src/HOL/FixedPoint.thy

 *)
 
 header {* Fixed Points and the Knaster-Tarski Theorem*}
 
 theory FixedPoint
-imports Fun
+imports Lattices
 begin
-
-subsection {* Complete lattices *}
-
-class complete_lattice = lattice +
-  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
-  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
-  assumes Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
-begin
-
-definition
-  Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
-where
-  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<^loc>\<le> b}"
-
-lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<^loc>\<le> a}"
-  unfolding Sup_def by (auto intro: Inf_greatest Inf_lower)
-
-lemma Sup_upper: "x \<in> A \<Longrightarrow> x \<^loc>\<le> \<Squnion>A"
-  by (auto simp: Sup_def intro: Inf_greatest)
-
-lemma Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<^loc>\<le> z) \<Longrightarrow> \<Squnion>A \<^loc>\<le> z"
-  by (auto simp: Sup_def intro: Inf_lower)
-
-lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
-  unfolding Sup_def by auto
-
-lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
-  unfolding Inf_Sup by auto
-
-lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
-  apply (rule antisym)
-  apply (rule le_infI)
-  apply (rule Inf_lower)
-  apply simp
-  apply (rule Inf_greatest)
-  apply (rule Inf_lower)
-  apply simp
-  apply (rule Inf_greatest)
-  apply (erule insertE)
-  apply (rule le_infI1)
-  apply simp
-  apply (rule le_infI2)
-  apply (erule Inf_lower)
-  done
-
-lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
-  apply (rule antisym)
-  apply (rule Sup_least)
-  apply (erule insertE)
-  apply (rule le_supI1)
-  apply simp
-  apply (rule le_supI2)
-  apply (erule Sup_upper)
-  apply (rule le_supI)
-  apply (rule Sup_upper)
-  apply simp
-  apply (rule Sup_least)
-  apply (rule Sup_upper)
-  apply simp
-  done
-
-lemma Inf_singleton [simp]:
-  "\<Sqinter>{a} = a"
-  by (auto intro: antisym Inf_lower Inf_greatest)
-
-lemma Sup_singleton [simp]:
-  "\<Squnion>{a} = a"
-  by (auto intro: antisym Sup_upper Sup_least)
-
-lemma Inf_insert_simp:
-  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
-  by (cases "A = {}") (simp_all, simp add: Inf_insert)
-
-lemma Sup_insert_simp:
-  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
-  by (cases "A = {}") (simp_all, simp add: Sup_insert)
-
-lemma Inf_binary:
-  "\<Sqinter>{a, b} = a \<sqinter> b"
-  by (simp add: Inf_insert_simp)
-
-lemma Sup_binary:
-  "\<Squnion>{a, b} = a \<squnion> b"
-  by (simp add: Sup_insert_simp)
-
-end
-
-lemmas Sup_def = Sup_def [folded complete_lattice_class.Sup]
-lemmas Sup_upper = Sup_upper [folded complete_lattice_class.Sup]
-lemmas Sup_least = Sup_least [folded complete_lattice_class.Sup]
-
-lemmas Sup_insert [code func] = Sup_insert [folded complete_lattice_class.Sup]
-lemmas Sup_singleton [simp, code func] = Sup_singleton [folded complete_lattice_class.Sup]
-lemmas Sup_insert_simp = Sup_insert_simp [folded complete_lattice_class.Sup]
-lemmas Sup_binary = Sup_binary [folded complete_lattice_class.Sup]
-
-(* FIXME: definition inside class does not work *)
-definition
-  top :: "'a::complete_lattice"
-where
-  "top = Inf {}"
-
-definition
-  bot :: "'a::complete_lattice"
-where
-  "bot = Sup {}"
-
-lemma top_greatest [simp]: "x \<le> top"
-  by (unfold top_def, rule Inf_greatest, simp)
-
-lemma bot_least [simp]: "bot \<le> x"
-  by (unfold bot_def, rule Sup_least, simp)
-
-definition
-  SUPR :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b" where
-  "SUPR A f == Sup (f ` A)"
-
-definition
-  INFI :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b" where
-  "INFI A f == Inf (f ` A)"
-
-syntax
-  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
-  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
-  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
-  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
-
-translations
-  "SUP x y. B"   == "SUP x. SUP y. B"
-  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
-  "SUP x. B"     == "SUP x:UNIV. B"
-  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
-  "INF x y. B"   == "INF x. INF y. B"
-  "INF x. B"     == "CONST INFI UNIV (%x. B)"
-  "INF x. B"     == "INF x:UNIV. B"
-  "INF x:A. B"   == "CONST INFI A (%x. B)"
-
-(* To avoid eta-contraction of body: *)
-print_translation {*
-let
