src/HOL/Lattices.thy
author haftmann
Thu May 10 10:21:44 2007 +0200 (2007-05-10)
changeset 22916 8caf6da610e2
parent 22737 d87ccbcc2702
child 23018 1d29bc31b0cb
permissions -rw-r--r--
tuned
     1 (*  Title:      HOL/Lattices.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4 *)
     5 
     6 header {* Abstract lattices *}
     7 
     8 theory Lattices
     9 imports Orderings
    10 begin
    11 
    12 subsection{* Lattices *}
    13 
    14 text{*
    15   This theory of lattices only defines binary sup and inf
    16   operations. The extension to complete lattices is done in theory
    17   @{text FixedPoint}.
    18 *}
    19 
    20 class lower_semilattice = order +
    21   fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
    22   assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
    23   and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
    24   and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
    25 
    26 class upper_semilattice = order +
    27   fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
    28   assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
    29   and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
    30   and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
    31 
    32 class lattice = lower_semilattice + upper_semilattice
    33 
    34 subsubsection{* Intro and elim rules*}
    35 
    36 context lower_semilattice
    37 begin
    38 
    39 lemmas antisym_intro [intro!] = antisym
    40 lemmas (in -) [rule del] = antisym_intro
    41 
    42 lemma le_infI1[intro]: "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
    43 apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> a")
    44  apply(blast intro: order_trans)
    45 apply simp
    46 done
    47 lemmas (in -) [rule del] = le_infI1
    48 
    49 lemma le_infI2[intro]: "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
    50 apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> b")
    51  apply(blast intro: order_trans)
    52 apply simp
    53 done
    54 lemmas (in -) [rule del] = le_infI2
    55 
    56 lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
    57 by(blast intro: inf_greatest)
    58 lemmas (in -) [rule del] = le_infI
    59 
    60 lemma le_infE [elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
    61   by (blast intro: order_trans)
    62 lemmas (in -) [rule del] = le_infE
    63 
    64 lemma le_inf_iff [simp]:
    65  "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
    66 by blast
    67 
    68 lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
    69 by(blast dest:eq_iff[THEN iffD1])
    70 
    71 end
    72 
    73 
    74 context upper_semilattice
    75 begin
    76 
    77 lemmas antisym_intro [intro!] = antisym
    78 lemmas (in -) [rule del] = antisym_intro
    79 
    80 lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
    81 apply(subgoal_tac "a \<sqsubseteq> a \<squnion> b")
    82  apply(blast intro: order_trans)
    83 apply simp
    84 done
    85 lemmas (in -) [rule del] = le_supI1
    86 
    87 lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
    88 apply(subgoal_tac "b \<sqsubseteq> a \<squnion> b")
    89  apply(blast intro: order_trans)
    90 apply simp
    91 done
    92 lemmas (in -) [rule del] = le_supI2
    93 
    94 lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
    95 by(blast intro: sup_least)
    96 lemmas (in -) [rule del] = le_supI
    97 
    98 lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
    99   by (blast intro: order_trans)
   100 lemmas (in -) [rule del] = le_supE
   101 
   102 
   103 lemma ge_sup_conv[simp]:
   104  "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
   105 by blast
   106 
   107 lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
   108 by(blast dest:eq_iff[THEN iffD1])
   109 
   110 end
   111 
   112 
   113 subsubsection{* Equational laws *}
   114 
   115 
   116 context lower_semilattice
   117 begin
   118 
   119 lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
   120 by blast
   121 
   122 lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
   123 by blast
   124 
   125 lemma inf_idem[simp]: "x \<sqinter> x = x"
   126 by blast
   127 
   128 lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
   129 by blast
   130 
   131 lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
   132 by blast
   133 
   134 lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
   135 by blast
   136 
   137 lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
   138 by blast
   139 
   140 lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
   141 
   142 end
   143 
   144 
   145 context upper_semilattice
   146 begin
   147 
   148 lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
   149 by blast
   150 
   151 lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
   152 by blast
   153 
   154 lemma sup_idem[simp]: "x \<squnion> x = x"
   155 by blast
   156 
   157 lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
   158 by blast
   159 
   160 lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
   161 by blast
   162 
   163 lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
   164 by blast
   165 
   166 lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
   167 by blast
   168 
   169 lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
   170 
   171 end
   172 
   173 context lattice
   174 begin
   175 
   176 lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
   177 by(blast intro: antisym inf_le1 inf_greatest sup_ge1)
   178 
   179 lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
   180 by(blast intro: antisym sup_ge1 sup_least inf_le1)
   181 
   182 lemmas ACI = inf_ACI sup_ACI
   183 
   184 lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
   185 
   186 text{* Towards distributivity *}
   187 
   188 lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   189 by blast
   190 
   191 lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
   192 by blast
   193 
   194 
   195 text{* If you have one of them, you have them all. *}
   196 
   197 lemma distrib_imp1:
   198 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
   199 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   200 proof-
   201   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
   202   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
   203   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
   204     by(simp add:inf_sup_absorb inf_commute)
   205   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
   206   finally show ?thesis .
