src/HOL/ex/Classpackage.thy
author urbanc
Tue Jun 05 09:56:19 2007 +0200 (2007-06-05)
changeset 23243 a37d3e6e8323
parent 22845 5f9138bcb3d7
child 23810 f5e6932d0500
permissions -rw-r--r--
included Class.thy in the compiling process for Nominal/Examples
     1 (*  ID:         $Id$
     2     Author:     Florian Haftmann, TU Muenchen
     3 *)
     4 
     5 header {* Test and examples for Isar class package *}
     6 
     7 theory Classpackage
     8 imports Main
     9 begin
    10 
    11 class semigroup = type +
    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 mult_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 mult_int_def show "k \<otimes> l \<otimes> j = k \<otimes> (l \<otimes> j)" by simp
    27 qed
    28 
    29 instance list :: (type) 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 mult_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 nat :: monoidl and int :: monoidl
    45   "\<one> \<equiv> 0"
    46   "\<one> \<equiv> 0"
    47 proof
    48   fix n :: nat
    49   from mult_nat_def one_nat_def show "\<one> \<otimes> n = n" by simp
    50 next
    51   fix k :: int
    52   from mult_int_def one_int_def show "\<one> \<otimes> k = k" by simp
    53 qed
    54 
    55 instance list :: (type) 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 one_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 list :: (type) 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 one_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 nat :: monoid_comm and int :: monoid_comm
    85 proof
    86   fix n :: nat
    87   from mult_nat_def one_nat_def show "n \<otimes> \<one> = n" by simp
    88 next
    89   fix n m :: nat
    90   from mult_nat_def show "n \<otimes> m = m \<otimes> n" by simp
    91 next
    92   fix k :: int
    93   from mult_int_def one_int_def show "k \<otimes> \<one> = k" by simp
    94 next
    95   fix k l :: int
    96   from mult_int_def show "k \<otimes> l = l \<otimes> k" by simp
    97 qed
    98 
    99 context monoid
   100 begin
   101 
   102 definition
   103   units :: "'a set" where
   104   "units = {y. \<exists>x. x \<^loc>\<otimes> y = \<^loc>\<one> \<and> y \<^loc>\<otimes> x = \<^loc>\<one>}"
   105 
   106 end
   107 
   108 context monoid
   109 begin
   110 
   111 lemma inv_obtain:
   112   assumes "x \<in> units"
   113   obtains y where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>"
   114 proof -
   115   from assms units_def obtain y
   116     where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>" by auto
   117   thus ?thesis ..
   118 qed
   119 
   120 lemma inv_unique:
   121   assumes "y \<^loc>\<otimes> x = \<^loc>\<one>" "x \<^loc>\<otimes> y' = \<^loc>\<one>"
   122   shows "y = y'"
   123 proof -
   124   from assms neutr have "y = y \<^loc>\<otimes> (x \<^loc>\<otimes> y')" by simp
   125   also with assoc have "... = (y \<^loc>\<otimes> x) \<^loc>\<otimes> y'" by simp
   126   also with assms neutl have "... = y'" by simp
   127   finally show ?thesis .
   128 qed
   129 
   130 lemma units_inv_comm:
   131   assumes inv: "x \<^loc>\<otimes> y = \<^loc>\<one>"
   132     and G: "x \<in> units"
   133   shows "y \<^loc>\<otimes> x = \<^loc>\<one>"
   134 proof -
   135   from G inv_obtain obtain z
   136     where z_choice: "z \<^loc>\<otimes> x = \<^loc>\<one>" by blast
   137   from inv neutl neutr have "x \<^loc>\<otimes> y \<^loc>\<otimes> x = x \<^loc>\<otimes> \<^loc>\<one>" by simp
   138   with assoc have "z \<^loc>\<otimes> x \<^loc>\<otimes> y \<^loc>\<otimes> x = z \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<one>" by simp
   139   with neutl z_choice show ?thesis by simp
   140 qed
   141 
   142 end
   143 
   144 consts
   145   reduce :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
   146 
   147 primrec
   148   "reduce f g 0 x = g"
   149   "reduce f g (Suc n) x = f x (reduce f g n x)"
   150 
   151 context monoid
   152 begin
   153 
   154 definition
   155   npow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" where
   156   npow_def_prim: "npow n x = reduce (op \<^loc>\<otimes>) \<^loc>\<one> n x"
   157 
   158 end
   159 
   160 context monoid
   161 begin
   162 
   163 abbreviation
   164   npow_syn :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infix "\<^loc>\<up>" 75) where
   165   "x \<^loc>\<up> n \<equiv> npow n x"
   166 
