src/HOL/LOrder.thy
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 15524 2ef571f80a55
permissions -rw-r--r--
import -> imports
     1 (*  Title:   HOL/LOrder.thy
     2     ID:      $Id$
     3     Author:  Steven Obua, TU Muenchen
     4 *)
     5 
     6 header {* Lattice Orders *}
     7 
     8 theory LOrder
     9 imports HOL
    10 begin
    11 
    12 text {*
    13   The theory of lattices developed here is taken from the book:
    14   \begin{itemize}
    15   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979. 
    16   \end{itemize}
    17 *}
    18 
    19 constdefs
    20   is_meet :: "(('a::order) \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
    21   "is_meet m == ! a b x. m a b \<le> a \<and> m a b \<le> b \<and> (x \<le> a \<and> x \<le> b \<longrightarrow> x \<le> m a b)"
    22   is_join :: "(('a::order) \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
    23   "is_join j == ! a b x. a \<le> j a b \<and> b \<le> j a b \<and> (a \<le> x \<and> b \<le> x \<longrightarrow> j a b \<le> x)"  
    24 
    25 lemma is_meet_unique: 
    26   assumes "is_meet u" "is_meet v" shows "u = v"
    27 proof -
    28   {
    29     fix a b :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    30     assume a: "is_meet a"
    31     assume b: "is_meet b"
    32     {
    33       fix x y 
    34       let ?za = "a x y"
    35       let ?zb = "b x y"
    36       from a have za_le: "?za <= x & ?za <= y" by (auto simp add: is_meet_def)
    37       with b have "?za <= ?zb" by (auto simp add: is_meet_def)
    38     }
    39   }
    40   note f_le = this
    41   show "u = v" by ((rule ext)+, simp_all add: order_antisym prems f_le) 
    42 qed
    43 
    44 lemma is_join_unique: 
    45   assumes "is_join u" "is_join v" shows "u = v"
    46 proof -
    47   {
    48     fix a b :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    49     assume a: "is_join a"
    50     assume b: "is_join b"
    51     {
    52       fix x y 
    53       let ?za = "a x y"
    54       let ?zb = "b x y"
    55       from a have za_le: "x <= ?za & y <= ?za" by (auto simp add: is_join_def)
    56       with b have "?zb <= ?za" by (auto simp add: is_join_def)
    57     }
    58   }
    59   note f_le = this
    60   show "u = v" by ((rule ext)+, simp_all add: order_antisym prems f_le) 
    61 qed
    62 
    63 axclass join_semilorder < order
    64   join_exists: "? j. is_join j"
    65 
    66 axclass meet_semilorder < order
    67   meet_exists: "? m. is_meet m"
    68 
    69 axclass lorder < join_semilorder, meet_semilorder
    70 
    71 constdefs
    72   meet :: "('a::meet_semilorder) \<Rightarrow> 'a \<Rightarrow> 'a"
    73   "meet == THE m. is_meet m"
    74   join :: "('a::join_semilorder) \<Rightarrow> 'a \<Rightarrow> 'a"
    75   "join ==  THE j. is_join j"
    76 
    77 lemma is_meet_meet: "is_meet (meet::'a \<Rightarrow> 'a \<Rightarrow> ('a::meet_semilorder))"
    78 proof -
    79   from meet_exists obtain k::"'a \<Rightarrow> 'a \<Rightarrow> 'a" where "is_meet k" ..
    80   with is_meet_unique[of _ k] show ?thesis
    81     by (simp add: meet_def theI[of is_meet])    
    82 qed
    83 
    84 lemma meet_unique: "(is_meet m) = (m = meet)" 
    85 by (insert is_meet_meet, auto simp add: is_meet_unique)
    86 
    87 lemma is_join_join: "is_join (join::'a \<Rightarrow> 'a \<Rightarrow> ('a::join_semilorder))"
    88 proof -
    89   from join_exists obtain k::"'a \<Rightarrow> 'a \<Rightarrow> 'a" where "is_join k" ..
