6.95  
    6.96  - In locale expressions, instantiation replaces renaming.  Parameters
    6.97  must be declared in a for clause.  To aid compatibility with previous
    6.98 @@ -161,15 +166,15 @@
    6.99  
   6.100  - More flexible mechanisms to qualify names generated by locale
   6.101  expressions.  Qualifiers (prefixes) may be specified in locale
   6.102 -expressions.  Available are normal qualifiers (syntax "name:") and strict
   6.103 -qualifiers (syntax "name!:").  The latter must occur in name references
   6.104 -and are useful to avoid accidental hiding of names, the former are
   6.105 -optional.  Qualifiers derived from the parameter names of a locale are no
   6.106 -longer generated.
   6.107 -
   6.108 -- "sublocale l < e" replaces "interpretation l < e".  The instantiation
   6.109 -clause in "interpretation" and "interpret" (square brackets) is no
   6.110 -longer available.  Use locale expressions.
   6.111 +expressions.  Available are normal qualifiers (syntax "name:") and
   6.112 +strict qualifiers (syntax "name!:").  The latter must occur in name
   6.113 +references and are useful to avoid accidental hiding of names, the
   6.114 +former are optional.  Qualifiers derived from the parameter names of a
   6.115 +locale are no longer generated.
   6.116 +
   6.117 +- "sublocale l < e" replaces "interpretation l < e".  The
   6.118 +instantiation clause in "interpretation" and "interpret" (square
   6.119 +brackets) is no longer available.  Use locale expressions.
   6.120  
   6.121  - When converting proof scripts, be sure to replace qualifiers in
   6.122  "interpretation" and "interpret" by strict qualifiers.  Qualifiers in
   6.123 @@ -183,8 +188,8 @@
   6.124  * The 'axiomatization' command now only works within a global theory
   6.125  context.  INCOMPATIBILITY.
   6.126  
   6.127 -* New find_theorems criterion "solves" matching theorems that
   6.128 -directly solve the current goal. Try "find_theorems solves".
   6.129 +* New find_theorems criterion "solves" matching theorems that directly
   6.130 +solve the current goal. Try "find_theorems solves".
   6.131  
   6.132  * Added an auto solve option, which can be enabled through the
   6.133  ProofGeneral Isabelle settings menu (disabled by default).
   6.134 @@ -193,14 +198,15 @@
   6.135  stated. Any theorems that could solve the lemma directly are listed
   6.136  underneath the goal.
   6.137  
   6.138 -* New command find_consts searches for constants based on type and name 
   6.139 -patterns, e.g.
   6.140 +* New command find_consts searches for constants based on type and
   6.141 +name patterns, e.g.
   6.142  
   6.143      find_consts "_ => bool"
   6.144  
   6.145 -By default, matching is against subtypes, but it may be restricted to the
   6.146 -whole type. Searching by name is possible. Multiple queries are conjunctive
   6.147 -and queries may be negated by prefixing them with a hyphen:
   6.148 +By default, matching is against subtypes, but it may be restricted to
   6.149 +the whole type. Searching by name is possible. Multiple queries are
   6.150 +conjunctive and queries may be negated by prefixing them with a
   6.151 +hyphen:
   6.152  
   6.153      find_consts strict: "_ => bool" name: "Int" -"int => int"
   6.154  
   6.155 @@ -240,7 +246,7 @@
   6.156      src/HOL/Library/Order_Relation.thy ~> src/HOL/
   6.157      src/HOL/Library/Parity.thy ~> src/HOL/
   6.158      src/HOL/Library/Univ_Poly.thy ~> src/HOL/
   6.159 -    src/HOL/Real/ContNotDenum.thy ~> src/HOL/
   6.160 +    src/HOL/Real/ContNotDenum.thy ~> src/HOL/Library/
   6.161      src/HOL/Real/Lubs.thy ~> src/HOL/
   6.162      src/HOL/Real/PReal.thy ~> src/HOL/
   6.163      src/HOL/Real/Rational.thy ~> src/HOL/
   6.164 @@ -250,8 +256,8 @@
   6.165      src/HOL/Real/Real.thy ~> src/HOL/
   6.166      src/HOL/Complex/Complex_Main.thy ~> src/HOL/
   6.167      src/HOL/Complex/Complex.thy ~> src/HOL/
   6.168 -    src/HOL/Complex/FrechetDeriv.thy ~> src/HOL/
   6.169 -    src/HOL/Complex/Fundamental_Theorem_Algebra.thy ~> src/HOL/
   6.170 +    src/HOL/Complex/FrechetDeriv.thy ~> src/HOL/Library/
   6.171 +    src/HOL/Complex/Fundamental_Theorem_Algebra.thy ~> src/HOL/Library/
   6.172      src/HOL/Hyperreal/Deriv.thy ~> src/HOL/
   6.173      src/HOL/Hyperreal/Fact.thy ~> src/HOL/
   6.174      src/HOL/Hyperreal/Integration.thy ~> src/HOL/
   6.175 @@ -312,7 +318,7 @@
   6.176  process.  New thread-based implementation also works on non-Unix
   6.177  platforms (Cygwin).  Provers are no longer hardwired, but defined
   6.178  within the theory via plain ML wrapper functions.  Basic Sledgehammer
   6.179 -commands are covered in the isar-ref manual
   6.180 +commands are covered in the isar-ref manual.
   6.181  
   6.182  * Wrapper scripts for remote SystemOnTPTP service allows to use
   6.183  sledgehammer without local ATP installation (Vampire etc.).  See also
   6.184 @@ -342,6 +348,9 @@
   6.185  etc. slightly changed.  Some theorems named order_class.* now named
   6.186  preorder_class.*.
   6.187  
   6.188 +* HOL/Relation:
   6.189 +Renamed "refl" to "refl_on", "reflexive" to "refl, "diag" to "Id_on".
   6.190 +
   6.191  * HOL/Finite_Set: added a new fold combinator of type
   6.192    ('a => 'b => 'b) => 'b => 'a set => 'b
   6.193  Occasionally this is more convenient than the old fold combinator which is
   6.194 @@ -367,10 +376,60 @@
   6.195      mult_div ~>             div_mult_self2_is_id
   6.196      mult_mod ~>             mod_mult_self2_is_0
   6.197  
   6.198 +* HOL/IntDiv: removed many lemmas that are instances of class-based
   6.199 +generalizations (from Divides and Ring_and_Field).
   6.200 +INCOMPATIBILITY. Rename old lemmas as follows:
   6.201 +
   6.202 +dvd_diff               -> nat_dvd_diff
   6.203 +dvd_zminus_iff         -> dvd_minus_iff
   6.204 +mod_add1_eq            -> mod_add_eq
   6.205 +mod_mult1_eq           -> mod_mult_right_eq
   6.206 +mod_mult1_eq'          -> mod_mult_left_eq
   6.207 +mod_mult_distrib_mod   -> mod_mult_eq
   6.208 +nat_mod_add_left_eq    -> mod_add_left_eq
   6.209 +nat_mod_add_right_eq   -> mod_add_right_eq
   6.210 +nat_mod_div_trivial    -> mod_div_trivial
   6.211 +nat_mod_mod_trivial    -> mod_mod_trivial
   6.212 +zdiv_zadd_self1        -> div_add_self1
   6.213 +zdiv_zadd_self2        -> div_add_self2
   6.214 +zdiv_zmult_self1       -> div_mult_self2_is_id
   6.215 +zdiv_zmult_self2       -> div_mult_self1_is_id
   6.216 +zdvd_triv_left         -> dvd_triv_left
   6.217 +zdvd_triv_right        -> dvd_triv_right
   6.218 +zdvd_zmult_cancel_disj -> dvd_mult_cancel_left
   6.219 +zmod_eq0_zdvd_iff      -> dvd_eq_mod_eq_0[symmetric]
   6.220 +zmod_zadd_left_eq      -> mod_add_left_eq
   6.221 +zmod_zadd_right_eq     -> mod_add_right_eq
   6.222 +zmod_zadd_self1        -> mod_add_self1
   6.223 +zmod_zadd_self2        -> mod_add_self2
   6.224 +zmod_zadd1_eq          -> mod_add_eq
   6.225 +zmod_zdiff1_eq         -> mod_diff_eq
   6.226 +zmod_zdvd_zmod         -> mod_mod_cancel
   6.227 +zmod_zmod_cancel       -> mod_mod_cancel
   6.228 +zmod_zmult_self1       -> mod_mult_self2_is_0
   6.229 +zmod_zmult_self2       -> mod_mult_self1_is_0
   6.230 +zmod_1                 -> mod_by_1
   6.231 +zdiv_1                 -> div_by_1
   6.232 +zdvd_abs1              -> abs_dvd_iff
   6.233 +zdvd_abs2              -> dvd_abs_iff
   6.234 +zdvd_refl              -> dvd_refl
   6.235 +zdvd_trans             -> dvd_trans
   6.236 +zdvd_zadd              -> dvd_add
   6.237 +zdvd_zdiff             -> dvd_diff
   6.238 +zdvd_zminus_iff        -> dvd_minus_iff
   6.239 +zdvd_zminus2_iff       -> minus_dvd_iff
   6.240 +zdvd_zmultD            -> dvd_mult_right
   6.241 +zdvd_zmultD2           -> dvd_mult_left
   6.242 +zdvd_zmult_mono        -> mult_dvd_mono
   6.243 +zdvd_0_right           -> dvd_0_right
   6.244 +zdvd_0_left            -> dvd_0_left_iff
   6.245 +zdvd_1_left            -> one_dvd
   6.246 +zminus_dvd_iff         -> minus_dvd_iff
   6.247 +
   6.248  * HOL/Library/GCD: Curried operations gcd, lcm (for nat) and zgcd,
   6.249  zlcm (for int); carried together from various gcd/lcm developements in
   6.250  the HOL Distribution.  zgcd and zlcm replace former igcd and ilcm;
   6.251 -corresponding theorems renamed accordingly.  INCOMPATIBILY.  To
   6.252 +corresponding theorems renamed accordingly.  INCOMPATIBILITY.  To
   6.253  recover tupled syntax, use syntax declarations like:
   6.254  
   6.255      hide (open) const gcd
   6.256 @@ -384,7 +443,7 @@
   6.257  * HOL/Real/Rational: 'Fract k 0' now equals '0'.  INCOMPATIBILITY.
   6.258  
   6.259  * The real numbers offer decimal input syntax: 12.34 is translated into
   6.260 -  1234/10^4. This translation is not reversed upon output.
   6.261 +  1234/10^2. This translation is not reversed upon output.
   6.262  
   6.263  * New ML antiquotation @{code}: takes constant as argument, generates
   6.264  corresponding code in background and inserts name of the corresponding
   6.265 @@ -441,6 +500,9 @@
   6.266      Suc_Suc_eq                  ~> nat.inject
   6.267      Suc_not_Zero Zero_not_Suc   ~> nat.distinct
   6.268  
   6.269 +* The option datatype has been moved to a new theory HOL/Option.thy.
   6.270 +Renamed option_map to Option.map.
   6.271 +
   6.272  * Library/Nat_Infinity: added addition, numeral syntax and more
   6.273  instantiations for algebraic structures.  Removed some duplicate
   6.274  theorems.  Changes in simp rules.  INCOMPATIBILITY.
   6.275 @@ -452,9 +514,8 @@
   6.276  *** HOL-Algebra ***
   6.277  
   6.278  * New locales for orders and lattices where the equivalence relation
   6.279 -  is not restricted to equality.  INCOMPATIBILITY: all order and
   6.280 -  lattice locales use a record structure with field eq for the
   6.281 -  equivalence.
   6.282 +is not restricted to equality.  INCOMPATIBILITY: all order and lattice
   6.283 +locales use a record structure with field eq for the equivalence.
   6.284  
   6.285  * New theory of factorial domains.
   6.286  
   6.287 @@ -485,6 +546,23 @@
   6.288  * Proof of Zorn's Lemma for partial orders.
   6.289  
   6.290  
   6.291 +*** HOLCF ***
   6.292 +
   6.293 +* Reimplemented the simplification procedure for proving continuity
   6.294 +subgoals.  The new simproc is extensible; users can declare additional
   6.295 +continuity introduction rules with the attribute [cont2cont].
   6.296 +
   6.297 +* The continuity simproc now uses a different introduction rule for
   6.298 +solving continuity subgoals on terms with lambda abstractions.  In
   6.299 +some rare cases the new simproc may fail to solve subgoals that the
   6.300 +old one could solve, and "simp add: cont2cont_LAM" may be necessary.
   6.301 +Potential INCOMPATIBILITY.
   6.302 +
   6.303 +* The syntax of the fixrec package has changed.  The specification
   6.304 +syntax now conforms in style to definition, primrec, function, etc.
   6.305 +See HOLCF/ex/Fixrec_ex.thy for examples.  INCOMPATIBILITY.
   6.306 +
   6.307 +
   6.308  *** ML ***
   6.309  
   6.310  * High-level support for concurrent ML programming, see

     7.1 --- a/README_REPOSITORY	Wed Mar 04 11:05:02 2009 +0100
     7.2 +++ b/README_REPOSITORY	Wed Mar 04 11:05:29 2009 +0100
     7.3 @@ -32,9 +32,9 @@
     7.4  Initial configuration
     7.5  ---------------------
     7.6  
     7.7 -Always use Mercurial version 1.0 or later, such as 1.0.1 or 1.0.2.
     7.8 +Always use Mercurial versions from the 1.0 or 1.1 branch, or later.
     7.9  The old 0.9.x versions do not work in a multi-user environment with
    7.10 -shared file spaces.
    7.11 +shared file spaces!
    7.12  
    7.13  
    7.14  The official Isabelle repository can be cloned like this:
    7.15 @@ -62,7 +62,8 @@
    7.16  
    7.17  In principle, user names can be chosen freely, but for longterm
    7.18  committers of the Isabelle repository the obvious choice is to keep
    7.19 -with the old CVS naming scheme.
    7.20 +with the old CVS naming scheme.  Others should use their regular "full
    7.21 +name"; including an email address is optional.
    7.22  
    7.23  
    7.24  There are other useful configuration to go into $HOME/.hgrc,
    7.25 @@ -135,6 +136,29 @@
    7.26    hg clone ssh://wenzelm@atbroy100//home/isabelle-repository/repos/isabelle
    7.27  
    7.28  
    7.29 +Simplified merges
    7.30 +-----------------
    7.31 +
    7.32 +The main idea of Mercurial is to let individual users produce
    7.33 +independent branches of development first, but merge with others
    7.34 +frequently.  The basic hg merge operation is more general than
    7.35 +required for the mode of operation with a shared pull/push area.  The
    7.36 +hg fetch extension accommodates this case nicely, automating trivial
    7.37 +merges and requiring manual intervention for actual conflicts only.
    7.38 +
    7.39 +The fetch extension can be configured via the user's ~/.hgrc like
    7.40 +this:
    7.41 +
    7.42 +  [extensions]
    7.43 +  hgext.fetch =
    7.44 +
    7.45 +  [defaults]
    7.46 +  fetch = -m "merged"
    7.47 +
    7.48 +Note that the potential for merge conflicts can be greatly reduced by
    7.49 +doing "hg fetch" before any starting local changes!
    7.50 +
    7.51 +
    7.52  Content discipline
    7.53  ------------------
    7.54  
    7.55 @@ -172,7 +196,9 @@
    7.56      Mercurial provides nice web presentation of incoming changes with
    7.57      a digest of log entries; this also includes RSS/Atom news feeds.
    7.58      Users should be aware that others will actually read what is
    7.59 -    written into log messages.
    7.60 +    written into log messages.  There are also add-on browsers,
    7.61 +    notably hgtk that is part of the TortoiseHg distribution and works
    7.62 +    for generic Python/GTk platforms.
    7.63  
    7.64      The usual changelog presentation style for the Isabelle repository
    7.65      admits log entries that consist of several lines, but without the
    7.66 @@ -194,6 +220,3 @@
    7.67  
    7.68  Needless to say, the results from the build process must not be added
    7.69  to the repository!
    7.70 -
    7.71 -
    7.72 -Makarius 30-Nov-2008

