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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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cpodef (open) 'a lower_pd =
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"{S::'a pd_basis cset. lower_le.ideal (Rep_cset S)}"
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by (rule lower_le.cpodef_ideal_lemma)
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lemma ideal_Rep_lower_pd: "lower_le.ideal (Rep_cset (Rep_lower_pd xs))"
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by (rule Rep_lower_pd [unfolded mem_Collect_eq])
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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 (Abs_cset {u. u \<le>\<flat> t})"
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lemma Rep_lower_principal:
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"Rep_cset (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
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[lower_le approx_pd lower_principal "\<lambda>x. Rep_cset (Rep_lower_pd x)"]
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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 (simp add: cont2contlubE [OF cont_Rep_lower_pd] Rep_cset_lub)
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apply (rule Rep_lower_principal)
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apply (simp only: less_lower_pd_def sq_le_cset_def)
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done
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text {* Lower powerdomain is pointed *}
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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)
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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 *}
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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 ..
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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_cset (Rep_lower_pd xs).
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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 *}
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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)
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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
|
huffman@25904
|
258 |
|
huffman@25904
|
259 |
lemma lower_plus_less_iff:
|
huffman@26927
|
260 |
"xs +\<flat> ys \<sqsubseteq> zs \<longleftrightarrow> xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs"
|
huffman@25904
|
261 |
apply safe
|
huffman@25904
|
262 |
apply (erule trans_less [OF lower_plus_less1])
|
huffman@25904
|
263 |
apply (erule trans_less [OF lower_plus_less2])
|
huffman@25904
|
264 |
apply (erule (1) lower_plus_least)
|
huffman@25904
|
265 |
done
|
huffman@25904
|
266 |
|
huffman@25904
|
267 |
lemma lower_unit_less_plus_iff:
|
huffman@26927
|
268 |
"{x}\<flat> \<sqsubseteq> ys +\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs"
|
huffman@25904
|
269 |
apply (rule iffI)
|
huffman@25904
|
270 |
apply (subgoal_tac
|
huffman@26927
|
271 |
"adm (\<lambda>f. f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>ys \<or> f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>zs)")
|
huffman@25925
|
272 |
apply (drule admD, rule chain_approx)
|
huffman@25904
|
273 |
apply (drule_tac f="approx i" in monofun_cfun_arg)
|
huffman@27289
|
274 |
apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp)
|
huffman@27289
|
275 |
apply (cut_tac x="approx i\<cdot>ys" in lower_pd.compact_imp_principal, simp)
|
huffman@27289
|
276 |
apply (cut_tac x="approx i\<cdot>zs" in lower_pd.compact_imp_principal, simp)
|
huffman@25904
|
277 |
apply (clarify, simp add: lower_le_PDUnit_PDPlus_iff)
|
