src/HOL/Lattices.thy
author huffman
Sat Jun 06 09:11:12 2009 -0700 (2009-06-06)
changeset 31488 5691ccb8d6b5
parent 30729 461ee3e49ad3
child 31991 37390299214a
permissions -rw-r--r--
generalize tendsto to class topological_space
     1 (*  Title:      HOL/Lattices.thy
     2     Author:     Tobias Nipkow
     3 *)
     4 
     5 header {* Abstract lattices *}
     6 
     7 theory Lattices
     8 imports Orderings
     9 begin
    10 
    11 subsection {* Lattices *}
    12 
    13 notation
    14   less_eq  (infix "\<sqsubseteq>" 50) and
    15   less  (infix "\<sqsubset>" 50)
    16 
    17 class lower_semilattice = order +
    18   fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
    19   assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
    20   and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
    21   and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
    22 
    23 class upper_semilattice = order +
    24   fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
    25   assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
    26   and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
    27   and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
    28 begin
    29 
    30 text {* Dual lattice *}
    31 
    32 lemma dual_lattice:
    33   "lower_semilattice (op \<ge>) (op >) sup"
    34 by (rule lower_semilattice.intro, rule dual_order)
    35   (unfold_locales, simp_all add: sup_least)
    36 
    37 end
    38 
    39 class lattice = lower_semilattice + upper_semilattice
    40 
    41 
    42 subsubsection {* Intro and elim rules*}
    43 
    44 context lower_semilattice
    45 begin
    46 
    47 lemma le_infI1[intro]:
    48   assumes "a \<sqsubseteq> x"
    49   shows "a \<sqinter> b \<sqsubseteq> x"
    50 proof (rule order_trans)
    51   from assms show "a \<sqsubseteq> x" .
    52   show "a \<sqinter> b \<sqsubseteq> a" by simp 
    53 qed
    54 lemmas (in -) [rule del] = le_infI1
    55 
    56 lemma le_infI2[intro]:
    57   assumes "b \<sqsubseteq> x"
    58   shows "a \<sqinter> b \<sqsubseteq> x"
    59 proof (rule order_trans)
    60   from assms show "b \<sqsubseteq> x" .
    61   show "a \<sqinter> b \<sqsubseteq> b" by simp
    62 qed
    63 lemmas (in -) [rule del] = le_infI2
    64 
    65 lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
    66 by(blast intro: inf_greatest)
    67 lemmas (in -) [rule del] = le_infI
    68 
    69 lemma le_infE [elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
    70   by (blast intro: order_trans)
    71 lemmas (in -) [rule del] = le_infE
    72 
    73 lemma le_inf_iff [simp]:
    74   "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
    75 by blast
    76 
    77 lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
    78   by (blast intro: antisym dest: eq_iff [THEN iffD1])
    79 
    80 lemma mono_inf:
    81   fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice"
    82   shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B"
    83   by (auto simp add: mono_def intro: Lattices.inf_greatest)
    84 
    85 end
    86 
    87 context upper_semilattice
    88 begin
    89 
    90 lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
    91   by (rule order_trans) auto
    92 lemmas (in -) [rule del] = le_supI1
    93 
    94 lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
    95   by (rule order_trans) auto 
    96 lemmas (in -) [rule del] = le_supI2
    97 
    98 lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
    99   by (blast intro: sup_least)
   100 lemmas (in -) [rule del] = le_supI
   101 
   102 lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
   103   by (blast intro: order_trans)
   104 lemmas (in -) [rule del] = le_supE
   105 
   106 lemma ge_sup_conv[simp]:
   107   "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
   108 by blast
   109 
   110 lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
   111   by (blast intro: antisym dest: eq_iff [THEN iffD1])
   112 
   113 lemma mono_sup:
   114   fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice"
   115   shows "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)"
   116   by (auto simp add: mono_def intro: Lattices.sup_least)
   117 
   118 end
   119 
   120 
   121 subsubsection{* Equational laws *}
   122 
   123 context lower_semilattice
   124 begin
   125 
   126 lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
   127   by (blast intro: antisym)
   128 
   129 lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
   130   by (blast intro: antisym)
   131 
   132 lemma inf_idem[simp]: "x \<sqinter> x = x"
   133   by (blast intro: antisym)
   134 
   135 lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
   136   by (blast intro: antisym)
   137 
   138 lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
   139   by (blast intro: antisym)
   140 
   141 lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
   142   by (blast intro: antisym)
   143 
   144 lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
   145   by (blast intro: antisym)
   146 
   147 lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
   148 
   149 end
   150 
   151 
   152 context upper_semilattice
   153 begin
   154 
   155 lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
   156   by (blast intro: antisym)
   157 
   158 lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
   159   by (blast intro: antisym)
   160 
   161 lemma sup_idem[simp]: "x \<squnion> x = x"
   162   by (blast intro: antisym)
   163 
   164 lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
   165   by (blast intro: antisym)
   166 
   167 lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
   168   by (blast intro: antisym)
   169 
   170 lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
   171   by (blast intro: antisym)
   172 
   173 lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
   174   by (blast intro: antisym)
   175 
   176 lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
   177 
   178 end
   179 
   180 context lattice
   181 begin
   182 
   183 lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
   184   by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
   185 
   186 lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
   187   by (blast intro: antisym sup_ge1 sup_least inf_le1)
   188 
   189 lemmas ACI = inf_ACI sup_ACI
   190 
   191 lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
   192 
   193 text{* Towards distributivity *}
   194 
   195 lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   196   by blast
   197 
   198 lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
   199   by blast
   200 
   201 
   202 text{* If you have one of them, you have them all. *}
   203 
   204 lemma distrib_imp1:
   205 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
   206 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   207 proof-
   208   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
   209   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
   210   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
   211     by(simp add:inf_sup_absorb inf_commute)
   212   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
   213   finally show ?thesis .