     8.1 --- a/doc-src/AxClass/Group/Group.thy	Wed Mar 04 11:05:02 2009 +0100
     8.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     8.3 @@ -1,322 +0,0 @@
     8.4 -
     8.5 -header {* Basic group theory *}
     8.6 -
     8.7 -theory Group imports Main begin
     8.8 -
     8.9 -text {*
    8.10 -  \medskip\noindent The meta-level type system of Isabelle supports
    8.11 -  \emph{intersections} and \emph{inclusions} of type classes. These
    8.12 -  directly correspond to intersections and inclusions of type
    8.13 -  predicates in a purely set theoretic sense. This is sufficient as a
    8.14 -  means to describe simple hierarchies of structures.  As an
    8.15 -  illustration, we use the well-known example of semigroups, monoids,
    8.16 -  general groups and Abelian groups.
    8.17 -*}
    8.18 -
    8.19 -subsection {* Monoids and Groups *}
    8.20 -
    8.21 -text {*
    8.22 -  First we declare some polymorphic constants required later for the
    8.23 -  signature parts of our structures.
    8.24 -*}
    8.25 -
    8.26 -consts
    8.27 -  times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<odot>" 70)
    8.28 -  invers :: "'a \<Rightarrow> 'a"    ("(_\<inv>)" [1000] 999)
    8.29 -  one :: 'a    ("\<one>")
    8.30 -
    8.31 -text {*
    8.32 -  \noindent Next we define class @{text monoid} of monoids with
    8.33 -  operations @{text \<odot>} and @{text \<one>}.  Note that multiple class
    8.34 -  axioms are allowed for user convenience --- they simply represent
    8.35 -  the conjunction of their respective universal closures.
    8.36 -*}
    8.37 -
    8.38 -axclass monoid \<subseteq> type
    8.39 -  assoc: "(x \<odot> y) \<odot> z = x \<odot> (y \<odot> z)"
    8.40 -  left_unit: "\<one> \<odot> x = x"
    8.41 -  right_unit: "x \<odot> \<one> = x"
    8.42 -
    8.43 -text {*
    8.44 -  \noindent So class @{text monoid} contains exactly those types
    8.45 -  @{text \<tau>} where @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow> \<tau>"} and @{text "\<one> \<Colon> \<tau>"}
    8.46 -  are specified appropriately, such that @{text \<odot>} is associative and
    8.47 -  @{text \<one>} is a left and right unit element for the @{text \<odot>}
    8.48 -  operation.
    8.49 -*}
    8.50 -
    8.51 -text {*
    8.52 -  \medskip Independently of @{text monoid}, we now define a linear
    8.53 -  hierarchy of semigroups, general groups and Abelian groups.  Note
    8.54 -  that the names of class axioms are automatically qualified with each
    8.55 -  class name, so we may re-use common names such as @{text assoc}.
    8.56 -*}
    8.57 -
    8.58 -axclass semigroup \<subseteq> type
    8.59 -  assoc: "(x \<odot> y) \<odot> z = x \<odot> (y \<odot> z)"
    8.60 -
    8.61 -axclass group \<subseteq> semigroup
    8.62 -  left_unit: "\<one> \<odot> x = x"
    8.63 -  left_inverse: "x\<inv> \<odot> x = \<one>"
    8.64 -
    8.65 -axclass agroup \<subseteq> group
    8.66 -  commute: "x \<odot> y = y \<odot> x"
    8.67 -
    8.68 -text {*
    8.69 -  \noindent Class @{text group} inherits associativity of @{text \<odot>}
    8.70 -  from @{text semigroup} and adds two further group axioms. Similarly,
    8.71 -  @{text agroup} is defined as the subset of @{text group} such that
    8.72 -  for all of its elements @{text \<tau>}, the operation @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow>
    8.73 -  \<tau>"} is even commutative.
    8.74 -*}
    8.75 -
    8.76 -
    8.77 -subsection {* Abstract reasoning *}
    8.78 -
    8.79 -text {*
    8.80 -  In a sense, axiomatic type classes may be viewed as \emph{abstract
    8.81 -  theories}.  Above class definitions gives rise to abstract axioms
    8.82 -  @{text assoc}, @{text left_unit}, @{text left_inverse}, @{text
    8.83 -  commute}, where any of these contain a type variable @{text "'a \<Colon>
    8.84 -  c"} that is restricted to types of the corresponding class @{text
    8.85 -  c}.  \emph{Sort constraints} like this express a logical
    8.86 -  precondition for the whole formula.  For example, @{text assoc}
    8.87 -  states that for all @{text \<tau>}, provided that @{text "\<tau> \<Colon>
    8.88 -  semigroup"}, the operation @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow> \<tau>"} is associative.
    8.89 -
    8.90 -  \medskip From a technical point of view, abstract axioms are just
    8.91 -  ordinary Isabelle theorems, which may be used in proofs without
    8.92 -  special treatment.  Such ``abstract proofs'' usually yield new
    8.93 -  ``abstract theorems''.  For example, we may now derive the following
    8.94 -  well-known laws of general groups.
    8.95 -*}
    8.96 -
    8.97 -theorem group_right_inverse: "x \<odot> x\<inv> = (\<one>\<Colon>'a\<Colon>group)"
    8.98 -proof -
    8.99 -  have "x \<odot> x\<inv> = \<one> \<odot> (x \<odot> x\<inv>)"
   8.100 -    by (simp only: group_class.left_unit)
   8.101 -  also have "... = \<one> \<odot> x \<odot> x\<inv>"
   8.102 -    by (simp only: semigroup_class.assoc)
   8.103 -  also have "... = (x\<inv>)\<inv> \<odot> x\<inv> \<odot> x \<odot> x\<inv>"
   8.104 -    by (simp only: group_class.left_inverse)
   8.105 -  also have "... = (x\<inv>)\<inv> \<odot> (x\<inv> \<odot> x) \<odot> x\<inv>"
   8.106 -    by (simp only: semigroup_class.assoc)
   8.107 -  also have "... = (x\<inv>)\<inv> \<odot> \<one> \<odot> x\<inv>"
   8.108 -    by (simp only: group_class.left_inverse)
   8.109 -  also have "... = (x\<inv>)\<inv> \<odot> (\<one> \<odot> x\<inv>)"
   8.110 -    by (simp only: semigroup_class.assoc)
   8.111 -  also have "... = (x\<inv>)\<inv> \<odot> x\<inv>"
   8.112 -    by (simp only: group_class.left_unit)
   8.113 -  also have "... = \<one>"
   8.114 -    by (simp only: group_class.left_inverse)
   8.115 -  finally show ?thesis .
   8.116 -qed
   8.117 -
   8.118 -text {*
   8.119 -  \noindent With @{text group_right_inverse} already available, @{text
   8.120 -  group_right_unit}\label{thm:group-right-unit} is now established
   8.121 -  much easier.
   8.122 -*}
   8.123 -
   8.124 -theorem group_right_unit: "x \<odot> \<one> = (x\<Colon>'a\<Colon>group)"
   8.125 -proof -
   8.126 -  have "x \<odot> \<one> = x \<odot> (x\<inv> \<odot> x)"
   8.127 -    by (simp only: group_class.left_inverse)
   8.128 -  also have "... = x \<odot> x\<inv> \<odot> x"
   8.129 -    by (simp only: semigroup_class.assoc)
   8.130 -  also have "... = \<one> \<odot> x"
   8.131 -    by (simp only: group_right_inverse)
   8.132 -  also have "... = x"
   8.133 -    by (simp only: group_class.left_unit)
   8.134 -  finally show ?thesis .
   8.135 -qed
   8.136 -
   8.137 -text {*
   8.138 -  \medskip Abstract theorems may be instantiated to only those types
   8.139 -  @{text \<tau>} where the appropriate class membership @{text "\<tau> \<Colon> c"} is
   8.140 -  known at Isabelle's type signature level.  Since we have @{text
   8.141 -  "agroup \<subseteq> group \<subseteq> semigroup"} by definition, all theorems of @{text
   8.142 -  semigroup} and @{text group} are automatically inherited by @{text
   8.143 -  group} and @{text agroup}.
   8.144 -*}
   8.145 -
   8.146 -
   8.147 -subsection {* Abstract instantiation *}
   8.148 -
   8.149 -text {*
   8.150 -  From the definition, the @{text monoid} and @{text group} classes
   8.151 -  have been independent.  Note that for monoids, @{text right_unit}
   8.152 -  had to be included as an axiom, but for groups both @{text
   8.153 -  right_unit} and @{text right_inverse} are derivable from the other
   8.154 -  axioms.  With @{text group_right_unit} derived as a theorem of group
   8.155 -  theory (see page~\pageref{thm:group-right-unit}), we may now
   8.156 -  instantiate @{text "monoid \<subseteq> semigroup"} and @{text "group \<subseteq>
   8.157 -  monoid"} properly as follows (cf.\ \figref{fig:monoid-group}).
   8.158 -
   8.159 - \begin{figure}[htbp]
   8.160 -   \begin{center}
   8.161 -     \small
   8.162 -     \unitlength 0.6mm
   8.163 -     \begin{picture}(65,90)(0,-10)
   8.164 -       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
   8.165 -       \put(15,50){\line(1,1){10}} \put(35,60){\line(1,-1){10}}
   8.166 -       \put(15,5){\makebox(0,0){@{text agroup}}}
   8.167 -       \put(15,25){\makebox(0,0){@{text group}}}
   8.168 -       \put(15,45){\makebox(0,0){@{text semigroup}}}
   8.169 -       \put(30,65){\makebox(0,0){@{text type}}} \put(50,45){\makebox(0,0){@{text monoid}}}
   8.170 -     \end{picture}
   8.171 -     \hspace{4em}
   8.172 -     \begin{picture}(30,90)(0,0)
   8.173 -       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
   8.174 -       \put(15,50){\line(0,1){10}} \put(15,70){\line(0,1){10}}
   8.175 -       \put(15,5){\makebox(0,0){@{text agroup}}}
   8.176 -       \put(15,25){\makebox(0,0){@{text group}}}
   8.177 -       \put(15,45){\makebox(0,0){@{text monoid}}}
   8.178 -       \put(15,65){\makebox(0,0){@{text semigroup}}}
   8.179 -       \put(15,85){\makebox(0,0){@{text type}}}
   8.180 -     \end{picture}
   8.181 -     \caption{Monoids and groups: according to definition, and by proof}
   8.182 -     \label{fig:monoid-group}
   8.183 -   \end{center}
   8.184 - \end{figure}
   8.185 -*}
   8.186 -
   8.187 -instance monoid \<subseteq> semigroup
   8.188 -proof
   8.189 -  fix x y z :: "'a\<Colon>monoid"
   8.190 -  show "x \<odot> y \<odot> z = x \<odot> (y \<odot> z)"
   8.191 -    by (rule monoid_class.assoc)
   8.192 -qed
   8.193 -
   8.194 -instance group \<subseteq> monoid
   8.195 -proof
   8.196 -  fix x y z :: "'a\<Colon>group"
   8.197 -  show "x \<odot> y \<odot> z = x \<odot> (y \<odot> z)"
   8.198 -    by (rule semigroup_class.assoc)
   8.199 -  show "\<one> \<odot> x = x"
   8.200 -    by (rule group_class.left_unit)
   8.201 -  show "x \<odot> \<one> = x"
   8.202 -    by (rule group_right_unit)
   8.203 -qed
   8.204 -
   8.205 -text {*
   8.206 -  \medskip The \isakeyword{instance} command sets up an appropriate
   8.207 -  goal that represents the class inclusion (or type arity, see
   8.208 -  \secref{sec:inst-arity}) to be proven (see also
   8.209 -  \cite{isabelle-isar-ref}).  The initial proof step causes
   8.210 -  back-chaining of class membership statements wrt.\ the hierarchy of
   8.211 -  any classes defined in the current theory; the effect is to reduce
   8.212 -  to the initial statement to a number of goals that directly
   8.213 -  correspond to any class axioms encountered on the path upwards
   8.214 -  through the class hierarchy.
   8.215 -*}
   8.216 -
   8.217 -
   8.218 -subsection {* Concrete instantiation \label{sec:inst-arity} *}
   8.219 -
   8.220 -text {*
   8.221 -  So far we have covered the case of the form
   8.222 -  \isakeyword{instance}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"}, namely
   8.223 -  \emph{abstract instantiation} --- $c@1$ is more special than @{text
   8.224 -  "c\<^sub>1"} and thus an instance of @{text "c\<^sub>2"}.  Even more
   8.225 -  interesting for practical applications are \emph{concrete
   8.226 -  instantiations} of axiomatic type classes.  That is, certain simple
   8.227 -  schemes @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t \<Colon> c"} of class
   8.228 -  membership may be established at the logical level and then
   8.229 -  transferred to Isabelle's type signature level.
   8.230 -
   8.231 -  \medskip As a typical example, we show that type @{typ bool} with
   8.232 -  exclusive-or as @{text \<odot>} operation, identity as @{text \<inv>}, and
   8.233 -  @{term False} as @{text \<one>} forms an Abelian group.
   8.234 -*}
   8.235 -
   8.236 -defs (overloaded)
   8.237 -  times_bool_def: "x \<odot> y \<equiv> x \<noteq> (y\<Colon>bool)"
   8.238 -  inverse_bool_def: "x\<inv> \<equiv> x\<Colon>bool"
   8.239 -  unit_bool_def: "\<one> \<equiv> False"
   8.240 -
   8.241 -text {*
   8.242 -  \medskip It is important to note that above \isakeyword{defs} are
   8.243 -  just overloaded meta-level constant definitions, where type classes
   8.244 -  are not yet involved at all.  This form of constant definition with
   8.245 -  overloading (and optional recursion over the syntactic structure of
   8.246 -  simple types) are admissible as definitional extensions of plain HOL
   8.247 -  \cite{Wenzel:1997:TPHOL}.  The Haskell-style type system is not
   8.248 -  required for overloading.  Nevertheless, overloaded definitions are
   8.249 -  best applied in the context of type classes.
   8.250 -
   8.251 -  \medskip Since we have chosen above \isakeyword{defs} of the generic
   8.252 -  group operations on type @{typ bool} appropriately, the class
   8.253 -  membership @{text "bool \<Colon> agroup"} may be now derived as follows.
   8.254 -*}
   8.255 -
   8.256 -instance bool :: agroup
   8.257 -proof (intro_classes,
   8.258 -    unfold times_bool_def inverse_bool_def unit_bool_def)
   8.259 -  fix x y z
   8.260 -  show "((x \<noteq> y) \<noteq> z) = (x \<noteq> (y \<noteq> z))" by blast
   8.261 -  show "(False \<noteq> x) = x" by blast
   8.262 -  show "(x \<noteq> x) = False" by blast
   8.263 -  show "(x \<noteq> y) = (y \<noteq> x)" by blast
   8.264 -qed
   8.265 -
   8.266 -text {*
   8.267 -  The result of an \isakeyword{instance} statement is both expressed
   8.268 -  as a theorem of Isabelle's meta-logic, and as a type arity of the
   8.269 -  type signature.  The latter enables type-inference system to take
   8.270 -  care of this new instance automatically.
   8.271 -
   8.272 -  \medskip We could now also instantiate our group theory classes to
   8.273 -  many other concrete types.  For example, @{text "int \<Colon> agroup"}
   8.274 -  (e.g.\ by defining @{text \<odot>} as addition, @{text \<inv>} as negation
   8.275 -  and @{text \<one>} as zero) or @{text "list \<Colon> (type) semigroup"}
   8.276 -  (e.g.\ if @{text \<odot>} is defined as list append).  Thus, the
   8.277 -  characteristic constants @{text \<odot>}, @{text \<inv>}, @{text \<one>}
   8.278 -  really become overloaded, i.e.\ have different meanings on different
   8.279 -  types.
   8.280 -*}
   8.281 -
   8.282 -
   8.283 -subsection {* Lifting and Functors *}
   8.284 -
   8.285 -text {*
   8.286 -  As already mentioned above, overloading in the simply-typed HOL
   8.287 -  systems may include recursion over the syntactic structure of types.
   8.288 -  That is, definitional equations @{text "c\<^sup>\<tau> \<equiv> t"} may also
   8.289 -  contain constants of name @{text c} on the right-hand side --- if
   8.290 -  these have types that are structurally simpler than @{text \<tau>}.
   8.291 -
   8.292 -  This feature enables us to \emph{lift operations}, say to Cartesian
   8.293 -  products, direct sums or function spaces.  Subsequently we lift
   8.294 -  @{text \<odot>} component-wise to binary products @{typ "'a \<times> 'b"}.
   8.295 -*}
   8.296 -
   8.297 -defs (overloaded)
   8.298 -  times_prod_def: "p \<odot> q \<equiv> (fst p \<odot> fst q, snd p \<odot> snd q)"
   8.299 -
   8.300 -text {*
   8.301 -  It is very easy to see that associativity of @{text \<odot>} on @{typ 'a}
   8.302 -  and @{text \<odot>} on @{typ 'b} transfers to @{text \<odot>} on @{typ "'a \<times>
   8.303 -  'b"}.  Hence the binary type constructor @{text \<odot>} maps semigroups
   8.304 -  to semigroups.  This may be established formally as follows.
   8.305 -*}
   8.306 -
   8.307 -instance * :: (semigroup, semigroup) semigroup
   8.308 -proof (intro_classes, unfold times_prod_def)
   8.309 -  fix p q r :: "'a\<Colon>semigroup \<times> 'b\<Colon>semigroup"
   8.310 -  show
   8.311 -    "(fst (fst p \<odot> fst q, snd p \<odot> snd q) \<odot> fst r,
   8.312 -      snd (fst p \<odot> fst q, snd p \<odot> snd q) \<odot> snd r) =
   8.313 -       (fst p \<odot> fst (fst q \<odot> fst r, snd q \<odot> snd r),
   8.314 -        snd p \<odot> snd (fst q \<odot> fst r, snd q \<odot> snd r))"
   8.315 -    by (simp add: semigroup_class.assoc)
   8.316 -qed
   8.317 -
   8.318 -text {*
   8.319 -  Thus, if we view class instances as ``structures'', then overloaded
   8.320 -  constant definitions with recursion over types indirectly provide
   8.321 -  some kind of ``functors'' --- i.e.\ mappings between abstract
   8.322 -  theories.
   8.323 -*}
   8.324 -
   8.325 -end

     9.1 --- a/doc-src/AxClass/Group/Product.thy	Wed Mar 04 11:05:02 2009 +0100
     9.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     9.3 @@ -1,85 +0,0 @@
     9.4 -
     9.5 -header {* Syntactic classes *}
     9.6 -
     9.7 -theory Product imports Main begin
     9.8 -
     9.9 -text {*
    9.10 -  \medskip\noindent There is still a feature of Isabelle's type system
    9.11 -  left that we have not yet discussed.  When declaring polymorphic
    9.12 -  constants @{text "c \<Colon> \<sigma>"}, the type variables occurring in @{text \<sigma>}
    9.13 -  may be constrained by type classes (or even general sorts) in an
    9.14 -  arbitrary way.  Note that by default, in Isabelle/HOL the
    9.15 -  declaration @{text "\<odot> \<Colon> 'a \<Rightarrow> 'a \<Rightarrow> 'a"} is actually an abbreviation
    9.16 -  for @{text "\<odot> \<Colon> 'a\<Colon>type \<Rightarrow> 'a \<Rightarrow> 'a"} Since class @{text type} is the
    9.17 -  universal class of HOL, this is not really a constraint at all.
    9.18 -
    9.19 - The @{text product} class below provides a less degenerate example of
    9.20 - syntactic type classes.
    9.21 -*}
    9.22 -
    9.23 -axclass
    9.24 -  product \<subseteq> type
    9.25 -consts
    9.26 -  product :: "'a\<Colon>product \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<odot>" 70)
    9.27 -
    9.28 -text {*
    9.29 -  Here class @{text product} is defined as subclass of @{text type}
    9.30 -  without any additional axioms.  This effects in logical equivalence
    9.31 -  of @{text product} and @{text type}, as is reflected by the trivial
    9.32 -  introduction rule generated for this definition.
    9.33 -
    9.34 -  \medskip So what is the difference of declaring @{text "\<odot> \<Colon>
    9.35 -  'a\<Colon>product \<Rightarrow> 'a \<Rightarrow> 'a"} vs.\ declaring @{text "\<odot> \<Colon> 'a\<Colon>type \<Rightarrow> 'a \<Rightarrow>
    9.36 -  'a"} anyway?  In this particular case where @{text "product \<equiv>
    9.37 -  type"}, it should be obvious that both declarations are the same
    9.38 -  from the logic's point of view.  It even makes the most sense to
    9.39 -  remove sort constraints from constant declarations, as far as the
    9.40 -  purely logical meaning is concerned \cite{Wenzel:1997:TPHOL}.
    9.41 -
    9.42 -  On the other hand there are syntactic differences, of course.
    9.43 - Constants @{text \<odot>} on some type @{text \<tau>} are rejected by the
    9.44 - type-checker, unless the arity @{text "\<tau> \<Colon> product"} is part of the
    9.45 - type signature.  In our example, this arity may be always added when
    9.46 - required by means of an \isakeyword{instance} with the default proof
    9.47 - (double-dot).
    9.48 -
    9.49 -  \medskip Thus, we may observe the following discipline of using
    9.50 -  syntactic classes.  Overloaded polymorphic constants have their type
    9.51 -  arguments restricted to an associated (logically trivial) class
    9.52 -  @{text c}.  Only immediately before \emph{specifying} these
    9.53 -  constants on a certain type @{text \<tau>} do we instantiate @{text "\<tau> \<Colon>
    9.54 -  c"}.
    9.55 -
    9.56 -  This is done for class @{text product} and type @{typ bool} as
    9.57 -  follows.
    9.58 -*}
    9.59 -
    9.60 -instance bool :: product ..
    9.61 -defs (overloaded)
    9.62 -  product_bool_def: "x \<odot> y \<equiv> x \<and> y"
    9.63 -
    9.64 -text {*
    9.65 - The definition @{text prod_bool_def} becomes syntactically
    9.66 - well-formed only after the arity @{text "bool \<Colon> product"} is made
    9.67 - known to the type checker.
    9.68 -
    9.69 - \medskip It is very important to see that above \isakeyword{defs} are
    9.70 - not directly connected with \isakeyword{instance} at all!  We were
    9.71 - just following our convention to specify @{text \<odot>} on @{typ bool}
    9.72 - after having instantiated @{text "bool \<Colon> product"}.  Isabelle does
    9.73 - not require these definitions, which is in contrast to programming
    9.74 - languages like Haskell \cite{haskell-report}.
    9.75 -
    9.76 - \medskip While Isabelle type classes and those of Haskell are almost
    9.77 - the same as far as type-checking and type inference are concerned,
    9.78 - there are important semantic differences.  Haskell classes require
    9.79 - their instances to \emph{provide operations} of certain \emph{names}.
    9.80 - Therefore, its \texttt{instance} has a \texttt{where} part that tells
    9.81 - the system what these ``member functions'' should be.
    9.82 -
    9.83 - This style of \texttt{instance} would not make much sense in
    9.84 - Isabelle's meta-logic, because there is no internal notion of
    9.85 - ``providing operations'' or even ``names of functions''.
    9.86 -*}
    9.87 -
    9.88 -end