huffman@25904
|
278 |
apply simp
|
huffman@25904
|
279 |
apply simp
|
huffman@25904
|
280 |
apply (erule disjE)
|
huffman@25904
|
281 |
apply (erule trans_less [OF _ lower_plus_less1])
|
huffman@25904
|
282 |
apply (erule trans_less [OF _ lower_plus_less2])
|
huffman@25904
|
283 |
done
|
huffman@25904
|
284 |
|
huffman@26927
|
285 |
lemma lower_unit_less_iff [simp]: "{x}\<flat> \<sqsubseteq> {y}\<flat> \<longleftrightarrow> x \<sqsubseteq> y"
|
huffman@26927
|
286 |
apply (rule iffI)
|
huffman@27309
|
287 |
apply (rule profinite_less_ext)
|
huffman@26927
|
288 |
apply (drule_tac f="approx i" in monofun_cfun_arg, simp)
|
huffman@27289
|
289 |
apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp)
|
huffman@27289
|
290 |
apply (cut_tac x="approx i\<cdot>y" in compact_basis.compact_imp_principal, simp)
|
huffman@27289
|
291 |
apply clarsimp
|
huffman@26927
|
292 |
apply (erule monofun_cfun_arg)
|
huffman@26927
|
293 |
done
|
huffman@26927
|
294 |
|
huffman@25904
|
295 |
lemmas lower_pd_less_simps =
|
huffman@25904
|
296 |
lower_unit_less_iff
|
huffman@25904
|
297 |
lower_plus_less_iff
|
huffman@25904
|
298 |
lower_unit_less_plus_iff
|
huffman@25904
|
299 |
|
huffman@27289
|
300 |
lemma fooble:
|
huffman@27289
|
301 |
fixes f :: "'a::po \<Rightarrow> 'b::po"
|
huffman@27289
|
302 |
assumes f: "\<And>x y. f x \<sqsubseteq> f y \<longleftrightarrow> x \<sqsubseteq> y"
|
huffman@27289
|
303 |
shows "f x = f y \<longleftrightarrow> x = y"
|
huffman@27289
|
304 |
unfolding po_eq_conv by (simp add: f)
|
huffman@27289
|
305 |
|
huffman@26927
|
306 |
lemma lower_unit_eq_iff [simp]: "{x}\<flat> = {y}\<flat> \<longleftrightarrow> x = y"
|
huffman@27289
|
307 |
by (rule lower_unit_less_iff [THEN fooble])
|
huffman@26927
|
308 |
|
huffman@26927
|
309 |
lemma lower_unit_strict [simp]: "{\<bottom>}\<flat> = \<bottom>"
|
huffman@26927
|
310 |
unfolding inst_lower_pd_pcpo Rep_compact_bot [symmetric] by simp
|
huffman@26927
|
311 |
|
huffman@26927
|
312 |
lemma lower_unit_strict_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>"
|
huffman@26927
|
313 |
unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff)
|
huffman@26927
|
314 |
|
huffman@26927
|
315 |
lemma lower_plus_strict_iff [simp]:
|
huffman@26927
|
316 |
"xs +\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>"
|
huffman@26927
|
317 |
apply safe
|
huffman@26927
|
318 |
apply (rule UU_I, erule subst, rule lower_plus_less1)
|
huffman@26927
|
319 |
apply (rule UU_I, erule subst, rule lower_plus_less2)
|
huffman@26927
|
320 |
apply (rule lower_plus_absorb)
|
huffman@26927
|
321 |
done
|
huffman@26927
|
322 |
|
huffman@26927
|
323 |
lemma lower_plus_strict1 [simp]: "\<bottom> +\<flat> ys = ys"
|
huffman@26927
|
324 |
apply (rule antisym_less [OF _ lower_plus_less2])
|
huffman@26927
|
325 |
apply (simp add: lower_plus_least)
|
huffman@26927
|
326 |
done
|
huffman@26927
|
327 |
|
huffman@26927
|
328 |
lemma lower_plus_strict2 [simp]: "xs +\<flat> \<bottom> = xs"
|
huffman@26927
|
329 |
apply (rule antisym_less [OF _ lower_plus_less1])
|
huffman@26927
|
330 |
apply (simp add: lower_plus_least)
|
huffman@26927
|
331 |
done
|
huffman@26927
|
332 |
|
huffman@26927
|
333 |
lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x"
|
huffman@27309
|
334 |
unfolding profinite_compact_iff by simp
|
huffman@26927
|
335 |
|
huffman@26927
|
336 |
lemma compact_lower_plus [simp]:
|
huffman@26927
|
337 |
"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<flat> ys)"
|
huffman@27289
|
338 |
by (auto dest!: lower_pd.compact_imp_principal)
|
huffman@26927
|
339 |
|
huffman@25904
|
340 |
|
huffman@25904
|
341 |
subsection {* Induction rules *}
|
huffman@25904
|
342 |
|
huffman@25904
|
343 |
lemma lower_pd_induct1:
|
huffman@25904
|
344 |
assumes P: "adm P"
|
huffman@26927
|
345 |
assumes unit: "\<And>x. P {x}\<flat>"