   167 lemma npow_def:
   168   "x \<^loc>\<up> 0 = \<^loc>\<one>"
   169   "x \<^loc>\<up> Suc n = x \<^loc>\<otimes> x \<^loc>\<up> n"
   170 using npow_def_prim by simp_all
   171 
   172 lemma nat_pow_one:
   173   "\<^loc>\<one> \<^loc>\<up> n = \<^loc>\<one>"
   174 using npow_def neutl by (induct n) simp_all
   175 
   176 lemma nat_pow_mult: "x \<^loc>\<up> n \<^loc>\<otimes> x \<^loc>\<up> m = x \<^loc>\<up> (n + m)"
   177 proof (induct n)
   178   case 0 with neutl npow_def show ?case by simp
   179 next
   180   case (Suc n) with Suc.hyps assoc npow_def show ?case by simp
   181 qed
   182 
   183 lemma nat_pow_pow: "(x \<^loc>\<up> m) \<^loc>\<up> n = x \<^loc>\<up> (n * m)"
   184 using npow_def nat_pow_mult by (induct n) simp_all
   185 
   186 end
   187 
   188 class group = monoidl +
   189   fixes inv :: "'a \<Rightarrow> 'a" ("\<^loc>\<div> _" [81] 80)
   190   assumes invl: "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>"
   191 
   192 class group_comm = group + monoid_comm
   193 
   194 instance int :: group_comm
   195   "\<div> k \<equiv> - (k\<Colon>int)"
   196 proof
   197   fix k :: int
   198   from mult_int_def one_int_def inv_int_def show "\<div> k \<otimes> k = \<one>" by simp
   199 qed
   200 
   201 lemma (in group) cancel:
   202   "(x \<^loc>\<otimes> y = x \<^loc>\<otimes> z) = (y = z)"
   203 proof
   204   fix x y z :: 'a
   205   assume eq: "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z"
   206   hence "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> y) = \<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> z)" by simp
   207   with assoc have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> y = \<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> z" by simp
   208   with neutl invl show "y = z" by simp
   209 next
   210   fix x y z :: 'a
   211   assume eq: "y = z"
   212   thus "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z" by simp
   213 qed
   214 
   215 lemma (in group) neutr:
   216   "x \<^loc>\<otimes> \<^loc>\<one> = x"
   217 proof -
   218   from invl have "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>" by simp
   219   with assoc [symmetric] neutl invl have "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<one>) = \<^loc>\<div> x \<^loc>\<otimes> x" by simp
   220   with cancel show ?thesis by simp
   221 qed
   222 
   223 lemma (in group) invr:
   224   "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<one>"
   225 proof -
   226   from neutl invl have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x" by simp
   227   with neutr have "\<^loc>\<div> x \<^loc>\<otimes> x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x \<^loc>\<otimes> \<^loc>\<one> " by simp
   228   with assoc have "\<^loc>\<div> x \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<div> x) = \<^loc>\<div> x \<^loc>\<otimes> \<^loc>\<one> " by simp
   229   with cancel show ?thesis ..
   230 qed
   231 
   232 instance advanced group < monoid
   233 proof unfold_locales
   234   fix x
   235   from neutr show "x \<^loc>\<otimes> \<^loc>\<one> = x" .
   236 qed
   237 
   238 lemma (in group) all_inv [intro]:
   239   "(x\<Colon>'a) \<in> monoid.units (op \<^loc>\<otimes>) \<^loc>\<one>"
   240   unfolding units_def
   241 proof -
   242   fix x :: "'a"
   243   from invl invr have "\<^loc>\<div> x \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<one>" . 
   244   then obtain y where "y \<^loc>\<otimes> x = \<^loc>\<one>" and "x \<^loc>\<otimes> y = \<^loc>\<one>" ..
   245   hence "\<exists>y\<Colon>'a. y \<^loc>\<otimes> x = \<^loc>\<one> \<and> x \<^loc>\<otimes> y = \<^loc>\<one>" by blast
   246   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
   247 qed
   248 
   249 lemma (in group) cancer:
   250   "(y \<^loc>\<otimes> x = z \<^loc>\<otimes> x) = (y = z)"
   251 proof
   252   assume eq: "y \<^loc>\<otimes> x = z \<^loc>\<otimes> x"
   253   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)
   254   with invr neutr show "y = z" by simp
   255 next
   256   assume eq: "y = z"
   257   thus "y \<^loc>\<otimes> x = z \<^loc>\<otimes> x" by simp
   258 qed
   259 
   260 lemma (in group) inv_one:
   261   "\<^loc>\<div> \<^loc>\<one> = \<^loc>\<one>"
   262 proof -
   263   from neutl have "\<^loc>\<div> \<^loc>\<one> = \<^loc>\<one> \<^loc>\<otimes> (\<^loc>\<div> \<^loc>\<one>)" ..