    10.1 --- a/doc-src/AxClass/Group/ROOT.ML	Wed Mar 04 11:05:02 2009 +0100
    10.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    10.3 @@ -1,4 +0,0 @@
    10.4 -
    10.5 -use_thy "Semigroups";
    10.6 -use_thy "Group";
    10.7 -use_thy "Product";

    11.1 --- a/doc-src/AxClass/Group/Semigroups.thy	Wed Mar 04 11:05:02 2009 +0100
    11.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    11.3 @@ -1,54 +0,0 @@
    11.4 -
    11.5 -header {* Semigroups *}
    11.6 -
    11.7 -theory Semigroups imports Main begin
    11.8 -
    11.9 -text {*
   11.10 -  \medskip\noindent An axiomatic type class is simply a class of types
   11.11 -  that all meet certain properties, which are also called \emph{class
   11.12 -  axioms}. Thus, type classes may be also understood as type
   11.13 -  predicates --- i.e.\ abstractions over a single type argument @{typ
   11.14 -  'a}.  Class axioms typically contain polymorphic constants that
   11.15 -  depend on this type @{typ 'a}.  These \emph{characteristic
   11.16 -  constants} behave like operations associated with the ``carrier''
   11.17 -  type @{typ 'a}.
   11.18 -
   11.19 -  We illustrate these basic concepts by the following formulation of
   11.20 -  semigroups.
   11.21 -*}
   11.22 -
   11.23 -consts
   11.24 -  times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<odot>" 70)
   11.25 -axclass semigroup \<subseteq> type
   11.26 -  assoc: "(x \<odot> y) \<odot> z = x \<odot> (y \<odot> z)"
   11.27 -
   11.28 -text {*
   11.29 -  \noindent Above we have first declared a polymorphic constant @{text
   11.30 -  "\<odot> \<Colon> 'a \<Rightarrow> 'a \<Rightarrow> 'a"} and then defined the class @{text semigroup} of
   11.31 -  all types @{text \<tau>} such that @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow> \<tau>"} is indeed an
   11.32 -  associative operator.  The @{text assoc} axiom contains exactly one
   11.33 -  type variable, which is invisible in the above presentation, though.
   11.34 -  Also note that free term variables (like @{term x}, @{term y},
   11.35 -  @{term z}) are allowed for user convenience --- conceptually all of
   11.36 -  these are bound by outermost universal quantifiers.
   11.37 -
   11.38 -  \medskip In general, type classes may be used to describe
   11.39 -  \emph{structures} with exactly one carrier @{typ 'a} and a fixed
   11.40 -  \emph{signature}.  Different signatures require different classes.
   11.41 -  Below, class @{text plus_semigroup} represents semigroups @{text
   11.42 -  "(\<tau>, \<oplus>\<^sup>\<tau>)"}, while the original @{text semigroup} would
   11.43 -  correspond to semigroups of the form @{text "(\<tau>, \<odot>\<^sup>\<tau>)"}.
   11.44 -*}
   11.45 -
   11.46 -consts
   11.47 -  plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<oplus>" 70)
   11.48 -axclass plus_semigroup \<subseteq> type
   11.49 -  assoc: "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
   11.50 -
   11.51 -text {*
   11.52 -  \noindent Even if classes @{text plus_semigroup} and @{text
   11.53 -  semigroup} both represent semigroups in a sense, they are certainly
   11.54 -  not quite the same.
   11.55 -*}
   11.56 -
   11.57 -end

