author  huffman 
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permissions  rwrr 
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(* Title: HOLCF/LowerPD.thy 
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ID: $Id$ 

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Author: Brian Huffman 

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*) 

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header {* Lower powerdomain *} 

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theory LowerPD 

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imports CompactBasis 

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begin 

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subsection {* Basis preorder *} 

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definition 

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lower_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<flat>" 50) where 

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"lower_le = (\<lambda>u v. \<forall>x\<in>Rep_pd_basis u. \<exists>y\<in>Rep_pd_basis v. x \<sqsubseteq> y)" 
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lemma lower_le_refl [simp]: "t \<le>\<flat> t" 

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unfolding lower_le_def by fast 
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lemma lower_le_trans: "\<lbrakk>t \<le>\<flat> u; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> t \<le>\<flat> v" 

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unfolding lower_le_def 

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apply (rule ballI) 

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apply (drule (1) bspec, erule bexE) 

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apply (drule (1) bspec, erule bexE) 

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apply (erule rev_bexI) 

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apply (erule (1) trans_less) 
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done 
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interpretation lower_le: preorder [lower_le] 

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by (rule preorder.intro, rule lower_le_refl, rule lower_le_trans) 

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lemma lower_le_minimal [simp]: "PDUnit compact_bot \<le>\<flat> t" 

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unfolding lower_le_def Rep_PDUnit 

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by (simp, rule Rep_pd_basis_nonempty [folded ex_in_conv]) 

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lemma PDUnit_lower_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<flat> PDUnit y" 
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unfolding lower_le_def Rep_PDUnit by fast 
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lemma PDPlus_lower_mono: "\<lbrakk>s \<le>\<flat> t; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<flat> PDPlus t v" 

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unfolding lower_le_def Rep_PDPlus by fast 

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lemma PDPlus_lower_less: "t \<le>\<flat> PDPlus t u" 

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unfolding lower_le_def Rep_PDPlus by fast 
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lemma lower_le_PDUnit_PDUnit_iff [simp]: 

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"(PDUnit a \<le>\<flat> PDUnit b) = a \<sqsubseteq> b" 
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unfolding lower_le_def Rep_PDUnit by fast 
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lemma lower_le_PDUnit_PDPlus_iff: 

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"(PDUnit a \<le>\<flat> PDPlus t u) = (PDUnit a \<le>\<flat> t \<or> PDUnit a \<le>\<flat> u)" 

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unfolding lower_le_def Rep_PDPlus Rep_PDUnit by fast 

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lemma lower_le_PDPlus_iff: "(PDPlus t u \<le>\<flat> v) = (t \<le>\<flat> v \<and> u \<le>\<flat> v)" 

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unfolding lower_le_def Rep_PDPlus by fast 

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lemma lower_le_induct [induct set: lower_le]: 

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assumes le: "t \<le>\<flat> u" 

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assumes 1: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)" 
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assumes 2: "\<And>t u a. P (PDUnit a) t \<Longrightarrow> P (PDUnit a) (PDPlus t u)" 
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assumes 3: "\<And>t u v. \<lbrakk>P t v; P u v\<rbrakk> \<Longrightarrow> P (PDPlus t u) v" 

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shows "P t u" 

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using le 

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apply (induct t arbitrary: u rule: pd_basis_induct) 

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apply (erule rev_mp) 

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apply (induct_tac u rule: pd_basis_induct) 

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apply (simp add: 1) 

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apply (simp add: lower_le_PDUnit_PDPlus_iff) 

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apply (simp add: 2) 

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apply (subst PDPlus_commute) 

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apply (simp add: 2) 

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apply (simp add: lower_le_PDPlus_iff 3) 

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done 

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lemma approx_pd_lower_chain: 
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"approx_pd n t \<le>\<flat> approx_pd (Suc n) t" 

