src/HOL/ex/Classpackage.thy
author haftmann
Fri Jul 21 14:47:44 2006 +0200 (2006-07-21)
changeset 20178 e56fa3c8b1f1
parent 20106 a3d4b4eb35b9
child 20187 af47971ea304
permissions -rw-r--r--
adaption to changes in class_package
     1 (*  ID:         $Id$
     2     Author:     Florian Haftmann, TU Muenchen
     3 *)
     4 
     5 header {* Test and Examples for Pure/Tools/class_package.ML *}
     6 
     7 theory Classpackage
     8 imports Main
     9 begin
    10 
    11 class semigroup =
    12   fixes mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>\<otimes>" 70)
    13   assumes assoc: "x \<^loc>\<otimes> y \<^loc>\<otimes> z = x \<^loc>\<otimes> (y \<^loc>\<otimes> z)"
    14 
    15 instance nat :: semigroup
    16   "m \<otimes> n \<equiv> m + n"
    17 proof
    18   fix m n q :: nat 
    19   from semigroup_nat_def show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)" by simp
    20 qed
    21 
    22 instance int :: semigroup
    23   "k \<otimes> l \<equiv> k + l"
    24 proof
    25   fix k l j :: int
    26   from semigroup_int_def show "k \<otimes> l \<otimes> j = k \<otimes> (l \<otimes> j)" by simp
    27 qed
    28 
    29 instance (type) list :: semigroup
    30   "xs \<otimes> ys \<equiv> xs @ ys"
    31 proof
    32   fix xs ys zs :: "'a list"
    33   show "xs \<otimes> ys \<otimes> zs = xs \<otimes> (ys \<otimes> zs)"
    34   proof -
    35     from semigroup_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
    36     thus ?thesis by simp
    37   qed
    38 qed
    39 
    40 class monoidl = semigroup +
    41   fixes one :: 'a ("\<^loc>\<one>")
    42   assumes neutl: "\<^loc>\<one> \<^loc>\<otimes> x = x"
    43 
    44 instance monoidl_num_def: nat :: monoidl and int :: monoidl
    45   "\<one> \<equiv> 0"
    46   "\<one> \<equiv> 0"
    47 proof
    48   fix n :: nat
    49   from monoidl_num_def show "\<one> \<otimes> n = n" by simp
    50 next
    51   fix k :: int
    52   from monoidl_num_def show "\<one> \<otimes> k = k" by simp
    53 qed
    54 
    55 instance (type) list :: monoidl
    56   "\<one> \<equiv> []"
    57 proof
    58   fix xs :: "'a list"
    59   show "\<one> \<otimes> xs = xs"
    60   proof -
    61     from mult_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
    62     moreover from monoidl_list_def have "\<one> \<equiv> []\<Colon>'a list" by simp
    63     ultimately show ?thesis by simp
    64   qed
    65 qed  
    66 
    67 class monoid = monoidl +
    68   assumes neutr: "x \<^loc>\<otimes> \<^loc>\<one> = x"
    69 
    70 instance monoid_list_def: (type) list :: monoid
    71 proof
    72   fix xs :: "'a list"
    73   show "xs \<otimes> \<one> = xs"
    74   proof -
    75     from mult_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
    76     moreover from monoid_list_def have "\<one> \<equiv> []\<Colon>'a list" by simp
    77     ultimately show ?thesis by simp
    78   qed
    79 qed  
    80 
    81 class monoid_comm = monoid +
    82   assumes comm: "x \<^loc>\<otimes> y = y \<^loc>\<otimes> x"
    83 
    84 instance monoid_comm_num_def: nat :: monoid_comm and int :: monoid_comm
    85 proof
    86   fix n :: nat
    87   from monoid_comm_num_def show "n \<otimes> \<one> = n" by simp
    88 next
    89   fix n m :: nat
    90   from monoid_comm_num_def show "n \<otimes> m = m \<otimes> n" by simp
    91 next
    92   fix k :: int
    93   from monoid_comm_num_def show "k \<otimes> \<one> = k" by simp
    94 next
    95   fix k l :: int
    96   from monoid_comm_num_def show "k \<otimes> l = l \<otimes> k" by simp
    97 qed
    98 
    99 definition (in monoid)
   100   units :: "'a set"
   101   units_def: "units = { y. \<exists>x. x \<^loc>\<otimes> y = \<^loc>\<one> \<and> y \<^loc>\<otimes> x = \<^loc>\<one> }"
   102 
   103 lemma (in monoid) inv_obtain:
   104   assumes ass: "x \<in> units"
   105   obtains y where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>"
   106 proof -
   107   from ass units_def obtain y
   108     where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>" by auto
   109   thus ?thesis ..