    90   with is_join_unique[of _ k] show ?thesis
    91     by (simp add: join_def theI[of is_join])    
    92 qed
    93 
    94 lemma join_unique: "(is_join j) = (j = join)"
    95 by (insert is_join_join, auto simp add: is_join_unique)
    96 
    97 lemma meet_left_le: "meet a b \<le> (a::'a::meet_semilorder)"
    98 by (insert is_meet_meet, auto simp add: is_meet_def)
    99 
   100 lemma meet_right_le: "meet a b \<le> (b::'a::meet_semilorder)"
   101 by (insert is_meet_meet, auto simp add: is_meet_def)
   102 
   103 lemma meet_imp_le: "x \<le> a \<Longrightarrow> x \<le> b \<Longrightarrow> x \<le> meet a (b::'a::meet_semilorder)"
   104 by (insert is_meet_meet, auto simp add: is_meet_def)
   105 
   106 lemma join_left_le: "a \<le> join a (b::'a::join_semilorder)"
   107 by (insert is_join_join, auto simp add: is_join_def)
   108 
   109 lemma join_right_le: "b \<le> join a (b::'a::join_semilorder)"
   110 by (insert is_join_join, auto simp add: is_join_def)
   111 
   112 lemma join_imp_le: "a \<le> x \<Longrightarrow> b \<le> x \<Longrightarrow> join a b \<le> (x::'a::join_semilorder)"
   113 by (insert is_join_join, auto simp add: is_join_def)
   114 
   115 lemmas meet_join_le = meet_left_le meet_right_le join_left_le join_right_le
   116 
   117 lemma is_meet_min: "is_meet (min::'a \<Rightarrow> 'a \<Rightarrow> ('a::linorder))"
   118 by (auto simp add: is_meet_def min_def)
   119 
   120 lemma is_join_max: "is_join (max::'a \<Rightarrow> 'a \<Rightarrow> ('a::linorder))"
   121 by (auto simp add: is_join_def max_def)
   122 
   123 instance linorder \<subseteq> meet_semilorder
   124 proof
   125   from is_meet_min show "? (m::'a\<Rightarrow>'a\<Rightarrow>('a::linorder)). is_meet m" by auto
   126 qed
   127 
   128 instance linorder \<subseteq> join_semilorder
   129 proof
   130   from is_join_max show "? (j::'a\<Rightarrow>'a\<Rightarrow>('a::linorder)). is_join j" by auto 
   131 qed
   132     
   133 instance linorder \<subseteq> lorder ..
   134 
   135 lemma meet_min: "meet = (min :: 'a\<Rightarrow>'a\<Rightarrow>('a::linorder))" 
   136 by (simp add: is_meet_meet is_meet_min is_meet_unique)
   137 
   138 lemma join_max: "join = (max :: 'a\<Rightarrow>'a\<Rightarrow>('a::linorder))"
   139 by (simp add: is_join_join is_join_max is_join_unique)
   140 
   141 lemma meet_idempotent[simp]: "meet x x = x"
   142 by (rule order_antisym, simp_all add: meet_left_le meet_imp_le)
   143 
   144 lemma join_idempotent[simp]: "join x x = x"
   145 by (rule order_antisym, simp_all add: join_left_le join_imp_le)
   146 
   147 lemma meet_comm: "meet x y = meet y x" 
   148 by (rule order_antisym, (simp add: meet_left_le meet_right_le meet_imp_le)+)
   149 
   150 lemma join_comm: "join x y = join y x"
   151 by (rule order_antisym, (simp add: join_right_le join_left_le join_imp_le)+)
   152 
   153 lemma meet_assoc: "meet (meet x y) z = meet x (meet y z)" (is "?l=?r")
   154 proof - 
   155   have "?l <= meet x y & meet x y <= x & ?l <= z & meet x y <= y" by (simp add: meet_left_le meet_right_le)
   156   hence "?l <= x & ?l <= y & ?l <= z" by auto
   157   hence "?l <= ?r" by (simp add: meet_imp_le)
   158   hence a:"?l <= meet x (meet y z)" by (simp add: meet_imp_le)
   159   have "?r <= meet y z & meet y z <= y & meet y z <= z & ?r <= x" by (simp add: meet_left_le meet_right_le)  
   160   hence "?r <= x & ?r <= y & ?r <= z" by (auto) 
   161   hence "?r <= meet x y & ?r <= z" by (simp add: meet_imp_le)
   162   hence b:"?r <= ?l" by (simp add: meet_imp_le)
   163   from a b show "?l = ?r" by auto
   164 qed
   165 
   166 lemma join_assoc: "join (join x y) z = join x (join y z)" (is "?l=?r")
