author | wenzelm |
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permissions | -rw-r--r-- |
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(* Title: HOL/HOLCF/UpperPD.thy |
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Author: Brian Huffman |
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*) |
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section \<open>Upper powerdomain\<close> |
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theory UpperPD |
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imports Compact_Basis |
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begin |
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subsection \<open>Basis preorder\<close> |
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definition |
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upper_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<sharp>" 50) where |
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"upper_le = (\<lambda>u v. \<forall>y\<in>Rep_pd_basis v. \<exists>x\<in>Rep_pd_basis u. x \<sqsubseteq> y)" |
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lemma upper_le_refl [simp]: "t \<le>\<sharp> t" |
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unfolding upper_le_def by fast |
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lemma upper_le_trans: "\<lbrakk>t \<le>\<sharp> u; u \<le>\<sharp> v\<rbrakk> \<Longrightarrow> t \<le>\<sharp> v" |
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unfolding upper_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 upper_le: preorder upper_le |
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by (rule preorder.intro, rule upper_le_refl, rule upper_le_trans) |
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lemma upper_le_minimal [simp]: "PDUnit compact_bot \<le>\<sharp> t" |
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unfolding upper_le_def Rep_PDUnit by simp |
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lemma PDUnit_upper_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<sharp> PDUnit y" |
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unfolding upper_le_def Rep_PDUnit by simp |
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lemma PDPlus_upper_mono: "\<lbrakk>s \<le>\<sharp> t; u \<le>\<sharp> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<sharp> PDPlus t v" |
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unfolding upper_le_def Rep_PDPlus by fast |
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lemma PDPlus_upper_le: "PDPlus t u \<le>\<sharp> t" |
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unfolding upper_le_def Rep_PDPlus by fast |
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lemma upper_le_PDUnit_PDUnit_iff [simp]: |
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"(PDUnit a \<le>\<sharp> PDUnit b) = (a \<sqsubseteq> b)" |
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unfolding upper_le_def Rep_PDUnit by fast |
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lemma upper_le_PDPlus_PDUnit_iff: |
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"(PDPlus t u \<le>\<sharp> PDUnit a) = (t \<le>\<sharp> PDUnit a \<or> u \<le>\<sharp> PDUnit a)" |
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unfolding upper_le_def Rep_PDPlus Rep_PDUnit by fast |
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lemma upper_le_PDPlus_iff: "(t \<le>\<sharp> PDPlus u v) = (t \<le>\<sharp> u \<and> t \<le>\<sharp> v)" |
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unfolding upper_le_def Rep_PDPlus by fast |
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lemma upper_le_induct [induct set: upper_le]: |
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assumes le: "t \<le>\<sharp> 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 t (PDUnit a) \<Longrightarrow> P (PDPlus t u) (PDUnit a)" |
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assumes 3: "\<And>t u v. \<lbrakk>P t u; P t v\<rbrakk> \<Longrightarrow> P t (PDPlus u v)" |
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shows "P t u" |
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using le apply (induct u arbitrary: t rule: pd_basis_induct) |
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apply (erule rev_mp) |
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apply (induct_tac t rule: pd_basis_induct) |
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apply (simp add: 1) |
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apply (simp add: upper_le_PDPlus_PDUnit_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: upper_le_PDPlus_iff 3) |
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done |
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subsection \<open>Type definition\<close> |
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typedef 'a upper_pd ("('(_')\<sharp>)") = |
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"{S::'a pd_basis set. upper_le.ideal S}" |
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by (rule upper_le.ex_ideal) |
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instantiation upper_pd :: (bifinite) below |
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begin |