|
huffman@25904
|
346 |
assumes insert:
|
huffman@26927
|
347 |
"\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> +\<flat> ys)"
|
huffman@25904
|
348 |
shows "P (xs::'a lower_pd)"
|
huffman@27289
|
349 |
apply (induct xs rule: lower_pd.principal_induct, rule P)
|
huffman@27289
|
350 |
apply (induct_tac a rule: pd_basis_induct1)
|
huffman@25904
|
351 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric])
|
huffman@25904
|
352 |
apply (rule unit)
|
huffman@25904
|
353 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric]
|
huffman@25904
|
354 |
lower_plus_principal [symmetric])
|
huffman@25904
|
355 |
apply (erule insert [OF unit])
|
huffman@25904
|
356 |
done
|
huffman@25904
|
357 |
|
huffman@25904
|
358 |
lemma lower_pd_induct:
|
huffman@25904
|
359 |
assumes P: "adm P"
|
huffman@26927
|
360 |
assumes unit: "\<And>x. P {x}\<flat>"
|
huffman@26927
|
361 |
assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<flat> ys)"
|
huffman@25904
|
362 |
shows "P (xs::'a lower_pd)"
|
huffman@27289
|
363 |
apply (induct xs rule: lower_pd.principal_induct, rule P)
|
huffman@27289
|
364 |
apply (induct_tac a rule: pd_basis_induct)
|
huffman@25904
|
365 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit)
|
huffman@25904
|
366 |
apply (simp only: lower_plus_principal [symmetric] plus)
|
huffman@25904
|
367 |
done
|
huffman@25904
|
368 |
|
huffman@25904
|
369 |
|
huffman@25904
|
370 |
subsection {* Monadic bind *}
|
huffman@25904
|
371 |
|
huffman@25904
|
372 |
definition
|
huffman@25904
|
373 |
lower_bind_basis ::
|
huffman@25904
|
374 |
"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
|
huffman@25904
|
375 |
"lower_bind_basis = fold_pd
|
huffman@25904
|
376 |
(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
|
huffman@26927
|
377 |
(\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)"
|
huffman@25904
|
378 |
|
huffman@26927
|
379 |
lemma ACI_lower_bind:
|
huffman@26927
|
380 |
"ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)"
|
huffman@25904
|
381 |
apply unfold_locales
|
haftmann@26041
|
382 |
apply (simp add: lower_plus_assoc)
|
huffman@25904
|
383 |
apply (simp add: lower_plus_commute)
|
huffman@25904
|
384 |
apply (simp add: lower_plus_absorb eta_cfun)
|
huffman@25904
|
385 |
done
|
huffman@25904
|
386 |
|
huffman@25904
|
387 |
lemma lower_bind_basis_simps [simp]:
|
huffman@25904
|
388 |
"lower_bind_basis (PDUnit a) =
|
huffman@25904
|
389 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
|
huffman@25904
|
390 |
"lower_bind_basis (PDPlus t u) =
|
huffman@26927
|
391 |
(\<Lambda> f. lower_bind_basis t\<cdot>f +\<flat> lower_bind_basis u\<cdot>f)"
|
huffman@25904
|
392 |
unfolding lower_bind_basis_def
|
huffman@25904
|
393 |
apply -
|
huffman@26927
|
394 |
apply (rule fold_pd_PDUnit [OF ACI_lower_bind])
|
huffman@26927
|
395 |
apply (rule fold_pd_PDPlus [OF ACI_lower_bind])
|
huffman@25904
|
396 |
done
|
huffman@25904
|
397 |
|
huffman@25904
|
398 |
lemma lower_bind_basis_mono:
|
huffman@25904
|
399 |
"t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u"
|
huffman@25904
|
400 |
unfolding expand_cfun_less
|
huffman@25904
|
401 |
apply (erule lower_le_induct, safe)
|
huffman@27289
|
402 |
apply (simp add: monofun_cfun)
|
huffman@25904
|
403 |
apply (simp add: rev_trans_less [OF lower_plus_less1])
|
huffman@25904
|
404 |
apply (simp add: lower_plus_less_iff)
|
huffman@25904
|
405 |
done
|
huffman@25904
|
406 |
|
huffman@25904
|
407 |
definition
|
huffman@25904
|
408 |
lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
|
huffman@25904
|
409 |
"lower_bind = lower_pd.basis_fun lower_bind_basis"
|
huffman@25904
|
410 |
|
huffman@25904
|
411 |
lemma lower_bind_principal [simp]:
|
huffman@25904
|
412 |
"lower_bind\<cdot>(lower_principal t) = lower_bind_basis t"
|
huffman@25904
|
413 |