   167 proof -
   168   have "join x y <= ?l & x <= join x y & z <= ?l & y <= join x y" by (simp add: join_left_le join_right_le)
   169   hence "x <= ?l & y <= ?l & z <= ?l" by auto
   170   hence "join y z <= ?l & x <= ?l" by (simp add: join_imp_le)
   171   hence a:"?r <= ?l" by (simp add: join_imp_le)
   172   have "join y z <= ?r & y <= join y z & z <= join y z & x <= ?r" by (simp add: join_left_le join_right_le)
   173   hence "y <= ?r & z <= ?r & x <= ?r" by auto
   174   hence "join x y <= ?r & z <= ?r" by (simp add: join_imp_le)
   175   hence b:"?l <= ?r" by (simp add: join_imp_le)
   176   from a b show "?l = ?r" by auto
   177 qed
   178 
   179 lemma meet_left_comm: "meet a (meet b c) = meet b (meet a c)"
   180 by (simp add: meet_assoc[symmetric, of a b c], simp add: meet_comm[of a b], simp add: meet_assoc)
   181 
   182 lemma meet_left_idempotent: "meet y (meet y x) = meet y x"
   183 by (simp add: meet_assoc meet_comm meet_left_comm)
   184 
   185 lemma join_left_comm: "join a (join b c) = join b (join a c)"
   186 by (simp add: join_assoc[symmetric, of a b c], simp add: join_comm[of a b], simp add: join_assoc)
   187 
   188 lemma join_left_idempotent: "join y (join y x) = join y x"
   189 by (simp add: join_assoc join_comm join_left_comm)
   190     
   191 lemmas meet_aci = meet_assoc meet_comm meet_left_comm meet_left_idempotent
   192 
   193 lemmas join_aci = join_assoc join_comm join_left_comm join_left_idempotent
   194 
   195 lemma le_def_meet: "(x <= y) = (meet x y = x)" 
   196 proof -
   197   have u: "x <= y \<longrightarrow> meet x y = x"
   198   proof 
   199     assume "x <= y"
   200     hence "x <= meet x y & meet x y <= x" by (simp add: meet_imp_le meet_left_le)
   201     thus "meet x y = x" by auto
   202   qed
   203   have v:"meet x y = x \<longrightarrow> x <= y" 
   204   proof 
   205     have a:"meet x y <= y" by (simp add: meet_right_le)
   206     assume "meet x y = x"
   207     hence "x = meet x y" by auto
   208     with a show "x <= y" by (auto)
   209   qed
   210   from u v show ?thesis by blast
   211 qed
   212 
   213 lemma le_def_join: "(x <= y) = (join x y = y)" 
   214 proof -
   215   have u: "x <= y \<longrightarrow> join x y = y"
   216   proof 
   217     assume "x <= y"
   218     hence "join x y <= y & y <= join x y" by (simp add: join_imp_le join_right_le)
   219     thus "join x y = y" by auto
   220   qed
   221   have v:"join x y = y \<longrightarrow> x <= y" 
   222   proof 
   223     have a:"x <= join x y" by (simp add: join_left_le)
   224     assume "join x y = y"
   225     hence "y = join x y" by auto
   226     with a show "x <= y" by (auto)
   227   qed
   228   from u v show ?thesis by blast
   229 qed
   230 
   231 lemma meet_join_absorp: "meet x (join x y) = x"
   232 proof -
   233   have a:"meet x (join x y) <= x" by (simp add: meet_left_le)
   234   have b:"x <= meet x (join x y)" by (rule meet_imp_le, simp_all add: join_left_le)
   235   from a b show ?thesis by auto
   236 qed
   237 
   238 lemma join_meet_absorp: "join x (meet x y) = x"
   239 proof - 
   240   have a:"x <= join x (meet x y)" by (simp add: join_left_le)
   241   have b:"join x (meet x y) <= x" by (rule join_imp_le, simp_all add: meet_left_le)
   242   from a b show ?thesis by auto
   243 qed
   244 
   245 lemma meet_mono: "y \<le> z \<Longrightarrow> meet x y \<le> meet x z"
   246 proof -
   247   assume a: "y <= z"
   248   have "meet x y <= x & meet x y <= y" by (simp add: meet_left_le meet_right_le)
   249   with a have "meet x y <= x & meet x y <= z" by auto 
   250   thus "meet x y <= meet x z" by (simp add: meet_imp_le)
   251 qed
   252 