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apply (induct t rule: pd_basis_induct) 
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apply (simp add: compact_basis.take_chain) 
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apply (simp add: PDPlus_lower_mono) 
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done 

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lemma approx_pd_lower_le: "approx_pd i t \<le>\<flat> t" 

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apply (induct t rule: pd_basis_induct) 

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apply (simp add: compact_basis.take_less) 
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apply (simp add: PDPlus_lower_mono) 
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done 

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lemma approx_pd_lower_mono: 

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"t \<le>\<flat> u \<Longrightarrow> approx_pd n t \<le>\<flat> approx_pd n u" 

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apply (erule lower_le_induct) 

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apply (simp add: compact_basis.take_mono) 
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apply (simp add: lower_le_PDUnit_PDPlus_iff) 
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apply (simp add: lower_le_PDPlus_iff) 

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done 

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subsection {* Type definition *} 

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typedef (open) 'a lower_pd = 
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"{S::'a pd_basis set. lower_le.ideal S}" 
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by (fast intro: lower_le.ideal_principal) 
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instantiation lower_pd :: (profinite) sq_ord 
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begin 
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definition 
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"x \<sqsubseteq> y \<longleftrightarrow> Rep_lower_pd x \<subseteq> Rep_lower_pd y" 
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instance .. 
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end 
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instance lower_pd :: (profinite) po 
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by (rule lower_le.typedef_ideal_po 
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[OF type_definition_lower_pd sq_le_lower_pd_def]) 
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instance lower_pd :: (profinite) cpo 
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by (rule lower_le.typedef_ideal_cpo 
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[OF type_definition_lower_pd sq_le_lower_pd_def]) 
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lemma Rep_lower_pd_lub: 
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"chain Y \<Longrightarrow> Rep_lower_pd (\<Squnion>i. Y i) = (\<Union>i. Rep_lower_pd (Y i))" 
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by (rule lower_le.typedef_ideal_rep_contlub 
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[OF type_definition_lower_pd sq_le_lower_pd_def]) 
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lemma ideal_Rep_lower_pd: "lower_le.ideal (Rep_lower_pd xs)" 
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by (rule Rep_lower_pd [unfolded mem_Collect_eq]) 
25904  127 

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definition 

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lower_principal :: "'a pd_basis \<Rightarrow> 'a lower_pd" where 

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"lower_principal t = Abs_lower_pd {u. u \<le>\<flat> t}" 
25904  131 

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lemma Rep_lower_principal: 

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"Rep_lower_pd (lower_principal t) = {u. u \<le>\<flat> t}" 
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unfolding lower_principal_def 
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by (simp add: Abs_lower_pd_inverse lower_le.ideal_principal) 
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interpretation lower_pd: 

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ideal_completion [lower_le approx_pd lower_principal Rep_lower_pd] 
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apply unfold_locales 
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apply (rule approx_pd_lower_le) 

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apply (rule approx_pd_idem) 

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apply (erule approx_pd_lower_mono) 

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apply (rule approx_pd_lower_chain) 
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apply (rule finite_range_approx_pd) 
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apply (rule approx_pd_covers) 
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apply (rule ideal_Rep_lower_pd) 
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apply (erule Rep_lower_pd_lub) 
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apply (rule Rep_lower_principal) 
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apply (simp only: sq_le_lower_pd_def) 
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done 
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text {* Lower powerdomain is pointed *} 
25904  153 

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lemma lower_pd_minimal: "lower_principal (PDUnit compact_bot) \<sqsubseteq> ys" 

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by (induct ys rule: lower_pd.principal_induct, simp, simp) 

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instance lower_pd :: (bifinite) pcpo 

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by intro_classes (fast intro: lower_pd_minimal) 
25904  159 

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lemma inst_lower_pd_pcpo: "\<bottom> = lower_principal (PDUnit compact_bot)" 

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by (rule lower_pd_minimal [THEN UU_I, symmetric]) 

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text {* Lower powerdomain is profinite *} 
25904  164 