    12.1 --- a/doc-src/AxClass/Group/document/Group.tex	Wed Mar 04 11:05:02 2009 +0100
    12.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    12.3 @@ -1,512 +0,0 @@
    12.4 -%
    12.5 -\begin{isabellebody}%
    12.6 -\def\isabellecontext{Group}%
    12.7 -%
    12.8 -\isamarkupheader{Basic group theory%
    12.9 -}
   12.10 -\isamarkuptrue%
   12.11 -%
   12.12 -\isadelimtheory
   12.13 -%
   12.14 -\endisadelimtheory
   12.15 -%
   12.16 -\isatagtheory
   12.17 -\isacommand{theory}\isamarkupfalse%
   12.18 -\ Group\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
   12.19 -\endisatagtheory
   12.20 -{\isafoldtheory}%
   12.21 -%
   12.22 -\isadelimtheory
   12.23 -%
   12.24 -\endisadelimtheory
   12.25 -%
   12.26 -\begin{isamarkuptext}%
   12.27 -\medskip\noindent The meta-level type system of Isabelle supports
   12.28 -  \emph{intersections} and \emph{inclusions} of type classes. These
   12.29 -  directly correspond to intersections and inclusions of type
   12.30 -  predicates in a purely set theoretic sense. This is sufficient as a
   12.31 -  means to describe simple hierarchies of structures.  As an
   12.32 -  illustration, we use the well-known example of semigroups, monoids,
   12.33 -  general groups and Abelian groups.%
   12.34 -\end{isamarkuptext}%
   12.35 -\isamarkuptrue%
   12.36 -%
   12.37 -\isamarkupsubsection{Monoids and Groups%
   12.38 -}
   12.39 -\isamarkuptrue%
   12.40 -%
   12.41 -\begin{isamarkuptext}%
   12.42 -First we declare some polymorphic constants required later for the
   12.43 -  signature parts of our structures.%
   12.44 -\end{isamarkuptext}%
   12.45 -\isamarkuptrue%
   12.46 -\isacommand{consts}\isamarkupfalse%
   12.47 -\isanewline
   12.48 -\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymodot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
   12.49 -\ \ invers\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}{\isachardoublequoteopen}{\isacharparenleft}{\isacharunderscore}{\isasyminv}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
   12.50 -\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}{\isacharparenright}%
   12.51 -\begin{isamarkuptext}%
   12.52 -\noindent Next we define class \isa{monoid} of monoids with
   12.53 -  operations \isa{{\isasymodot}} and \isa{{\isasymone}}.  Note that multiple class
   12.54 -  axioms are allowed for user convenience --- they simply represent
   12.55 -  the conjunction of their respective universal closures.%
   12.56 -\end{isamarkuptext}%
   12.57 -\isamarkuptrue%
   12.58 -\isacommand{axclass}\isamarkupfalse%
   12.59 -\ monoid\ {\isasymsubseteq}\ type\isanewline
   12.60 -\ \ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
   12.61 -\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
   12.62 -\ \ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}%
   12.63 -\begin{isamarkuptext}%
   12.64 -\noindent So class \isa{monoid} contains exactly those types
   12.65 -  \isa{{\isasymtau}} where \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} and \isa{{\isasymone}\ {\isasymColon}\ {\isasymtau}}
   12.66 -  are specified appropriately, such that \isa{{\isasymodot}} is associative and
   12.67 -  \isa{{\isasymone}} is a left and right unit element for the \isa{{\isasymodot}}
   12.68 -  operation.%
   12.69 -\end{isamarkuptext}%
   12.70 -\isamarkuptrue%
   12.71 -%
   12.72 -\begin{isamarkuptext}%
   12.73 -\medskip Independently of \isa{monoid}, we now define a linear
   12.74 -  hierarchy of semigroups, general groups and Abelian groups.  Note
   12.75 -  that the names of class axioms are automatically qualified with each
   12.76 -  class name, so we may re-use common names such as \isa{assoc}.%
   12.77 -\end{isamarkuptext}%
   12.78 -\isamarkuptrue%
   12.79 -\isacommand{axclass}\isamarkupfalse%
   12.80 -\ semigroup\ {\isasymsubseteq}\ type\isanewline
   12.81 -\ \ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
   12.82 -\isanewline
   12.83 -\isacommand{axclass}\isamarkupfalse%
   12.84 -\ group\ {\isasymsubseteq}\ semigroup\isanewline
   12.85 -\ \ left{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
   12.86 -\ \ left{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequoteopen}x{\isasyminv}\ {\isasymodot}\ x\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
   12.87 -\isanewline
   12.88 -\isacommand{axclass}\isamarkupfalse%
   12.89 -\ agroup\ {\isasymsubseteq}\ group\isanewline
   12.90 -\ \ commute{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymodot}\ y\ {\isacharequal}\ y\ {\isasymodot}\ x{\isachardoublequoteclose}%
   12.91 -\begin{isamarkuptext}%
   12.92 -\noindent Class \isa{group} inherits associativity of \isa{{\isasymodot}}
   12.93 -  from \isa{semigroup} and adds two further group axioms. Similarly,
   12.94 -  \isa{agroup} is defined as the subset of \isa{group} such that
   12.95 -  for all of its elements \isa{{\isasymtau}}, the operation \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is even commutative.%
   12.96 -\end{isamarkuptext}%
   12.97 -\isamarkuptrue%
   12.98 -%
   12.99 -\isamarkupsubsection{Abstract reasoning%
  12.100 -}
  12.101 -\isamarkuptrue%
  12.102 -%
  12.103 -\begin{isamarkuptext}%
  12.104 -In a sense, axiomatic type classes may be viewed as \emph{abstract
  12.105 -  theories}.  Above class definitions gives rise to abstract axioms
  12.106 -  \isa{assoc}, \isa{left{\isacharunderscore}unit}, \isa{left{\isacharunderscore}inverse}, \isa{commute}, where any of these contain a type variable \isa{{\isacharprime}a\ {\isasymColon}\ c} that is restricted to types of the corresponding class \isa{c}.  \emph{Sort constraints} like this express a logical
  12.107 -  precondition for the whole formula.  For example, \isa{assoc}
  12.108 -  states that for all \isa{{\isasymtau}}, provided that \isa{{\isasymtau}\ {\isasymColon}\ semigroup}, the operation \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is associative.
  12.109 -
  12.110 -  \medskip From a technical point of view, abstract axioms are just
  12.111 -  ordinary Isabelle theorems, which may be used in proofs without
  12.112 -  special treatment.  Such ``abstract proofs'' usually yield new
  12.113 -  ``abstract theorems''.  For example, we may now derive the following
  12.114 -  well-known laws of general groups.%
  12.115 -\end{isamarkuptext}%
  12.116 -\isamarkuptrue%
  12.117 -\isacommand{theorem}\isamarkupfalse%
  12.118 -\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isacharparenleft}{\isasymone}{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.119 -%
  12.120 -\isadelimproof
  12.121 -%
  12.122 -\endisadelimproof
  12.123 -%
  12.124 -\isatagproof
  12.125 -\isacommand{proof}\isamarkupfalse%
  12.126 -\ {\isacharminus}\isanewline
  12.127 -\ \ \isacommand{have}\isamarkupfalse%
  12.128 -\ {\isachardoublequoteopen}x\ {\isasymodot}\ x{\isasyminv}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ {\isacharparenleft}x\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.129 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.130 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
  12.131 -\ \ \isacommand{also}\isamarkupfalse%
  12.132 -\ \isacommand{have}\isamarkupfalse%
  12.133 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequoteclose}\isanewline
  12.134 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.135 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isacharunderscore}class{\isachardot}assoc{\isacharparenright}\isanewline
  12.136 -\ \ \isacommand{also}\isamarkupfalse%
  12.137 -\ \isacommand{have}\isamarkupfalse%
  12.138 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x\ {\isasymodot}\ x{\isasyminv}{\isachardoublequoteclose}\isanewline
  12.139 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.140 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
  12.141 -\ \ \isacommand{also}\isamarkupfalse%
  12.142 -\ \isacommand{have}\isamarkupfalse%
  12.143 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequoteclose}\isanewline
  12.144 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.145 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isacharunderscore}class{\isachardot}assoc{\isacharparenright}\isanewline
  12.146 -\ \ \isacommand{also}\isamarkupfalse%
  12.147 -\ \isacommand{have}\isamarkupfalse%
  12.148 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequoteclose}\isanewline
  12.149 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.150 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
  12.151 -\ \ \isacommand{also}\isamarkupfalse%
  12.152 -\ \isacommand{have}\isamarkupfalse%
  12.153 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ {\isacharparenleft}{\isasymone}\ {\isasymodot}\ x{\isasyminv}{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.154 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.155 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isacharunderscore}class{\isachardot}assoc{\isacharparenright}\isanewline
  12.156 -\ \ \isacommand{also}\isamarkupfalse%
  12.157 -\ \isacommand{have}\isamarkupfalse%
  12.158 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminv}{\isacharparenright}{\isasyminv}\ {\isasymodot}\ x{\isasyminv}{\isachardoublequoteclose}\isanewline
  12.159 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.160 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
  12.161 -\ \ \isacommand{also}\isamarkupfalse%
  12.162 -\ \isacommand{have}\isamarkupfalse%
  12.163 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
  12.164 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.165 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
  12.166 -\ \ \isacommand{finally}\isamarkupfalse%
  12.167 -\ \isacommand{show}\isamarkupfalse%
  12.168 -\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
  12.169 -\isanewline
  12.170 -\isacommand{qed}\isamarkupfalse%
  12.171 -%
  12.172 -\endisatagproof
  12.173 -{\isafoldproof}%
  12.174 -%
  12.175 -\isadelimproof
  12.176 -%
  12.177 -\endisadelimproof
  12.178 -%
  12.179 -\begin{isamarkuptext}%
  12.180 -\noindent With \isa{group{\isacharunderscore}right{\isacharunderscore}inverse} already available, \isa{group{\isacharunderscore}right{\isacharunderscore}unit}\label{thm:group-right-unit} is now established
  12.181 -  much easier.%
  12.182 -\end{isamarkuptext}%
  12.183 -\isamarkuptrue%
  12.184 -\isacommand{theorem}\isamarkupfalse%
  12.185 -\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ {\isacharparenleft}x{\isasymColon}{\isacharprime}a{\isasymColon}group{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.186 -%
  12.187 -\isadelimproof
  12.188 -%
  12.189 -\endisadelimproof
  12.190 -%
  12.191 -\isatagproof
  12.192 -\isacommand{proof}\isamarkupfalse%
  12.193 -\ {\isacharminus}\isanewline
  12.194 -\ \ \isacommand{have}\isamarkupfalse%
  12.195 -\ {\isachardoublequoteopen}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}x{\isasyminv}\ {\isasymodot}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.196 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.197 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}inverse{\isacharparenright}\isanewline
  12.198 -\ \ \isacommand{also}\isamarkupfalse%
  12.199 -\ \isacommand{have}\isamarkupfalse%
  12.200 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymodot}\ x{\isasyminv}\ {\isasymodot}\ x{\isachardoublequoteclose}\isanewline
  12.201 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.202 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ semigroup{\isacharunderscore}class{\isachardot}assoc{\isacharparenright}\isanewline
  12.203 -\ \ \isacommand{also}\isamarkupfalse%
  12.204 -\ \isacommand{have}\isamarkupfalse%
  12.205 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isasymone}\ {\isasymodot}\ x{\isachardoublequoteclose}\isanewline
  12.206 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.207 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}right{\isacharunderscore}inverse{\isacharparenright}\isanewline
  12.208 -\ \ \isacommand{also}\isamarkupfalse%
  12.209 -\ \isacommand{have}\isamarkupfalse%
  12.210 -\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
  12.211 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.212 -\ {\isacharparenleft}simp\ only{\isacharcolon}\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
  12.213 -\ \ \isacommand{finally}\isamarkupfalse%
  12.214 -\ \isacommand{show}\isamarkupfalse%
  12.215 -\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse%
  12.216 -\isanewline
  12.217 -\isacommand{qed}\isamarkupfalse%
  12.218 -%
  12.219 -\endisatagproof
  12.220 -{\isafoldproof}%
  12.221 -%
  12.222 -\isadelimproof
  12.223 -%
  12.224 -\endisadelimproof
  12.225 -%
  12.226 -\begin{isamarkuptext}%
  12.227 -\medskip Abstract theorems may be instantiated to only those types
  12.228 -  \isa{{\isasymtau}} where the appropriate class membership \isa{{\isasymtau}\ {\isasymColon}\ c} is
  12.229 -  known at Isabelle's type signature level.  Since we have \isa{agroup\ {\isasymsubseteq}\ group\ {\isasymsubseteq}\ semigroup} by definition, all theorems of \isa{semigroup} and \isa{group} are automatically inherited by \isa{group} and \isa{agroup}.%
  12.230 -\end{isamarkuptext}%
  12.231 -\isamarkuptrue%
  12.232 -%
  12.233 -\isamarkupsubsection{Abstract instantiation%
  12.234 -}
  12.235 -\isamarkuptrue%
  12.236 -%
  12.237 -\begin{isamarkuptext}%
  12.238 -From the definition, the \isa{monoid} and \isa{group} classes
  12.239 -  have been independent.  Note that for monoids, \isa{right{\isacharunderscore}unit}
  12.240 -  had to be included as an axiom, but for groups both \isa{right{\isacharunderscore}unit} and \isa{right{\isacharunderscore}inverse} are derivable from the other
  12.241 -  axioms.  With \isa{group{\isacharunderscore}right{\isacharunderscore}unit} derived as a theorem of group
  12.242 -  theory (see page~\pageref{thm:group-right-unit}), we may now
  12.243 -  instantiate \isa{monoid\ {\isasymsubseteq}\ semigroup} and \isa{group\ {\isasymsubseteq}\ monoid} properly as follows (cf.\ \figref{fig:monoid-group}).
  12.244 -
  12.245 - \begin{figure}[htbp]
  12.246 -   \begin{center}
  12.247 -     \small
  12.248 -     \unitlength 0.6mm
  12.249 -     \begin{picture}(65,90)(0,-10)
  12.250 -       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
  12.251 -       \put(15,50){\line(1,1){10}} \put(35,60){\line(1,-1){10}}
  12.252 -       \put(15,5){\makebox(0,0){\isa{agroup}}}
  12.253 -       \put(15,25){\makebox(0,0){\isa{group}}}
  12.254 -       \put(15,45){\makebox(0,0){\isa{semigroup}}}
  12.255 -       \put(30,65){\makebox(0,0){\isa{type}}} \put(50,45){\makebox(0,0){\isa{monoid}}}
  12.256 -     \end{picture}
  12.257 -     \hspace{4em}
  12.258 -     \begin{picture}(30,90)(0,0)
  12.259 -       \put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
  12.260 -       \put(15,50){\line(0,1){10}} \put(15,70){\line(0,1){10}}
  12.261 -       \put(15,5){\makebox(0,0){\isa{agroup}}}
  12.262 -       \put(15,25){\makebox(0,0){\isa{group}}}
  12.263 -       \put(15,45){\makebox(0,0){\isa{monoid}}}
  12.264 -       \put(15,65){\makebox(0,0){\isa{semigroup}}}
  12.265 -       \put(15,85){\makebox(0,0){\isa{type}}}
  12.266 -     \end{picture}
  12.267 -     \caption{Monoids and groups: according to definition, and by proof}
  12.268 -     \label{fig:monoid-group}
  12.269 -   \end{center}
  12.270 - \end{figure}%
  12.271 -\end{isamarkuptext}%
  12.272 -\isamarkuptrue%
  12.273 -\isacommand{instance}\isamarkupfalse%
  12.274 -\ monoid\ {\isasymsubseteq}\ semigroup\isanewline
  12.275 -%
  12.276 -\isadelimproof
  12.277 -%
  12.278 -\endisadelimproof
  12.279 -%
  12.280 -\isatagproof
  12.281 -\isacommand{proof}\isamarkupfalse%
  12.282 -\isanewline
  12.283 -\ \ \isacommand{fix}\isamarkupfalse%
  12.284 -\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}monoid{\isachardoublequoteclose}\isanewline
  12.285 -\ \ \isacommand{show}\isamarkupfalse%
  12.286 -\ {\isachardoublequoteopen}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.287 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.288 -\ {\isacharparenleft}rule\ monoid{\isacharunderscore}class{\isachardot}assoc{\isacharparenright}\isanewline
  12.289 -\isacommand{qed}\isamarkupfalse%
  12.290 -%
  12.291 -\endisatagproof
  12.292 -{\isafoldproof}%
  12.293 -%
  12.294 -\isadelimproof
  12.295 -\isanewline
  12.296 -%
  12.297 -\endisadelimproof
  12.298 -\isanewline
  12.299 -\isacommand{instance}\isamarkupfalse%
  12.300 -\ group\ {\isasymsubseteq}\ monoid\isanewline
  12.301 -%
  12.302 -\isadelimproof
  12.303 -%
  12.304 -\endisadelimproof
  12.305 -%
  12.306 -\isatagproof
  12.307 -\isacommand{proof}\isamarkupfalse%
  12.308 -\isanewline
  12.309 -\ \ \isacommand{fix}\isamarkupfalse%
  12.310 -\ x\ y\ z\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}group{\isachardoublequoteclose}\isanewline
  12.311 -\ \ \isacommand{show}\isamarkupfalse%
  12.312 -\ {\isachardoublequoteopen}x\ {\isasymodot}\ y\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.313 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.314 -\ {\isacharparenleft}rule\ semigroup{\isacharunderscore}class{\isachardot}assoc{\isacharparenright}\isanewline
  12.315 -\ \ \isacommand{show}\isamarkupfalse%
  12.316 -\ {\isachardoublequoteopen}{\isasymone}\ {\isasymodot}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
  12.317 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.318 -\ {\isacharparenleft}rule\ group{\isacharunderscore}class{\isachardot}left{\isacharunderscore}unit{\isacharparenright}\isanewline
  12.319 -\ \ \isacommand{show}\isamarkupfalse%
  12.320 -\ {\isachardoublequoteopen}x\ {\isasymodot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
  12.321 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.322 -\ {\isacharparenleft}rule\ group{\isacharunderscore}right{\isacharunderscore}unit{\isacharparenright}\isanewline
  12.323 -\isacommand{qed}\isamarkupfalse%
  12.324 -%
  12.325 -\endisatagproof
  12.326 -{\isafoldproof}%
  12.327 -%
  12.328 -\isadelimproof
  12.329 -%
  12.330 -\endisadelimproof
  12.331 -%
  12.332 -\begin{isamarkuptext}%
  12.333 -\medskip The \isakeyword{instance} command sets up an appropriate
  12.334 -  goal that represents the class inclusion (or type arity, see
  12.335 -  \secref{sec:inst-arity}) to be proven (see also
  12.336 -  \cite{isabelle-isar-ref}).  The initial proof step causes
  12.337 -  back-chaining of class membership statements wrt.\ the hierarchy of
  12.338 -  any classes defined in the current theory; the effect is to reduce
  12.339 -  to the initial statement to a number of goals that directly
  12.340 -  correspond to any class axioms encountered on the path upwards
  12.341 -  through the class hierarchy.%
  12.342 -\end{isamarkuptext}%
  12.343 -\isamarkuptrue%
  12.344 -%
  12.345 -\isamarkupsubsection{Concrete instantiation \label{sec:inst-arity}%
  12.346 -}
  12.347 -\isamarkuptrue%
  12.348 -%
  12.349 -\begin{isamarkuptext}%
  12.350 -So far we have covered the case of the form
  12.351 -  \isakeyword{instance}~\isa{c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}}, namely
  12.352 -  \emph{abstract instantiation} --- $c@1$ is more special than \isa{c\isactrlsub {\isadigit{1}}} and thus an instance of \isa{c\isactrlsub {\isadigit{2}}}.  Even more
  12.353 -  interesting for practical applications are \emph{concrete
  12.354 -  instantiations} of axiomatic type classes.  That is, certain simple
  12.355 -  schemes \isa{{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isasymColon}\ c} of class
  12.356 -  membership may be established at the logical level and then
  12.357 -  transferred to Isabelle's type signature level.
  12.358 -
  12.359 -  \medskip As a typical example, we show that type \isa{bool} with
  12.360 -  exclusive-or as \isa{{\isasymodot}} operation, identity as \isa{{\isasyminv}}, and
  12.361 -  \isa{False} as \isa{{\isasymone}} forms an Abelian group.%
  12.362 -\end{isamarkuptext}%
  12.363 -\isamarkuptrue%
  12.364 -\isacommand{defs}\isamarkupfalse%
  12.365 -\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
  12.366 -\ \ times{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymnoteq}\ {\isacharparenleft}y{\isasymColon}bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.367 -\ \ inverse{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}x{\isasyminv}\ {\isasymequiv}\ x{\isasymColon}bool{\isachardoublequoteclose}\isanewline
  12.368 -\ \ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymequiv}\ False{\isachardoublequoteclose}%
  12.369 -\begin{isamarkuptext}%
  12.370 -\medskip It is important to note that above \isakeyword{defs} are
  12.371 -  just overloaded meta-level constant definitions, where type classes
  12.372 -  are not yet involved at all.  This form of constant definition with
  12.373 -  overloading (and optional recursion over the syntactic structure of
  12.374 -  simple types) are admissible as definitional extensions of plain HOL
  12.375 -  \cite{Wenzel:1997:TPHOL}.  The Haskell-style type system is not
  12.376 -  required for overloading.  Nevertheless, overloaded definitions are
  12.377 -  best applied in the context of type classes.
  12.378 -
  12.379 -  \medskip Since we have chosen above \isakeyword{defs} of the generic
  12.380 -  group operations on type \isa{bool} appropriately, the class
  12.381 -  membership \isa{bool\ {\isasymColon}\ agroup} may be now derived as follows.%
  12.382 -\end{isamarkuptext}%
  12.383 -\isamarkuptrue%
  12.384 -\isacommand{instance}\isamarkupfalse%
  12.385 -\ bool\ {\isacharcolon}{\isacharcolon}\ agroup\isanewline
  12.386 -%
  12.387 -\isadelimproof
  12.388 -%
  12.389 -\endisadelimproof
  12.390 -%
  12.391 -\isatagproof
  12.392 -\isacommand{proof}\isamarkupfalse%
  12.393 -\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\isanewline
  12.394 -\ \ \ \ unfold\ times{\isacharunderscore}bool{\isacharunderscore}def\ inverse{\isacharunderscore}bool{\isacharunderscore}def\ unit{\isacharunderscore}bool{\isacharunderscore}def{\isacharparenright}\isanewline
  12.395 -\ \ \isacommand{fix}\isamarkupfalse%
  12.396 -\ x\ y\ z\isanewline
  12.397 -\ \ \isacommand{show}\isamarkupfalse%
  12.398 -\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isasymnoteq}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymnoteq}\ {\isacharparenleft}y\ {\isasymnoteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
  12.399 -\ blast\isanewline
  12.400 -\ \ \isacommand{show}\isamarkupfalse%
  12.401 -\ {\isachardoublequoteopen}{\isacharparenleft}False\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
  12.402 -\ blast\isanewline
  12.403 -\ \ \isacommand{show}\isamarkupfalse%
  12.404 -\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymnoteq}\ x{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
  12.405 -\ blast\isanewline
  12.406 -\ \ \isacommand{show}\isamarkupfalse%
  12.407 -\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymnoteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymnoteq}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
  12.408 -\ blast\isanewline
  12.409 -\isacommand{qed}\isamarkupfalse%
  12.410 -%
  12.411 -\endisatagproof
  12.412 -{\isafoldproof}%
  12.413 -%
  12.414 -\isadelimproof
  12.415 -%
  12.416 -\endisadelimproof
  12.417 -%
  12.418 -\begin{isamarkuptext}%
  12.419 -The result of an \isakeyword{instance} statement is both expressed
  12.420 -  as a theorem of Isabelle's meta-logic, and as a type arity of the
  12.421 -  type signature.  The latter enables type-inference system to take
  12.422 -  care of this new instance automatically.
  12.423 -
  12.424 -  \medskip We could now also instantiate our group theory classes to
  12.425 -  many other concrete types.  For example, \isa{int\ {\isasymColon}\ agroup}
  12.426 -  (e.g.\ by defining \isa{{\isasymodot}} as addition, \isa{{\isasyminv}} as negation
  12.427 -  and \isa{{\isasymone}} as zero) or \isa{list\ {\isasymColon}\ {\isacharparenleft}type{\isacharparenright}\ semigroup}
  12.428 -  (e.g.\ if \isa{{\isasymodot}} is defined as list append).  Thus, the
  12.429 -  characteristic constants \isa{{\isasymodot}}, \isa{{\isasyminv}}, \isa{{\isasymone}}
  12.430 -  really become overloaded, i.e.\ have different meanings on different
  12.431 -  types.%
  12.432 -\end{isamarkuptext}%
  12.433 -\isamarkuptrue%
  12.434 -%
  12.435 -\isamarkupsubsection{Lifting and Functors%
  12.436 -}
  12.437 -\isamarkuptrue%
  12.438 -%
  12.439 -\begin{isamarkuptext}%
  12.440 -As already mentioned above, overloading in the simply-typed HOL
  12.441 -  systems may include recursion over the syntactic structure of types.
  12.442 -  That is, definitional equations \isa{c\isactrlsup {\isasymtau}\ {\isasymequiv}\ t} may also
  12.443 -  contain constants of name \isa{c} on the right-hand side --- if
  12.444 -  these have types that are structurally simpler than \isa{{\isasymtau}}.
  12.445 -
  12.446 -  This feature enables us to \emph{lift operations}, say to Cartesian
  12.447 -  products, direct sums or function spaces.  Subsequently we lift
  12.448 -  \isa{{\isasymodot}} component-wise to binary products \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.%
  12.449 -\end{isamarkuptext}%
  12.450 -\isamarkuptrue%
  12.451 -\isacommand{defs}\isamarkupfalse%
  12.452 -\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
  12.453 -\ \ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}p\ {\isasymodot}\ q\ {\isasymequiv}\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}{\isachardoublequoteclose}%
  12.454 -\begin{isamarkuptext}%
  12.455 -It is very easy to see that associativity of \isa{{\isasymodot}} on \isa{{\isacharprime}a}
  12.456 -  and \isa{{\isasymodot}} on \isa{{\isacharprime}b} transfers to \isa{{\isasymodot}} on \isa{{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b}.  Hence the binary type constructor \isa{{\isasymodot}} maps semigroups
  12.457 -  to semigroups.  This may be established formally as follows.%
  12.458 -\end{isamarkuptext}%
  12.459 -\isamarkuptrue%
  12.460 -\isacommand{instance}\isamarkupfalse%
  12.461 -\ {\isacharasterisk}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}semigroup{\isacharcomma}\ semigroup{\isacharparenright}\ semigroup\isanewline
  12.462 -%
  12.463 -\isadelimproof
  12.464 -%
  12.465 -\endisadelimproof
  12.466 -%
  12.467 -\isatagproof
  12.468 -\isacommand{proof}\isamarkupfalse%
  12.469 -\ {\isacharparenleft}intro{\isacharunderscore}classes{\isacharcomma}\ unfold\ times{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}\isanewline
  12.470 -\ \ \isacommand{fix}\isamarkupfalse%
  12.471 -\ p\ q\ r\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}semigroup\ {\isasymtimes}\ {\isacharprime}b{\isasymColon}semigroup{\isachardoublequoteclose}\isanewline
  12.472 -\ \ \isacommand{show}\isamarkupfalse%
  12.473 -\isanewline
  12.474 -\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}fst\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ fst\ r{\isacharcomma}\isanewline
  12.475 -\ \ \ \ \ \ snd\ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ q{\isacharcomma}\ snd\ p\ {\isasymodot}\ snd\ q{\isacharparenright}\ {\isasymodot}\ snd\ r{\isacharparenright}\ {\isacharequal}\isanewline
  12.476 -\ \ \ \ \ \ \ {\isacharparenleft}fst\ p\ {\isasymodot}\ fst\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharcomma}\isanewline
  12.477 -\ \ \ \ \ \ \ \ snd\ p\ {\isasymodot}\ snd\ {\isacharparenleft}fst\ q\ {\isasymodot}\ fst\ r{\isacharcomma}\ snd\ q\ {\isasymodot}\ snd\ r{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
  12.478 -\ \ \ \ \isacommand{by}\isamarkupfalse%
  12.479 -\ {\isacharparenleft}simp\ add{\isacharcolon}\ semigroup{\isacharunderscore}class{\isachardot}assoc{\isacharparenright}\isanewline
  12.480 -\isacommand{qed}\isamarkupfalse%
  12.481 -%
  12.482 -\endisatagproof
  12.483 -{\isafoldproof}%
  12.484 -%
  12.485 -\isadelimproof
  12.486 -%
  12.487 -\endisadelimproof
  12.488 -%
  12.489 -\begin{isamarkuptext}%
  12.490 -Thus, if we view class instances as ``structures'', then overloaded
  12.491 -  constant definitions with recursion over types indirectly provide
  12.492 -  some kind of ``functors'' --- i.e.\ mappings between abstract
  12.493 -  theories.%
  12.494 -\end{isamarkuptext}%
  12.495 -\isamarkuptrue%
  12.496 -%
  12.497 -\isadelimtheory
  12.498 -%
  12.499 -\endisadelimtheory
  12.500 -%
  12.501 -\isatagtheory
  12.502 -\isacommand{end}\isamarkupfalse%
  12.503 -%
  12.504 -\endisatagtheory
  12.505 -{\isafoldtheory}%
  12.506 -%
  12.507 -\isadelimtheory
  12.508 -%
  12.509 -\endisadelimtheory
  12.510 -\isanewline
  12.511 -\end{isabellebody}%
  12.512 -%%% Local Variables:
  12.513 -%%% mode: latex
  12.514 -%%% TeX-master: "root"
  12.515 -%%% End:

    13.1 --- a/doc-src/AxClass/Group/document/Product.tex	Wed Mar 04 11:05:02 2009 +0100
    13.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    13.3 @@ -1,133 +0,0 @@
    13.4 -%
    13.5 -\begin{isabellebody}%
    13.6 -\def\isabellecontext{Product}%
    13.7 -%
    13.8 -\isamarkupheader{Syntactic classes%
    13.9 -}
   13.10 -\isamarkuptrue%
   13.11 -%
   13.12 -\isadelimtheory
   13.13 -%
   13.14 -\endisadelimtheory
   13.15 -%
   13.16 -\isatagtheory
   13.17 -\isacommand{theory}\isamarkupfalse%
   13.18 -\ Product\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
   13.19 -\endisatagtheory
   13.20 -{\isafoldtheory}%
   13.21 -%
   13.22 -\isadelimtheory
   13.23 -%
   13.24 -\endisadelimtheory
   13.25 -%
   13.26 -\begin{isamarkuptext}%
   13.27 -\medskip\noindent There is still a feature of Isabelle's type system
   13.28 -  left that we have not yet discussed.  When declaring polymorphic
   13.29 -  constants \isa{c\ {\isasymColon}\ {\isasymsigma}}, the type variables occurring in \isa{{\isasymsigma}}
   13.30 -  may be constrained by type classes (or even general sorts) in an
   13.31 -  arbitrary way.  Note that by default, in Isabelle/HOL the
   13.32 -  declaration \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} is actually an abbreviation
   13.33 -  for \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} Since class \isa{type} is the
   13.34 -  universal class of HOL, this is not really a constraint at all.
   13.35 -
   13.36 - The \isa{product} class below provides a less degenerate example of
   13.37 - syntactic type classes.%
   13.38 -\end{isamarkuptext}%
   13.39 -\isamarkuptrue%
   13.40 -\isacommand{axclass}\isamarkupfalse%
   13.41 -\isanewline
   13.42 -\ \ product\ {\isasymsubseteq}\ type\isanewline
   13.43 -\isacommand{consts}\isamarkupfalse%
   13.44 -\isanewline
   13.45 -\ \ product\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymodot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}%
   13.46 -\begin{isamarkuptext}%
   13.47 -Here class \isa{product} is defined as subclass of \isa{type}
   13.48 -  without any additional axioms.  This effects in logical equivalence
   13.49 -  of \isa{product} and \isa{type}, as is reflected by the trivial
   13.50 -  introduction rule generated for this definition.
   13.51 -
   13.52 -  \medskip So what is the difference of declaring \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}product\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} vs.\ declaring \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}type\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} anyway?  In this particular case where \isa{product\ {\isasymequiv}\ type}, it should be obvious that both declarations are the same
   13.53 -  from the logic's point of view.  It even makes the most sense to
   13.54 -  remove sort constraints from constant declarations, as far as the
   13.55 -  purely logical meaning is concerned \cite{Wenzel:1997:TPHOL}.
   13.56 -
   13.57 -  On the other hand there are syntactic differences, of course.
   13.58 - Constants \isa{{\isasymodot}} on some type \isa{{\isasymtau}} are rejected by the
   13.59 - type-checker, unless the arity \isa{{\isasymtau}\ {\isasymColon}\ product} is part of the
   13.60 - type signature.  In our example, this arity may be always added when
   13.61 - required by means of an \isakeyword{instance} with the default proof
   13.62 - (double-dot).
   13.63 -
   13.64 -  \medskip Thus, we may observe the following discipline of using
   13.65 -  syntactic classes.  Overloaded polymorphic constants have their type
   13.66 -  arguments restricted to an associated (logically trivial) class
   13.67 -  \isa{c}.  Only immediately before \emph{specifying} these
   13.68 -  constants on a certain type \isa{{\isasymtau}} do we instantiate \isa{{\isasymtau}\ {\isasymColon}\ c}.
   13.69 -
   13.70 -  This is done for class \isa{product} and type \isa{bool} as
   13.71 -  follows.%
   13.72 -\end{isamarkuptext}%
   13.73 -\isamarkuptrue%
   13.74 -\isacommand{instance}\isamarkupfalse%
   13.75 -\ bool\ {\isacharcolon}{\isacharcolon}\ product%
   13.76 -\isadelimproof
   13.77 -\ %
   13.78 -\endisadelimproof
   13.79 -%
   13.80 -\isatagproof
   13.81 -\isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
   13.82 -%
   13.83 -\endisatagproof
   13.84 -{\isafoldproof}%
   13.85 -%
   13.86 -\isadelimproof
   13.87 -%
   13.88 -\endisadelimproof
   13.89 -\isanewline
   13.90 -\isacommand{defs}\isamarkupfalse%
   13.91 -\ {\isacharparenleft}\isakeyword{overloaded}{\isacharparenright}\isanewline
   13.92 -\ \ product{\isacharunderscore}bool{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymodot}\ y\ {\isasymequiv}\ x\ {\isasymand}\ y{\isachardoublequoteclose}%
   13.93 -\begin{isamarkuptext}%
   13.94 -The definition \isa{prod{\isacharunderscore}bool{\isacharunderscore}def} becomes syntactically
   13.95 - well-formed only after the arity \isa{bool\ {\isasymColon}\ product} is made
   13.96 - known to the type checker.
   13.97 -
   13.98 - \medskip It is very important to see that above \isakeyword{defs} are
   13.99 - not directly connected with \isakeyword{instance} at all!  We were
  13.100 - just following our convention to specify \isa{{\isasymodot}} on \isa{bool}
  13.101 - after having instantiated \isa{bool\ {\isasymColon}\ product}.  Isabelle does
  13.102 - not require these definitions, which is in contrast to programming
  13.103 - languages like Haskell \cite{haskell-report}.
  13.104 -
  13.105 - \medskip While Isabelle type classes and those of Haskell are almost
  13.106 - the same as far as type-checking and type inference are concerned,
  13.107 - there are important semantic differences.  Haskell classes require
  13.108 - their instances to \emph{provide operations} of certain \emph{names}.
  13.109 - Therefore, its \texttt{instance} has a \texttt{where} part that tells
  13.110 - the system what these ``member functions'' should be.
  13.111 -
  13.112 - This style of \texttt{instance} would not make much sense in
  13.113 - Isabelle's meta-logic, because there is no internal notion of
  13.114 - ``providing operations'' or even ``names of functions''.%
  13.115 -\end{isamarkuptext}%
  13.116 -\isamarkuptrue%
  13.117 -%
  13.118 -\isadelimtheory
  13.119 -%
  13.120 -\endisadelimtheory
  13.121 -%
  13.122 -\isatagtheory
  13.123 -\isacommand{end}\isamarkupfalse%
  13.124 -%
  13.125 -\endisatagtheory
  13.126 -{\isafoldtheory}%
  13.127 -%
  13.128 -\isadelimtheory
  13.129 -%
  13.130 -\endisadelimtheory
  13.131 -\isanewline
  13.132 -\end{isabellebody}%
  13.133 -%%% Local Variables:
  13.134 -%%% mode: latex
  13.135 -%%% TeX-master: "root"
  13.136 -%%% End:

    14.1 --- a/doc-src/AxClass/Group/document/Semigroups.tex	Wed Mar 04 11:05:02 2009 +0100
    14.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    14.3 @@ -1,88 +0,0 @@
    14.4 -%
    14.5 -\begin{isabellebody}%
    14.6 -\def\isabellecontext{Semigroups}%
    14.7 -%
    14.8 -\isamarkupheader{Semigroups%
    14.9 -}
   14.10 -\isamarkuptrue%
   14.11 -%
   14.12 -\isadelimtheory
   14.13 -%
   14.14 -\endisadelimtheory
   14.15 -%
   14.16 -\isatagtheory
   14.17 -\isacommand{theory}\isamarkupfalse%
   14.18 -\ Semigroups\ \isakeyword{imports}\ Main\ \isakeyword{begin}%
   14.19 -\endisatagtheory
   14.20 -{\isafoldtheory}%
   14.21 -%
   14.22 -\isadelimtheory
   14.23 -%
   14.24 -\endisadelimtheory
   14.25 -%
   14.26 -\begin{isamarkuptext}%
   14.27 -\medskip\noindent An axiomatic type class is simply a class of types
   14.28 -  that all meet certain properties, which are also called \emph{class
   14.29 -  axioms}. Thus, type classes may be also understood as type
   14.30 -  predicates --- i.e.\ abstractions over a single type argument \isa{{\isacharprime}a}.  Class axioms typically contain polymorphic constants that
   14.31 -  depend on this type \isa{{\isacharprime}a}.  These \emph{characteristic
   14.32 -  constants} behave like operations associated with the ``carrier''
   14.33 -  type \isa{{\isacharprime}a}.
   14.34 -
   14.35 -  We illustrate these basic concepts by the following formulation of
   14.36 -  semigroups.%
   14.37 -\end{isamarkuptext}%
   14.38 -\isamarkuptrue%
   14.39 -\isacommand{consts}\isamarkupfalse%
   14.40 -\isanewline
   14.41 -\ \ times\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymodot}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
   14.42 -\isacommand{axclass}\isamarkupfalse%
   14.43 -\ semigroup\ {\isasymsubseteq}\ type\isanewline
   14.44 -\ \ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequoteclose}%
   14.45 -\begin{isamarkuptext}%
   14.46 -\noindent Above we have first declared a polymorphic constant \isa{{\isasymodot}\ {\isasymColon}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a} and then defined the class \isa{semigroup} of
   14.47 -  all types \isa{{\isasymtau}} such that \isa{{\isasymodot}\ {\isasymColon}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}\ {\isasymRightarrow}\ {\isasymtau}} is indeed an
   14.48 -  associative operator.  The \isa{assoc} axiom contains exactly one
   14.49 -  type variable, which is invisible in the above presentation, though.
   14.50 -  Also note that free term variables (like \isa{x}, \isa{y},
   14.51 -  \isa{z}) are allowed for user convenience --- conceptually all of
   14.52 -  these are bound by outermost universal quantifiers.
   14.53 -
   14.54 -  \medskip In general, type classes may be used to describe
   14.55 -  \emph{structures} with exactly one carrier \isa{{\isacharprime}a} and a fixed
   14.56 -  \emph{signature}.  Different signatures require different classes.
   14.57 -  Below, class \isa{plus{\isacharunderscore}semigroup} represents semigroups \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymoplus}\isactrlsup {\isasymtau}{\isacharparenright}}, while the original \isa{semigroup} would
   14.58 -  correspond to semigroups of the form \isa{{\isacharparenleft}{\isasymtau}{\isacharcomma}\ {\isasymodot}\isactrlsup {\isasymtau}{\isacharparenright}}.%
   14.59 -\end{isamarkuptext}%
   14.60 -\isamarkuptrue%
   14.61 -\isacommand{consts}\isamarkupfalse%
   14.62 -\isanewline
   14.63 -\ \ plus\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymoplus}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
   14.64 -\isacommand{axclass}\isamarkupfalse%
   14.65 -\ plus{\isacharunderscore}semigroup\ {\isasymsubseteq}\ type\isanewline
   14.66 -\ \ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isasymoplus}\ z\ {\isacharequal}\ x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isachardoublequoteclose}%
   14.67 -\begin{isamarkuptext}%
   14.68 -\noindent Even if classes \isa{plus{\isacharunderscore}semigroup} and \isa{semigroup} both represent semigroups in a sense, they are certainly
   14.69 -  not quite the same.%
   14.70 -\end{isamarkuptext}%
   14.71 -\isamarkuptrue%
   14.72 -%
   14.73 -\isadelimtheory
   14.74 -%
   14.75 -\endisadelimtheory
   14.76 -%
   14.77 -\isatagtheory
   14.78 -\isacommand{end}\isamarkupfalse%
   14.79 -%
   14.80 -\endisatagtheory
   14.81 -{\isafoldtheory}%
   14.82 -%
   14.83 -\isadelimtheory
   14.84 -%
   14.85 -\endisadelimtheory
   14.86 -\isanewline
   14.87 -\end{isabellebody}%
   14.88 -%%% Local Variables:
   14.89 -%%% mode: latex
   14.90 -%%% TeX-master: "root"
   14.91 -%%% End:

    15.1 --- a/doc-src/AxClass/IsaMakefile	Wed Mar 04 11:05:02 2009 +0100
    15.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    15.3 @@ -1,47 +0,0 @@
    15.4 -
    15.5 -## targets
    15.6 -
    15.7 -default: Group Nat
    15.8 -images: 
    15.9 -test: Group Nat
   15.10 -
   15.11 -all: images test
   15.12 -
   15.13 -
   15.14 -## global settings
   15.15 -
   15.16 -SRC = $(ISABELLE_HOME)/src
   15.17 -OUT = $(ISABELLE_OUTPUT)
   15.18 -LOG = $(OUT)/log
   15.19 -USEDIR = $(ISABELLE_TOOL) usedir -d false -D document
   15.20 -
   15.21 -
   15.22 -## Group
   15.23 -
   15.24 -Group: HOL $(LOG)/HOL-Group.gz
   15.25 -
   15.26 -HOL:
   15.27 -	@cd $(SRC)/HOL; $(ISABELLE_TOOL) make HOL
   15.28 -
   15.29 -$(LOG)/HOL-Group.gz: $(OUT)/HOL Group/ROOT.ML Group/Group.thy \
   15.30 -  Group/Product.thy Group/Semigroups.thy
   15.31 -	@$(USEDIR) $(OUT)/HOL Group
   15.32 -	@rm -f Group/document/pdfsetup.sty Group/document/session.tex
   15.33 -
   15.34 -
   15.35 -## Nat
   15.36 -
   15.37 -Nat: FOL $(LOG)/FOL-Nat.gz
   15.38 -
   15.39 -FOL:
   15.40 -	@cd $(SRC)/FOL; $(ISABELLE_TOOL) make FOL
   15.41 -
   15.42 -$(LOG)/FOL-Nat.gz: $(OUT)/FOL Nat/ROOT.ML Nat/NatClass.thy
   15.43 -	@$(USEDIR) $(OUT)/FOL Nat
   15.44 -	@rm -f Nat/document/*.sty Nat/document/session.tex
   15.45 -
   15.46 -
   15.47 -## clean
   15.48 -
   15.49 -clean:
   15.50 -	@rm -f $(LOG)/HOL-Group.gz $(LOG)/FOL-Nat.gz

    16.1 --- a/doc-src/AxClass/Makefile	Wed Mar 04 11:05:02 2009 +0100
    16.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    16.3 @@ -1,36 +0,0 @@
    16.4 -#
    16.5 -# $Id$
    16.6 -#
    16.7 -
    16.8 -## targets
    16.9 -
   16.10 -default: dvi
   16.11 -
   16.12 -
   16.13 -## dependencies
   16.14 -
   16.15 -include ../Makefile.in
   16.16 -
   16.17 -NAME = axclass
   16.18 -
   16.19 -FILES = axclass.tex body.tex ../iman.sty ../extra.sty ../isar.sty	\
   16.20 -  ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty			\
   16.21 -  Group/document/Group.tex Nat/document/NatClass.tex			\
   16.22 -  Group/document/Product.tex Group/document/Semigroups.tex
   16.23 -
   16.24 -dvi: $(NAME).dvi
   16.25 -
   16.26 -$(NAME).dvi: $(FILES) isabelle_isar.eps
   16.27 -	$(LATEX) $(NAME)
   16.28 -	$(BIBTEX) $(NAME)
   16.29 -	$(LATEX) $(NAME)
   16.30 -	$(LATEX) $(NAME)
   16.31 -
   16.32 -pdf: $(NAME).pdf
   16.33 -
   16.34 -$(NAME).pdf: $(FILES) isabelle_isar.pdf
   16.35 -	$(PDFLATEX) $(NAME)
   16.36 -	$(FIXBOOKMARKS) $(NAME).out
   16.37 -	$(BIBTEX) $(NAME)
   16.38 -	$(PDFLATEX) $(NAME)
   16.39 -	$(PDFLATEX) $(NAME)

    17.1 --- a/doc-src/AxClass/Nat/NatClass.thy	Wed Mar 04 11:05:02 2009 +0100
    17.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    17.3 @@ -1,117 +0,0 @@
    17.4 -
    17.5 -header {* Defining natural numbers in FOL \label{sec:ex-natclass} *}
    17.6 -
    17.7 -theory NatClass imports FOL begin
    17.8 -
    17.9 -text {*
   17.10 - \medskip\noindent Axiomatic type classes abstract over exactly one
   17.11 - type argument. Thus, any \emph{axiomatic} theory extension where each
   17.12 - axiom refers to at most one type variable, may be trivially turned
   17.13 - into a \emph{definitional} one.
   17.14 -
   17.15 - We illustrate this with the natural numbers in
   17.16 - Isabelle/FOL.\footnote{See also
   17.17 - \url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}}
   17.18 -*}
   17.19 -
   17.20 -consts
   17.21 -  zero :: 'a    ("\<zero>")
   17.22 -  Suc :: "'a \<Rightarrow> 'a"
   17.23 -  rec :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a"
   17.24 -
   17.25 -axclass nat \<subseteq> "term"
   17.26 -  induct: "P(\<zero>) \<Longrightarrow> (\<And>x. P(x) \<Longrightarrow> P(Suc(x))) \<Longrightarrow> P(n)"
   17.27 -  Suc_inject: "Suc(m) = Suc(n) \<Longrightarrow> m = n"
   17.28 -  Suc_neq_0: "Suc(m) = \<zero> \<Longrightarrow> R"
   17.29 -  rec_0: "rec(\<zero>, a, f) = a"
   17.30 -  rec_Suc: "rec(Suc(m), a, f) = f(m, rec(m, a, f))"
   17.31 -
   17.32 -constdefs
   17.33 -  add :: "'a::nat \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "+" 60)
   17.34 -  "m + n \<equiv> rec(m, n, \<lambda>x y. Suc(y))"
   17.35 -
   17.36 -text {*
   17.37 - This is an abstract version of the plain @{text Nat} theory in
   17.38 - FOL.\footnote{See
   17.39 - \url{http://isabelle.in.tum.de/library/FOL/ex/Nat.html}} Basically,
   17.40 - we have just replaced all occurrences of type @{text nat} by @{typ
   17.41 - 'a} and used the natural number axioms to define class @{text nat}.
   17.42 - There is only a minor snag, that the original recursion operator
   17.43 - @{term rec} had to be made monomorphic.
   17.44 -
   17.45 - Thus class @{text nat} contains exactly those types @{text \<tau>} that
   17.46 - are isomorphic to ``the'' natural numbers (with signature @{term
   17.47 - \<zero>}, @{term Suc}, @{term rec}).
   17.48 -
   17.49 - \medskip What we have done here can be also viewed as \emph{type
   17.50 - specification}.  Of course, it still remains open if there is some
   17.51 - type at all that meets the class axioms.  Now a very nice property of
   17.52 - axiomatic type classes is that abstract reasoning is always possible
   17.53 - --- independent of satisfiability.  The meta-logic won't break, even
   17.54 - if some classes (or general sorts) turns out to be empty later ---
   17.55 - ``inconsistent'' class definitions may be useless, but do not cause
   17.56 - any harm.
   17.57 -
   17.58 - Theorems of the abstract natural numbers may be derived in the same
   17.59 - way as for the concrete version.  The original proof scripts may be
   17.60 - re-used with some trivial changes only (mostly adding some type
   17.61 - constraints).
   17.62 -*}
   17.63 -
   17.64 -(*<*)
   17.65 -lemma Suc_n_not_n: "Suc(k) ~= (k::'a::nat)"
   17.66 -apply (rule_tac n = k in induct)
   17.67 -apply (rule notI)
   17.68 -apply (erule Suc_neq_0)
   17.69 -apply (rule notI)
   17.70 -apply (erule notE)
   17.71 -apply (erule Suc_inject)
   17.72 -done
   17.73 -
   17.74 -lemma "(k+m)+n = k+(m+n)"
   17.75 -apply (rule induct)
   17.76 -back
   17.77 -back
   17.78 -back
   17.79 -back
   17.80 -back
   17.81 -back
   17.82 -oops
   17.83 -
   17.84 -lemma add_0 [simp]: "\<zero>+n = n"
   17.85 -apply (unfold add_def)
   17.86 -apply (rule rec_0)
   17.87 -done
   17.88 -
   17.89 -lemma add_Suc [simp]: "Suc(m)+n = Suc(m+n)"
   17.90 -apply (unfold add_def)
   17.91 -apply (rule rec_Suc)
   17.92 -done
   17.93 -
   17.94 -lemma add_assoc: "(k+m)+n = k+(m+n)"
   17.95 -apply (rule_tac n = k in induct)
   17.96 -apply simp
   17.97 -apply simp
   17.98 -done
   17.99 -
  17.100 -lemma add_0_right: "m+\<zero> = m"
  17.101 -apply (rule_tac n = m in induct)
  17.102 -apply simp
  17.103 -apply simp
  17.104 -done
  17.105 -
  17.106 -lemma add_Suc_right: "m+Suc(n) = Suc(m+n)"
  17.107 -apply (rule_tac n = m in induct)
  17.108 -apply simp_all
  17.109 -done
  17.110 -
  17.111 -lemma
  17.112 -  assumes prem: "!!n. f(Suc(n)) = Suc(f(n))"
  17.113 -  shows "f(i+j) = i+f(j)"
  17.114 -apply (rule_tac n = i in induct)
  17.115 -apply simp
  17.116 -apply (simp add: prem)
  17.117 -done
  17.118 -(*>*)
  17.119 -
  17.120 -end
  17.121 \ No newline at end of file