   264   moreover from invr have "... = \<^loc>\<one>" by simp
   265   finally show ?thesis .
   266 qed
   267 
   268 lemma (in group) inv_inv:
   269   "\<^loc>\<div> (\<^loc>\<div> x) = x"
   270 proof -
   271   from invl invr neutr
   272     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
   273   with assoc [symmetric]
   274     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
   275   with invl neutr show ?thesis by simp
   276 qed
   277 
   278 lemma (in group) inv_mult_group:
   279   "\<^loc>\<div> (x \<^loc>\<otimes> y) = \<^loc>\<div> y \<^loc>\<otimes> \<^loc>\<div> x"
   280 proof -
   281   from invl have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> (x \<^loc>\<otimes> y) = \<^loc>\<one>" by simp
   282   with assoc have "\<^loc>\<div> (x \<^loc>\<otimes> y) \<^loc>\<otimes> x \<^loc>\<otimes> y = \<^loc>\<one>" by simp
   283   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
   284   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
   285   with invr neutr show ?thesis by simp
   286 qed
   287 
   288 lemma (in group) inv_comm:
   289   "x \<^loc>\<otimes> \<^loc>\<div> x = \<^loc>\<div> x \<^loc>\<otimes> x"
   290 using invr invl by simp
   291 
   292 definition (in group)
   293   pow :: "int \<Rightarrow> 'a \<Rightarrow> 'a" where
   294   "pow k x = (if k < 0 then \<^loc>\<div> (monoid.npow (op \<^loc>\<otimes>) \<^loc>\<one> (nat (-k)) x)
   295     else (monoid.npow (op \<^loc>\<otimes>) \<^loc>\<one> (nat k) x))"
   296 
   297 abbreviation (in group)
   298   pow_syn :: "'a \<Rightarrow> int \<Rightarrow> 'a" (infix "\<^loc>\<up>" 75) where
   299   "x \<^loc>\<up> k \<equiv> pow k x"
   300 
   301 lemma (in group) int_pow_zero:
   302   "x \<^loc>\<up> (0\<Colon>int) = \<^loc>\<one>"
   303 using npow_def pow_def by simp
   304 
   305 lemma (in group) int_pow_one:
   306   "\<^loc>\<one> \<^loc>\<up> (k\<Colon>int) = \<^loc>\<one>"
   307 using pow_def nat_pow_one inv_one by simp
   308 
   309 instance * :: (semigroup, semigroup) semigroup
   310   mult_prod_def: "x \<otimes> y \<equiv> let (x1, x2) = x; (y1, y2) = y in
   311               (x1 \<otimes> y1, x2 \<otimes> y2)"
   312 by default (simp_all add: split_paired_all mult_prod_def assoc)
   313 
   314 instance * :: (monoidl, monoidl) monoidl
   315   one_prod_def: "\<one> \<equiv> (\<one>, \<one>)"
   316 by default (simp_all add: split_paired_all mult_prod_def one_prod_def neutl)
   317 
   318 instance * :: (monoid, monoid) monoid
   319 by default (simp_all add: split_paired_all mult_prod_def one_prod_def monoid_class.mult_one.neutr)
   320 
   321 instance * :: (monoid_comm, monoid_comm) monoid_comm
   322 by default (simp_all add: split_paired_all mult_prod_def comm)
   323 
   324 instance * :: (group, group) group
   325   inv_prod_def: "\<div> x \<equiv> let (x1, x2) = x in (\<div> x1, \<div> x2)"
   326 by default (simp_all add: split_paired_all mult_prod_def one_prod_def inv_prod_def invl)
   327 
   328 instance * :: (group_comm, group_comm) group_comm
   329 by default (simp_all add: split_paired_all mult_prod_def comm)
   330 
   331  
   332 definition
   333   "X a b c = (a \<otimes> \<one> \<otimes> b, a \<otimes> \<one> \<otimes> b, [a, b] \<otimes> \<one> \<otimes> [a, b, c])"
   334 definition
   335   "Y a b c = (a, \<div> a) \<otimes> \<one> \<otimes> \<div> (b, \<div> c)"
   336 
   337 definition "x1 = X (1::nat) 2 3"
   338 definition "x2 = X (1::int) 2 3"
   339 definition "y2 = Y (1::int) 2 3"
   340 
   341 code_gen x1 x2 y2 in SML in OCaml file - in Haskell file -
   342 
   343 end