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instantiation lower_pd :: (profinite) profinite 
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begin 
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definition 
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approx_lower_pd_def: "approx = lower_pd.completion_approx" 
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instance 
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apply (intro_classes, unfold approx_lower_pd_def) 
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apply (rule lower_pd.chain_completion_approx) 
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apply (rule lower_pd.lub_completion_approx) 
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apply (rule lower_pd.completion_approx_idem) 

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apply (rule lower_pd.finite_fixes_completion_approx) 

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done 

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end 
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instance lower_pd :: (bifinite) bifinite .. 
25904  182 

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lemma approx_lower_principal [simp]: 

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"approx n\<cdot>(lower_principal t) = lower_principal (approx_pd n t)" 

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unfolding approx_lower_pd_def 

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by (rule lower_pd.completion_approx_principal) 
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lemma approx_eq_lower_principal: 

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"\<exists>t\<in>Rep_lower_pd xs. approx n\<cdot>xs = lower_principal (approx_pd n t)" 
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unfolding approx_lower_pd_def 
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by (rule lower_pd.completion_approx_eq_principal) 
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subsection {* Monadic unit and plus *} 
25904  195 

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definition 

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lower_unit :: "'a \<rightarrow> 'a lower_pd" where 

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"lower_unit = compact_basis.basis_fun (\<lambda>a. lower_principal (PDUnit a))" 

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definition 

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lower_plus :: "'a lower_pd \<rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd" where 

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"lower_plus = lower_pd.basis_fun (\<lambda>t. lower_pd.basis_fun (\<lambda>u. 

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lower_principal (PDPlus t u)))" 

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abbreviation 

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lower_add :: "'a lower_pd \<Rightarrow> 'a lower_pd \<Rightarrow> 'a lower_pd" 

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(infixl "+\<flat>" 65) where 

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"xs +\<flat> ys == lower_plus\<cdot>xs\<cdot>ys" 

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syntax 
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"_lower_pd" :: "args \<Rightarrow> 'a lower_pd" ("{_}\<flat>") 

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translations 

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"{x,xs}\<flat>" == "{x}\<flat> +\<flat> {xs}\<flat>" 

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"{x}\<flat>" == "CONST lower_unit\<cdot>x" 

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lemma lower_unit_Rep_compact_basis [simp]: 

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"{Rep_compact_basis a}\<flat> = lower_principal (PDUnit a)" 

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unfolding lower_unit_def 

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by (simp add: compact_basis.basis_fun_principal PDUnit_lower_mono) 
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lemma lower_plus_principal [simp]: 
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"lower_principal t +\<flat> lower_principal u = lower_principal (PDPlus t u)" 
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unfolding lower_plus_def 
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by (simp add: lower_pd.basis_fun_principal 

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lower_pd.basis_fun_mono PDPlus_lower_mono) 

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lemma approx_lower_unit [simp]: 
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"approx n\<cdot>{x}\<flat> = {approx n\<cdot>x}\<flat>" 

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apply (induct x rule: compact_basis.principal_induct, simp) 
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apply (simp add: approx_Rep_compact_basis) 
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done 

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lemma approx_lower_plus [simp]: 
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"approx n\<cdot>(xs +\<flat> ys) = (approx n\<cdot>xs) +\<flat> (approx n\<cdot>ys)" 
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by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp) 
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lemma lower_plus_assoc: "(xs +\<flat> ys) +\<flat> zs = xs +\<flat> (ys +\<flat> zs)" 
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apply (induct xs ys arbitrary: zs rule: lower_pd.principal_induct2, simp, simp) 
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apply (rule_tac x=zs in lower_pd.principal_induct, simp) 

25904  241 
apply (simp add: PDPlus_assoc) 
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done 

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lemma lower_plus_commute: "xs +\<flat> ys = ys +\<flat> xs" 
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apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) 
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apply (simp add: PDPlus_commute) 
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done 