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definition |
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"x \<sqsubseteq> y \<longleftrightarrow> Rep_upper_pd x \<subseteq> Rep_upper_pd y" |
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instance .. |
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end |
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instance upper_pd :: (bifinite) po |
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using type_definition_upper_pd below_upper_pd_def |
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by (rule upper_le.typedef_ideal_po) |
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instance upper_pd :: (bifinite) cpo |
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using type_definition_upper_pd below_upper_pd_def |
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by (rule upper_le.typedef_ideal_cpo) |
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definition |
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upper_principal :: "'a pd_basis \<Rightarrow> 'a upper_pd" where |
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"upper_principal t = Abs_upper_pd {u. u \<le>\<sharp> t}" |
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interpretation upper_pd: |
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ideal_completion upper_le upper_principal Rep_upper_pd |
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using type_definition_upper_pd below_upper_pd_def |
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using upper_principal_def pd_basis_countable |
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by (rule upper_le.typedef_ideal_completion) |
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text \<open>Upper powerdomain is pointed\<close> |
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lemma upper_pd_minimal: "upper_principal (PDUnit compact_bot) \<sqsubseteq> ys" |
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by (induct ys rule: upper_pd.principal_induct, simp, simp) |
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instance upper_pd :: (bifinite) pcpo |
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by intro_classes (fast intro: upper_pd_minimal) |
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lemma inst_upper_pd_pcpo: "\<bottom> = upper_principal (PDUnit compact_bot)" |
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by (rule upper_pd_minimal [THEN bottomI, symmetric]) |
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subsection \<open>Monadic unit and plus\<close> |
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definition |
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upper_unit :: "'a \<rightarrow> 'a upper_pd" where |
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"upper_unit = compact_basis.extension (\<lambda>a. upper_principal (PDUnit a))" |
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definition |
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upper_plus :: "'a upper_pd \<rightarrow> 'a upper_pd \<rightarrow> 'a upper_pd" where |
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"upper_plus = upper_pd.extension (\<lambda>t. upper_pd.extension (\<lambda>u. |
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upper_principal (PDPlus t u)))" |
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abbreviation |
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upper_add :: "'a upper_pd \<Rightarrow> 'a upper_pd \<Rightarrow> 'a upper_pd" |
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(infixl "\<union>\<sharp>" 65) where |
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"xs \<union>\<sharp> ys == upper_plus\<cdot>xs\<cdot>ys" |
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syntax |
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"_upper_pd" :: "args \<Rightarrow> logic" ("{_}\<sharp>") |
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translations |
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"{x,xs}\<sharp>" == "{x}\<sharp> \<union>\<sharp> {xs}\<sharp>" |
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"{x}\<sharp>" == "CONST upper_unit\<cdot>x" |
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lemma upper_unit_Rep_compact_basis [simp]: |
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"{Rep_compact_basis a}\<sharp> = upper_principal (PDUnit a)" |
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unfolding upper_unit_def |
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by (simp add: compact_basis.extension_principal PDUnit_upper_mono) |
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lemma upper_plus_principal [simp]: |
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"upper_principal t \<union>\<sharp> upper_principal u = upper_principal (PDPlus t u)" |
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by (simp add: upper_pd.extension_principal |
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upper_pd.extension_mono PDPlus_upper_mono) |
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interpretation upper_add: semilattice upper_add proof |
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fix xs ys zs :: "'a upper_pd" |
