
oheimb@11351: (*  Title:      HOL/Library/Continuity.thy
wenzelm@11355:     Author:     David von Oheimb, TU Muenchen
oheimb@11351: *)
oheimb@11351: 
wenzelm@14706: header {* Continuity and iterations (of set transformers) *}
oheimb@11351: 
nipkow@15131: theory Continuity
wenzelm@46508: imports Main
nipkow@15131: begin
oheimb@11351: 
wenzelm@22367: subsection {* Continuity for complete lattices *}
nipkow@21312: 
wenzelm@22367: definition
haftmann@22452:   chain :: "(nat \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
wenzelm@22367:   "chain M \<longleftrightarrow> (\<forall>i. M i \<le> M (Suc i))"
wenzelm@22367: 
wenzelm@22367: definition
haftmann@22452:   continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
wenzelm@22367:   "continuous F \<longleftrightarrow> (\<forall>M. chain M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
nipkow@21312: 
nipkow@21312: lemma SUP_nat_conv:
berghofe@22431:   "(SUP n. M n) = sup (M 0) (SUP n. M(Suc n))"
nipkow@21312: apply(rule order_antisym)
hoelzl@44928:  apply(rule SUP_least)
nipkow@21312:  apply(case_tac n)
nipkow@21312:   apply simp
hoelzl@44928:  apply (fast intro:SUP_upper le_supI2)
nipkow@21312: apply(simp)
hoelzl@44928: apply (blast intro:SUP_least SUP_upper)
nipkow@21312: done
nipkow@21312: 
haftmann@22452: lemma continuous_mono: fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
nipkow@21312:   assumes "continuous F" shows "mono F"
nipkow@21312: proof
nipkow@21312:   fix A B :: "'a" assume "A <= B"
nipkow@21312:   let ?C = "%i::nat. if i=0 then A else B"
nipkow@21312:   have "chain ?C" using `A <= B` by(simp add:chain_def)
haftmann@22422:   have "F B = sup (F A) (F B)"
nipkow@21312:   proof -
haftmann@22422:     have "sup A B = B" using `A <= B` by (simp add:sup_absorb2)
berghofe@22431:     hence "F B = F(SUP i. ?C i)" by (subst SUP_nat_conv) simp
nipkow@21312:     also have "\<dots> = (SUP i. F(?C i))"
nipkow@21312:       using `chain ?C` `continuous F` by(simp add:continuous_def)
berghofe@22431:     also have "\<dots> = sup (F A) (F B)" by (subst SUP_nat_conv) simp
nipkow@21312:     finally show ?thesis .
nipkow@21312:   qed
haftmann@22422:   thus "F A \<le> F B" by(subst le_iff_sup, simp)
nipkow@21312: qed
nipkow@21312: 
nipkow@21312: lemma continuous_lfp:
haftmann@30971:  assumes "continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
nipkow@21312: proof -
nipkow@21312:   note mono = continuous_mono[OF `continuous F`]
haftmann@30971:   { fix i have "(F ^^ i) bot \<le> lfp F"
nipkow@21312:     proof (induct i)
haftmann@30971:       show "(F ^^ 0) bot \<le> lfp F" by simp
nipkow@21312:     next
nipkow@21312:       case (Suc i)
haftmann@30971:       have "(F ^^ Suc i) bot = F((F ^^ i) bot)" by simp
nipkow@21312:       also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
nipkow@21312:       also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
nipkow@21312:       finally show ?case .