    18.1 --- a/doc-src/AxClass/Nat/ROOT.ML	Wed Mar 04 11:05:02 2009 +0100
    18.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    18.3 @@ -1,2 +0,0 @@
    18.4 -
    18.5 -use_thy "NatClass";

    19.1 --- a/doc-src/AxClass/Nat/document/NatClass.tex	Wed Mar 04 11:05:02 2009 +0100
    19.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    19.3 @@ -1,201 +0,0 @@
    19.4 -%
    19.5 -\begin{isabellebody}%
    19.6 -\def\isabellecontext{NatClass}%
    19.7 -%
    19.8 -\isamarkupheader{Defining natural numbers in FOL \label{sec:ex-natclass}%
    19.9 -}
   19.10 -\isamarkuptrue%
   19.11 -%
   19.12 -\isadelimtheory
   19.13 -%
   19.14 -\endisadelimtheory
   19.15 -%
   19.16 -\isatagtheory
   19.17 -\isacommand{theory}\isamarkupfalse%
   19.18 -\ NatClass\ \isakeyword{imports}\ FOL\ \isakeyword{begin}%
   19.19 -\endisatagtheory
   19.20 -{\isafoldtheory}%
   19.21 -%
   19.22 -\isadelimtheory
   19.23 -%
   19.24 -\endisadelimtheory
   19.25 -%
   19.26 -\begin{isamarkuptext}%
   19.27 -\medskip\noindent Axiomatic type classes abstract over exactly one
   19.28 - type argument. Thus, any \emph{axiomatic} theory extension where each
   19.29 - axiom refers to at most one type variable, may be trivially turned
   19.30 - into a \emph{definitional} one.
   19.31 -
   19.32 - We illustrate this with the natural numbers in
   19.33 - Isabelle/FOL.\footnote{See also
   19.34 - \url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}}%
   19.35 -\end{isamarkuptext}%
   19.36 -\isamarkuptrue%
   19.37 -\isacommand{consts}\isamarkupfalse%
   19.38 -\isanewline
   19.39 -\ \ zero\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ \ \ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymzero}{\isachardoublequoteclose}{\isacharparenright}\isanewline
   19.40 -\ \ Suc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\isanewline
   19.41 -\ \ rec\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\isanewline
   19.42 -\isanewline
   19.43 -\isacommand{axclass}\isamarkupfalse%
   19.44 -\ nat\ {\isasymsubseteq}\ {\isachardoublequoteopen}term{\isachardoublequoteclose}\isanewline
   19.45 -\ \ induct{\isacharcolon}\ {\isachardoublequoteopen}P{\isacharparenleft}{\isasymzero}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ P{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}Suc{\isacharparenleft}x{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharparenleft}n{\isacharparenright}{\isachardoublequoteclose}\isanewline
   19.46 -\ \ Suc{\isacharunderscore}inject{\isacharcolon}\ {\isachardoublequoteopen}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ Suc{\isacharparenleft}n{\isacharparenright}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
   19.47 -\ \ Suc{\isacharunderscore}neq{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequoteopen}Suc{\isacharparenleft}m{\isacharparenright}\ {\isacharequal}\ {\isasymzero}\ {\isasymLongrightarrow}\ R{\isachardoublequoteclose}\isanewline
   19.48 -\ \ rec{\isacharunderscore}{\isadigit{0}}{\isacharcolon}\ {\isachardoublequoteopen}rec{\isacharparenleft}{\isasymzero}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ a{\isachardoublequoteclose}\isanewline
   19.49 -\ \ rec{\isacharunderscore}Suc{\isacharcolon}\ {\isachardoublequoteopen}rec{\isacharparenleft}Suc{\isacharparenleft}m{\isacharparenright}{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}m{\isacharcomma}\ rec{\isacharparenleft}m{\isacharcomma}\ a{\isacharcomma}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
   19.50 -\isanewline
   19.51 -\isacommand{constdefs}\isamarkupfalse%
   19.52 -\isanewline
   19.53 -\ \ add\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a{\isacharcolon}{\isacharcolon}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isacharplus}{\isachardoublequoteclose}\ {\isadigit{6}}{\isadigit{0}}{\isacharparenright}\isanewline
   19.54 -\ \ {\isachardoublequoteopen}m\ {\isacharplus}\ n\ {\isasymequiv}\ rec{\isacharparenleft}m{\isacharcomma}\ n{\isacharcomma}\ {\isasymlambda}x\ y{\isachardot}\ Suc{\isacharparenleft}y{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
   19.55 -\begin{isamarkuptext}%
   19.56 -This is an abstract version of the plain \isa{Nat} theory in
   19.57 - FOL.\footnote{See
   19.58 - \url{http://isabelle.in.tum.de/library/FOL/ex/Nat.html}} Basically,
   19.59 - we have just replaced all occurrences of type \isa{nat} by \isa{{\isacharprime}a} and used the natural number axioms to define class \isa{nat}.
   19.60 - There is only a minor snag, that the original recursion operator
   19.61 - \isa{rec} had to be made monomorphic.
   19.62 -
   19.63 - Thus class \isa{nat} contains exactly those types \isa{{\isasymtau}} that
   19.64 - are isomorphic to ``the'' natural numbers (with signature \isa{{\isasymzero}}, \isa{Suc}, \isa{rec}).
   19.65 -
   19.66 - \medskip What we have done here can be also viewed as \emph{type
   19.67 - specification}.  Of course, it still remains open if there is some
   19.68 - type at all that meets the class axioms.  Now a very nice property of
   19.69 - axiomatic type classes is that abstract reasoning is always possible
   19.70 - --- independent of satisfiability.  The meta-logic won't break, even
   19.71 - if some classes (or general sorts) turns out to be empty later ---
   19.72 - ``inconsistent'' class definitions may be useless, but do not cause
   19.73 - any harm.
   19.74 -
   19.75 - Theorems of the abstract natural numbers may be derived in the same
   19.76 - way as for the concrete version.  The original proof scripts may be
   19.77 - re-used with some trivial changes only (mostly adding some type
   19.78 - constraints).%
   19.79 -\end{isamarkuptext}%
   19.80 -\isamarkuptrue%
   19.81 -%
   19.82 -\isadelimproof
   19.83 -%
   19.84 -\endisadelimproof
   19.85 -%
   19.86 -\isatagproof
   19.87 -%
   19.88 -\endisatagproof
   19.89 -{\isafoldproof}%
   19.90 -%
   19.91 -\isadelimproof
   19.92 -%
   19.93 -\endisadelimproof
   19.94 -%
   19.95 -\isadelimproof
   19.96 -%
   19.97 -\endisadelimproof
   19.98 -%
   19.99 -\isatagproof
  19.100 -%
  19.101 -\endisatagproof
  19.102 -{\isafoldproof}%
  19.103 -%
  19.104 -\isadelimproof
  19.105 -%
  19.106 -\endisadelimproof
  19.107 -%
  19.108 -\isadelimproof
  19.109 -%
  19.110 -\endisadelimproof
  19.111 -%
  19.112 -\isatagproof
  19.113 -%
  19.114 -\endisatagproof
  19.115 -{\isafoldproof}%
  19.116 -%
  19.117 -\isadelimproof
  19.118 -%
  19.119 -\endisadelimproof
  19.120 -%
  19.121 -\isadelimproof
  19.122 -%
  19.123 -\endisadelimproof
  19.124 -%
  19.125 -\isatagproof
  19.126 -%
  19.127 -\endisatagproof
  19.128 -{\isafoldproof}%
  19.129 -%
  19.130 -\isadelimproof
  19.131 -%
  19.132 -\endisadelimproof
  19.133 -%
  19.134 -\isadelimproof
  19.135 -%
  19.136 -\endisadelimproof
  19.137 -%
  19.138 -\isatagproof
  19.139 -%
  19.140 -\endisatagproof
  19.141 -{\isafoldproof}%
  19.142 -%
  19.143 -\isadelimproof
  19.144 -%
  19.145 -\endisadelimproof
  19.146 -%
  19.147 -\isadelimproof
  19.148 -%
  19.149 -\endisadelimproof
  19.150 -%
  19.151 -\isatagproof
  19.152 -%
  19.153 -\endisatagproof
  19.154 -{\isafoldproof}%
  19.155 -%
  19.156 -\isadelimproof
  19.157 -%
  19.158 -\endisadelimproof
  19.159 -%
  19.160 -\isadelimproof
  19.161 -%
  19.162 -\endisadelimproof
  19.163 -%
  19.164 -\isatagproof
  19.165 -%
  19.166 -\endisatagproof
  19.167 -{\isafoldproof}%
  19.168 -%
  19.169 -\isadelimproof
  19.170 -%
  19.171 -\endisadelimproof
  19.172 -%
  19.173 -\isadelimproof
  19.174 -%
  19.175 -\endisadelimproof
  19.176 -%
  19.177 -\isatagproof
  19.178 -%
  19.179 -\endisatagproof
  19.180 -{\isafoldproof}%
  19.181 -%
  19.182 -\isadelimproof
  19.183 -\isanewline
  19.184 -%
  19.185 -\endisadelimproof
  19.186 -%
  19.187 -\isadelimtheory
  19.188 -%
  19.189 -\endisadelimtheory
  19.190 -%
  19.191 -\isatagtheory
  19.192 -\isacommand{end}\isamarkupfalse%
  19.193 -%
  19.194 -\endisatagtheory
  19.195 -{\isafoldtheory}%
  19.196 -%
  19.197 -\isadelimtheory
  19.198 -%
  19.199 -\endisadelimtheory
  19.200 -\end{isabellebody}%
  19.201 -%%% Local Variables:
  19.202 -%%% mode: latex
  19.203 -%%% TeX-master: "root"
  19.204 -%%% End:

    20.1 --- a/doc-src/AxClass/axclass.tex	Wed Mar 04 11:05:02 2009 +0100
    20.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    20.3 @@ -1,80 +0,0 @@
    20.4 -
    20.5 -\documentclass[12pt,a4paper,fleqn]{report}
    20.6 -\usepackage{graphicx,../iman,../extra,../isar}
    20.7 -\usepackage{../isabelle,../isabellesym}
    20.8 -\usepackage{../pdfsetup}  % last one!
    20.9 -
   20.10 -\isabellestyle{it}
   20.11 -\newcommand{\isasyminv}{\isamath{{}^{-1}}}
   20.12 -\renewcommand{\isasymzero}{\isamath{0}}
   20.13 -\renewcommand{\isasymone}{\isamath{1}}
   20.14 -
   20.15 -\newcommand{\secref}[1]{\S\ref{#1}}
   20.16 -\newcommand{\figref}[1]{figure~\ref{#1}}
   20.17 -
   20.18 -\hyphenation{Isabelle}
   20.19 -\hyphenation{Isar}
   20.20 -\hyphenation{Haskell}
   20.21 -
   20.22 -\title{\includegraphics[scale=0.5]{isabelle_isar}
   20.23 -  \\[4ex] Using Axiomatic Type Classes in Isabelle}
   20.24 -\author{\emph{Markus Wenzel} \\ TU M\"unchen}
   20.25 -
   20.26 -
   20.27 -\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
   20.28 -
   20.29 -\pagestyle{headings}
   20.30 -\sloppy
   20.31 -\binperiod     %%%treat . like a binary operator
   20.32 -
   20.33 -
   20.34 -\begin{document}
   20.35 -
   20.36 -\underscoreoff
   20.37 -
   20.38 -\maketitle 
   20.39 -
   20.40 -\begin{abstract}
   20.41 -  Isabelle offers order-sorted type classes on top of the simple types of
   20.42 -  plain Higher-Order Logic.  The resulting type system is similar to that of
   20.43 -  the programming language Haskell.  Its interpretation within the logic
   20.44 -  enables further application, though, apart from restricting polymorphism
   20.45 -  syntactically.  In particular, the concept of \emph{Axiomatic Type Classes}
   20.46 -  provides a useful light-weight mechanism for hierarchically-structured
   20.47 -  abstract theories. Subsequently, we demonstrate typical uses of Isabelle's
   20.48 -  axiomatic type classes to model basic algebraic structures.
   20.49 -  
   20.50 -  This document describes axiomatic type classes using Isabelle/Isar theories,
   20.51 -  with proofs expressed via Isar proof language elements.  The new theory
   20.52 -  format greatly simplifies the arrangement of the overall development, since
   20.53 -  definitions and proofs may be freely intermixed.  Users who prefer tactic
   20.54 -  scripts over structured proofs do not need to fall back on separate ML
   20.55 -  scripts, though, but may refer to Isar's tactic emulation commands.
   20.56 -\end{abstract}
   20.57 -
   20.58 -
   20.59 -\pagenumbering{roman} \tableofcontents \clearfirst
   20.60 -
   20.61 -\include{body}
   20.62 -
   20.63 -%FIXME
   20.64 -\nocite{nipkow-types93}
   20.65 -\nocite{nipkow-sorts93}
   20.66 -\nocite{Wenzel:1997:TPHOL}
   20.67 -\nocite{paulson-isa-book}
   20.68 -\nocite{isabelle-isar-ref}
   20.69 -\nocite{Wenzel:1999:TPHOL}
   20.70 -
   20.71 -\begingroup
   20.72 -  \bibliographystyle{plain} \small\raggedright\frenchspacing
   20.73 -  \bibliography{../manual}
   20.74 -\endgroup
   20.75 -
   20.76 -\end{document}
   20.77 -
   20.78 -
   20.79 -%%% Local Variables: 
   20.80 -%%% mode: latex
   20.81 -%%% TeX-master: t
   20.82 -%%% End: 
   20.83 -% LocalWords:  Isabelle FIXME

    21.1 --- a/doc-src/AxClass/body.tex	Wed Mar 04 11:05:02 2009 +0100
    21.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
    21.3 @@ -1,166 +0,0 @@
    21.4 -
    21.5 -\chapter{Introduction}
    21.6 -
    21.7 -A Haskell-style type-system \cite{haskell-report} with ordered type-classes
    21.8 -has been present in Isabelle since 1991 already \cite{nipkow-sorts93}.
    21.9 -Initially, classes have mainly served as a \emph{purely syntactic} tool to
   21.10 -formulate polymorphic object-logics in a clean way, such as the standard
   21.11 -Isabelle formulation of many-sorted FOL \cite{paulson-isa-book}.
   21.12 -
   21.13 -Applying classes at the \emph{logical level} to provide a simple notion of
   21.14 -abstract theories and instantiations to concrete ones, has been long proposed
   21.15 -as well \cite{nipkow-types93,nipkow-sorts93}.  At that time, Isabelle still
   21.16 -lacked built-in support for these \emph{axiomatic type classes}. More
   21.17 -importantly, their semantics was not yet fully fleshed out (and unnecessarily
   21.18 -complicated, too).
   21.19 -
   21.20 -Since Isabelle94, actual axiomatic type classes have been an integral part of
   21.21 -Isabelle's meta-logic.  This very simple implementation is based on a
   21.22 -straight-forward extension of traditional simply-typed Higher-Order Logic, by
   21.23 -including types qualified by logical predicates and overloaded constant
   21.24 -definitions (see \cite{Wenzel:1997:TPHOL} for further details).
   21.25 -
   21.26 -Yet even until Isabelle99, there used to be still a fundamental methodological
   21.27 -problem in using axiomatic type classes conveniently, due to the traditional
   21.28 -distinction of Isabelle theory files vs.\ ML proof scripts.  This has been
   21.29 -finally overcome with the advent of Isabelle/Isar theories
   21.30 -\cite{isabelle-isar-ref}: now definitions and proofs may be freely intermixed.
   21.31 -This nicely accommodates the usual procedure of defining axiomatic type
   21.32 -classes, proving abstract properties, defining operations on concrete types,
   21.33 -proving concrete properties for instantiation of classes etc.
   21.34 -
   21.35 -\medskip
   21.36 -
   21.37 -So to cut a long story short, the present version of axiomatic type classes
   21.38 -now provides an even more useful and convenient mechanism for light-weight
   21.39 -abstract theories, without any special technical provisions to be observed by
   21.40 -the user.
   21.41 -
   21.42 -
   21.43 -\chapter{Examples}\label{sec:ex}
   21.44 -
   21.45 -Axiomatic type classes are a concept of Isabelle's meta-logic
   21.46 -\cite{paulson-isa-book,Wenzel:1997:TPHOL}.  They may be applied to any
   21.47 -object-logic that directly uses the meta type system, such as Isabelle/HOL
   21.48 -\cite{isabelle-HOL}.  Subsequently, we present various examples that are all
   21.49 -formulated within HOL, except the one of \secref{sec:ex-natclass} which is in
   21.50 -FOL.  See also \url{http://isabelle.in.tum.de/library/HOL/AxClasses/} and
   21.51 -\url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}.
   21.52 -
   21.53 -\input{Group/document/Semigroups}
   21.54 -
   21.55 -\input{Group/document/Group}
   21.56 -
   21.57 -\input{Group/document/Product}
   21.58 -
   21.59 -\input{Nat/document/NatClass}
   21.60 -
   21.61 -
   21.62 -%% FIXME move some parts to ref or isar-ref manual (!?);
   21.63 -
   21.64 -% \chapter{The user interface of Isabelle's axclass package}
   21.65 -
   21.66 -% The actual axiomatic type class package of Isabelle/Pure mainly consists
   21.67 -% of two new theory sections: \texttt{axclass} and \texttt{instance}.  Some
   21.68 -% typical applications of these have already been demonstrated in
   21.69 -% \secref{sec:ex}, below their syntax and semantics are presented more
   21.70 -% completely.
   21.71 -
   21.72 -
   21.73 -% \section{The axclass section}
   21.74 -
   21.75 -% Within theory files, \texttt{axclass} introduces an axiomatic type class
   21.76 -% definition. Its concrete syntax is:
   21.77 -
   21.78 -% \begin{matharray}{l}
   21.79 -%   \texttt{axclass} \\
   21.80 -%   \ \ c \texttt{ < } c@1\texttt, \ldots\texttt, c@n \\
   21.81 -%   \ \ id@1\ axm@1 \\
   21.82 -%   \ \ \vdots \\
   21.83 -%   \ \ id@m\ axm@m
   21.84 -% \emphnd{matharray}
   21.85 -
   21.86 -% Where $c, c@1, \ldots, c@n$ are classes (category $id$ or
   21.87 -% $string$) and $axm@1, \ldots, axm@m$ (with $m \geq
   21.88 -% 0$) are formulas (category $string$).
   21.89 -
   21.90 -% Class $c$ has to be new, and sort $\{c@1, \ldots, c@n\}$ a subsort of
   21.91 -% \texttt{logic}. Each class axiom $axm@j$ may contain any term
   21.92 -% variables, but at most one type variable (which need not be the same
   21.93 -% for all axioms). The sort of this type variable has to be a supersort
   21.94 -% of $\{c@1, \ldots, c@n\}$.
   21.95 -
   21.96 -% \medskip
   21.97 -
   21.98 -% The \texttt{axclass} section declares $c$ as subclass of $c@1, \ldots,
   21.99 -% c@n$ to the type signature.
  21.100 -
  21.101 -% Furthermore, $axm@1, \ldots, axm@m$ are turned into the
  21.102 -% ``abstract axioms'' of $c$ with names $id@1, \ldots,
  21.103 -% id@m$.  This is done by replacing all occurring type variables
  21.104 -% by $\alpha :: c$. Original axioms that do not contain any type
  21.105 -% variable will be prefixed by the logical precondition
  21.106 -% $\texttt{OFCLASS}(\alpha :: \texttt{logic}, c\texttt{_class})$.
  21.107 -
  21.108 -% Another axiom of name $c\texttt{I}$ --- the ``class $c$ introduction
  21.109 -% rule'' --- is built from the respective universal closures of
  21.110 -% $axm@1, \ldots, axm@m$ appropriately.
  21.111 -
  21.112 -
  21.113 -% \section{The instance section}
  21.114 -
  21.115 -% Section \texttt{instance} proves class inclusions or type arities at the
  21.116 -% logical level and then transfers these into the type signature.
  21.117 -
  21.118 -% Its concrete syntax is:
  21.119 -
  21.120 -% \begin{matharray}{l}
  21.121 -%   \texttt{instance} \\
  21.122 -%   \ \ [\ c@1 \texttt{ < } c@2 \ |\
  21.123 -%       t \texttt{ ::\ (}sort@1\texttt, \ldots \texttt, sort@n\texttt) sort\ ] \\
  21.124 -%   \ \ [\ \texttt(name@1 \texttt, \ldots\texttt, name@m\texttt)\ ] \\
  21.125 -%   \ \ [\ \texttt{\{|} text \texttt{|\}}\ ]
  21.126 -% \emphnd{matharray}
  21.127 -
  21.128 -% Where $c@1, c@2$ are classes and $t$ is an $n$-place type constructor
  21.129 -% (all of category $id$ or $string)$. Furthermore,
  21.130 -% $sort@i$ are sorts in the usual Isabelle-syntax.
  21.131 -
  21.132 -% \medskip
  21.133 -
  21.134 -% Internally, \texttt{instance} first sets up an appropriate goal that
  21.135 -% expresses the class inclusion or type arity as a meta-proposition.
  21.136 -% Then tactic \texttt{AxClass.axclass_tac} is applied with all preceding
  21.137 -% meta-definitions of the current theory file and the user-supplied
  21.138 -% witnesses. The latter are $name@1, \ldots, name@m$, where
  21.139 -% $id$ refers to an \ML-name of a theorem, and $string$ to an
  21.140 -% axiom of the current theory node\footnote{Thus, the user may reference
  21.141 -%   axioms from above this \texttt{instance} in the theory file. Note
  21.142 -%   that new axioms appear at the \ML-toplevel only after the file is
  21.143 -%   processed completely.}.
  21.144 -
  21.145 -% Tactic \texttt{AxClass.axclass_tac} first unfolds the class definition by
  21.146 -% resolving with rule $c\texttt\texttt{I}$, and then applies the witnesses
  21.147 -% according to their form: Meta-definitions are unfolded, all other
  21.148 -% formulas are repeatedly resolved\footnote{This is done in a way that
  21.149 -%   enables proper object-\emph{rules} to be used as witnesses for
  21.150 -%   corresponding class axioms.} with.
  21.151 -
  21.152 -% The final optional argument $text$ is \ML-code of an arbitrary
  21.153 -% user tactic which is applied last to any remaining goals.
  21.154 -
  21.155 -% \medskip
  21.156 -
  21.157 -% Because of the complexity of \texttt{instance}'s witnessing mechanisms,
  21.158 -% new users of the axclass package are advised to only use the simple
  21.159 -% form $\texttt{instance}\ \ldots\ (id@1, \ldots, id@!m)$, where
  21.160 -% the identifiers refer to theorems that are appropriate type instances
  21.161 -% of the class axioms. This typically requires an auxiliary theory,
  21.162 -% though, which defines some constants and then proves these witnesses.
  21.163 -
  21.164 -
  21.165 -%%% Local Variables: 
  21.166 -%%% mode: latex
  21.167 -%%% TeX-master: "axclass"
  21.168 -%%% End: 
  21.169 -% LocalWords:  Isabelle FOL