   207 qed
   208 
   209 lemma distrib_imp2:
   210 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   211 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
   212 proof-
   213   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
   214   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
   215   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
   216     by(simp add:sup_inf_absorb sup_commute)
   217   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
   218   finally show ?thesis .
   219 qed
   220 
   221 (* seems unused *)
   222 lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
   223 by blast
   224 
   225 end
   226 
   227 
   228 subsection{* Distributive lattices *}
   229 
   230 class distrib_lattice = lattice +
   231   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
   232 
   233 context distrib_lattice
   234 begin
   235 
   236 lemma sup_inf_distrib2:
   237  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
   238 by(simp add:ACI sup_inf_distrib1)
   239 
   240 lemma inf_sup_distrib1:
   241  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
   242 by(rule distrib_imp2[OF sup_inf_distrib1])
   243 
   244 lemma inf_sup_distrib2:
   245  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
   246 by(simp add:ACI inf_sup_distrib1)
   247 
   248 lemmas distrib =
   249   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
   250 
   251 end
   252 
   253 
   254 subsection {* Uniqueness of inf and sup *}
   255 
   256 lemma (in lower_semilattice) inf_unique:
   257   fixes f (infixl "\<triangle>" 70)
   258   assumes le1: "\<And>x y. x \<triangle> y \<^loc>\<le> x" and le2: "\<And>x y. x \<triangle> y \<^loc>\<le> y"
   259   and greatest: "\<And>x y z. x \<^loc>\<le> y \<Longrightarrow> x \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> y \<triangle> z"
   260   shows "x \<sqinter> y = x \<triangle> y"
   261 proof (rule antisym)
   262   show "x \<triangle> y \<^loc>\<le> x \<sqinter> y" by (rule le_infI) (rule le1 le2)
   263 next
   264   have leI: "\<And>x y z. x \<^loc>\<le> y \<Longrightarrow> x \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> y \<triangle> z" by (blast intro: greatest)
   265   show "x \<sqinter> y \<^loc>\<le> x \<triangle> y" by (rule leI) simp_all
   266 qed
   267 
   268 lemma (in upper_semilattice) sup_unique:
   269   fixes f (infixl "\<nabla>" 70)
   270   assumes ge1 [simp]: "\<And>x y. x \<^loc>\<le> x \<nabla> y" and ge2: "\<And>x y. y \<^loc>\<le> x \<nabla> y"
   271   and least: "\<And>x y z. y \<^loc>\<le> x \<Longrightarrow> z \<^loc>\<le> x \<Longrightarrow> y \<nabla> z \<^loc>\<le> x"
   272   shows "x \<squnion> y = x \<nabla> y"
   273 proof (rule antisym)
   274   show "x \<squnion> y \<^loc>\<le> x \<nabla> y" by (rule le_supI) (rule ge1 ge2)
   275 next
   276   have leI: "\<And>x y z. x \<^loc>\<le> z \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<nabla> y \<^loc>\<le> z" by (blast intro: least)
   277   show "x \<nabla> y \<^loc>\<le> x \<squnion> y" by (rule leI) simp_all
   278 qed
   279   
   280 
   281 subsection {* @{const min}/@{const max} on linear orders as
   282   special case of @{const inf}/@{const sup} *}
   283 
   284 lemma (in linorder) distrib_lattice_min_max:
   285   "distrib_lattice_pred (op \<^loc>\<le>) (op \<^loc><) min max"
   286 proof unfold_locales
   287   have aux: "\<And>x y \<Colon> 'a. x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x \<Longrightarrow> x = y"
   288     by (auto simp add: less_le antisym)
   289   fix x y z
   290   show "max x (min y z) = min (max x y) (max x z)"
   291   unfolding min_def max_def
   292     by (auto simp add: intro: antisym, unfold not_le,
   293       auto intro: less_trans le_less_trans aux)
   294 qed (auto simp add: min_def max_def not_le less_imp_le)
   295 
   296 interpretation min_max:
   297   distrib_lattice ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max]
   298   by (rule distrib_lattice_min_max [unfolded linorder_class_min linorder_class_max])
   299 
   300 lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
   301   by (rule ext)+ auto
   302 
   303 lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
   304   by (rule ext)+ auto
   305 
   306 lemmas le_maxI1 = min_max.sup_ge1
   307 lemmas le_maxI2 = min_max.sup_ge2
   308  
   309 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
   310   mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
   311 
   312 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
   313   mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
   314 
   315 text {*
   316   Now we have inherited antisymmetry as an intro-rule on all
   317   linear orders. This is a problem because it applies to bool, which is
   318   undesirable.
   319 *}
   320 
   321 lemmas [rule del] = min_max.antisym_intro min_max.le_infI min_max.le_supI
   322   min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
   323   min_max.le_infI1 min_max.le_infI2
   324 
   325 
   326 subsection {* Bool as lattice *}
   327 
   328 instance bool :: distrib_lattice
   329   inf_bool_eq: "inf P Q \<equiv> P \<and> Q"
   330   sup_bool_eq: "sup P Q \<equiv> P \<or> Q"
   331   by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
   332 
   333 
   334 text {* duplicates *}
   335 
   336 lemmas inf_aci = inf_ACI
   337 lemmas sup_aci = sup_ACI
   338 
   339 end