src/HOL/HOLCF/LowerPD.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 51489 f738e6dbd844
child 58880 0baae4311a9f
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
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(*  Title:      HOL/HOLCF/LowerPD.thy
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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 Compact_Basis
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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) below_trans)
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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_le: "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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subsection {* Type definition *}
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typedef 'a lower_pd =
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  "{S::'a pd_basis set. lower_le.ideal S}"
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by (rule lower_le.ex_ideal)
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type_notation (xsymbols) lower_pd ("('(_')\<flat>)")
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instantiation lower_pd :: (bifinite) below
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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 :: (bifinite) po
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using type_definition_lower_pd below_lower_pd_def
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by (rule lower_le.typedef_ideal_po)
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instance lower_pd :: (bifinite) cpo
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using type_definition_lower_pd below_lower_pd_def
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by (rule lower_le.typedef_ideal_cpo)
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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}"
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interpretation lower_pd:
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  ideal_completion lower_le lower_principal Rep_lower_pd
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using type_definition_lower_pd below_lower_pd_def
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using lower_principal_def pd_basis_countable
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by (rule lower_le.typedef_ideal_completion)
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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 bottomI, symmetric])
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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.extension (\<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.extension (\<lambda>t. lower_pd.extension (\<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 "\<union>\<flat>" 65) where
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  "xs \<union>\<flat> ys == lower_plus\<cdot>xs\<cdot>ys"
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syntax
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  "_lower_pd" :: "args \<Rightarrow> logic" ("{_}\<flat>")
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translations
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  "{x,xs}\<flat>" == "{x}\<flat> \<union>\<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.extension_principal PDUnit_lower_mono)
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lemma lower_plus_principal [simp]:
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  "lower_principal t \<union>\<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.extension_principal
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    lower_pd.extension_mono PDPlus_lower_mono)
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interpretation lower_add: semilattice lower_add proof
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  fix xs ys zs :: "'a lower_pd"
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  show "(xs \<union>\<flat> ys) \<union>\<flat> zs = xs \<union>\<flat> (ys \<union>\<flat> zs)"
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    apply (induct xs rule: lower_pd.principal_induct, simp)
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    apply (induct ys rule: lower_pd.principal_induct, simp)
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    apply (induct zs rule: lower_pd.principal_induct, simp)
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    apply (simp add: PDPlus_assoc)
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    done
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  show "xs \<union>\<flat> ys = ys \<union>\<flat> xs"
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    apply (induct xs rule: lower_pd.principal_induct, simp)
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    apply (induct ys rule: lower_pd.principal_induct, simp)
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    apply (simp add: PDPlus_commute)
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    done
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  show "xs \<union>\<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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qed
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lemmas lower_plus_assoc = lower_add.assoc
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lemmas lower_plus_commute = lower_add.commute
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lemmas lower_plus_absorb = lower_add.idem
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lemmas lower_plus_left_commute = lower_add.left_commute