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lemma lower_plus_absorb: "xs +\<flat> xs = xs" 

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apply (induct xs rule: lower_pd.principal_induct, simp) 
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apply (simp add: PDPlus_absorb) 
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done 

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interpretation aci_lower_plus: ab_semigroup_idem_mult ["op +\<flat>"] 
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by unfold_locales 

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(rule lower_plus_assoc lower_plus_commute lower_plus_absorb)+ 

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lemma lower_plus_left_commute: "xs +\<flat> (ys +\<flat> zs) = ys +\<flat> (xs +\<flat> zs)" 

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by (rule aci_lower_plus.mult_left_commute) 

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lemma lower_plus_left_absorb: "xs +\<flat> (xs +\<flat> ys) = xs +\<flat> ys" 

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by (rule aci_lower_plus.mult_left_idem) 

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lemmas lower_plus_aci = aci_lower_plus.mult_ac_idem 

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lemma lower_plus_less1: "xs \<sqsubseteq> xs +\<flat> ys" 

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apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) 
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apply (simp add: PDPlus_lower_less) 
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done 

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lemma lower_plus_less2: "ys \<sqsubseteq> xs +\<flat> ys" 
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by (subst lower_plus_commute, rule lower_plus_less1) 
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lemma lower_plus_least: "\<lbrakk>xs \<sqsubseteq> zs; ys \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs +\<flat> ys \<sqsubseteq> zs" 
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apply (subst lower_plus_absorb [of zs, symmetric]) 
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apply (erule (1) monofun_cfun [OF monofun_cfun_arg]) 

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done 

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lemma lower_plus_less_iff: 

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"xs +\<flat> ys \<sqsubseteq> zs \<longleftrightarrow> xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs" 
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apply safe 
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apply (erule trans_less [OF lower_plus_less1]) 

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apply (erule trans_less [OF lower_plus_less2]) 

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apply (erule (1) lower_plus_least) 

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done 

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lemma lower_unit_less_plus_iff: 

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"{x}\<flat> \<sqsubseteq> ys +\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs" 
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apply (rule iffI) 
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apply (subgoal_tac 

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"adm (\<lambda>f. f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>ys \<or> f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>zs)") 
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apply (drule admD, rule chain_approx) 
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apply (drule_tac f="approx i" in monofun_cfun_arg) 
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apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp) 
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apply (cut_tac x="approx i\<cdot>ys" in lower_pd.compact_imp_principal, simp) 

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apply (cut_tac x="approx i\<cdot>zs" in lower_pd.compact_imp_principal, simp) 

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apply (clarify, simp add: lower_le_PDUnit_PDPlus_iff) 
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apply simp 

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apply simp 

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apply (erule disjE) 

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apply (erule trans_less [OF _ lower_plus_less1]) 

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apply (erule trans_less [OF _ lower_plus_less2]) 

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done 

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lemma lower_unit_less_iff [simp]: "{x}\<flat> \<sqsubseteq> {y}\<flat> \<longleftrightarrow> x \<sqsubseteq> y" 
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apply (rule iffI) 

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apply (rule profinite_less_ext) 
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apply (drule_tac f="approx i" in monofun_cfun_arg, simp) 
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apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp) 
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apply (cut_tac x="approx i\<cdot>y" in compact_basis.compact_imp_principal, simp) 

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apply clarsimp 

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apply (erule monofun_cfun_arg) 
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done 

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lemmas lower_pd_less_simps = 
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lower_unit_less_iff 

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lower_plus_less_iff 

318 
lower_unit_less_plus_iff 

319 

27289  320 
lemma fooble: 
321 
fixes f :: "'a::po \<Rightarrow> 'b::po" 

322 
assumes f: "\<And>x y. f x \<sqsubseteq> f y \<longleftrightarrow> x \<sqsubseteq> y" 