    22.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    22.2 +++ b/doc-src/Classes/IsaMakefile	Wed Mar 04 11:05:29 2009 +0100
    22.3 @@ -0,0 +1,33 @@
    22.4 +
    22.5 +## targets
    22.6 +
    22.7 +default: Thy
    22.8 +images: 
    22.9 +test: Thy
   22.10 +
   22.11 +all: images test
   22.12 +
   22.13 +
   22.14 +## global settings
   22.15 +
   22.16 +SRC = $(ISABELLE_HOME)/src
   22.17 +OUT = $(ISABELLE_OUTPUT)
   22.18 +LOG = $(OUT)/log
   22.19 +
   22.20 +USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
   22.21 +
   22.22 +
   22.23 +## Thy
   22.24 +
   22.25 +THY = $(LOG)/HOL-Thy.gz
   22.26 +
   22.27 +Thy: $(THY)
   22.28 +
   22.29 +$(THY): Thy/ROOT.ML Thy/Setup.thy Thy/Classes.thy ../antiquote_setup.ML ../more_antiquote.ML
   22.30 +	@$(USEDIR) HOL Thy
   22.31 +
   22.32 +
   22.33 +## clean
   22.34 +
   22.35 +clean:
   22.36 +	@rm -f $(THY)

    23.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    23.2 +++ b/doc-src/Classes/Makefile	Wed Mar 04 11:05:29 2009 +0100
    23.3 @@ -0,0 +1,35 @@
    23.4 +
    23.5 +## targets
    23.6 +
    23.7 +default: dvi
    23.8 +
    23.9 +
   23.10 +## dependencies
   23.11 +
   23.12 +include ../Makefile.in
   23.13 +
   23.14 +NAME = classes
   23.15 +
   23.16 +FILES = $(NAME).tex classes.tex Thy/document/Classes.tex \
   23.17 +  style.sty ../iman.sty ../extra.sty ../isar.sty \
   23.18 +  ../isabelle.sty ../isabellesym.sty ../pdfsetup.sty \
   23.19 +  ../manual.bib ../proof.sty
   23.20 +
   23.21 +dvi: $(NAME).dvi
   23.22 +
   23.23 +$(NAME).dvi: $(FILES) isabelle_isar.eps
   23.24 +	$(LATEX) $(NAME)
   23.25 +	$(BIBTEX) $(NAME)
   23.26 +	$(LATEX) $(NAME)
   23.27 +	$(LATEX) $(NAME)
   23.28 +
   23.29 +pdf: $(NAME).pdf
   23.30 +
   23.31 +$(NAME).pdf: $(FILES) isabelle_isar.pdf
   23.32 +	$(PDFLATEX) $(NAME)
   23.33 +	$(BIBTEX) $(NAME)
   23.34 +	$(PDFLATEX) $(NAME)
   23.35 +	$(PDFLATEX) $(NAME)
   23.36 +	$(FIXBOOKMARKS) $(NAME).out
   23.37 +	$(PDFLATEX) $(NAME)
   23.38 +	$(PDFLATEX) $(NAME)