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show "(xs \<union>\<sharp> ys) \<union>\<sharp> zs = xs \<union>\<sharp> (ys \<union>\<sharp> zs)" |
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apply (induct xs rule: upper_pd.principal_induct, simp) |
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apply (induct ys rule: upper_pd.principal_induct, simp) |
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apply (induct zs rule: upper_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>\<sharp> ys = ys \<union>\<sharp> xs" |
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apply (induct xs rule: upper_pd.principal_induct, simp) |
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apply (induct ys rule: upper_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>\<sharp> xs = xs" |
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apply (induct xs rule: upper_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 upper_plus_assoc = upper_add.assoc |
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lemmas upper_plus_commute = upper_add.commute |
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lemmas upper_plus_absorb = upper_add.idem |
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lemmas upper_plus_left_commute = upper_add.left_commute |
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lemmas upper_plus_left_absorb = upper_add.left_idem |
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text \<open>Useful for \<open>simp add: upper_plus_ac\<close>\<close> |
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lemmas upper_plus_ac = |
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upper_plus_assoc upper_plus_commute upper_plus_left_commute |
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text \<open>Useful for \<open>simp only: upper_plus_aci\<close>\<close> |
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lemmas upper_plus_aci = |
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upper_plus_ac upper_plus_absorb upper_plus_left_absorb |
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lemma upper_plus_below1: "xs \<union>\<sharp> ys \<sqsubseteq> xs" |
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apply (induct xs rule: upper_pd.principal_induct, simp) |
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apply (induct ys rule: upper_pd.principal_induct, simp) |
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apply (simp add: PDPlus_upper_le) |
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done |
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lemma upper_plus_below2: "xs \<union>\<sharp> ys \<sqsubseteq> ys" |
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by (subst upper_plus_commute, rule upper_plus_below1) |
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lemma upper_plus_greatest: "\<lbrakk>xs \<sqsubseteq> ys; xs \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs \<sqsubseteq> ys \<union>\<sharp> zs" |
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apply (subst upper_plus_absorb [of xs, symmetric]) |
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apply (erule (1) monofun_cfun [OF monofun_cfun_arg]) |
|
197 |
done |
|
198 |
||
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lemma upper_below_plus_iff [simp]: |
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"xs \<sqsubseteq> ys \<union>\<sharp> zs \<longleftrightarrow> xs \<sqsubseteq> ys \<and> xs \<sqsubseteq> zs" |
25904 | 201 |
apply safe |
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apply (erule below_trans [OF _ upper_plus_below1]) |
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apply (erule below_trans [OF _ upper_plus_below2]) |
25904 | 204 |
apply (erule (1) upper_plus_greatest) |
205 |
done |
|
206 |
||
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lemma upper_plus_below_unit_iff [simp]: |
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"xs \<union>\<sharp> ys \<sqsubseteq> {z}\<sharp> \<longleftrightarrow> xs \<sqsubseteq> {z}\<sharp> \<or> ys \<sqsubseteq> {z}\<sharp>" |
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apply (induct xs rule: upper_pd.principal_induct, simp) |
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apply (induct ys rule: upper_pd.principal_induct, simp) |
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apply (induct z rule: compact_basis.principal_induct, simp) |
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apply (simp add: upper_le_PDPlus_PDUnit_iff) |
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done |
214 |
||
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lemma upper_unit_below_iff [simp]: "{x}\<sharp> \<sqsubseteq> {y}\<sharp> \<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 |
26927 | 219 |
done |
220 |
||
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lemmas upper_pd_below_simps = |
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upper_unit_below_iff |
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upper_below_plus_iff |
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224 |
upper_plus_below_unit_iff |
25904 | 225 |
|
26927 | 226 |
lemma upper_unit_eq_iff [simp]: "{x}\<sharp> = {y}\<sharp> \<longleftrightarrow> x = y" |
227 |
unfolding po_eq_conv by simp |
|
228 |
||
229 |