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lemmas lower_plus_left_absorb = lower_add.left_idem
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text {* Useful for @{text "simp add: lower_plus_ac"} *}
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lemmas lower_plus_ac =
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  lower_plus_assoc lower_plus_commute lower_plus_left_commute
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text {* Useful for @{text "simp only: lower_plus_aci"} *}
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lemmas lower_plus_aci =
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  lower_plus_ac lower_plus_absorb lower_plus_left_absorb
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lemma lower_plus_below1: "xs \<sqsubseteq> xs \<union>\<flat> ys"
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apply (induct xs rule: lower_pd.principal_induct, simp)
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apply (induct ys rule: lower_pd.principal_induct, simp)
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apply (simp add: PDPlus_lower_le)
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done
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lemma lower_plus_below2: "ys \<sqsubseteq> xs \<union>\<flat> ys"
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by (subst lower_plus_commute, rule lower_plus_below1)
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lemma lower_plus_least: "\<lbrakk>xs \<sqsubseteq> zs; ys \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs \<union>\<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_below_iff [simp]:
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  "xs \<union>\<flat> ys \<sqsubseteq> zs \<longleftrightarrow> xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs"
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apply safe
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apply (erule below_trans [OF lower_plus_below1])
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apply (erule below_trans [OF lower_plus_below2])
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apply (erule (1) lower_plus_least)
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done
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lemma lower_unit_below_plus_iff [simp]:
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  "{x}\<flat> \<sqsubseteq> ys \<union>\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs"
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apply (induct x rule: compact_basis.principal_induct, simp)
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apply (induct ys rule: lower_pd.principal_induct, simp)
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apply (induct zs rule: lower_pd.principal_induct, simp)
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apply (simp add: lower_le_PDUnit_PDPlus_iff)
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done
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lemma lower_unit_below_iff [simp]: "{x}\<flat> \<sqsubseteq> {y}\<flat> \<longleftrightarrow> x \<sqsubseteq> y"
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apply (induct x rule: compact_basis.principal_induct, simp)
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apply (induct y rule: compact_basis.principal_induct, simp)
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apply simp
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done
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lemmas lower_pd_below_simps =
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  lower_unit_below_iff
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  lower_plus_below_iff
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  lower_unit_below_plus_iff
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lemma lower_unit_eq_iff [simp]: "{x}\<flat> = {y}\<flat> \<longleftrightarrow> x = y"
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by (simp add: po_eq_conv)
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lemma lower_unit_strict [simp]: "{\<bottom>}\<flat> = \<bottom>"
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using lower_unit_Rep_compact_basis [of compact_bot]
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by (simp add: inst_lower_pd_pcpo)
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lemma lower_unit_bottom_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>"
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unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff)
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lemma lower_plus_bottom_iff [simp]:
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  "xs \<union>\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>"
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apply safe
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apply (rule bottomI, erule subst, rule lower_plus_below1)
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apply (rule bottomI, erule subst, rule lower_plus_below2)
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apply (rule lower_plus_absorb)
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done
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lemma lower_plus_strict1 [simp]: "\<bottom> \<union>\<flat> ys = ys"
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apply (rule below_antisym [OF _ lower_plus_below2])
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apply (simp add: lower_plus_least)
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done