323 
shows "f x = f y \<longleftrightarrow> x = y" 

324 
unfolding po_eq_conv by (simp add: f) 

325 

26927  326 
lemma lower_unit_eq_iff [simp]: "{x}\<flat> = {y}\<flat> \<longleftrightarrow> x = y" 
27289  327 
by (rule lower_unit_less_iff [THEN fooble]) 
26927  328 

329 
lemma lower_unit_strict [simp]: "{\<bottom>}\<flat> = \<bottom>" 

330 
unfolding inst_lower_pd_pcpo Rep_compact_bot [symmetric] by simp 

331 

332 
lemma lower_unit_strict_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>" 

333 
unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff) 

334 

335 
lemma lower_plus_strict_iff [simp]: 

336 
"xs +\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>" 

337 
apply safe 

338 
apply (rule UU_I, erule subst, rule lower_plus_less1) 

339 
apply (rule UU_I, erule subst, rule lower_plus_less2) 

340 
apply (rule lower_plus_absorb) 

341 
done 

342 

343 
lemma lower_plus_strict1 [simp]: "\<bottom> +\<flat> ys = ys" 

344 
apply (rule antisym_less [OF _ lower_plus_less2]) 

345 
apply (simp add: lower_plus_least) 

346 
done 

347 

348 
lemma lower_plus_strict2 [simp]: "xs +\<flat> \<bottom> = xs" 

349 
apply (rule antisym_less [OF _ lower_plus_less1]) 

350 
apply (simp add: lower_plus_least) 

351 
done 

352 

353 
lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x" 

27309  354 
unfolding profinite_compact_iff by simp 
26927  355 

356 
lemma compact_lower_plus [simp]: 

357 
"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<flat> ys)" 

27289  358 
by (auto dest!: lower_pd.compact_imp_principal) 
26927  359 

25904  360 

361 
subsection {* Induction rules *} 

362 

363 
lemma lower_pd_induct1: 

364 
assumes P: "adm P" 

26927  365 
assumes unit: "\<And>x. P {x}\<flat>" 
25904  366 
assumes insert: 
26927  367 
"\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> +\<flat> ys)" 
25904  368 
shows "P (xs::'a lower_pd)" 
27289  369 
apply (induct xs rule: lower_pd.principal_induct, rule P) 
370 
apply (induct_tac a rule: pd_basis_induct1) 

25904  371 
apply (simp only: lower_unit_Rep_compact_basis [symmetric]) 
372 
apply (rule unit) 

373 
apply (simp only: lower_unit_Rep_compact_basis [symmetric] 

374 
lower_plus_principal [symmetric]) 

375 
apply (erule insert [OF unit]) 

376 
done 

377 

378 
lemma lower_pd_induct: 

379 
assumes P: "adm P" 

26927  380 
assumes unit: "\<And>x. P {x}\<flat>" 
381 
assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<flat> ys)" 

25904  382 
shows "P (xs::'a lower_pd)" 
27289  383 
apply (induct xs rule: lower_pd.principal_induct, rule P) 
384 
apply (induct_tac a rule: pd_basis_induct) 

25904  385 
apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit) 
386 
apply (simp only: lower_plus_principal [symmetric] plus) 

387 
done 

388 

389 

390 
subsection {* Monadic bind *} 

391 

392 
definition 

393 
lower_bind_basis :: 

394 
"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where 

395 
"lower_bind_basis = fold_pd 

396 
(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a)) 

26927  397 
(\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" 
25904  398 

26927  399 
lemma ACI_lower_bind: 
400 
"ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" 

25904  401 
apply unfold_locales 
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25925
diff
changeset

402 
apply (simp add: lower_plus_assoc) 
25904  403 
apply (simp add: lower_plus_commute) 
404 
apply (simp add: lower_plus_absorb eta_cfun) 

405 
done 

406 

407 
lemma lower_bind_basis_simps [simp]: 