unfolding lower_bind_def
|
huffman@25904
|
414 |
apply (rule lower_pd.basis_fun_principal)
|
huffman@25904
|
415 |
apply (erule lower_bind_basis_mono)
|
huffman@25904
|
416 |
done
|
huffman@25904
|
417 |
|
huffman@25904
|
418 |
lemma lower_bind_unit [simp]:
|
huffman@26927
|
419 |
"lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x"
|
huffman@27289
|
420 |
by (induct x rule: compact_basis.principal_induct, simp, simp)
|
huffman@25904
|
421 |
|
huffman@25904
|
422 |
lemma lower_bind_plus [simp]:
|
huffman@26927
|
423 |
"lower_bind\<cdot>(xs +\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f +\<flat> lower_bind\<cdot>ys\<cdot>f"
|
huffman@27289
|
424 |
by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp)
|
huffman@25904
|
425 |
|
huffman@25904
|
426 |
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
|
huffman@25904
|
427 |
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit)
|
huffman@25904
|
428 |
|
huffman@25904
|
429 |
|
huffman@25904
|
430 |
subsection {* Map and join *}
|
huffman@25904
|
431 |
|
huffman@25904
|
432 |
definition
|
huffman@25904
|
433 |
lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where
|
huffman@26927
|
434 |
"lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))"
|
huffman@25904
|
435 |
|
huffman@25904
|
436 |
definition
|
huffman@25904
|
437 |
lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where
|
huffman@25904
|
438 |
"lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"
|
huffman@25904
|
439 |
|
huffman@25904
|
440 |
lemma lower_map_unit [simp]:
|
huffman@26927
|
441 |
"lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>"
|
huffman@25904
|
442 |
unfolding lower_map_def by simp
|
huffman@25904
|
443 |
|
huffman@25904
|
444 |
lemma lower_map_plus [simp]:
|
huffman@26927
|
445 |
"lower_map\<cdot>f\<cdot>(xs +\<flat> ys) = lower_map\<cdot>f\<cdot>xs +\<flat> lower_map\<cdot>f\<cdot>ys"
|
huffman@25904
|
446 |
unfolding lower_map_def by simp
|
huffman@25904
|
447 |
|
huffman@25904
|
448 |
lemma lower_join_unit [simp]:
|
huffman@26927
|
449 |
"lower_join\<cdot>{xs}\<flat> = xs"
|
huffman@25904
|
450 |
unfolding lower_join_def by simp
|
huffman@25904
|
451 |
|
huffman@25904
|
452 |
lemma lower_join_plus [simp]:
|
huffman@26927
|
453 |
"lower_join\<cdot>(xss +\<flat> yss) = lower_join\<cdot>xss +\<flat> lower_join\<cdot>yss"
|
huffman@25904
|
454 |
unfolding lower_join_def by simp
|
huffman@25904
|
455 |
|
huffman@25904
|
456 |
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
|
huffman@25904
|
457 |
by (induct xs rule: lower_pd_induct, simp_all)
|
huffman@25904
|
458 |
|
huffman@25904
|
459 |
lemma lower_map_map:
|
huffman@25904
|
460 |
"lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
|
huffman@25904
|
461 |
by (induct xs rule: lower_pd_induct, simp_all)
|
huffman@25904
|
462 |
|
huffman@25904
|
463 |
lemma lower_join_map_unit:
|
huffman@25904
|
464 |
"lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs"
|
huffman@25904
|
465 |
by (induct xs rule: lower_pd_induct, simp_all)
|
huffman@25904
|
466 |
|
huffman@25904
|
467 |
lemma lower_join_map_join:
|
huffman@25904
|
468 |
"lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)"
|
huffman@25904
|
469 |
by (induct xsss rule: lower_pd_induct, simp_all)
|
huffman@25904
|
470 |
|
huffman@25904
|
471 |
lemma lower_join_map_map:
|
huffman@25904
|
472 |
"lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) =
|
huffman@25904
|
473 |
lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)"
|
huffman@25904
|
474 |
by (induct xss rule: lower_pd_induct, simp_all)
|
huffman@25904
|
475 |
|
huffman@25904
|
476 |
lemma lower_map_approx: "lower_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs"
|
huffman@25904
|
477 |
by (induct xs rule: lower_pd_induct, simp_all)
|
huffman@25904
|
478 |
|
huffman@25904
|
479 |
end
|