lemma upper_unit_strict [simp]: "{\<bottom>}\<sharp> = \<bottom>" |
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using upper_unit_Rep_compact_basis [of compact_bot] |
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by (simp add: inst_upper_pd_pcpo) |
26927 | 232 |
|
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lemma upper_plus_strict1 [simp]: "\<bottom> \<union>\<sharp> ys = \<bottom>" |
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by (rule bottomI, rule upper_plus_below1) |
26927 | 235 |
|
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lemma upper_plus_strict2 [simp]: "xs \<union>\<sharp> \<bottom> = \<bottom>" |
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by (rule bottomI, rule upper_plus_below2) |
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|
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lemma upper_unit_bottom_iff [simp]: "{x}\<sharp> = \<bottom> \<longleftrightarrow> x = \<bottom>" |
26927 | 240 |
unfolding upper_unit_strict [symmetric] by (rule upper_unit_eq_iff) |
241 |
||
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242 |
lemma upper_plus_bottom_iff [simp]: |
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"xs \<union>\<sharp> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<or> ys = \<bottom>" |
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apply (induct xs rule: upper_pd.principal_induct, simp) |
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apply (induct ys rule: upper_pd.principal_induct, simp) |
27289 | 246 |
apply (simp add: inst_upper_pd_pcpo upper_pd.principal_eq_iff |
26927 | 247 |
upper_le_PDPlus_PDUnit_iff) |
248 |
done |
|
249 |
||
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lemma compact_upper_unit: "compact x \<Longrightarrow> compact {x}\<sharp>" |
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by (auto dest!: compact_basis.compact_imp_principal) |
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|
26927 | 253 |
lemma compact_upper_unit_iff [simp]: "compact {x}\<sharp> \<longleftrightarrow> compact x" |
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apply (safe elim!: compact_upper_unit) |
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apply (simp only: compact_def upper_unit_below_iff [symmetric]) |
40327 | 256 |
apply (erule adm_subst [OF cont_Rep_cfun2]) |
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257 |
done |
26927 | 258 |
|
259 |
lemma compact_upper_plus [simp]: |
|
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"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs \<union>\<sharp> ys)" |
27289 | 261 |
by (auto dest!: upper_pd.compact_imp_principal) |
26927 | 262 |
|
25904 | 263 |
|
62175 | 264 |
subsection \<open>Induction rules\<close> |
25904 | 265 |
|
266 |
lemma upper_pd_induct1: |
|
267 |
assumes P: "adm P" |
|
26927 | 268 |
assumes unit: "\<And>x. P {x}\<sharp>" |
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assumes insert: "\<And>x ys. \<lbrakk>P {x}\<sharp>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<sharp> \<union>\<sharp> ys)" |
25904 | 270 |
shows "P (xs::'a upper_pd)" |
27289 | 271 |
apply (induct xs rule: upper_pd.principal_induct, rule P) |
272 |
apply (induct_tac a rule: pd_basis_induct1) |
|
25904 | 273 |
apply (simp only: upper_unit_Rep_compact_basis [symmetric]) |
274 |
apply (rule unit) |
|
275 |
apply (simp only: upper_unit_Rep_compact_basis [symmetric] |
|
276 |
upper_plus_principal [symmetric]) |
|
277 |
apply (erule insert [OF unit]) |
|
278 |
done |
|
279 |
||
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lemma upper_pd_induct |
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[case_names adm upper_unit upper_plus, induct type: upper_pd]: |
25904 | 282 |
assumes P: "adm P" |
26927 | 283 |
assumes unit: "\<And>x. P {x}\<sharp>" |
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assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs \<union>\<sharp> ys)" |
25904 | 285 |
shows "P (xs::'a upper_pd)" |
27289 | 286 |
apply (induct xs rule: upper_pd.principal_induct, rule P) |
287 |
apply (induct_tac a rule: pd_basis_induct) |
|
25904 | 288 |
apply (simp only: upper_unit_Rep_compact_basis [symmetric] unit) |
289 |
apply (simp only: upper_plus_principal [symmetric] plus) |
|
290 |
done |
|
291 |
||
292 |
||
62175 | 293 |
subsection \<open>Monadic bind\<close> |
25904 | 294 |
|
295 |
definition |
|
296 |
upper_bind_basis :: |
|
297 |
"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b upper_pd) \<rightarrow> 'b upper_pd" where |
|
298 |
"upper_bind_basis = fold_pd |
|
299 |
(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a)) |
|
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(\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<sharp> y\<cdot>f)" |
25904 | 301 |
|
26927 | 302 |
lemma ACI_upper_bind: |
51489 | 303 |
"semilattice (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<sharp> y\<cdot>f)" |
25904 | 304 |
apply unfold_locales |
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apply (simp add: upper_plus_assoc) |
25904 | 306 |
apply (simp add: upper_plus_commute) |
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|
307 |
apply (simp add: eta_cfun) |
25904 | 308 |
done |
309 |
||
310 |
lemma upper_bind_basis_simps [simp]: |
|
311 |