   110 qed
   111 
   112 lemma (in monoid) inv_unique:
   113   assumes eq: "y \<^loc>\<otimes> x = \<^loc>\<one>" "x \<^loc>\<otimes> y' = \<^loc>\<one>"
   114   shows "y = y'"
   115 proof -
   116   from eq neutr have "y = y \<^loc>\<otimes> (x \<^loc>\<otimes> y')" by simp
   117   also with assoc have "... = (y \<^loc>\<otimes> x) \<^loc>\<otimes> y'" by simp
   118   also with eq neutl have "... = y'" by simp
   119   finally show ?thesis .
   120 qed
   121 
   122 lemma (in monoid) units_inv_comm:
   123   assumes inv: "x \<^loc>\<otimes> y = \<^loc>\<one>"
   124     and G: "x \<in> units"
   125   shows "y \<^loc>\<otimes> x = \<^loc>\<one>"
   126 proof -
   127   from G inv_obtain obtain z
   128     where z_choice: "z \<^loc>\<otimes> x = \<^loc>\<one>" by blast
   129   from inv neutl neutr have "x \<^loc>\<otimes> y \<^loc>\<otimes> x = x \<^loc>\<otimes> \<^loc>\<one>" by simp
   130   with assoc have "z \<^loc>\<otimes> x \<^loc>\<otimes> y \<^loc>\<otimes> x = z \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<one>" by simp
   131   with neutl z_choice show ?thesis by simp
   132 qed
   133 
   134 consts
   135   reduce :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
   136 
   137 primrec
   138   "reduce f g 0 x = g"
   139   "reduce f g (Suc n) x = f x (reduce f g n x)"
   140 
   141 definition (in monoid)
   142   npow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
   143   npow_def_prim: "npow n x = reduce (op \<^loc>\<otimes>) \<^loc>\<one> n x"
   144 
   145 abbreviation (in monoid)
   146   abbrev_npow :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infix "\<^loc>\<up>" 75)
   147   "x \<^loc>\<up> n \<equiv> npow n x"
   148 
   149 lemma (in monoid) npow_def:
   150   "x \<^loc>\<up> 0 = \<^loc>\<one>"
   151   "x \<^loc>\<up> Suc n = x \<^loc>\<otimes> x \<^loc>\<up> n"
   152 using npow_def_prim by simp_all
   153 
   154 lemma (in monoid) nat_pow_one:
   155   "\<^loc>\<one> \<^loc>\<up> n = \<^loc>\<one>"
   156 using npow_def neutl by (induct n) simp_all
   157 
   158 lemma (in monoid) nat_pow_mult:
   159   "npow n x \<^loc>\<otimes> npow m x = npow (n + m) x"
   160 proof (induct n)
   161   case 0 with neutl npow_def show ?case by simp
   162 next
   163   case (Suc n) with Suc.hyps assoc npow_def show ?case by simp
   164 qed
   165 
   166 lemma (in monoid) nat_pow_pow:
   167   "npow n (npow m x) = npow (n * m) x"
   168 using npow_def nat_pow_mult by (induct n) simp_all
   169 
   170 class group = monoidl +
   171   fixes inv :: "'a \<Rightarrow> 'a" ("\<^loc>\<div> _" [81] 80)
   172   assumes invl: "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>"
   173 
   174 class group_comm = group + monoid_comm
   175 
   176 instance group_comm_int_def: int :: group_comm
   177   "\<div> k \<equiv> - (k\<Colon>int)"
   178 proof
   179   fix k :: int
   180   from group_comm_int_def show "\<div> k \<otimes> k = \<one>" by simp
   181 qed
   182 
   183 lemma (in group) cancel:
   184   "(x \<^loc>\<otimes> y = x \<^loc>\<otimes> z) = (y = z)"
   185 proof
   186   fix x y z :: 'a
   187   assume eq: "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z"
