src/HOL/Lattices.thy
 changeset 23878 bd651ecd4b8a parent 23389 aaca6a8e5414 child 23948 261bd4678076
```     1.1 --- a/src/HOL/Lattices.thy	Fri Jul 20 00:01:40 2007 +0200
1.2 +++ b/src/HOL/Lattices.thy	Fri Jul 20 14:27:56 2007 +0200
1.3 @@ -11,12 +11,6 @@
1.4
1.5  subsection{* Lattices *}
1.6
1.7 -text{*
1.8 -  This theory of lattices only defines binary sup and inf
1.9 -  operations. The extension to complete lattices is done in theory
1.10 -  @{text FixedPoint}.
1.11 -*}
1.12 -
1.13  class lower_semilattice = order +
1.14    fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
1.15    assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
1.16 @@ -70,6 +64,9 @@
1.17
1.18  end
1.19
1.20 +lemma mono_inf: "mono f \<Longrightarrow> f (inf A B) \<le> inf (f A) (f B)"
1.21 +  by (auto simp add: mono_def)
1.22 +
1.23
1.24  context upper_semilattice
1.25  begin
1.26 @@ -109,6 +106,9 @@
1.27
1.28  end
1.29
1.30 +lemma mono_sup: "mono f \<Longrightarrow> sup (f A) (f B) \<le> f (sup A B)"
1.31 +  by (auto simp add: mono_def)
1.32 +
1.33
1.34  subsubsection{* Equational laws *}
1.35
1.36 @@ -323,6 +323,174 @@
1.37    min_max.le_infI1 min_max.le_infI2
1.38
1.39
1.40 +subsection {* Complete lattices *}
1.41 +
1.42 +class complete_lattice = lattice +
1.43 +  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
1.44 +  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
1.45 +  assumes Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
1.46 +begin
1.47 +
1.48 +definition
1.49 +  Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
1.50 +where
1.51 +  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<^loc>\<le> b}"
1.52 +
1.53 +lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<^loc>\<le> a}"
1.54 +  unfolding Sup_def by (auto intro: Inf_greatest Inf_lower)
1.55 +
1.56 +lemma Sup_upper: "x \<in> A \<Longrightarrow> x \<^loc>\<le> \<Squnion>A"
1.57 +  by (auto simp: Sup_def intro: Inf_greatest)
1.58 +
1.59 +lemma Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<^loc>\<le> z) \<Longrightarrow> \<Squnion>A \<^loc>\<le> z"
1.60 +  by (auto simp: Sup_def intro: Inf_lower)
1.61 +
1.62 +lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
1.63 +  unfolding Sup_def by auto
1.64 +
1.65 +lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
1.66 +  unfolding Inf_Sup by auto
1.67 +
1.68 +lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
1.69 +  apply (rule antisym)
1.70 +  apply (rule le_infI)
1.71 +  apply (rule Inf_lower)
1.72 +  apply simp
1.73 +  apply (rule Inf_greatest)
1.74 +  apply (rule Inf_lower)
1.75 +  apply simp
1.76 +  apply (rule Inf_greatest)
1.77 +  apply (erule insertE)
1.78 +  apply (rule le_infI1)
1.79 +  apply simp
1.80 +  apply (rule le_infI2)
1.81 +  apply (erule Inf_lower)
1.82 +  done
1.83 +
1.84 +lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
1.85 +  apply (rule antisym)
1.86 +  apply (rule Sup_least)
1.87 +  apply (erule insertE)
1.88 +  apply (rule le_supI1)
1.89 +  apply simp
1.90 +  apply (rule le_supI2)
1.91 +  apply (erule Sup_upper)
1.92 +  apply (rule le_supI)
1.93 +  apply (rule Sup_upper)
1.94 +  apply simp
1.95 +  apply (rule Sup_least)