"upper_bind_basis (PDUnit a) = |
|
312 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" |
|
313 |
"upper_bind_basis (PDPlus t u) = |
|
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314 |
(\<Lambda> f. upper_bind_basis t\<cdot>f \<union>\<sharp> upper_bind_basis u\<cdot>f)" |
25904 | 315 |
unfolding upper_bind_basis_def |
316 |
apply - |
|
26927 | 317 |
apply (rule fold_pd_PDUnit [OF ACI_upper_bind]) |
318 |
apply (rule fold_pd_PDPlus [OF ACI_upper_bind]) |
|
25904 | 319 |
done |
320 |
||
321 |
lemma upper_bind_basis_mono: |
|
322 |
"t \<le>\<sharp> u \<Longrightarrow> upper_bind_basis t \<sqsubseteq> upper_bind_basis u" |
|
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323 |
unfolding cfun_below_iff |
25904 | 324 |
apply (erule upper_le_induct, safe) |
27289 | 325 |
apply (simp add: monofun_cfun) |
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326 |
apply (simp add: below_trans [OF upper_plus_below1]) |
40734 | 327 |
apply simp |
25904 | 328 |
done |
329 |
||
330 |
definition |
|
331 |
upper_bind :: "'a upper_pd \<rightarrow> ('a \<rightarrow> 'b upper_pd) \<rightarrow> 'b upper_pd" where |
|
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332 |
"upper_bind = upper_pd.extension upper_bind_basis" |
25904 | 333 |
|
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334 |
syntax |
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|
335 |
"_upper_bind" :: "[logic, logic, logic] \<Rightarrow> logic" |
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336 |
("(3\<Union>\<sharp>_\<in>_./ _)" [0, 0, 10] 10) |
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|
337 |
|
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338 |
translations |
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339 |
"\<Union>\<sharp>x\<in>xs. e" == "CONST upper_bind\<cdot>xs\<cdot>(\<Lambda> x. e)" |
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340 |
|
25904 | 341 |
lemma upper_bind_principal [simp]: |
342 |
"upper_bind\<cdot>(upper_principal t) = upper_bind_basis t" |
|
343 |
unfolding upper_bind_def |
|
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|
344 |
apply (rule upper_pd.extension_principal) |
25904 | 345 |
apply (erule upper_bind_basis_mono) |
346 |
done |
|
347 |
||
348 |
lemma upper_bind_unit [simp]: |
|
26927 | 349 |
"upper_bind\<cdot>{x}\<sharp>\<cdot>f = f\<cdot>x" |
27289 | 350 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 351 |
|
352 |
lemma upper_bind_plus [simp]: |
|
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|
353 |
"upper_bind\<cdot>(xs \<union>\<sharp> ys)\<cdot>f = upper_bind\<cdot>xs\<cdot>f \<union>\<sharp> upper_bind\<cdot>ys\<cdot>f" |
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|
354 |
by (induct xs rule: upper_pd.principal_induct, simp, |
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|
355 |
induct ys rule: upper_pd.principal_induct, simp, simp) |
25904 | 356 |
|
357 |
lemma upper_bind_strict [simp]: "upper_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" |
|
358 |
unfolding upper_unit_strict [symmetric] by (rule upper_bind_unit) |
|
359 |
||
40589 | 360 |
lemma upper_bind_bind: |
361 |
"upper_bind\<cdot>(upper_bind\<cdot>xs\<cdot>f)\<cdot>g = upper_bind\<cdot>xs\<cdot>(\<Lambda> x. upper_bind\<cdot>(f\<cdot>x)\<cdot>g)" |
|
362 |
by (induct xs, simp_all) |
|
363 |
||
25904 | 364 |
|
62175 | 365 |
subsection \<open>Map\<close> |
25904 | 366 |
|
367 |
definition |
|
368 |
upper_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a upper_pd \<rightarrow> 'b upper_pd" where |
|
26927 | 369 |
"upper_map = (\<Lambda> f xs. upper_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<sharp>))" |
25904 | 370 |
|
371 |
lemma upper_map_unit [simp]: |
|
26927 | 372 |
"upper_map\<cdot>f\<cdot>{x}\<sharp> = {f\<cdot>x}\<sharp>" |
25904 | 373 |
unfolding upper_map_def by simp |
374 |
||
375 |
lemma upper_map_plus [simp]: |
|
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|
376 |
"upper_map\<cdot>f\<cdot>(xs \<union>\<sharp> ys) = upper_map\<cdot>f\<cdot>xs \<union>\<sharp> upper_map\<cdot>f\<cdot>ys" |
25904 | 377 |
unfolding upper_map_def by simp |
378 |
||
40577 | 379 |
lemma upper_map_bottom [simp]: "upper_map\<cdot>f\<cdot>\<bottom> = {f\<cdot>\<bottom>}\<sharp>" |
380 |
unfolding upper_map_def by simp |
|
381 |
||
25904 | 382 |
lemma upper_map_ident: "upper_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" |
383 |
by (induct xs rule: upper_pd_induct, simp_all) |
|
384 |
||
33808 | 385 |
lemma upper_map_ID: "upper_map\<cdot>ID = ID" |
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|
386 |
by (simp add: cfun_eq_iff ID_def upper_map_ident) |
33808 | 387 |
|
25904 | 388 |
lemma upper_map_map: |
389 |
"upper_map\<cdot>f\<cdot>(upper_map\<cdot>g\<cdot>xs) = upper_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" |
|
390 |
by (induct xs rule: upper_pd_induct, simp_all) |
|
391 |
||
41110 | 392 |
lemma upper_bind_map: |
393 |
"upper_bind\<cdot>(upper_map\<cdot>f\<cdot>xs)\<cdot>g = upper_bind\<cdot>xs\<cdot>(\<Lambda> x. g\<cdot>(f\<cdot>x))" |
|
394 |
by (simp add: upper_map_def upper_bind_bind) |
|
395 |
||
396 |
lemma upper_map_bind: |
|
397 |
"upper_map\<cdot>f\<cdot>(upper_bind\<cdot>xs\<cdot>g) = upper_bind\<cdot>xs\<cdot>(\<Lambda> x. upper_map\<cdot>f\<cdot>(g\<cdot>x))" |
