src/HOL/Library/Continuity.thy
author wenzelm
Tue, 27 Feb 2007 00:33:49 +0100
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tuned document;
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(*  Title:      HOL/Library/Continuity.thy
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    ID:         $Id$
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    Author:     David von Oheimb, TU Muenchen
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*)
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header {* Continuity and iterations (of set transformers) *}
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theory Continuity
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imports Main
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begin
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subsection {* Continuity for complete lattices *}
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definition
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  chain :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
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  "chain M \<longleftrightarrow> (\<forall>i. M i \<le> M (Suc i))"
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definition
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  continuous :: "('a::order \<Rightarrow> 'a::order) \<Rightarrow> bool" where
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  "continuous F \<longleftrightarrow> (\<forall>M. chain M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
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abbreviation
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  bot :: "'a::order" where
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  "bot \<equiv> Sup {}"
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lemma SUP_nat_conv:
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  "(SUP n::nat. M n::'a::comp_lat) = join (M 0) (SUP n. M(Suc n))"
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apply(rule order_antisym)
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 apply(rule SUP_leI)
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 apply(case_tac n)
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  apply simp
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 apply (blast intro:le_SUPI le_joinI2)
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apply(simp)
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apply (blast intro:SUP_leI le_SUPI)
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done
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lemma continuous_mono: fixes F :: "'a::comp_lat \<Rightarrow> 'a::comp_lat"
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  assumes "continuous F" shows "mono F"
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proof
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  fix A B :: "'a" assume "A <= B"
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  let ?C = "%i::nat. if i=0 then A else B"
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  have "chain ?C" using `A <= B` by(simp add:chain_def)
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  have "F B = join (F A) (F B)"
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  proof -
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    have "join A B = B" using `A <= B` by (simp add:join_absorp2)
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    hence "F B = F(SUP i. ?C i)" by(simp add: SUP_nat_conv)
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    also have "\<dots> = (SUP i. F(?C i))"
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      using `chain ?C` `continuous F` by(simp add:continuous_def)
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    also have "\<dots> = join (F A) (F B)" by(simp add: SUP_nat_conv)
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    finally show ?thesis .
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  qed
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  thus "F A \<le> F B" by(subst le_def_join, simp)
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qed
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lemma continuous_lfp:
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 assumes "continuous F" shows "lfp F = (SUP i. (F^i) bot)"
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proof -
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  note mono = continuous_mono[OF `continuous F`]
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  { fix i have "(F^i) bot \<le> lfp F"
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    proof (induct i)
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      show "(F^0) bot \<le> lfp F" by simp
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    next
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      case (Suc i)
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      have "(F^(Suc i)) bot = F((F^i) bot)" by simp
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      also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
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      also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
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      finally show ?case .
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    qed }
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  hence "(SUP i. (F^i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
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  moreover have "lfp F \<le> (SUP i. (F^i) bot)" (is "_ \<le> ?U")
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  proof (rule lfp_lowerbound)
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    have "chain(%i. (F^i) bot)"
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    proof -
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      { fix i have "(F^i) bot \<le> (F^(Suc i)) bot"
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	proof (induct i)
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	  case 0 show ?case by simp
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	next
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	  case Suc thus ?case using monoD[OF mono Suc] by auto
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	qed }
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      thus ?thesis by(auto simp add:chain_def)
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    qed
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    hence "F ?U = (SUP i. (F^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
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    also have "\<dots> \<le> ?U" by(blast intro:SUP_leI le_SUPI)
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    finally show "F ?U \<le> ?U" .
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  qed
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  ultimately show ?thesis by (blast intro:order_antisym)
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qed
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text{* The following development is just for sets but presents an up
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and a down version of chains and continuity and covers @{const gfp}. *}
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subsection "Chains"
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definition
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  up_chain :: "(nat => 'a set) => bool" where
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  "up_chain F = (\<forall>i. F i \<subseteq> F (Suc i))"
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lemma up_chainI: "(!!i. F i \<subseteq> F (Suc i)) ==> up_chain F"
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  by (simp add: up_chain_def)
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lemma up_chainD: "up_chain F ==> F i \<subseteq> F (Suc i)"
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  by (simp add: up_chain_def)
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lemma up_chain_less_mono:
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    "up_chain F ==> x < y ==> F x \<subseteq> F y"
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  apply (induct y)
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   apply (blast dest: up_chainD elim: less_SucE)+
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  done
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lemma up_chain_mono: "up_chain F ==> x \<le> y ==> F x \<subseteq> F y"
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  apply (drule le_imp_less_or_eq)
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  apply (blast dest: up_chain_less_mono)
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  done
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definition
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  down_chain :: "(nat => 'a set) => bool" where
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  "down_chain F = (\<forall>i. F (Suc i) \<subseteq> F i)"
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lemma down_chainI: "(!!i. F (Suc i) \<subseteq> F i) ==> down_chain F"
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  by (simp add: down_chain_def)
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lemma down_chainD: "down_chain F ==> F (Suc i) \<subseteq> F i"
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  by (simp add: down_chain_def)
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lemma down_chain_less_mono:
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    "down_chain F ==> x < y ==> F y \<subseteq> F x"
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  apply (induct y)
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   apply (blast dest: down_chainD elim: less_SucE)+
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  done
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lemma down_chain_mono: "down_chain F ==> x \<le> y ==> F y \<subseteq> F x"
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  apply (drule le_imp_less_or_eq)
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  apply (blast dest: down_chain_less_mono)
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  done
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subsection "Continuity"
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definition
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  up_cont :: "('a set => 'a set) => bool" where
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  "up_cont f = (\<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F))"
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lemma up_contI:
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    "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
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  apply (unfold up_cont_def)