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lemma lower_plus_strict2 [simp]: "xs \<union>\<flat> \<bottom> = xs"
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apply (rule below_antisym [OF _ lower_plus_below1])
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apply (simp add: lower_plus_least)
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done
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lemma compact_lower_unit: "compact x \<Longrightarrow> compact {x}\<flat>"
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by (auto dest!: compact_basis.compact_imp_principal)
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lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x"
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apply (safe elim!: compact_lower_unit)
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apply (simp only: compact_def lower_unit_below_iff [symmetric])
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apply (erule adm_subst [OF cont_Rep_cfun2])
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done
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lemma compact_lower_plus [simp]:
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  "\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs \<union>\<flat> ys)"
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by (auto dest!: lower_pd.compact_imp_principal)
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subsection {* Induction rules *}
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lemma lower_pd_induct1:
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  assumes P: "adm P"
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  assumes unit: "\<And>x. P {x}\<flat>"
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  assumes insert:
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    "\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> \<union>\<flat> ys)"
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  shows "P (xs::'a lower_pd)"
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apply (induct xs rule: lower_pd.principal_induct, rule P)
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apply (induct_tac a rule: pd_basis_induct1)
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apply (simp only: lower_unit_Rep_compact_basis [symmetric])
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apply (rule unit)
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apply (simp only: lower_unit_Rep_compact_basis [symmetric]
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                  lower_plus_principal [symmetric])
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apply (erule insert [OF unit])
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done
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lemma lower_pd_induct
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  [case_names adm lower_unit lower_plus, induct type: lower_pd]:
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  assumes P: "adm P"
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  assumes unit: "\<And>x. P {x}\<flat>"
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  assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs \<union>\<flat> ys)"
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  shows "P (xs::'a lower_pd)"
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apply (induct xs rule: lower_pd.principal_induct, rule P)
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apply (induct_tac a rule: pd_basis_induct)
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apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit)
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apply (simp only: lower_plus_principal [symmetric] plus)
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done
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subsection {* Monadic bind *}
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definition
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  lower_bind_basis ::
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  "'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
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  "lower_bind_basis = fold_pd
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    (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
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   309
    (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)"
huffman@25904
   310
huffman@26927
   311
lemma ACI_lower_bind:
haftmann@51489
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  "semilattice (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)"
huffman@25904
   313
apply unfold_locales
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   314
apply (simp add: lower_plus_assoc)
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   315
apply (simp add: lower_plus_commute)
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   316
apply (simp add: eta_cfun)
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   317
done
huffman@25904
   318
huffman@25904
   319
lemma lower_bind_basis_simps [simp]:
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   320
  "lower_bind_basis (PDUnit a) =
huffman@25904
   321
    (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
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   322
  "lower_bind_basis (PDPlus t u) =
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   323
    (\<Lambda> f. lower_bind_basis t\<cdot>f \<union>\<flat> lower_bind_basis u\<cdot>f)"
huffman@25904
   324
unfolding lower_bind_basis_def
