author | wenzelm |
Wed, 30 Dec 2015 21:23:38 +0100 | |
changeset 61998 | b66d2ca1f907 |
parent 61169 | 4de9ff3ea29a |
child 62175 | 8ffc4d0e652d |
permissions | -rw-r--r-- |
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(* Title: HOL/HOLCF/LowerPD.thy |
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Author: Brian Huffman |
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*) |
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section {* 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 ("('(_')\<flat>)") = |
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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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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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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) |
207 |
done |
|
208 |
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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 |
216 |
||
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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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||
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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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|
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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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|
231 |
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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|
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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) |
237 |
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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>" |
26927 | 240 |
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) |
26927 | 243 |
apply (rule lower_plus_absorb) |
244 |
done |
|
245 |
||
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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) |
249 |
done |
|
250 |
||
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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) |
254 |
done |
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255 |
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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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258 |
|
26927 | 259 |
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]) |
40327 | 262 |
apply (erule adm_subst [OF cont_Rep_cfun2]) |
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263 |
done |
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|
265 |
lemma compact_lower_plus [simp]: |
|
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"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs \<union>\<flat> ys)" |
27289 | 267 |
by (auto dest!: lower_pd.compact_imp_principal) |
26927 | 268 |
|
25904 | 269 |
|
270 |
subsection {* Induction rules *} |
|
271 |
||
272 |
lemma lower_pd_induct1: |
|
273 |
assumes P: "adm P" |
|
26927 | 274 |
assumes unit: "\<And>x. P {x}\<flat>" |
25904 | 275 |
assumes insert: |
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"\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> \<union>\<flat> ys)" |
25904 | 277 |
shows "P (xs::'a lower_pd)" |
27289 | 278 |
apply (induct xs rule: lower_pd.principal_induct, rule P) |
279 |
apply (induct_tac a rule: pd_basis_induct1) |
|
25904 | 280 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric]) |
281 |
apply (rule unit) |
|
282 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] |
|
283 |
lower_plus_principal [symmetric]) |
|
284 |
apply (erule insert [OF unit]) |
|
285 |
done |
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286 |
||
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lemma lower_pd_induct |
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[case_names adm lower_unit lower_plus, induct type: lower_pd]: |
25904 | 289 |
assumes P: "adm P" |
26927 | 290 |
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)" |
25904 | 292 |
shows "P (xs::'a lower_pd)" |
27289 | 293 |
apply (induct xs rule: lower_pd.principal_induct, rule P) |
294 |
apply (induct_tac a rule: pd_basis_induct) |
|
25904 | 295 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit) |
296 |
apply (simp only: lower_plus_principal [symmetric] plus) |
|
297 |
done |
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298 |
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299 |
||
300 |
subsection {* Monadic bind *} |
|
301 |
||
302 |
definition |
|
303 |
lower_bind_basis :: |
|
304 |
"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
|
305 |
"lower_bind_basis = fold_pd |
|
306 |
(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a)) |
|
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(\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)" |
25904 | 308 |
|
26927 | 309 |
lemma ACI_lower_bind: |
51489 | 310 |
"semilattice (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)" |
25904 | 311 |
apply unfold_locales |
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apply (simp add: lower_plus_assoc) |
25904 | 313 |
apply (simp add: lower_plus_commute) |
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apply (simp add: eta_cfun) |
25904 | 315 |
done |
316 |
||
317 |
lemma lower_bind_basis_simps [simp]: |
|
318 |
"lower_bind_basis (PDUnit a) = |
|
319 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" |
|
320 |
"lower_bind_basis (PDPlus t u) = |
|
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321 |
(\<Lambda> f. lower_bind_basis t\<cdot>f \<union>\<flat> lower_bind_basis u\<cdot>f)" |
25904 | 322 |
unfolding lower_bind_basis_def |
323 |
apply - |
|
26927 | 324 |
apply (rule fold_pd_PDUnit [OF ACI_lower_bind]) |
325 |
apply (rule fold_pd_PDPlus [OF ACI_lower_bind]) |
|
25904 | 326 |
done |
327 |
||
328 |
lemma lower_bind_basis_mono: |
|
329 |
"t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u" |
|
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|
330 |
unfolding cfun_below_iff |
25904 | 331 |
apply (erule lower_le_induct, safe) |
27289 | 332 |
apply (simp add: monofun_cfun) |
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333 |
apply (simp add: rev_below_trans [OF lower_plus_below1]) |
40734 | 334 |
apply simp |
25904 | 335 |
done |
336 |
||
337 |
definition |
|
338 |
lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
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339 |
"lower_bind = lower_pd.extension lower_bind_basis" |
25904 | 340 |
|
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341 |
syntax |
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342 |
"_lower_bind" :: "[logic, logic, logic] \<Rightarrow> logic" |
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|
343 |
("(3\<Union>\<flat>_\<in>_./ _)" [0, 0, 10] 10) |
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344 |
|
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345 |
translations |
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346 |
"\<Union>\<flat>x\<in>xs. e" == "CONST lower_bind\<cdot>xs\<cdot>(\<Lambda> x. e)" |