   253 lemma join_mono: "y \<le> z \<Longrightarrow> join x y \<le> join x z"
   254 proof -
   255   assume a: "y \<le> z"
   256   have "x <= join x z & z <= join x z" by (simp add: join_left_le join_right_le)
   257   with a have "x <= join x z & y <= join x z" by auto
   258   thus "join x y <= join x z" by (simp add: join_imp_le)
   259 qed
   260 
   261 lemma distrib_join_le: "join x (meet y z) \<le> meet (join x y) (join x z)" (is "_ <= ?r")
   262 proof -
   263   have a: "x <= ?r" by (rule meet_imp_le, simp_all add: join_left_le)
   264   from meet_join_le have b: "meet y z <= ?r" 
   265     by (rule_tac meet_imp_le, (blast intro: order_trans)+)
   266   from a b show ?thesis by (simp add: join_imp_le)
   267 qed
   268   
   269 lemma distrib_meet_le: "join (meet x y) (meet x z) \<le> meet x (join y z)" (is "?l <= _") 
   270 proof -
   271   have a: "?l <= x" by (rule join_imp_le, simp_all add: meet_left_le)
   272   from meet_join_le have b: "?l <= join y z" 
   273     by (rule_tac join_imp_le, (blast intro: order_trans)+)
   274   from a b show ?thesis by (simp add: meet_imp_le)
   275 qed
   276 
   277 lemma meet_join_eq_imp_le: "a = c \<or> a = d \<or> b = c \<or> b = d \<Longrightarrow> meet a b \<le> join c d"
   278 by (insert meet_join_le, blast intro: order_trans)
   279 
   280 lemma modular_le: "x \<le> z \<Longrightarrow> join x (meet y z) \<le> meet (join x y) z" (is "_ \<Longrightarrow> ?t <= _")
   281 proof -
   282   assume a: "x <= z"
   283   have b: "?t <= join x y" by (rule join_imp_le, simp_all add: meet_join_le meet_join_eq_imp_le)
   284   have c: "?t <= z" by (rule join_imp_le, simp_all add: meet_join_le a)
   285   from b c show ?thesis by (simp add: meet_imp_le)
   286 qed
   287 
   288 ML {*
   289 val is_meet_unique = thm "is_meet_unique";
   290 val is_join_unique = thm "is_join_unique";
   291 val join_exists = thm "join_exists";
   292 val meet_exists = thm "meet_exists";
   293 val is_meet_meet = thm "is_meet_meet";
   294 val meet_unique = thm "meet_unique";
   295 val is_join_join = thm "is_join_join";
   296 val join_unique = thm "join_unique";
   297 val meet_left_le = thm "meet_left_le";
   298 val meet_right_le = thm "meet_right_le";
   299 val meet_imp_le = thm "meet_imp_le";
   300 val join_left_le = thm "join_left_le";
   301 val join_right_le = thm "join_right_le";
   302 val join_imp_le = thm "join_imp_le";
   303 val meet_join_le = thms "meet_join_le";
   304 val is_meet_min = thm "is_meet_min";
   305 val is_join_max = thm "is_join_max";
   306 val meet_min = thm "meet_min";
   307 val join_max = thm "join_max";
   308 val meet_idempotent = thm "meet_idempotent";
   309 val join_idempotent = thm "join_idempotent";
   310 val meet_comm = thm "meet_comm";
   311 val join_comm = thm "join_comm";
   312 val meet_assoc = thm "meet_assoc";
   313 val join_assoc = thm "join_assoc";
   314 val meet_left_comm = thm "meet_left_comm";
   315 val meet_left_idempotent = thm "meet_left_idempotent";
   316 val join_left_comm = thm "join_left_comm";
   317 val join_left_idempotent = thm "join_left_idempotent";
   318 val meet_aci = thms "meet_aci";
   319 val join_aci = thms "join_aci";
   320 val le_def_meet = thm "le_def_meet";
   321 val le_def_join = thm "le_def_join";
   322 val meet_join_absorp = thm "meet_join_absorp";
   323 val join_meet_absorp = thm "join_meet_absorp";
   324 val meet_mono = thm "meet_mono";
   325 val join_mono = thm "join_mono";
   326 val distrib_join_le = thm "distrib_join_le";
   327 val distrib_meet_le = thm "distrib_meet_le";
   328 val meet_join_eq_imp_le = thm "meet_join_eq_imp_le";
   329 val modular_le = thm "modular_le";
   330 *}
   331 
   332 end