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   325
apply -
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   326
apply (rule fold_pd_PDUnit [OF ACI_lower_bind])
huffman@26927
   327
apply (rule fold_pd_PDPlus [OF ACI_lower_bind])
huffman@25904
   328
done
huffman@25904
   329
huffman@25904
   330
lemma lower_bind_basis_mono:
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   331
  "t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u"
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   332
unfolding cfun_below_iff
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   333
apply (erule lower_le_induct, safe)
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   334
apply (simp add: monofun_cfun)
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   335
apply (simp add: rev_below_trans [OF lower_plus_below1])
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   336
apply simp
huffman@25904
   337
done
huffman@25904
   338
huffman@25904
   339
definition
huffman@25904
   340
  lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
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   341
  "lower_bind = lower_pd.extension lower_bind_basis"
huffman@25904
   342
huffman@41036
   343
syntax
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   344
  "_lower_bind" :: "[logic, logic, logic] \<Rightarrow> logic"
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   345
    ("(3\<Union>\<flat>_\<in>_./ _)" [0, 0, 10] 10)
huffman@41036
   346
huffman@41036
   347
translations
huffman@41036
   348
  "\<Union>\<flat>x\<in>xs. e" == "CONST lower_bind\<cdot>xs\<cdot>(\<Lambda> x. e)"
huffman@41036
   349
huffman@25904
   350
lemma lower_bind_principal [simp]:
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   351
  "lower_bind\<cdot>(lower_principal t) = lower_bind_basis t"
huffman@25904
   352
unfolding lower_bind_def
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   353
apply (rule lower_pd.extension_principal)
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   354
apply (erule lower_bind_basis_mono)
huffman@25904
   355
done
huffman@25904
   356
huffman@25904
   357
lemma lower_bind_unit [simp]:
huffman@26927
   358
  "lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x"
huffman@27289
   359
by (induct x rule: compact_basis.principal_induct, simp, simp)
huffman@25904
   360
huffman@25904
   361
lemma lower_bind_plus [simp]:
huffman@41399
   362
  "lower_bind\<cdot>(xs \<union>\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f \<union>\<flat> lower_bind\<cdot>ys\<cdot>f"
huffman@41402
   363
by (induct xs rule: lower_pd.principal_induct, simp,
huffman@41402
   364
    induct ys rule: lower_pd.principal_induct, simp, simp)
huffman@25904
   365
huffman@25904
   366
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
huffman@25904
   367
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit)
huffman@25904
   368
huffman@40589
   369
lemma lower_bind_bind:
huffman@40589
   370
  "lower_bind\<cdot>(lower_bind\<cdot>xs\<cdot>f)\<cdot>g = lower_bind\<cdot>xs\<cdot>(\<Lambda> x. lower_bind\<cdot>(f\<cdot>x)\<cdot>g)"
huffman@40589
   371
by (induct xs, simp_all)
huffman@40589
   372
huffman@25904
   373
huffman@39974
   374
subsection {* Map *}
huffman@25904
   375
huffman@25904
   376
definition
huffman@25904
   377
  lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where
huffman@26927
   378
  "lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))"
huffman@25904
   379
huffman@25904
   380
lemma lower_map_unit [simp]:
huffman@26927
   381
  "lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>"
huffman@25904
   382
unfolding lower_map_def by simp
huffman@25904
   383
huffman@25904
   384
lemma lower_map_plus [simp]:
huffman@41399
   385
  "lower_map\<cdot>f\<cdot>(xs \<union>\<flat> ys) = lower_map\<cdot>f\<cdot>xs \<union>\<flat> lower_map\<cdot>f\<cdot>ys"
huffman@25904
   386
unfolding lower_map_def by simp
huffman@25904
   387
huffman@40577
   388
lemma lower_map_bottom [simp]: "lower_map\<cdot>f\<cdot>\<bottom> = {f\<cdot>\<bottom>}\<flat>"
huffman@40577
   389
unfolding lower_map_def by simp
huffman@40577
   390
huffman@25904
   391
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
huffman@25904
   392
by (induct xs rule: lower_pd_induct, simp_all)
huffman@25904
   393
huffman@33808
   394
lemma lower_map_ID: "lower_map\<cdot>ID = ID"
huffman@40002
   395
by (simp add: cfun_eq_iff ID_def lower_map_ident)
huffman@33808
   396
huffman@25904
   397
lemma lower_map_map:
huffman@25904
   398
  "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
   399
by (induct xs rule: lower_pd_induct, simp_all)
huffman@25904
   400
huffman@41110
   401
lemma lower_bind_map:
huffman@41110
   402
  "lower_bind\<cdot>(lower_map\<cdot>f\<cdot>xs)\<cdot>g = lower_bind\<cdot>xs\<cdot>(\<Lambda> x. g\<cdot>(f\<cdot>x))"
huffman@41110
   403
by (simp add: lower_map_def lower_bind_bind)
huffman@41110
   404
huffman@41110
   405
lemma lower_map_bind:
huffman@41110
   406
  "lower_map\<cdot>f\<cdot>(lower_bind\<cdot>xs\<cdot>g) = lower_bind\<cdot>xs\<cdot>(\<Lambda> x. lower_map\<cdot>f\<cdot>(g\<cdot>x))"
huffman@41110
   407
by (simp add: lower_map_def lower_bind_bind)
huffman@41110
   408
huffman@33585
   409
lemma ep_pair_lower_map: "ep_pair e p \<Longrightarrow> ep_pair (lower_map\<cdot>e) (lower_map\<cdot>p)"
huffman@33585
   410
apply default
huffman@33585
   411
apply (induct_tac x rule: lower_pd_induct, simp_all add: ep_pair.e_inverse)
huffman@35901
   412
apply (induct_tac y rule: lower_pd_induct)
huffman@40734
   413
apply (simp_all add: ep_pair.e_p_below monofun_cfun del: lower_plus_below_iff)