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347 |
|
25904 | 348 |
lemma lower_bind_principal [simp]: |
349 |
"lower_bind\<cdot>(lower_principal t) = lower_bind_basis t" |
|
350 |
unfolding lower_bind_def |
|
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|
351 |
apply (rule lower_pd.extension_principal) |
25904 | 352 |
apply (erule lower_bind_basis_mono) |
353 |
done |
|
354 |
||
355 |
lemma lower_bind_unit [simp]: |
|
26927 | 356 |
"lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x" |
27289 | 357 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 358 |
|
359 |
lemma lower_bind_plus [simp]: |
|
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|
360 |
"lower_bind\<cdot>(xs \<union>\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f \<union>\<flat> lower_bind\<cdot>ys\<cdot>f" |
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|
361 |
by (induct xs rule: lower_pd.principal_induct, simp, |
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|
362 |
induct ys rule: lower_pd.principal_induct, simp, simp) |
25904 | 363 |
|
364 |
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" |
|
365 |
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit) |
|
366 |
||
40589 | 367 |
lemma lower_bind_bind: |
368 |
"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)" |
|
369 |
by (induct xs, simp_all) |
|
370 |
||
25904 | 371 |
|
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372 |
subsection {* Map *} |
25904 | 373 |
|
374 |
definition |
|
375 |
lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where |
|
26927 | 376 |
"lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))" |
25904 | 377 |
|
378 |
lemma lower_map_unit [simp]: |
|
26927 | 379 |
"lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>" |
25904 | 380 |
unfolding lower_map_def by simp |
381 |
||
382 |
lemma lower_map_plus [simp]: |
|
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|
383 |
"lower_map\<cdot>f\<cdot>(xs \<union>\<flat> ys) = lower_map\<cdot>f\<cdot>xs \<union>\<flat> lower_map\<cdot>f\<cdot>ys" |
25904 | 384 |
unfolding lower_map_def by simp |
385 |
||
40577 | 386 |
lemma lower_map_bottom [simp]: "lower_map\<cdot>f\<cdot>\<bottom> = {f\<cdot>\<bottom>}\<flat>" |
387 |
unfolding lower_map_def by simp |
|
388 |
||
25904 | 389 |
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" |
390 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
391 |
||
33808 | 392 |
lemma lower_map_ID: "lower_map\<cdot>ID = ID" |
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|
393 |
by (simp add: cfun_eq_iff ID_def lower_map_ident) |
33808 | 394 |
|
25904 | 395 |
lemma lower_map_map: |
396 |
"lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" |
|
397 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
398 |
||
41110 | 399 |
lemma lower_bind_map: |
400 |
"lower_bind\<cdot>(lower_map\<cdot>f\<cdot>xs)\<cdot>g = lower_bind\<cdot>xs\<cdot>(\<Lambda> x. g\<cdot>(f\<cdot>x))" |
|
401 |
by (simp add: lower_map_def lower_bind_bind) |
|
402 |
||
403 |
lemma lower_map_bind: |
|
404 |
"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))" |
|
405 |
by (simp add: lower_map_def lower_bind_bind) |
|
406 |
||
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|
407 |
lemma ep_pair_lower_map: "ep_pair e p \<Longrightarrow> ep_pair (lower_map\<cdot>e) (lower_map\<cdot>p)" |
61169 | 408 |
apply standard |
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|
409 |
apply (induct_tac x rule: lower_pd_induct, simp_all add: ep_pair.e_inverse) |
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|
410 |
apply (induct_tac y rule: lower_pd_induct) |
40734 | 411 |
apply (simp_all add: ep_pair.e_p_below monofun_cfun del: lower_plus_below_iff) |
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|
412 |
done |
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|
413 |
|
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|
414 |
lemma deflation_lower_map: "deflation d \<Longrightarrow> deflation (lower_map\<cdot>d)" |
61169 | 415 |
apply standard |
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ep_pair and deflation lemmas for powerdomain map functions
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|
416 |
apply (induct_tac x rule: lower_pd_induct, simp_all add: deflation.idem) |
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fix LaTeX overfull hbox warnings in HOLCF document
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|
417 |
apply (induct_tac x rule: lower_pd_induct) |
40734 | 418 |
apply (simp_all add: deflation.below monofun_cfun del: lower_plus_below_iff) |
33585
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parents:
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changeset
|
419 |
done |
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parents:
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changeset
|
420 |
|
39974
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|
421 |
(* FIXME: long proof! *) |
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|
422 |
lemma finite_deflation_lower_map: |
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|
423 |
assumes "finite_deflation d" shows "finite_deflation (lower_map\<cdot>d)" |
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|
424 |
proof (rule finite_deflation_intro) |
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|
425 |
interpret d: finite_deflation d by fact |
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|
426 |
have "deflation d" by fact |
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|
427 |
thus "deflation (lower_map\<cdot>d)" by (rule deflation_lower_map) |
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|
428 |
have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range) |
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|
429 |
hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))" |
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|
430 |
by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject) |
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|
431 |
hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp |
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|
432 |
hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" |
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|
433 |
by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject) |
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|
434 |
hence *: "finite (lower_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp |
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|
435 |
hence "finite (range (\<lambda>xs. lower_map\<cdot>d\<cdot>xs))" |
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changeset
|
436 |
apply (rule rev_finite_subset) |
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changeset
|
437 |
apply clarsimp |
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changeset
|
438 |
apply (induct_tac xs rule: lower_pd.principal_induct) |
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changeset
|
439 |
apply (simp add: adm_mem_finite *) |
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changeset
|
440 |
apply (rename_tac t, induct_tac t rule: pd_basis_induct) |
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changeset