1.96 +  apply (rule Sup_upper)
1.97 +  apply simp
1.98 +  done
1.99 +
1.100 +lemma Inf_singleton [simp]:
1.101 +  "\<Sqinter>{a} = a"
1.102 +  by (auto intro: antisym Inf_lower Inf_greatest)
1.103 +
1.104 +lemma Sup_singleton [simp]:
1.105 +  "\<Squnion>{a} = a"
1.106 +  by (auto intro: antisym Sup_upper Sup_least)
1.107 +
1.108 +lemma Inf_insert_simp:
1.109 +  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
1.110 +  by (cases "A = {}") (simp_all, simp add: Inf_insert)
1.111 +
1.112 +lemma Sup_insert_simp:
1.113 +  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
1.114 +  by (cases "A = {}") (simp_all, simp add: Sup_insert)
1.115 +
1.116 +lemma Inf_binary:
1.117 +  "\<Sqinter>{a, b} = a \<sqinter> b"
1.118 +  by (simp add: Inf_insert_simp)
1.119 +
1.120 +lemma Sup_binary:
1.121 +  "\<Squnion>{a, b} = a \<squnion> b"
1.122 +  by (simp add: Sup_insert_simp)
1.123 +
1.124 +end
1.125 +
1.126 +lemmas Sup_def = Sup_def [folded complete_lattice_class.Sup]
1.127 +lemmas Sup_upper = Sup_upper [folded complete_lattice_class.Sup]
1.128 +lemmas Sup_least = Sup_least [folded complete_lattice_class.Sup]
1.129 +
1.130 +lemmas Sup_insert [code func] = Sup_insert [folded complete_lattice_class.Sup]
1.131 +lemmas Sup_singleton [simp, code func] = Sup_singleton [folded complete_lattice_class.Sup]
1.132 +lemmas Sup_insert_simp = Sup_insert_simp [folded complete_lattice_class.Sup]
1.133 +lemmas Sup_binary = Sup_binary [folded complete_lattice_class.Sup]
1.134 +
1.135 +definition
1.136 +  top :: "'a::complete_lattice"
1.137 +where
1.138 +  "top = Inf {}"
1.139 +
1.140 +definition
1.141 +  bot :: "'a::complete_lattice"
1.142 +where
1.143 +  "bot = Sup {}"
1.144 +
1.145 +lemma top_greatest [simp]: "x \<le> top"
1.146 +  by (unfold top_def, rule Inf_greatest, simp)
1.147 +
1.148 +lemma bot_least [simp]: "bot \<le> x"
1.149 +  by (unfold bot_def, rule Sup_least, simp)
1.150 +
1.151 +definition
1.152 +  SUPR :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b"
1.153 +where
1.154 +  "SUPR A f == Sup (f ` A)"
1.155 +
1.156 +definition
1.157 +  INFI :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b"
1.158 +where
1.159 +  "INFI A f == Inf (f ` A)"
1.160 +
1.161 +syntax
1.162 +  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
1.163 +  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
1.164 +  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
1.165 +  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
1.166 +
1.167 +translations
1.168 +  "SUP x y. B"   == "SUP x. SUP y. B"
1.169 +  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
1.170 +  "SUP x. B"     == "SUP x:UNIV. B"
1.171 +  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
1.172 +  "INF x y. B"   == "INF x. INF y. B"
1.173 +  "INF x. B"     == "CONST INFI UNIV (%x. B)"
1.174 +  "INF x. B"     == "INF x:UNIV. B"
1.175 +  "INF x:A. B"   == "CONST INFI A (%x. B)"
1.176 +
1.177 +(* To avoid eta-contraction of body: *)
1.178 +print_translation {*
1.179 +let
1.180 +  fun btr' syn (A :: Abs abs :: ts) =
1.181 +    let val (x,t) = atomic_abs_tr' abs
1.182 +    in list_comb (Syntax.const syn \$ x \$ A \$ t, ts) end