   188   hence "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> y) = \<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> z)" by simp
   189   with assoc have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> y = \<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> z" by simp
   190   with neutl invl show "y = z" by simp
   191 next
   192   fix x y z :: 'a
   193   assume eq: "y = z"
   194   thus "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z" by simp
   195 qed
   196 
   197 lemma (in group) neutr:
   198   "x \<^loc>\<otimes> \<^loc>\<one> = x"
   199 proof -
   200   from invl have "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>" by simp
   201   with assoc [symmetric] neutl invl have "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<one>) = \<^loc>\<div> x \<^loc>\<otimes> x" by simp
   202   with cancel show ?thesis by simp
   203 qed
   204 
   205 lemma (in group) invr:
   206   "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<one>"
   207 proof -
   208   from neutl invl have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x" by simp
   209   with neutr have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x \<^loc>\<otimes> \<^loc>\<one> " by simp
   210   with assoc have "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<div> x) = \<^loc>\<div> x \<^loc>\<otimes> \<^loc>\<one> " by simp
   211   with cancel show ?thesis ..
   212 qed
   213 
   214 interpretation group < monoid
   215 proof -
   216   fix x :: "'a"
   217   from neutr show "x \<^loc>\<otimes> \<^loc>\<one> = x" .
   218 qed
   219 
   220 instance group < monoid
   221 proof
   222   fix x :: "'a\<Colon>group"
   223   from group.neutr show "x \<otimes> \<one> = x" .
   224 qed
   225 
   226 lemma (in group) all_inv [intro]:
   227   "(x\<Colon>'a) \<in> monoid.units (op \<^loc>\<otimes>) \<^loc>\<one>"
   228   unfolding units_def
   229 proof -
   230   fix x :: "'a"
   231   from invl invr have "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<one>" . 
   232   then obtain y where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>" ..
   233   hence "\<exists>y\<Colon>'a. y \<^loc>\<otimes> x = \<^loc>\<one> \<and> x \<^loc>\<otimes> y = \<^loc>\<one>" by blast
   234   thus "x \<in> {y\<Colon>'a. \<exists>x\<Colon>'a. x \<^loc>\<otimes> y = \<^loc>\<one> \<and> y \<^loc>\<otimes> x = \<^loc>\<one>}" by simp
   235 qed
   236 
   237 lemma (in group) cancer:
   238   "(y \<^loc>\<otimes> x = z \<^loc>\<otimes> x) = (y = z)"
   239 proof
   240   assume eq: "y \<^loc>\<otimes> x = z \<^loc>\<otimes> x"
   241   with assoc [symmetric] have "y \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<div> x) = z \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<div> x)" by (simp del: invr)
   242   with invr neutr show "y = z" by simp
   243 next
   244   assume eq: "y = z"
   245   thus "y \<^loc>\<otimes> x = z \<^loc>\<otimes> x" by simp
   246 qed
   247 
   248 lemma (in group) inv_one:
   249   "\<^loc>\<div> \<^loc>\<one> = \<^loc>\<one>"
   250 proof -
   251   from neutl have "\<^loc>\<div> \<^loc>\<one> = \<^loc>\<one> \<^loc>\<otimes> (\<^loc>\<div> \<^loc>\<one>)" ..
   252   moreover from invr have "... = \<^loc>\<one>" by simp
   253   finally show ?thesis .