nipkow@21312:     qed }
hoelzl@44928:   hence "(SUP i. (F ^^ i) bot) \<le> lfp F" by (blast intro!:SUP_least)
haftmann@30971:   moreover have "lfp F \<le> (SUP i. (F ^^ i) bot)" (is "_ \<le> ?U")
nipkow@21312:   proof (rule lfp_lowerbound)
haftmann@30971:     have "chain(%i. (F ^^ i) bot)"
nipkow@21312:     proof -
haftmann@30971:       { fix i have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot"
wenzelm@32960:         proof (induct i)
wenzelm@32960:           case 0 show ?case by simp
wenzelm@32960:         next
wenzelm@32960:           case Suc thus ?case using monoD[OF mono Suc] by auto
wenzelm@32960:         qed }
nipkow@21312:       thus ?thesis by(auto simp add:chain_def)
nipkow@21312:     qed
haftmann@30971:     hence "F ?U = (SUP i. (F ^^ (i+1)) bot)" using `continuous F` by (simp add:continuous_def)
hoelzl@44928:     also have "\<dots> \<le> ?U" by(fast intro:SUP_least SUP_upper)
nipkow@21312:     finally show "F ?U \<le> ?U" .
nipkow@21312:   qed
nipkow@21312:   ultimately show ?thesis by (blast intro:order_antisym)
nipkow@21312: qed
nipkow@21312: 
nipkow@21312: text{* The following development is just for sets but presents an up
nipkow@21312: and a down version of chains and continuity and covers @{const gfp}. *}
nipkow@21312: 
nipkow@21312: 
oheimb@11351: subsection "Chains"
oheimb@11351: 
wenzelm@19736: definition
wenzelm@21404:   up_chain :: "(nat => 'a set) => bool" where
wenzelm@19736:   "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@19736: lemma up_chain_less_mono:
wenzelm@19736:     "up_chain F ==> x < y ==> F x \<subseteq> F y"
wenzelm@19736:   apply (induct y)
wenzelm@19736:    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: 
wenzelm@19736: definition
wenzelm@21404:   down_chain :: "(nat => 'a set) => bool" where
wenzelm@19736:   "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@19736: lemma down_chain_less_mono:
wenzelm@19736:     "down_chain F ==> x < y ==> F y \<subseteq> F x"
wenzelm@19736:   apply (induct y)
wenzelm@19736:    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: 
wenzelm@19736: definition
wenzelm@21404:   up_cont :: "('a set => 'a set) => bool" where
wenzelm@19736:   "up_cont f = (\<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F))"
oheimb@11351: 
wenzelm@11355: lemma up_contI:
nipkow@24331:   "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
nipkow@24331: apply (unfold up_cont_def)
nipkow@24331: apply blast
nipkow@24331: done
oheimb@11351: 
wenzelm@11355: lemma up_contD:
nipkow@24331:   "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
nipkow@24331: apply (unfold up_cont_def)
nipkow@24331: apply auto
nipkow@24331: done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: lemma up_cont_mono: "up_cont f ==> mono f"
nipkow@24331: apply (rule monoI)
haftmann@25076: apply (drule_tac F = "\<lambda>i. if i = 0 then x else y" in up_contD)
nipkow@24331:  apply (rule up_chainI)
nipkow@24331:  apply simp
nipkow@24331: apply (drule Un_absorb1)
nipkow@32456: apply (auto split:split_if_asm)
nipkow@24331: done
oheimb@11351: 
oheimb@11351: 
wenzelm@19736: definition
wenzelm@21404:   down_cont :: "('a set => 'a set) => bool" where
wenzelm@19736:   "down_cont f =
wenzelm@19736:     (\<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"
nipkow@24331: apply (rule monoI)
haftmann@25076: apply (drule_tac F = "\<lambda>i. if i = 0 then y else x" in down_contD)
nipkow@24331:  apply (rule down_chainI)
nipkow@24331:  apply simp
nipkow@24331: apply (drule Int_absorb1)
nipkow@32456: apply (auto split:split_if_asm)
nipkow@24331: done
oheimb@11351: 
oheimb@11351: 
oheimb@11351: subsection "Iteration"
oheimb@11351: 
wenzelm@19736: definition
wenzelm@21404:   up_iterate :: "('a set => 'a set) => nat => 'a set" where
haftmann@30971:   "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: 
wenzelm@19736: definition
wenzelm@21404:   down_iterate :: "('a set => 'a set) => nat => 'a set" where
haftmann@30971:   "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




