oheimb@11351: (*  Title:      HOL/Library/Continuity.thy
wenzelm@11355:     ID:         $Id$
wenzelm@11355:     Author:     David von Oheimb, TU Muenchen
oheimb@11351:     License:    GPL (GNU GENERAL PUBLIC LICENSE)
oheimb@11351: *)
oheimb@11351: 
oheimb@11351: header {*
wenzelm@11355:   \title{Continuity and iterations (of set transformers)}
oheimb@11351:   \author{David von Oheimb}
oheimb@11351: *}
oheimb@11351: 
wenzelm@11355: theory Continuity = Main:
oheimb@11351: 
oheimb@11351: subsection "Chains"
oheimb@11351: 
oheimb@11351: constdefs
wenzelm@11355:   up_chain :: "(nat => 'a set) => bool"
wenzelm@11355:   "up_chain F == \<forall>i. F i \<subseteq> F (Suc i)"
oheimb@11351: 
wenzelm@11355: lemma up_chainI: "(!!i. F i \<subseteq> F (Suc i)) ==> up_chain F"
wenzelm@11355:   by (simp add: up_chain_def)
oheimb@11351: 
wenzelm@11355: lemma up_chainD: "up_chain F ==> F i \<subseteq> F (Suc i)"
wenzelm@11355:   by (simp add: up_chain_def)
oheimb@11351: 
wenzelm@11355: lemma up_chain_less_mono [rule_format]:
wenzelm@11355:     "up_chain F ==> x < y --> F x \<subseteq> F y"
wenzelm@11355:   apply (induct_tac y)
wenzelm@11355:   apply (blast dest: up_chainD elim: less_SucE)+
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma up_chain_mono: "up_chain F ==> x \<le> y ==> F x \<subseteq> F y"
wenzelm@11355:   apply (drule le_imp_less_or_eq)
wenzelm@11355:   apply (blast dest: up_chain_less_mono)
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: constdefs
wenzelm@11355:   down_chain :: "(nat => 'a set) => bool"
wenzelm@11355:   "down_chain F == \<forall>i. F (Suc i) \<subseteq> F i"
oheimb@11351: 
wenzelm@11355: lemma down_chainI: "(!!i. F (Suc i) \<subseteq> F i) ==> down_chain F"
wenzelm@11355:   by (simp add: down_chain_def)
oheimb@11351: 
wenzelm@11355: lemma down_chainD: "down_chain F ==> F (Suc i) \<subseteq> F i"
wenzelm@11355:   by (simp add: down_chain_def)
oheimb@11351: 
wenzelm@11355: lemma down_chain_less_mono [rule_format]:
wenzelm@11355:     "down_chain F ==> x < y --> F y \<subseteq> F x"
wenzelm@11355:   apply (induct_tac y)
wenzelm@11355:   apply (blast dest: down_chainD elim: less_SucE)+
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma down_chain_mono: "down_chain F ==> x \<le> y ==> F y \<subseteq> F x"
wenzelm@11355:   apply (drule le_imp_less_or_eq)
wenzelm@11355:   apply (blast dest: down_chain_less_mono)
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: subsection "Continuity"
oheimb@11351: 
oheimb@11351: constdefs
oheimb@11351:   up_cont :: "('a set => 'a set) => bool"
wenzelm@11355:   "up_cont f == \<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F)"
oheimb@11351: 
wenzelm@11355: lemma up_contI:
wenzelm@11355:     "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
wenzelm@11355:   apply (unfold up_cont_def)
wenzelm@11355:   apply blast
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma up_contD:
wenzelm@11355:     "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
wenzelm@11355:   apply (unfold up_cont_def)
wenzelm@11355:   apply auto
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: lemma up_cont_mono: "up_cont f ==> mono f"
wenzelm@11355:   apply (rule monoI)
wenzelm@11355:   apply (drule_tac F = "\<lambda>i. if i = 0 then A else B" in up_contD)
wenzelm@11355:    apply (rule up_chainI)
wenzelm@11355:    apply  simp+
wenzelm@11355:   apply (drule Un_absorb1)
wenzelm@11355:   apply auto
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: constdefs
oheimb@11351:   down_cont :: "('a set => 'a set) => bool"
wenzelm@11355:   "down_cont f ==
wenzelm@11355:     \<forall>F. down_chain F --> f (Inter (range F)) = Inter (f ` range F)"
oheimb@11351: 
wenzelm@11355: lemma down_contI:
wenzelm@11355:   "(!!F. down_chain F ==> f (Inter (range F)) = Inter (f ` range F)) ==>
wenzelm@11355:     down_cont f"
wenzelm@11355:   apply (unfold down_cont_def)
wenzelm@11355:   apply blast
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma down_contD: "down_cont f ==> down_chain F ==>
wenzelm@11355:     f (Inter (range F)) = Inter (f ` range F)"
wenzelm@11355:   apply (unfold down_cont_def)
wenzelm@11355:   apply auto
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: lemma down_cont_mono: "down_cont f ==> mono f"
wenzelm@11355:   apply (rule monoI)
wenzelm@11355:   apply (drule_tac F = "\<lambda>i. if i = 0 then B else A" in down_contD)
wenzelm@11355:    apply (rule down_chainI)
wenzelm@11355:    apply simp+
wenzelm@11355:   apply (drule Int_absorb1)
wenzelm@11355:   apply auto
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: subsection "Iteration"