|
398 |
by (simp add: upper_map_def upper_bind_bind) |
|
399 |
||
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|
400 |
lemma ep_pair_upper_map: "ep_pair e p \<Longrightarrow> ep_pair (upper_map\<cdot>e) (upper_map\<cdot>p)" |
61169 | 401 |
apply standard |
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|
402 |
apply (induct_tac x rule: upper_pd_induct, simp_all add: ep_pair.e_inverse) |
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|
403 |
apply (induct_tac y rule: upper_pd_induct) |
40734 | 404 |
apply (simp_all add: ep_pair.e_p_below monofun_cfun del: upper_below_plus_iff) |
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|
405 |
done |
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|
406 |
|
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|
407 |
lemma deflation_upper_map: "deflation d \<Longrightarrow> deflation (upper_map\<cdot>d)" |
61169 | 408 |
apply standard |
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|
409 |
apply (induct_tac x rule: upper_pd_induct, simp_all add: deflation.idem) |
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|
410 |
apply (induct_tac x rule: upper_pd_induct) |
40734 | 411 |
apply (simp_all add: deflation.below monofun_cfun del: upper_below_plus_iff) |
33585
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parents:
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|
412 |
done |
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parents:
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|
413 |
|
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|
414 |
(* FIXME: long proof! *) |
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|
415 |
lemma finite_deflation_upper_map: |
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|
416 |
assumes "finite_deflation d" shows "finite_deflation (upper_map\<cdot>d)" |
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|
417 |
proof (rule finite_deflation_intro) |
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|
418 |
interpret d: finite_deflation d by fact |
67682
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tuned proofs -- prefer explicit names for facts from 'interpret';
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|
419 |
from d.deflation_axioms show "deflation (upper_map\<cdot>d)" |
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|
420 |
by (rule deflation_upper_map) |
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|
421 |
have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range) |
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|
422 |
hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))" |
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|
423 |
by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject) |
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|
424 |
hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp |
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|
425 |
hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" |
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|
426 |
by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject) |
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|
427 |
hence *: "finite (upper_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp |
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|
428 |
hence "finite (range (\<lambda>xs. upper_map\<cdot>d\<cdot>xs))" |
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|
429 |
apply (rule rev_finite_subset) |
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|
430 |
apply clarsimp |
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|
431 |
apply (induct_tac xs rule: upper_pd.principal_induct) |
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|
432 |
apply (simp add: adm_mem_finite *) |
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|
433 |
apply (rename_tac t, induct_tac t rule: pd_basis_induct) |
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|
434 |
apply (simp only: upper_unit_Rep_compact_basis [symmetric] upper_map_unit) |
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|
435 |
apply simp |
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|
436 |
apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b") |
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|
437 |
apply clarsimp |
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|
438 |
apply (rule imageI) |
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|
439 |
apply (rule vimageI2) |
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|
440 |
apply (simp add: Rep_PDUnit) |
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|
441 |
apply (rule range_eqI) |
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|
442 |
apply (erule sym) |
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|
443 |
apply (rule exI) |
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|
444 |
apply (rule Abs_compact_basis_inverse [symmetric]) |
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|
445 |
apply (simp add: d.compact) |
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|
446 |
apply (simp only: upper_plus_principal [symmetric] upper_map_plus) |
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|
447 |
apply clarsimp |
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|