   254 qed
   255 
   256 lemma (in group) inv_inv:
   257   "\<^loc>\<div> (\<^loc>\<div> x) = x"
   258 proof -
   259   from invl invr neutr
   260     have "\<^loc>\<div> (\<^loc>\<div> x) \<^loc>\<otimes> \<^loc>\<div> x \<^loc>\<otimes> x = x \<^loc>\<otimes> \<^loc>\<div> x \<^loc>\<otimes> x" by simp
   261   with assoc [symmetric]
   262     have "\<^loc>\<div> (\<^loc>\<div> x) \<^loc>\<otimes> (\<^loc>\<div> x \<^loc>\<otimes> x) = x \<^loc>\<otimes> (\<^loc>\<div> x \<^loc>\<otimes> x)" by simp
   263   with invl neutr show ?thesis by simp
   264 qed
   265 
   266 lemma (in group) inv_mult_group:
   267   "\<^loc>\<div> (x \<^loc>\<otimes> y) = \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x"
   268 proof -
   269   from invl have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> (x \<^loc>\<otimes> y) = \<^loc>\<one>" by simp
   270   with assoc have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> x \<^loc>\<otimes> y = \<^loc>\<one>" by simp
   271   with neutl have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> x \<^loc>\<otimes> y \<^loc>\<otimes> \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x" by simp
   272   with assoc have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> (x \<^loc>\<otimes> (y \<^loc>\<otimes> \<^loc>\<div> y) \<^loc>\<otimes> \<^loc>\<div> x) = \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x" by simp
   273   with invr neutr show ?thesis by simp
   274 qed
   275 
   276 lemma (in group) inv_comm:
   277   "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x \<^loc>\<otimes> x"
   278 using invr invl by simp
   279 
   280 definition (in group)
   281   pow :: "int \<Rightarrow> 'a \<Rightarrow> 'a"
   282   pow_def: "pow k x = (if k < 0 then \<^loc>\<div> (monoid.npow (op \<^loc>\<otimes>) \<^loc>\<one> (nat (-k)) x)
   283     else (monoid.npow (op \<^loc>\<otimes>) \<^loc>\<one> (nat k) x))"
   284 
   285 abbreviation (in group)
   286   abbrev_pow :: "'a \<Rightarrow> int \<Rightarrow> 'a" (infix "\<^loc>\<up>" 75)
   287   "x \<^loc>\<up> k \<equiv> pow k x"
   288 
   289 lemma (in group) int_pow_zero:
   290   "x \<^loc>\<up> (0\<Colon>int) = \<^loc>\<one>"
   291 using npow_def pow_def by simp
   292 
   293 lemma (in group) int_pow_one:
   294   "\<^loc>\<one> \<^loc>\<up> (k\<Colon>int) = \<^loc>\<one>"
   295 using pow_def nat_pow_one inv_one by simp
   296 
   297 instance group_prod_def: (group, group) * :: group
   298   mult_prod_def: "x \<otimes> y \<equiv> let (x1, x2) = x; (y1, y2) = y in
   299               (x1 \<otimes> y1, x2 \<otimes> y2)"
   300   mult_one_def: "\<one> \<equiv> (\<one>, \<one>)"
   301   mult_inv_def: "\<div> x \<equiv> let (x1, x2) = x in (\<div> x1, \<div> x2)"
   302 by default (simp_all add: split_paired_all group_prod_def assoc neutl invl)
   303 
   304 instance group_comm_prod_def: (group_comm, group_comm) * :: group_comm
   305 by default (simp_all add: split_paired_all group_prod_def assoc neutl invl comm)
   306 
   307 definition
   308   "x = ((2\<Colon>nat) \<otimes> \<one> \<otimes> 3, (2\<Colon>int) \<otimes> \<one> \<otimes> \<div> 3, [1\<Colon>nat, 2] \<otimes> \<one> \<otimes> [1, 2, 3])"
   309   "y = (2 \<Colon> int, \<div> 2 \<Colon> int) \<otimes> \<one> \<otimes> (3, \<div> 3)"
   310 
   311 code_generate "op \<otimes>" \<one> inv
   312 code_generate (ml, haskell) x
   313 code_generate (ml, haskell) y
   314 
   315 code_serialize ml (_)
   316 code_serialize ml (-)
   317 
   318 end