1.183 +  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
1.184 +in
1.185 +[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
1.186 +end
1.187 +*}
1.188 +
1.189 +lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
1.190 +  by (auto simp add: SUPR_def intro: Sup_upper)
1.191 +
1.192 +lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
1.193 +  by (auto simp add: SUPR_def intro: Sup_least)
1.194 +
1.195 +lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
1.196 +  by (auto simp add: INFI_def intro: Inf_lower)
1.197 +
1.198 +lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
1.199 +  by (auto simp add: INFI_def intro: Inf_greatest)
1.200 +
1.201 +lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
1.202 +  by (auto intro: order_antisym SUP_leI le_SUPI)
1.203 +
1.204 +lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
1.205 +  by (auto intro: order_antisym INF_leI le_INFI)
1.206 +
1.207 +
1.208  subsection {* Bool as lattice *}
1.209
1.210  instance bool :: distrib_lattice
1.211 @@ -330,10 +498,156 @@
1.212    sup_bool_eq: "sup P Q \<equiv> P \<or> Q"
1.213    by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
1.214
1.215 +instance bool :: complete_lattice
1.216 +  Inf_bool_def: "Inf A \<equiv> \<forall>x\<in>A. x"
1.217 +  apply intro_classes
1.218 +  apply (unfold Inf_bool_def)
1.219 +  apply (iprover intro!: le_boolI elim: ballE)
1.220 +  apply (iprover intro!: ballI le_boolI elim: ballE le_boolE)
1.221 +  done
1.222
1.223 -text {* duplicates *}
1.224 +theorem Sup_bool_eq: "Sup A \<longleftrightarrow> (\<exists>x\<in>A. x)"
1.225 +  apply (rule order_antisym)
1.226 +  apply (rule Sup_least)
1.227 +  apply (rule le_boolI)
1.228 +  apply (erule bexI, assumption)
1.229 +  apply (rule le_boolI)
1.230 +  apply (erule bexE)
1.231 +  apply (rule le_boolE)
1.232 +  apply (rule Sup_upper)
1.233 +  apply assumption+
1.234 +  done
1.235 +
1.236 +lemma Inf_empty_bool [simp]:
1.237 +  "Inf {}"
1.238 +  unfolding Inf_bool_def by auto
1.239 +
1.240 +lemma not_Sup_empty_bool [simp]:
1.241 +  "\<not> Sup {}"
1.242 +  unfolding Sup_def Inf_bool_def by auto
1.243 +
1.244 +lemma top_bool_eq: "top = True"
1.245 +  by (iprover intro!: order_antisym le_boolI top_greatest)
1.246 +
1.247 +lemma bot_bool_eq: "bot = False"
1.248 +  by (iprover intro!: order_antisym le_boolI bot_least)
1.249 +
1.250 +
1.251 +subsection {* Set as lattice *}
1.252 +
1.253 +instance set :: (type) distrib_lattice
1.254 +  inf_set_eq: "inf A B \<equiv> A \<inter> B"
1.255 +  sup_set_eq: "sup A B \<equiv> A \<union> B"
1.256 +  by intro_classes (auto simp add: inf_set_eq sup_set_eq)
1.257 +
1.258 +lemmas [code func del] = inf_set_eq sup_set_eq
1.259 +
1.260 +lemmas mono_Int = mono_inf
1.261 +  [where ?'a="?'a set", where ?'b="?'b set", unfolded inf_set_eq sup_set_eq]
1.262 +
1.263 +lemmas mono_Un = mono_sup
1.264 +  [where ?'a="?'a set", where ?'b="?'b set", unfolded inf_set_eq sup_set_eq]
1.265 +
1.266 +instance set :: (type) complete_lattice
1.267 +  Inf_set_def: "Inf S \<equiv> \<Inter>S"
1.268 +  by intro_classes (auto simp add: Inf_set_def)
1.269 +
1.270 +lemmas [code func del] = Inf_set_def