huffman@33585
   414
done
huffman@33585
   415
huffman@33585
   416
lemma deflation_lower_map: "deflation d \<Longrightarrow> deflation (lower_map\<cdot>d)"
huffman@33585
   417
apply default
huffman@33585
   418
apply (induct_tac x rule: lower_pd_induct, simp_all add: deflation.idem)
huffman@35901
   419
apply (induct_tac x rule: lower_pd_induct)
huffman@40734
   420
apply (simp_all add: deflation.below monofun_cfun del: lower_plus_below_iff)
huffman@33585
   421
done
huffman@33585
   422
huffman@39974
   423
(* FIXME: long proof! *)
huffman@39974
   424
lemma finite_deflation_lower_map:
huffman@39974
   425
  assumes "finite_deflation d" shows "finite_deflation (lower_map\<cdot>d)"
huffman@39974
   426
proof (rule finite_deflation_intro)
huffman@39974
   427
  interpret d: finite_deflation d by fact
huffman@39974
   428
  have "deflation d" by fact
huffman@39974
   429
  thus "deflation (lower_map\<cdot>d)" by (rule deflation_lower_map)
huffman@39974
   430
  have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range)
huffman@39974
   431
  hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))"
huffman@39974
   432
    by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
huffman@39974
   433
  hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp
huffman@39974
   434
  hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))"
huffman@39974
   435
    by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject)
huffman@39974
   436
  hence *: "finite (lower_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp
huffman@39974
   437
  hence "finite (range (\<lambda>xs. lower_map\<cdot>d\<cdot>xs))"
huffman@39974
   438
    apply (rule rev_finite_subset)
huffman@39974
   439
    apply clarsimp
huffman@39974
   440
    apply (induct_tac xs rule: lower_pd.principal_induct)
huffman@39974
   441
    apply (simp add: adm_mem_finite *)
huffman@39974
   442
    apply (rename_tac t, induct_tac t rule: pd_basis_induct)
huffman@39974
   443
    apply (simp only: lower_unit_Rep_compact_basis [symmetric] lower_map_unit)
huffman@39974
   444
    apply simp
huffman@39974
   445
    apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b")
huffman@39974
   446
    apply clarsimp
huffman@39974
   447
    apply (rule imageI)
huffman@39974
   448
    apply (rule vimageI2)
huffman@39974
   449
    apply (simp add: Rep_PDUnit)
huffman@39974
   450
    apply (rule range_eqI)
huffman@39974
   451
    apply (erule sym)
huffman@39974
   452
    apply (rule exI)
huffman@39974
   453
    apply (rule Abs_compact_basis_inverse [symmetric])
huffman@39974
   454
    apply (simp add: d.compact)
huffman@39974
   455
    apply (simp only: lower_plus_principal [symmetric] lower_map_plus)
huffman@39974
   456
    apply clarsimp
huffman@39974
   457
    apply (rule imageI)
huffman@39974
   458
    apply (rule vimageI2)
huffman@39974
   459
    apply (simp add: Rep_PDPlus)
huffman@39974
   460
    done
huffman@39974
   461
  thus "finite {xs. lower_map\<cdot>d\<cdot>xs = xs}"
huffman@39974
   462
    by (rule finite_range_imp_finite_fixes)
huffman@39974
   463
qed
huffman@39974
   464
huffman@41289
   465
subsection {* Lower powerdomain is bifinite *}
huffman@39974
   466
huffman@41286
   467
lemma approx_chain_lower_map:
huffman@41286
   468
  assumes "approx_chain a"
huffman@41286
   469
  shows "approx_chain (\<lambda>i. lower_map\<cdot>(a i))"
huffman@41286
   470
  using assms unfolding approx_chain_def
huffman@41286
   471
  by (simp add: lub_APP lower_map_ID finite_deflation_lower_map)
huffman@41286
   472
huffman@41288
   473
instance lower_pd :: (bifinite) bifinite
huffman@41286
   474
proof
huffman@41286
   475
  show "\<exists>(a::nat \<Rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd). approx_chain a"
huffman@41286
   476
    using bifinite [where 'a='a]
huffman@41286
   477
    by (fast intro!: approx_chain_lower_map)
huffman@41286
   478
qed
huffman@41286
   479
huffman@39974
   480
subsection {* Join *}
huffman@39974
   481
huffman@39974
   482
definition
huffman@39974
   483
  lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where
huffman@39974
   484
  "lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"
huffman@39974
   485
huffman@39974
   486
lemma lower_join_unit [simp]:
huffman@39974
   487
  "lower_join\<cdot>{xs}\<flat> = xs"
huffman@39974
   488
unfolding lower_join_def by simp
huffman@39974
   489
huffman@39974
   490
lemma lower_join_plus [simp]:
huffman@41399
   491
  "lower_join\<cdot>(xss \<union>\<flat> yss) = lower_join\<cdot>xss \<union>\<flat> lower_join\<cdot>yss"
huffman@39974
   492
unfolding lower_join_def by simp
huffman@39974
   493
huffman@40577
   494
lemma lower_join_bottom [simp]: "lower_join\<cdot>\<bottom> = \<bottom>"
huffman@40577
   495
unfolding lower_join_def by simp
huffman@40577
   496
huffman@39974
   497
lemma lower_join_map_unit:
huffman@39974
   498
  "lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs"
huffman@39974
   499
by (induct xs rule: lower_pd_induct, simp_all)
huffman@39974
   500
huffman@39974
   501
lemma lower_join_map_join:
huffman@39974
   502
  "lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)"
huffman@39974
   503
by (induct xsss rule: lower_pd_induct, simp_all)
huffman@39974
   504
huffman@39974
   505
lemma lower_join_map_map:
huffman@39974
   506
  "lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) =
huffman@39974
   507
   lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)"
huffman@39974
   508
by (induct xss rule: lower_pd_induct, simp_all)
huffman@39974
   509
huffman@39974
   510
end