    24.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    24.2 +++ b/doc-src/Classes/Thy/Classes.thy	Wed Mar 04 11:05:29 2009 +0100
    24.3 @@ -0,0 +1,634 @@
    24.4 +theory Classes
    24.5 +imports Main Setup
    24.6 +begin
    24.7 +
    24.8 +section {* Introduction *}
    24.9 +
   24.10 +text {*
   24.11 +  Type classes were introduces by Wadler and Blott \cite{wadler89how}
   24.12 +  into the Haskell language, to allow for a reasonable implementation
   24.13 +  of overloading\footnote{throughout this tutorial, we are referring
   24.14 +  to classical Haskell 1.0 type classes, not considering
   24.15 +  later additions in expressiveness}.
   24.16 +  As a canonical example, a polymorphic equality function
   24.17 +  @{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} which is overloaded on different
   24.18 +  types for @{text "\<alpha>"}, which is achieved by splitting introduction
   24.19 +  of the @{text eq} function from its overloaded definitions by means
   24.20 +  of @{text class} and @{text instance} declarations:
   24.21 +  \footnote{syntax here is a kind of isabellized Haskell}
   24.22 +
   24.23 +  \begin{quote}
   24.24 +
   24.25 +  \noindent@{text "class eq where"} \\
   24.26 +  \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
   24.27 +
   24.28 +  \medskip\noindent@{text "instance nat \<Colon> eq where"} \\
   24.29 +  \hspace*{2ex}@{text "eq 0 0 = True"} \\
   24.30 +  \hspace*{2ex}@{text "eq 0 _ = False"} \\
   24.31 +  \hspace*{2ex}@{text "eq _ 0 = False"} \\
   24.32 +  \hspace*{2ex}@{text "eq (Suc n) (Suc m) = eq n m"}
   24.33 +
   24.34 +  \medskip\noindent@{text "instance (\<alpha>\<Colon>eq, \<beta>\<Colon>eq) pair \<Colon> eq where"} \\
   24.35 +  \hspace*{2ex}@{text "eq (x1, y1) (x2, y2) = eq x1 x2 \<and> eq y1 y2"}
   24.36 +
   24.37 +  \medskip\noindent@{text "class ord extends eq where"} \\
   24.38 +  \hspace*{2ex}@{text "less_eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
   24.39 +  \hspace*{2ex}@{text "less \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
   24.40 +
   24.41 +  \end{quote}
   24.42 +
   24.43 +  \noindent Type variables are annotated with (finitely many) classes;
   24.44 +  these annotations are assertions that a particular polymorphic type
   24.45 +  provides definitions for overloaded functions.
   24.46 +
   24.47 +  Indeed, type classes not only allow for simple overloading
   24.48 +  but form a generic calculus, an instance of order-sorted
   24.49 +  algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
   24.50 +
   24.51 +  From a software engeneering point of view, type classes
   24.52 +  roughly correspond to interfaces in object-oriented languages like Java;
   24.53 +  so, it is naturally desirable that type classes do not only
   24.54 +  provide functions (class parameters) but also state specifications
   24.55 +  implementations must obey.  For example, the @{text "class eq"}
   24.56 +  above could be given the following specification, demanding that
   24.57 +  @{text "class eq"} is an equivalence relation obeying reflexivity,
   24.58 +  symmetry and transitivity:
   24.59 +
   24.60 +  \begin{quote}
   24.61 +
   24.62 +  \noindent@{text "class eq where"} \\
   24.63 +  \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
   24.64 +  @{text "satisfying"} \\
   24.65 +  \hspace*{2ex}@{text "refl: eq x x"} \\
   24.66 +  \hspace*{2ex}@{text "sym: eq x y \<longleftrightarrow> eq x y"} \\
   24.67 +  \hspace*{2ex}@{text "trans: eq x y \<and> eq y z \<longrightarrow> eq x z"}
   24.68 +
   24.69 +  \end{quote}
   24.70 +
   24.71 +  \noindent From a theoretic point of view, type classes are lightweight
   24.72 +  modules; Haskell type classes may be emulated by
   24.73 +  SML functors \cite{classes_modules}. 
   24.74 +  Isabelle/Isar offers a discipline of type classes which brings
   24.75 +  all those aspects together:
   24.76 +
   24.77 +  \begin{enumerate}
   24.78 +    \item specifying abstract parameters together with
   24.79 +       corresponding specifications,
   24.80 +    \item instantiating those abstract parameters by a particular
   24.81 +       type
   24.82 +    \item in connection with a ``less ad-hoc'' approach to overloading,
   24.83 +    \item with a direct link to the Isabelle module system
   24.84 +      (aka locales \cite{kammueller-locales}).
   24.85 +  \end{enumerate}
   24.86 +
   24.87 +  \noindent Isar type classes also directly support code generation
   24.88 +  in a Haskell like fashion.
   24.89 +
   24.90 +  This tutorial demonstrates common elements of structured specifications
   24.91 +  and abstract reasoning with type classes by the algebraic hierarchy of
   24.92 +  semigroups, monoids and groups.  Our background theory is that of
   24.93 +  Isabelle/HOL \cite{isa-tutorial}, for which some
   24.94 +  familiarity is assumed.
   24.95 +
   24.96 +  Here we merely present the look-and-feel for end users.
   24.97 +  Internally, those are mapped to more primitive Isabelle concepts.
   24.98 +  See \cite{Haftmann-Wenzel:2006:classes} for more detail.
   24.99 +*}
  24.100 +
  24.101 +section {* A simple algebra example \label{sec:example} *}
  24.102 +
  24.103 +subsection {* Class definition *}
  24.104 +
  24.105 +text {*
  24.106 +  Depending on an arbitrary type @{text "\<alpha>"}, class @{text
  24.107 +  "semigroup"} introduces a binary operator @{text "(\<otimes>)"} that is
  24.108 +  assumed to be associative:
  24.109 +*}
  24.110 +
  24.111 +class %quote semigroup =
  24.112 +  fixes mult :: "\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"    (infixl "\<otimes>" 70)
  24.113 +  assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
  24.114 +
  24.115 +text {*
  24.116 +  \noindent This @{command class} specification consists of two
  24.117 +  parts: the \qn{operational} part names the class parameter
  24.118 +  (@{element "fixes"}), the \qn{logical} part specifies properties on them
  24.119 +  (@{element "assumes"}).  The local @{element "fixes"} and
  24.120 +  @{element "assumes"} are lifted to the theory toplevel,
  24.121 +  yielding the global
  24.122 +  parameter @{term [source] "mult \<Colon> \<alpha>\<Colon>semigroup \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"} and the
  24.123 +  global theorem @{fact "semigroup.assoc:"}~@{prop [source] "\<And>x y
  24.124 +  z \<Colon> \<alpha>\<Colon>semigroup. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"}.
  24.125 +*}
  24.126 +
  24.127 +
  24.128 +subsection {* Class instantiation \label{sec:class_inst} *}
  24.129 +
  24.130 +text {*
  24.131 +  The concrete type @{typ int} is made a @{class semigroup}
  24.132 +  instance by providing a suitable definition for the class parameter
  24.133 +  @{text "(\<otimes>)"} and a proof for the specification of @{fact assoc}.
  24.134 +  This is accomplished by the @{command instantiation} target:
  24.135 +*}
  24.136 +
  24.137 +instantiation %quote int :: semigroup
  24.138 +begin
  24.139 +
  24.140 +definition %quote
  24.141 +  mult_int_def: "i \<otimes> j = i + (j\<Colon>int)"
  24.142 +
  24.143 +instance %quote proof
  24.144 +  fix i j k :: int have "(i + j) + k = i + (j + k)" by simp
  24.145 +  then show "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
  24.146 +    unfolding mult_int_def .
  24.147 +qed
  24.148 +
  24.149 +end %quote
  24.150 +
  24.151 +text {*
  24.152 +  \noindent @{command instantiation} allows to define class parameters
  24.153 +  at a particular instance using common specification tools (here,
  24.154 +  @{command definition}).  The concluding @{command instance}
  24.155 +  opens a proof that the given parameters actually conform
  24.156 +  to the class specification.  Note that the first proof step
  24.157 +  is the @{method default} method,
  24.158 +  which for such instance proofs maps to the @{method intro_classes} method.
  24.159 +  This boils down an instance judgement to the relevant primitive
  24.160 +  proof goals and should conveniently always be the first method applied
  24.161 +  in an instantiation proof.
  24.162 +
  24.163 +  From now on, the type-checker will consider @{typ int}
  24.164 +  as a @{class semigroup} automatically, i.e.\ any general results
  24.165 +  are immediately available on concrete instances.
  24.166 +
  24.167 +  \medskip Another instance of @{class semigroup} are the natural numbers:
  24.168 +*}
  24.169 +
  24.170 +instantiation %quote nat :: semigroup
  24.171 +begin
  24.172 +
  24.173 +primrec %quote mult_nat where
  24.174 +  "(0\<Colon>nat) \<otimes> n = n"
  24.175 +  | "Suc m \<otimes> n = Suc (m \<otimes> n)"
  24.176 +
  24.177 +instance %quote proof
  24.178 +  fix m n q :: nat 
  24.179 +  show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
  24.180 +    by (induct m) auto
  24.181 +qed
  24.182 +
  24.183 +end %quote
  24.184 +
  24.185 +text {*
  24.186 +  \noindent Note the occurence of the name @{text mult_nat}
  24.187 +  in the primrec declaration;  by default, the local name of
  24.188 +  a class operation @{text f} to instantiate on type constructor
  24.189 +  @{text \<kappa>} are mangled as @{text f_\<kappa>}.  In case of uncertainty,
  24.190 +  these names may be inspected using the @{command "print_context"} command
  24.191 +  or the corresponding ProofGeneral button.
  24.192 +*}
  24.193 +
  24.194 +subsection {* Lifting and parametric types *}
  24.195 +
  24.196 +text {*
  24.197 +  Overloaded definitions giving on class instantiation
  24.198 +  may include recursion over the syntactic structure of types.
  24.199 +  As a canonical example, we model product semigroups
  24.200 +  using our simple algebra:
  24.201 +*}
  24.202 +
  24.203 +instantiation %quote * :: (semigroup, semigroup) semigroup
  24.204 +begin
  24.205 +
  24.206 +definition %quote
  24.207 +  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)"
  24.208 +
  24.209 +instance %quote proof
  24.210 +  fix p\<^isub>1 p\<^isub>2 p\<^isub>3 :: "\<alpha>\<Colon>semigroup \<times> \<beta>\<Colon>semigroup"
  24.211 +  show "p\<^isub>1 \<otimes> p\<^isub>2 \<otimes> p\<^isub>3 = p\<^isub>1 \<otimes> (p\<^isub>2 \<otimes> p\<^isub>3)"
  24.212 +    unfolding mult_prod_def by (simp add: assoc)
  24.213 +qed      
  24.214 +
  24.215 +end %quote
  24.216 +
  24.217 +text {*
  24.218 +  \noindent Associativity from product semigroups is
  24.219 +  established using
  24.220 +  the definition of @{text "(\<otimes>)"} on products and the hypothetical
  24.221 +  associativity of the type components;  these hypotheses
  24.222 +  are facts due to the @{class semigroup} constraints imposed
  24.223 +  on the type components by the @{command instance} proposition.
  24.224 +  Indeed, this pattern often occurs with parametric types
  24.225 +  and type classes.
  24.226 +*}
  24.227 +
  24.228 +
  24.229 +subsection {* Subclassing *}
  24.230 +
  24.231 +text {*
  24.232 +  We define a subclass @{text monoidl} (a semigroup with a left-hand neutral)
  24.233 +  by extending @{class semigroup}
  24.234 +  with one additional parameter @{text neutral} together
  24.235 +  with its property:
  24.236 +*}
  24.237 +
  24.238 +class %quote monoidl = semigroup +
  24.239 +  fixes neutral :: "\<alpha>" ("\<one>")
  24.240 +  assumes neutl: "\<one> \<otimes> x = x"
  24.241 +
  24.242 +text {*
  24.243 +  \noindent Again, we prove some instances, by
  24.244 +  providing suitable parameter definitions and proofs for the
  24.245 +  additional specifications.  Observe that instantiations
  24.246 +  for types with the same arity may be simultaneous:
  24.247 +*}
  24.248 +
  24.249 +instantiation %quote nat and int :: monoidl
  24.250 +begin
  24.251 +
  24.252 +definition %quote
  24.253 +  neutral_nat_def: "\<one> = (0\<Colon>nat)"
  24.254 +
  24.255 +definition %quote
  24.256 +  neutral_int_def: "\<one> = (0\<Colon>int)"
  24.257 +
  24.258 +instance %quote proof
  24.259 +  fix n :: nat
  24.260 +  show "\<one> \<otimes> n = n"
  24.261 +    unfolding neutral_nat_def by simp
  24.262 +next
  24.263 +  fix k :: int
  24.264 +  show "\<one> \<otimes> k = k"
  24.265 +    unfolding neutral_int_def mult_int_def by simp
  24.266 +qed
  24.267 +
  24.268 +end %quote
  24.269 +
  24.270 +instantiation %quote * :: (monoidl, monoidl) monoidl
  24.271 +begin
  24.272 +
  24.273 +definition %quote
  24.274 +  neutral_prod_def: "\<one> = (\<one>, \<one>)"
  24.275 +
  24.276 +instance %quote proof
  24.277 +  fix p :: "\<alpha>\<Colon>monoidl \<times> \<beta>\<Colon>monoidl"
  24.278 +  show "\<one> \<otimes> p = p"
  24.279 +    unfolding neutral_prod_def mult_prod_def by (simp add: neutl)
  24.280 +qed
  24.281 +
  24.282 +end %quote
  24.283 +
  24.284 +text {*
  24.285 +  \noindent Fully-fledged monoids are modelled by another subclass
  24.286 +  which does not add new parameters but tightens the specification:
  24.287 +*}
  24.288 +
  24.289 +class %quote monoid = monoidl +
  24.290 +  assumes neutr: "x \<otimes> \<one> = x"
  24.291 +
  24.292 +instantiation %quote nat and int :: monoid 
  24.293 +begin
  24.294 +
  24.295 +instance %quote proof
  24.296 +  fix n :: nat
  24.297 +  show "n \<otimes> \<one> = n"
  24.298 +    unfolding neutral_nat_def by (induct n) simp_all
  24.299 +next
  24.300 +  fix k :: int
  24.301 +  show "k \<otimes> \<one> = k"
  24.302 +    unfolding neutral_int_def mult_int_def by simp
  24.303 +qed
  24.304 +
  24.305 +end %quote
  24.306 +
  24.307 +instantiation %quote * :: (monoid, monoid) monoid
  24.308 +begin
  24.309 +
  24.310 +instance %quote proof 
  24.311 +  fix p :: "\<alpha>\<Colon>monoid \<times> \<beta>\<Colon>monoid"
  24.312 +  show "p \<otimes> \<one> = p"
  24.313 +    unfolding neutral_prod_def mult_prod_def by (simp add: neutr)
  24.314 +qed
  24.315 +
  24.316 +end %quote
  24.317 +
  24.318 +text {*
  24.319 +  \noindent To finish our small algebra example, we add a @{text group} class
  24.320 +  with a corresponding instance:
  24.321 +*}
  24.322 +
  24.323 +class %quote group = monoidl +
  24.324 +  fixes inverse :: "\<alpha> \<Rightarrow> \<alpha>"    ("(_\<div>)" [1000] 999)
  24.325 +  assumes invl: "x\<div> \<otimes> x = \<one>"
  24.326 +
  24.327 +instantiation %quote int :: group
  24.328 +begin
  24.329 +
  24.330 +definition %quote
  24.331 +  inverse_int_def: "i\<div> = - (i\<Colon>int)"
  24.332 +
  24.333 +instance %quote proof
  24.334 +  fix i :: int
  24.335 +  have "-i + i = 0" by simp
  24.336 +  then show "i\<div> \<otimes> i = \<one>"
  24.337 +    unfolding mult_int_def neutral_int_def inverse_int_def .
  24.338 +qed
  24.339 +
  24.340 +end %quote
  24.341 +
  24.342 +
  24.343 +section {* Type classes as locales *}
  24.344 +
  24.345 +subsection {* A look behind the scene *}
  24.346 +
  24.347 +text {*
  24.348 +  The example above gives an impression how Isar type classes work
  24.349 +  in practice.  As stated in the introduction, classes also provide
  24.350 +  a link to Isar's locale system.  Indeed, the logical core of a class
  24.351 +  is nothing else than a locale:
  24.352 +*}
  24.353 +
  24.354 +class %quote idem =
  24.355 +  fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
  24.356 +  assumes idem: "f (f x) = f x"
  24.357 +
  24.358 +text {*
  24.359 +  \noindent essentially introduces the locale
  24.360 +*} (*<*)setup %invisible {* Sign.add_path "foo" *}
  24.361 +(*>*)
  24.362 +locale %quote idem =
  24.363 +  fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
  24.364 +  assumes idem: "f (f x) = f x"
  24.365 +
  24.366 +text {* \noindent together with corresponding constant(s): *}
  24.367 +
  24.368 +consts %quote f :: "\<alpha> \<Rightarrow> \<alpha>"
  24.369 +
  24.370 +text {*
  24.371 +  \noindent The connection to the type system is done by means
  24.372 +  of a primitive axclass
  24.373 +*} (*<*)setup %invisible {* Sign.add_path "foo" *}
  24.374 +(*>*)
  24.375 +axclass %quote idem < type
  24.376 +  idem: "f (f x) = f x" (*<*)setup %invisible {* Sign.parent_path *}(*>*)
  24.377 +
  24.378 +text {* \noindent together with a corresponding interpretation: *}
  24.379 +
  24.380 +interpretation %quote idem_class:
  24.381 +  idem "f \<Colon> (\<alpha>\<Colon>idem) \<Rightarrow> \<alpha>"
  24.382 +proof qed (rule idem)
  24.383 +
  24.384 +text {*
  24.385 +  \noindent This gives you at hand the full power of the Isabelle module system;
  24.386 +  conclusions in locale @{text idem} are implicitly propagated
  24.387 +  to class @{text idem}.
  24.388 +*} (*<*)setup %invisible {* Sign.parent_path *}
  24.389 +(*>*)
  24.390 +subsection {* Abstract reasoning *}
  24.391 +
  24.392 +text {*
  24.393 +  Isabelle locales enable reasoning at a general level, while results
  24.394 +  are implicitly transferred to all instances.  For example, we can
  24.395 +  now establish the @{text "left_cancel"} lemma for groups, which
  24.396 +  states that the function @{text "(x \<otimes>)"} is injective:
  24.397 +*}
  24.398 +
  24.399 +lemma %quote (in group) left_cancel: "x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"
  24.400 +proof
  24.401 +  assume "x \<otimes> y = x \<otimes> z"
  24.402 +  then have "x\<div> \<otimes> (x \<otimes> y) = x\<div> \<otimes> (x \<otimes> z)" by simp
  24.403 +  then have "(x\<div> \<otimes> x) \<otimes> y = (x\<div> \<otimes> x) \<otimes> z" using assoc by simp
  24.404 +  then show "y = z" using neutl and invl by simp
  24.405 +next
  24.406 +  assume "y = z"
  24.407 +  then show "x \<otimes> y = x \<otimes> z" by simp
  24.408 +qed
  24.409 +
  24.410 +text {*
  24.411 +  \noindent Here the \qt{@{keyword "in"} @{class group}} target specification
  24.412 +  indicates that the result is recorded within that context for later
  24.413 +  use.  This local theorem is also lifted to the global one @{fact
  24.414 +  "group.left_cancel:"} @{prop [source] "\<And>x y z \<Colon> \<alpha>\<Colon>group. x \<otimes> y = x \<otimes>
  24.415 +  z \<longleftrightarrow> y = z"}.  Since type @{text "int"} has been made an instance of
  24.416 +  @{text "group"} before, we may refer to that fact as well: @{prop
  24.417 +  [source] "\<And>x y z \<Colon> int. x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"}.
  24.418 +*}
  24.419 +
  24.420 +
  24.421 +subsection {* Derived definitions *}
  24.422 +
  24.423 +text {*
  24.424 +  Isabelle locales support a concept of local definitions
  24.425 +  in locales:
  24.426 +*}
  24.427 +
  24.428 +primrec %quote (in monoid) pow_nat :: "nat \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
  24.429 +  "pow_nat 0 x = \<one>"
  24.430 +  | "pow_nat (Suc n) x = x \<otimes> pow_nat n x"
  24.431 +
  24.432 +text {*
  24.433 +  \noindent If the locale @{text group} is also a class, this local
  24.434 +  definition is propagated onto a global definition of
  24.435 +  @{term [source] "pow_nat \<Colon> nat \<Rightarrow> \<alpha>\<Colon>monoid \<Rightarrow> \<alpha>\<Colon>monoid"}
  24.436 +  with corresponding theorems
  24.437 +
  24.438 +  @{thm pow_nat.simps [no_vars]}.
  24.439 +
  24.440 +  \noindent As you can see from this example, for local
  24.441 +  definitions you may use any specification tool
  24.442 +  which works together with locales (e.g. \cite{krauss2006}).
  24.443 +*}
  24.444 +
  24.445 +
  24.446 +subsection {* A functor analogy *}
  24.447 +
  24.448 +text {*
  24.449 +  We introduced Isar classes by analogy to type classes
  24.450 +  functional programming;  if we reconsider this in the
  24.451 +  context of what has been said about type classes and locales,
  24.452 +  we can drive this analogy further by stating that type
  24.453 +  classes essentially correspond to functors which have
  24.454 +  a canonical interpretation as type classes.
  24.455 +  Anyway, there is also the possibility of other interpretations.
  24.456 +  For example, also @{text list}s form a monoid with
  24.457 +  @{text append} and @{term "[]"} as operations, but it
  24.458 +  seems inappropriate to apply to lists
  24.459 +  the same operations as for genuinely algebraic types.
  24.460 +  In such a case, we simply can do a particular interpretation
  24.461 +  of monoids for lists:
  24.462 +*}
  24.463 +
  24.464 +interpretation %quote list_monoid!: monoid append "[]"
  24.465 +  proof qed auto
  24.466 +
  24.467 +text {*
  24.468 +  \noindent This enables us to apply facts on monoids
  24.469 +  to lists, e.g. @{thm list_monoid.neutl [no_vars]}.
  24.470 +
  24.471 +  When using this interpretation pattern, it may also
  24.472 +  be appropriate to map derived definitions accordingly:
  24.473 +*}
  24.474 +
  24.475 +primrec %quote replicate :: "nat \<Rightarrow> \<alpha> list \<Rightarrow> \<alpha> list" where
  24.476 +  "replicate 0 _ = []"
  24.477 +  | "replicate (Suc n) xs = xs @ replicate n xs"
  24.478 +
  24.479 +interpretation %quote list_monoid!: monoid append "[]" where
  24.480 +  "monoid.pow_nat append [] = replicate"
  24.481 +proof -
  24.482 +  interpret monoid append "[]" ..
  24.483 +  show "monoid.pow_nat append [] = replicate"
  24.484 +  proof
  24.485 +    fix n
  24.486 +    show "monoid.pow_nat append [] n = replicate n"
  24.487 +      by (induct n) auto
  24.488 +  qed
  24.489 +qed intro_locales
  24.490 +
  24.491 +
  24.492 +subsection {* Additional subclass relations *}
  24.493 +
  24.494 +text {*
  24.495 +  Any @{text "group"} is also a @{text "monoid"};  this
  24.496 +  can be made explicit by claiming an additional
  24.497 +  subclass relation,
  24.498 +  together with a proof of the logical difference:
  24.499 +*}
  24.500 +
  24.501 +subclass %quote (in group) monoid
  24.502 +proof
  24.503 +  fix x
  24.504 +  from invl have "x\<div> \<otimes> x = \<one>" by simp
  24.505 +  with assoc [symmetric] neutl invl have "x\<div> \<otimes> (x \<otimes> \<one>) = x\<div> \<otimes> x" by simp
  24.506 +  with left_cancel show "x \<otimes> \<one> = x" by simp
  24.507 +qed
  24.508 +
  24.509 +text {*
  24.510 +  The logical proof is carried out on the locale level.
  24.511 +  Afterwards it is propagated
  24.512 +  to the type system, making @{text group} an instance of
  24.513 +  @{text monoid} by adding an additional edge
  24.514 +  to the graph of subclass relations
  24.515 +  (cf.\ \figref{fig:subclass}).
  24.516 +
  24.517 +  \begin{figure}[htbp]
  24.518 +   \begin{center}
  24.519 +     \small
  24.520 +     \unitlength 0.6mm
  24.521 +     \begin{picture}(40,60)(0,0)
  24.522 +       \put(20,60){\makebox(0,0){@{text semigroup}}}
  24.523 +       \put(20,40){\makebox(0,0){@{text monoidl}}}
  24.524 +       \put(00,20){\makebox(0,0){@{text monoid}}}
  24.525 +       \put(40,00){\makebox(0,0){@{text group}}}
  24.526 +       \put(20,55){\vector(0,-1){10}}
  24.527 +       \put(15,35){\vector(-1,-1){10}}
  24.528 +       \put(25,35){\vector(1,-3){10}}
  24.529 +     \end{picture}
  24.530 +     \hspace{8em}
  24.531 +     \begin{picture}(40,60)(0,0)
  24.532 +       \put(20,60){\makebox(0,0){@{text semigroup}}}
  24.533 +       \put(20,40){\makebox(0,0){@{text monoidl}}}
  24.534 +       \put(00,20){\makebox(0,0){@{text monoid}}}
  24.535 +       \put(40,00){\makebox(0,0){@{text group}}}
  24.536 +       \put(20,55){\vector(0,-1){10}}
  24.537 +       \put(15,35){\vector(-1,-1){10}}
  24.538 +       \put(05,15){\vector(3,-1){30}}
  24.539 +     \end{picture}
  24.540 +     \caption{Subclass relationship of monoids and groups:
  24.541 +        before and after establishing the relationship
  24.542 +        @{text "group \<subseteq> monoid"};  transitive edges are left out.}
  24.543 +     \label{fig:subclass}
  24.544 +   \end{center}
  24.545 +  \end{figure}
  24.546 +
  24.547 +  For illustration, a derived definition
  24.548 +  in @{text group} which uses @{text pow_nat}:
  24.549 +*}
  24.550 +
  24.551 +definition %quote (in group) pow_int :: "int \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
  24.552 +  "pow_int k x = (if k >= 0
  24.553 +    then pow_nat (nat k) x
  24.554 +    else (pow_nat (nat (- k)) x)\<div>)"
  24.555 +
  24.556 +text {*
  24.557 +  \noindent yields the global definition of
  24.558 +  @{term [source] "pow_int \<Colon> int \<Rightarrow> \<alpha>\<Colon>group \<Rightarrow> \<alpha>\<Colon>group"}
  24.559 +  with the corresponding theorem @{thm pow_int_def [no_vars]}.
  24.560 +*}
  24.561 +
  24.562 +subsection {* A note on syntax *}
  24.563 +
  24.564 +text {*
  24.565 +  As a commodity, class context syntax allows to refer
  24.566 +  to local class operations and their global counterparts
  24.567 +  uniformly;  type inference resolves ambiguities.  For example:
  24.568 +*}
  24.569 +
  24.570 +context %quote semigroup
  24.571 +begin
  24.572 +
  24.573 +term %quote "x \<otimes> y" -- {* example 1 *}
  24.574 +term %quote "(x\<Colon>nat) \<otimes> y" -- {* example 2 *}
  24.575 +
  24.576 +end  %quote
  24.577 +
  24.578 +term %quote "x \<otimes> y" -- {* example 3 *}
  24.579 +
  24.580 +text {*
  24.581 +  \noindent Here in example 1, the term refers to the local class operation
  24.582 +  @{text "mult [\<alpha>]"}, whereas in example 2 the type constraint
  24.583 +  enforces the global class operation @{text "mult [nat]"}.
  24.584 +  In the global context in example 3, the reference is
  24.585 +  to the polymorphic global class operation @{text "mult [?\<alpha> \<Colon> semigroup]"}.
  24.586 +*}
  24.587 +
  24.588 +section {* Further issues *}
  24.589 +
  24.590 +subsection {* Type classes and code generation *}
  24.591 +
  24.592 +text {*
  24.593 +  Turning back to the first motivation for type classes,
  24.594 +  namely overloading, it is obvious that overloading
  24.595 +  stemming from @{command class} statements and
  24.596 +  @{command instantiation}
  24.597 +  targets naturally maps to Haskell type classes.
  24.598 +  The code generator framework \cite{isabelle-codegen} 
  24.599 +  takes this into account.  Concerning target languages
  24.600 +  lacking type classes (e.g.~SML), type classes
  24.601 +  are implemented by explicit dictionary construction.
  24.602 +  As example, let's go back to the power function:
  24.603 +*}
  24.604 +
  24.605 +definition %quote example :: int where
  24.606 +  "example = pow_int 10 (-2)"
  24.607 +
  24.608 +text {*
  24.609 +  \noindent This maps to Haskell as:
  24.610 +*}
  24.611 +
  24.612 +text %quote {*@{code_stmts example (Haskell)}*}
  24.613 +
  24.614 +text {*
  24.615 +  \noindent The whole code in SML with explicit dictionary passing:
  24.616 +*}
  24.617 +
  24.618 +text %quote {*@{code_stmts example (SML)}*}
  24.619 +
  24.620 +subsection {* Inspecting the type class universe *}
  24.621 +
  24.622 +text {*
  24.623 +  To facilitate orientation in complex subclass structures,
  24.624 +  two diagnostics commands are provided:
  24.625 +
  24.626 +  \begin{description}
  24.627 +
  24.628 +    \item[@{command "print_classes"}] print a list of all classes
  24.629 +      together with associated operations etc.
  24.630 +
  24.631 +    \item[@{command "class_deps"}] visualizes the subclass relation
  24.632 +      between all classes as a Hasse diagram.
  24.633 +
  24.634 +  \end{description}
  24.635 +*}
  24.636 +
  24.637 +end