1.271 +
1.272 +theorem Sup_set_eq: "Sup S = \<Union>S"
1.273 +  apply (rule subset_antisym)
1.274 +  apply (rule Sup_least)
1.275 +  apply (erule Union_upper)
1.276 +  apply (rule Union_least)
1.277 +  apply (erule Sup_upper)
1.278 +  done
1.279 +
1.280 +lemma top_set_eq: "top = UNIV"
1.281 +  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
1.282 +
1.283 +lemma bot_set_eq: "bot = {}"
1.284 +  by (iprover intro!: subset_antisym empty_subsetI bot_least)
1.285 +
1.286 +
1.287 +subsection {* Fun as lattice *}
1.288 +
1.289 +instance "fun" :: (type, lattice) lattice
1.290 +  inf_fun_eq: "inf f g \<equiv> (\<lambda>x. inf (f x) (g x))"
1.291 +  sup_fun_eq: "sup f g \<equiv> (\<lambda>x. sup (f x) (g x))"
1.292 +apply intro_classes
1.293 +unfolding inf_fun_eq sup_fun_eq
1.294 +apply (auto intro: le_funI)
1.295 +apply (rule le_funI)
1.296 +apply (auto dest: le_funD)
1.297 +apply (rule le_funI)
1.298 +apply (auto dest: le_funD)
1.299 +done
1.300 +
1.301 +lemmas [code func del] = inf_fun_eq sup_fun_eq
1.302 +
1.303 +instance "fun" :: (type, distrib_lattice) distrib_lattice
1.304 +  by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
1.305 +
1.306 +instance "fun" :: (type, complete_lattice) complete_lattice
1.307 +  Inf_fun_def: "Inf A \<equiv> (\<lambda>x. Inf {y. \<exists>f\<in>A. y = f x})"
1.308 +  apply intro_classes
1.309 +  apply (unfold Inf_fun_def)
1.310 +  apply (rule le_funI)
1.311 +  apply (rule Inf_lower)
1.312 +  apply (rule CollectI)
1.313 +  apply (rule bexI)
1.314 +  apply (rule refl)
1.315 +  apply assumption
1.316 +  apply (rule le_funI)
1.317 +  apply (rule Inf_greatest)
1.318 +  apply (erule CollectE)
1.319 +  apply (erule bexE)
1.320 +  apply (iprover elim: le_funE)
1.321 +  done
1.322 +
1.323 +lemmas [code func del] = Inf_fun_def
1.324 +
1.325 +theorem Sup_fun_eq: "Sup A = (\<lambda>x. Sup {y. \<exists>f\<in>A. y = f x})"
1.326 +  apply (rule order_antisym)
1.327 +  apply (rule Sup_least)
1.328 +  apply (rule le_funI)
1.329 +  apply (rule Sup_upper)
1.330 +  apply fast
1.331 +  apply (rule le_funI)
1.332 +  apply (rule Sup_least)
1.333 +  apply (erule CollectE)
1.334 +  apply (erule bexE)
1.335 +  apply (drule le_funD [OF Sup_upper])
1.336 +  apply simp
1.337 +  done
1.338 +
1.339 +lemma Inf_empty_fun:
1.340 +  "Inf {} = (\<lambda>_. Inf {})"
1.341 +  by rule (auto simp add: Inf_fun_def)
1.342 +
1.343 +lemma Sup_empty_fun:
1.344 +  "Sup {} = (\<lambda>_. Sup {})"
1.345 +proof -
1.346 +  have aux: "\<And>x. {y. \<exists>f. y = f x} = UNIV" by auto
1.347 +  show ?thesis
1.348 +  by (auto simp add: Sup_def Inf_fun_def Inf_binary inf_bool_eq aux)
1.349 +qed
1.350 +
1.351 +lemma top_fun_eq: "top = (\<lambda>x. top)"
1.352 +  by (iprover intro!: order_antisym le_funI top_greatest)
1.353 +
1.354 +lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
1.355 +  by (iprover intro!: order_antisym le_funI bot_least)
1.356 +
1.357 +
1.358 +text {* redundant bindings *}
1.359
1.360  lemmas inf_aci = inf_ACI
1.361  lemmas sup_aci = sup_ACI
1.362
1.363 +ML {*
1.364 +val sup_fun_eq = @{thm sup_fun_eq}
1.365 +val sup_bool_eq = @{thm sup_bool_eq}
1.366 +*}
1.367 +
1.368  end
```