|
441 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] lower_map_unit) |
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|
442 |
apply simp |
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|
443 |
apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b") |
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changeset
|
444 |
apply clarsimp |
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changeset
|
445 |
apply (rule imageI) |
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|
446 |
apply (rule vimageI2) |
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changeset
|
447 |
apply (simp add: Rep_PDUnit) |
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changeset
|
448 |
apply (rule range_eqI) |
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changeset
|
449 |
apply (erule sym) |
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changeset
|
450 |
apply (rule exI) |
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changeset
|
451 |
apply (rule Abs_compact_basis_inverse [symmetric]) |
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changeset
|
452 |
apply (simp add: d.compact) |
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changeset
|
453 |
apply (simp only: lower_plus_principal [symmetric] lower_map_plus) |
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changeset
|
454 |
apply clarsimp |
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changeset
|
455 |
apply (rule imageI) |
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changeset
|
456 |
apply (rule vimageI2) |
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changeset
|
457 |
apply (simp add: Rep_PDPlus) |
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changeset
|
458 |
done |
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changeset
|
459 |
thus "finite {xs. lower_map\<cdot>d\<cdot>xs = xs}" |
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changeset
|
460 |
by (rule finite_range_imp_finite_fixes) |
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|
461 |
qed |
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|
462 |
|
41289
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minimize imports; move domain class instances for powerdomain types into Powerdomains.thy
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changeset
|
463 |
subsection {* Lower powerdomain is bifinite *} |
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|
464 |
|
41286
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changeset
|
465 |
lemma approx_chain_lower_map: |
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changeset
|
466 |
assumes "approx_chain a" |
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changeset
|
467 |
shows "approx_chain (\<lambda>i. lower_map\<cdot>(a i))" |
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changeset
|
468 |
using assms unfolding approx_chain_def |
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changeset
|
469 |
by (simp add: lub_APP lower_map_ID finite_deflation_lower_map) |
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changeset
|
470 |
|
41288
a19edebad961
powerdomain theories require class 'bifinite' instead of 'domain'
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changeset
|
471 |
instance lower_pd :: (bifinite) bifinite |
41286
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changeset
|
472 |
proof |
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changeset
|
473 |
show "\<exists>(a::nat \<Rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd). approx_chain a" |
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changeset
|
474 |
using bifinite [where 'a='a] |
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changeset
|
475 |
by (fast intro!: approx_chain_lower_map) |
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|
476 |
qed |
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parents:
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diff
changeset
|
477 |
|
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|
478 |
subsection {* Join *} |
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|
479 |
|
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|
480 |
definition |
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|
481 |
lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where |
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|
482 |
"lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" |
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changeset
|
483 |
|
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changeset
|
484 |
lemma lower_join_unit [simp]: |
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changeset
|
485 |
"lower_join\<cdot>{xs}\<flat> = xs" |
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diff
changeset
|
486 |
unfolding lower_join_def by simp |
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diff
changeset
|
487 |
|
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changeset
|
488 |
lemma lower_join_plus [simp]: |
41399
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changed syntax of powerdomain binary union operators
huffman
parents:
41394
diff
changeset
|
489 |
"lower_join\<cdot>(xss \<union>\<flat> yss) = lower_join\<cdot>xss \<union>\<flat> lower_join\<cdot>yss" |
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changeset
|
490 |
unfolding lower_join_def by simp |
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changeset
|
491 |
|
40577 | 492 |
lemma lower_join_bottom [simp]: "lower_join\<cdot>\<bottom> = \<bottom>" |
493 |
unfolding lower_join_def by simp |
|
494 |
||
39974
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|
495 |
lemma lower_join_map_unit: |
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changeset
|
496 |
"lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs" |
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changeset
|
497 |
by (induct xs rule: lower_pd_induct, simp_all) |
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changeset
|
498 |
|
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changeset
|
499 |
lemma lower_join_map_join: |
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changeset
|
500 |
"lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)" |
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changeset
|
501 |
by (induct xsss rule: lower_pd_induct, simp_all) |
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diff
changeset
|
502 |
|
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diff
changeset
|
503 |
lemma lower_join_map_map: |
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changeset
|
504 |
"lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) = |
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changeset
|
505 |
lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)" |
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changeset
|
506 |
by (induct xss rule: lower_pd_induct, simp_all) |
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parents:
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diff
changeset
|
507 |
|
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changeset
|
508 |
end |