   214 qed
   215 
   216 lemma distrib_imp2:
   217 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   218 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
   219 proof-
   220   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
   221   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
   222   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
   223     by(simp add:sup_inf_absorb sup_commute)
   224   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
   225   finally show ?thesis .
   226 qed
   227 
   228 (* seems unused *)
   229 lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
   230 by blast
   231 
   232 end
   233 
   234 
   235 subsection {* Distributive lattices *}
   236 
   237 class distrib_lattice = lattice +
   238   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   239 
   240 context distrib_lattice
   241 begin
   242 
   243 lemma sup_inf_distrib2:
   244  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
   245 by(simp add:ACI sup_inf_distrib1)
   246 
   247 lemma inf_sup_distrib1:
   248  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
   249 by(rule distrib_imp2[OF sup_inf_distrib1])
   250 
   251 lemma inf_sup_distrib2:
   252  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
   253 by(simp add:ACI inf_sup_distrib1)
   254 
   255 lemmas distrib =
   256   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
   257 
   258 end
   259 
   260 
   261 subsection {* Uniqueness of inf and sup *}
   262 
   263 lemma (in lower_semilattice) inf_unique:
   264   fixes f (infixl "\<triangle>" 70)
   265   assumes le1: "\<And>x y. x \<triangle> y \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y"
   266   and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z"
   267   shows "x \<sqinter> y = x \<triangle> y"
   268 proof (rule antisym)
   269   show "x \<triangle> y \<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
   270 next
   271   have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest)
   272   show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all
   273 qed
   274 
   275 lemma (in upper_semilattice) sup_unique:
   276   fixes f (infixl "\<nabla>" 70)
   277   assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y"
   278   and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x"
   279   shows "x \<squnion> y = x \<nabla> y"
   280 proof (rule antisym)
   281   show "x \<squnion> y \<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
   282 next
   283   have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least)
   284   show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all
   285 qed
   286   
   287 
   288 subsection {* @{const min}/@{const max} on linear orders as
   289   special case of @{const inf}/@{const sup} *}
   290 
   291 lemma (in linorder) distrib_lattice_min_max:
   292   "distrib_lattice (op \<le>) (op <) min max"
   293 proof
   294   have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
   295     by (auto simp add: less_le antisym)
   296   fix x y z
   297   show "max x (min y z) = min (max x y) (max x z)"
   298   unfolding min_def max_def
   299   by auto
   300 qed (auto simp add: min_def max_def not_le less_imp_le)
   301 
   302 interpretation min_max: distrib_lattice "op \<le> :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max
   303   by (rule distrib_lattice_min_max)
   304 
   305 lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
   306   by (rule ext)+ (auto intro: antisym)
   307 
   308 lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
   309   by (rule ext)+ (auto intro: antisym)
   310 
   311 lemmas le_maxI1 = min_max.sup_ge1
   312 lemmas le_maxI2 = min_max.sup_ge2
   313  
   314 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
   315   mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
   316 
   317 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
   318   mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
   319 
   320 text {*
   321   Now we have inherited antisymmetry as an intro-rule on all
   322   linear orders. This is a problem because it applies to bool, which is
   323   undesirable.
   324 *}
   325 
   326 lemmas [rule del] = min_max.le_infI min_max.le_supI
   327   min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
   328   min_max.le_infI1 min_max.le_infI2
   329 
   330 
   331 subsection {* Bool as lattice *}
   332 
   333 instantiation bool :: distrib_lattice
   334 begin
   335 
   336 definition
   337   inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
   338 
   339 definition
   340   sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
   341 
   342 instance
   343   by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
   344 
   345 end
   346 
   347 
   348 subsection {* Fun as lattice *}
   349 
   350 instantiation "fun" :: (type, lattice) lattice
   351 begin
   352 
   353 definition
   354   inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
   355 
   356 definition
   357   sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
   358 
   359 instance
   360 apply intro_classes
   361 unfolding inf_fun_eq sup_fun_eq
   362 apply (auto intro: le_funI)
   363 apply (rule le_funI)
   364 apply (auto dest: le_funD)
   365 apply (rule le_funI)
   366 apply (auto dest: le_funD)
   367 done
   368 
   369 end
   370 
   371 instance "fun" :: (type, distrib_lattice) distrib_lattice
   372   by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
   373 
   374 
   375 text {* redundant bindings *}
   376 
   377 lemmas inf_aci = inf_ACI
   378 lemmas sup_aci = sup_ACI
   379 
   380 no_notation
   381   less_eq  (infix "\<sqsubseteq>" 50) and
   382   less (infix "\<sqsubset>" 50) and
   383   inf  (infixl "\<sqinter>" 70) and
   384   sup  (infixl "\<squnion>" 65)
   385 
   386 end