-  fun btr' syn (A :: Abs abs :: ts) =
-    let val (x,t) = atomic_abs_tr' abs
-    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
-  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
-in
-[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
-end
-*}
-
-lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
-  by (auto simp add: SUPR_def intro: Sup_upper)
-
-lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
-  by (auto simp add: SUPR_def intro: Sup_least)
-
-lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
-  by (auto simp add: INFI_def intro: Inf_lower)
-
-lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
-  by (auto simp add: INFI_def intro: Inf_greatest)
-
-lemma mono_inf: "mono f \<Longrightarrow> f (inf A B) <= inf (f A) (f B)"
-  by (auto simp add: mono_def)
-
-lemma mono_sup: "mono f \<Longrightarrow> sup (f A) (f B) <= f (sup A B)"
-  by (auto simp add: mono_def)
-
-lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
-  by (auto intro: order_antisym SUP_leI le_SUPI)
-
-lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
-  by (auto intro: order_antisym INF_leI le_INFI)
-
-
-subsection {* Some instances of the type class of complete lattices *}
-
-subsubsection {* Booleans *}
-
-instance bool :: complete_lattice
-  Inf_bool_def: "Inf A \<equiv> \<forall>x\<in>A. x"
-  apply intro_classes
-  apply (unfold Inf_bool_def)
-  apply (iprover intro!: le_boolI elim: ballE)
-  apply (iprover intro!: ballI le_boolI elim: ballE le_boolE)
-  done
-
-theorem Sup_bool_eq: "Sup A \<longleftrightarrow> (\<exists>x\<in>A. x)"
-  apply (rule order_antisym)
-  apply (rule Sup_least)
-  apply (rule le_boolI)
-  apply (erule bexI, assumption)
-  apply (rule le_boolI)
-  apply (erule bexE)
-  apply (rule le_boolE)
-  apply (rule Sup_upper)
-  apply assumption+
-  done
-
-lemma Inf_empty_bool [simp]:
-  "Inf {}"
-  unfolding Inf_bool_def by auto
-
-lemma not_Sup_empty_bool [simp]:
-  "\<not> Sup {}"
-  unfolding Sup_def Inf_bool_def by auto
-
-lemma top_bool_eq: "top = True"
-  by (iprover intro!: order_antisym le_boolI top_greatest)
-
-lemma bot_bool_eq: "bot = False"
-  by (iprover intro!: order_antisym le_boolI bot_least)
-
-
-subsubsection {* Functions *}
-
-instance "fun" :: (type, complete_lattice) complete_lattice
-  Inf_fun_def: "Inf A \<equiv> (\<lambda>x. Inf {y. \<exists>f\<in>A. y = f x})"
-  apply intro_classes
-  apply (unfold Inf_fun_def)
-  apply (rule le_funI)
-  apply (rule Inf_lower)
-  apply (rule CollectI)
-  apply (rule bexI)
-  apply (rule refl)
-  apply assumption
-  apply (rule le_funI)
-  apply (rule Inf_greatest)
-  apply (erule CollectE)
-  apply (erule bexE)
-  apply (iprover elim: le_funE)
-  done
-
-lemmas [code func del] = Inf_fun_def
-
-theorem Sup_fun_eq: "Sup A = (\<lambda>x. Sup {y. \<exists>f\<in>A. y = f x})"
-  apply (rule order_antisym)
-  apply (rule Sup_least)
-  apply (rule le_funI)
-  apply (rule Sup_upper)
-  apply fast
-  apply (rule le_funI)
-  apply (rule Sup_least)
-  apply (erule CollectE)
-  apply (erule bexE)
-  apply (drule le_funD [OF Sup_upper])
-  apply simp
-  done
-
-lemma Inf_empty_fun:
-  "Inf {} = (\<lambda>_. Inf {})"
-  by rule (auto simp add: Inf_fun_def)
-
-lemma Sup_empty_fun:
-  "Sup {} = (\<lambda>_. Sup {})"
-proof -
-  have aux: "\<And>x. {y. \<exists>f. y = f x} = UNIV" by auto
-  show ?thesis
-  by (auto simp add: Sup_def Inf_fun_def Inf_binary inf_bool_eq aux)
-qed
-
-lemma top_fun_eq: "top = (\<lambda>x. top)"
-  by (iprover intro!: order_antisym le_funI top_greatest)
-
-lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
-  by (iprover intro!: order_antisym le_funI bot_least)
-
-
-subsubsection {* Sets *}
-
-instance set :: (type) complete_lattice
-  Inf_set_def: "Inf S \<equiv> \<Inter>S"
-  by intro_classes (auto simp add: Inf_set_def)
-
-lemmas [code func del] = Inf_set_def
-
-theorem Sup_set_eq: "Sup S = \<Union>S"
-  apply (rule subset_antisym)
-  apply (rule Sup_least)
-  apply (erule Union_upper)
-  apply (rule Union_least)
-  apply (erule Sup_upper)
-  done
-
-lemma top_set_eq: "top = UNIV"
-  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
-
-lemma bot_set_eq: "bot = {}"
-  by (iprover intro!: subset_antisym empty_subsetI bot_least)
-
 
 subsection {* Least and greatest fixed points *}
 
 definition
   lfp :: "('a\<Colon>complete_lattice \<Rightarrow> 'a) \<Rightarrow> 'a" where