408 
"lower_bind_basis (PDUnit a) = 

409 
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" 

410 
"lower_bind_basis (PDPlus t u) = 

26927  411 
(\<Lambda> f. lower_bind_basis t\<cdot>f +\<flat> lower_bind_basis u\<cdot>f)" 
25904  412 
unfolding lower_bind_basis_def 
413 
apply  

26927  414 
apply (rule fold_pd_PDUnit [OF ACI_lower_bind]) 
415 
apply (rule fold_pd_PDPlus [OF ACI_lower_bind]) 

25904  416 
done 
417 

418 
lemma lower_bind_basis_mono: 

419 
"t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u" 

420 
unfolding expand_cfun_less 

421 
apply (erule lower_le_induct, safe) 

27289  422 
apply (simp add: monofun_cfun) 
25904  423 
apply (simp add: rev_trans_less [OF lower_plus_less1]) 
424 
apply (simp add: lower_plus_less_iff) 

425 
done 

426 

427 
definition 

428 
lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where 

429 
"lower_bind = lower_pd.basis_fun lower_bind_basis" 

430 

431 
lemma lower_bind_principal [simp]: 

432 
"lower_bind\<cdot>(lower_principal t) = lower_bind_basis t" 

433 
unfolding lower_bind_def 

434 
apply (rule lower_pd.basis_fun_principal) 

435 
apply (erule lower_bind_basis_mono) 

436 
done 

437 

438 
lemma lower_bind_unit [simp]: 

26927  439 
"lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x" 
27289  440 
by (induct x rule: compact_basis.principal_induct, simp, simp) 
25904  441 

442 
lemma lower_bind_plus [simp]: 

26927  443 
"lower_bind\<cdot>(xs +\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f +\<flat> lower_bind\<cdot>ys\<cdot>f" 
27289  444 
by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp) 
25904  445 

446 
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" 

447 
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit) 

448 

449 

450 
subsection {* Map and join *} 

451 

452 
definition 

453 
lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where 

26927  454 
"lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))" 
25904  455 

456 
definition 

457 
lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where 

458 
"lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" 

459 

460 
lemma lower_map_unit [simp]: 

26927  461 
"lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>" 
25904  462 
unfolding lower_map_def by simp 
463 

464 
lemma lower_map_plus [simp]: 

26927  465 
"lower_map\<cdot>f\<cdot>(xs +\<flat> ys) = lower_map\<cdot>f\<cdot>xs +\<flat> lower_map\<cdot>f\<cdot>ys" 
25904  466 
unfolding lower_map_def by simp 
467 

468 
lemma lower_join_unit [simp]: 

26927  469 
"lower_join\<cdot>{xs}\<flat> = xs" 
25904  470 
unfolding lower_join_def by simp 
471 

472 
lemma lower_join_plus [simp]: 

26927  473 
"lower_join\<cdot>(xss +\<flat> yss) = lower_join\<cdot>xss +\<flat> lower_join\<cdot>yss" 
25904  474 
unfolding lower_join_def by simp 
475 

476 
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" 

477 
by (induct xs rule: lower_pd_induct, simp_all) 

478 

479 
lemma lower_map_map: 

480 
"lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" 

481 
by (induct xs rule: lower_pd_induct, simp_all) 

482 

483 
lemma lower_join_map_unit: 

484 
"lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs" 

485 
by (induct xs rule: lower_pd_induct, simp_all) 

486 

487 
lemma lower_join_map_join: 

488 
"lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)" 

489 
by (induct xsss rule: lower_pd_induct, simp_all) 

490 

491 
lemma lower_join_map_map: 

492 
"lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) = 

493 
lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)" 

494 
by (induct xss rule: lower_pd_induct, simp_all) 

495 

496 
lemma lower_map_approx: "lower_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs" 

497 
by (induct xs rule: lower_pd_induct, simp_all) 

498 

499 
end 