oheimb@11351: 
oheimb@11351: constdefs
oheimb@11351:   up_iterate :: "('a set => 'a set) => nat => 'a set"
wenzelm@11355:   "up_iterate f n == (f^n) {}"
oheimb@11351: 
oheimb@11351: lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
wenzelm@11355:   by (simp add: up_iterate_def)
oheimb@11351: 
wenzelm@11355: lemma up_iterate_Suc [simp]: "up_iterate f (Suc i) = f (up_iterate f i)"
wenzelm@11355:   by (simp add: up_iterate_def)
oheimb@11351: 
oheimb@11351: lemma up_iterate_chain: "mono F ==> up_chain (up_iterate F)"
wenzelm@11355:   apply (rule up_chainI)
wenzelm@11355:   apply (induct_tac i)
wenzelm@11355:    apply simp+
wenzelm@11355:   apply (erule (1) monoD)
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma UNION_up_iterate_is_fp:
wenzelm@11355:   "up_cont F ==>
wenzelm@11355:     F (UNION UNIV (up_iterate F)) = UNION UNIV (up_iterate F)"
wenzelm@11355:   apply (frule up_cont_mono [THEN up_iterate_chain])
wenzelm@11355:   apply (drule (1) up_contD)
wenzelm@11355:   apply simp
wenzelm@11355:   apply (auto simp del: up_iterate_Suc simp add: up_iterate_Suc [symmetric])
wenzelm@11355:   apply (case_tac xa)
wenzelm@11355:    apply auto
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma UNION_up_iterate_lowerbound:
wenzelm@11355:     "mono F ==> F P = P ==> UNION UNIV (up_iterate F) \<subseteq> P"
wenzelm@11355:   apply (subgoal_tac "(!!i. up_iterate F i \<subseteq> P)")
wenzelm@11355:    apply fast
wenzelm@11355:   apply (induct_tac i)
wenzelm@11355:   prefer 2 apply (drule (1) monoD)
wenzelm@11355:    apply auto
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma UNION_up_iterate_is_lfp:
wenzelm@11355:     "up_cont F ==> lfp F = UNION UNIV (up_iterate F)"
wenzelm@11355:   apply (rule set_eq_subset [THEN iffD2])
wenzelm@11355:   apply (rule conjI)
wenzelm@11355:    prefer 2
wenzelm@11355:    apply (drule up_cont_mono)
wenzelm@11355:    apply (rule UNION_up_iterate_lowerbound)
wenzelm@11355:     apply assumption
wenzelm@11355:    apply (erule lfp_unfold [symmetric])
wenzelm@11355:   apply (rule lfp_lowerbound)
wenzelm@11355:   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
wenzelm@11355:   apply (erule UNION_up_iterate_is_fp [symmetric])
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: constdefs
oheimb@11351:   down_iterate :: "('a set => 'a set) => nat => 'a set"
wenzelm@11355:   "down_iterate f n == (f^n) UNIV"
oheimb@11351: 
oheimb@11351: lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
wenzelm@11355:   by (simp add: down_iterate_def)
oheimb@11351: 
wenzelm@11355: lemma down_iterate_Suc [simp]:
wenzelm@11355:     "down_iterate f (Suc i) = f (down_iterate f i)"
wenzelm@11355:   by (simp add: down_iterate_def)
oheimb@11351: 
oheimb@11351: lemma down_iterate_chain: "mono F ==> down_chain (down_iterate F)"
wenzelm@11355:   apply (rule down_chainI)
wenzelm@11355:   apply (induct_tac i)
wenzelm@11355:    apply simp+
wenzelm@11355:   apply (erule (1) monoD)
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma INTER_down_iterate_is_fp:
wenzelm@11355:   "down_cont F ==>
wenzelm@11355:     F (INTER UNIV (down_iterate F)) = INTER UNIV (down_iterate F)"
wenzelm@11355:   apply (frule down_cont_mono [THEN down_iterate_chain])
wenzelm@11355:   apply (drule (1) down_contD)
wenzelm@11355:   apply simp
wenzelm@11355:   apply (auto simp del: down_iterate_Suc simp add: down_iterate_Suc [symmetric])
wenzelm@11355:   apply (case_tac xa)
wenzelm@11355:    apply auto
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma INTER_down_iterate_upperbound:
wenzelm@11355:     "mono F ==> F P = P ==> P \<subseteq> INTER UNIV (down_iterate F)"
wenzelm@11355:   apply (subgoal_tac "(!!i. P \<subseteq> down_iterate F i)")
wenzelm@11355:    apply fast
wenzelm@11355:   apply (induct_tac i)
wenzelm@11355:   prefer 2 apply (drule (1) monoD)
wenzelm@11355:    apply auto
wenzelm@11355:   done
oheimb@11351: 
wenzelm@11355: lemma INTER_down_iterate_is_gfp:
wenzelm@11355:     "down_cont F ==> gfp F = INTER UNIV (down_iterate F)"
wenzelm@11355:   apply (rule set_eq_subset [THEN iffD2])
wenzelm@11355:   apply (rule conjI)
wenzelm@11355:    apply (drule down_cont_mono)
wenzelm@11355:    apply (rule INTER_down_iterate_upperbound)
wenzelm@11355:     apply assumption
wenzelm@11355:    apply (erule gfp_unfold [symmetric])
wenzelm@11355:   apply (rule gfp_upperbound)
wenzelm@11355:   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
wenzelm@11355:   apply (erule INTER_down_iterate_is_fp)
wenzelm@11355:   done
oheimb@11351: 
oheimb@11351: end