448 |
apply (rule imageI) |
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|
449 |
apply (rule vimageI2) |
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|
450 |
apply (simp add: Rep_PDPlus) |
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|
451 |
done |
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|
452 |
thus "finite {xs. upper_map\<cdot>d\<cdot>xs = xs}" |
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|
453 |
by (rule finite_range_imp_finite_fixes) |
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|
454 |
qed |
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|
455 |
|
62175 | 456 |
subsection \<open>Upper powerdomain is bifinite\<close> |
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457 |
|
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|
458 |
lemma approx_chain_upper_map: |
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|
459 |
assumes "approx_chain a" |
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|
460 |
shows "approx_chain (\<lambda>i. upper_map\<cdot>(a i))" |
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|
461 |
using assms unfolding approx_chain_def |
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|
462 |
by (simp add: lub_APP upper_map_ID finite_deflation_upper_map) |
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|
463 |
|
41288
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powerdomain theories require class 'bifinite' instead of 'domain'
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|
464 |
instance upper_pd :: (bifinite) bifinite |
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|
465 |
proof |
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|
466 |
show "\<exists>(a::nat \<Rightarrow> 'a upper_pd \<rightarrow> 'a upper_pd). approx_chain a" |
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|
467 |
using bifinite [where 'a='a] |
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|
468 |
by (fast intro!: approx_chain_upper_map) |
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parents:
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|
469 |
qed |
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parents:
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diff
changeset
|
470 |
|
62175 | 471 |
subsection \<open>Join\<close> |
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|
472 |
|
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|
473 |
definition |
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|
474 |
upper_join :: "'a upper_pd upper_pd \<rightarrow> 'a upper_pd" where |
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|
475 |
"upper_join = (\<Lambda> xss. upper_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" |
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|
476 |
|
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|
477 |
lemma upper_join_unit [simp]: |
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|
478 |
"upper_join\<cdot>{xs}\<sharp> = xs" |
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|
479 |
unfolding upper_join_def by simp |
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|
480 |
|
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|
481 |
lemma upper_join_plus [simp]: |
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huffman
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diff
changeset
|
482 |
"upper_join\<cdot>(xss \<union>\<sharp> yss) = upper_join\<cdot>xss \<union>\<sharp> upper_join\<cdot>yss" |
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|
483 |
unfolding upper_join_def by simp |
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|
484 |
|
40577 | 485 |
lemma upper_join_bottom [simp]: "upper_join\<cdot>\<bottom> = \<bottom>" |
486 |
unfolding upper_join_def by simp |
|
487 |
||
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|
488 |
lemma upper_join_map_unit: |
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|
489 |
"upper_join\<cdot>(upper_map\<cdot>upper_unit\<cdot>xs) = xs" |
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changeset
|
490 |
by (induct xs rule: upper_pd_induct, simp_all) |
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changeset
|
491 |
|
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|
492 |
lemma upper_join_map_join: |
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|
493 |
"upper_join\<cdot>(upper_map\<cdot>upper_join\<cdot>xsss) = upper_join\<cdot>(upper_join\<cdot>xsss)" |
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changeset
|
494 |
by (induct xsss rule: upper_pd_induct, simp_all) |
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changeset
|
495 |
|
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|
496 |
lemma upper_join_map_map: |
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|
497 |
"upper_join\<cdot>(upper_map\<cdot>(upper_map\<cdot>f)\<cdot>xss) = |
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|
498 |
upper_map\<cdot>f\<cdot>(upper_join\<cdot>xss)" |
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changeset
|
499 |
by (induct xss rule: upper_pd_induct, simp_all) |
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changeset
|
500 |
|
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|
501 |
end |