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  apply blast
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  done
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lemma up_contD:
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    "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
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  apply (unfold up_cont_def)
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  apply auto
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  done
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lemma up_cont_mono: "up_cont f ==> mono f"
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  apply (rule monoI)
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  apply (drule_tac F = "\<lambda>i. if i = 0 then A else B" in up_contD)
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   apply (rule up_chainI)
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   apply  simp+
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  apply (drule Un_absorb1)
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  apply (auto simp add: nat_not_singleton)
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  done
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definition
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  down_cont :: "('a set => 'a set) => bool" where
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  "down_cont f =
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    (\<forall>F. down_chain F --> f (Inter (range F)) = Inter (f ` range F))"
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lemma down_contI:
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  "(!!F. down_chain F ==> f (Inter (range F)) = Inter (f ` range F)) ==>
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    down_cont f"
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  apply (unfold down_cont_def)
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  apply blast
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  done
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lemma down_contD: "down_cont f ==> down_chain F ==>
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    f (Inter (range F)) = Inter (f ` range F)"
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  apply (unfold down_cont_def)
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  apply auto
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  done
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lemma down_cont_mono: "down_cont f ==> mono f"
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  apply (rule monoI)
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  apply (drule_tac F = "\<lambda>i. if i = 0 then B else A" in down_contD)
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   apply (rule down_chainI)
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   apply simp+
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  apply (drule Int_absorb1)
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  apply (auto simp add: nat_not_singleton)
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  done
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subsection "Iteration"
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definition
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  up_iterate :: "('a set => 'a set) => nat => 'a set" where
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  "up_iterate f n = (f^n) {}"
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lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
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  by (simp add: up_iterate_def)
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lemma up_iterate_Suc [simp]: "up_iterate f (Suc i) = f (up_iterate f i)"
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  by (simp add: up_iterate_def)
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lemma up_iterate_chain: "mono F ==> up_chain (up_iterate F)"
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  apply (rule up_chainI)
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  apply (induct_tac i)
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   apply simp+
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  apply (erule (1) monoD)
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  done
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lemma UNION_up_iterate_is_fp:
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  "up_cont F ==>
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    F (UNION UNIV (up_iterate F)) = UNION UNIV (up_iterate F)"
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  apply (frule up_cont_mono [THEN up_iterate_chain])
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  apply (drule (1) up_contD)
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  apply simp
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  apply (auto simp del: up_iterate_Suc simp add: up_iterate_Suc [symmetric])
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  apply (case_tac xa)
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   apply auto
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  done
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lemma UNION_up_iterate_lowerbound:
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    "mono F ==> F P = P ==> UNION UNIV (up_iterate F) \<subseteq> P"
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  apply (subgoal_tac "(!!i. up_iterate F i \<subseteq> P)")
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   apply fast
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  apply (induct_tac i)
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  prefer 2 apply (drule (1) monoD)
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   apply auto
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  done
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lemma UNION_up_iterate_is_lfp:
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    "up_cont F ==> lfp F = UNION UNIV (up_iterate F)"
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  apply (rule set_eq_subset [THEN iffD2])
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  apply (rule conjI)
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   prefer 2
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   apply (drule up_cont_mono)
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   apply (rule UNION_up_iterate_lowerbound)
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    apply assumption
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   apply (erule lfp_unfold [symmetric])
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  apply (rule lfp_lowerbound)
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  apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
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  apply (erule UNION_up_iterate_is_fp [symmetric])
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  done
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definition
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  down_iterate :: "('a set => 'a set) => nat => 'a set" where
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  "down_iterate f n = (f^n) UNIV"
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lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
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  by (simp add: down_iterate_def)
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lemma down_iterate_Suc [simp]:
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    "down_iterate f (Suc i) = f (down_iterate f i)"
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  by (simp add: down_iterate_def)
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lemma down_iterate_chain: "mono F ==> down_chain (down_iterate F)"
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  apply (rule down_chainI)
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  apply (induct_tac i)
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   apply simp+
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  apply (erule (1) monoD)
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  done
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lemma INTER_down_iterate_is_fp:
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  "down_cont F ==>
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    F (INTER UNIV (down_iterate F)) = INTER UNIV (down_iterate F)"
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  apply (frule down_cont_mono [THEN down_iterate_chain])
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  apply (drule (1) down_contD)
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  apply simp
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  apply (auto simp del: down_iterate_Suc simp add: down_iterate_Suc [symmetric])
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  apply (case_tac xa)
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   apply auto
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  done
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lemma INTER_down_iterate_upperbound:
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    "mono F ==> F P = P ==> P \<subseteq> INTER UNIV (down_iterate F)"
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  apply (subgoal_tac "(!!i. P \<subseteq> down_iterate F i)")
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   apply fast
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  apply (induct_tac i)
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  prefer 2 apply (drule (1) monoD)
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   apply auto
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  done
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lemma INTER_down_iterate_is_gfp:
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    "down_cont F ==> gfp F = INTER UNIV (down_iterate F)"
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  apply (rule set_eq_subset [THEN iffD2])
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  apply (rule conjI)
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   apply (drule down_cont_mono)
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   apply (rule INTER_down_iterate_upperbound)
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    apply assumption
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   apply (erule gfp_unfold [symmetric])
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  apply (rule gfp_upperbound)
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  apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
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  apply (erule INTER_down_iterate_is_fp)
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  done
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end