author | huffman |
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(* Title: 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 CompactBasis |
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begin |
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subsection {* Basis preorder *} |
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definition |
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lower_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<flat>" 50) where |
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"lower_le = (\<lambda>u v. \<forall>x\<in>Rep_pd_basis u. \<exists>y\<in>Rep_pd_basis v. x \<sqsubseteq> y)" |
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lemma lower_le_refl [simp]: "t \<le>\<flat> t" |
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unfolding lower_le_def by fast |
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lemma lower_le_trans: "\<lbrakk>t \<le>\<flat> u; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> t \<le>\<flat> v" |
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unfolding lower_le_def |
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apply (rule ballI) |
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apply (drule (1) bspec, erule bexE) |
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apply (drule (1) bspec, erule bexE) |
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apply (erule rev_bexI) |
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apply (erule (1) 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 (open) 'a lower_pd = |
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"{S::'a pd_basis set. lower_le.ideal S}" |
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by (fast intro: lower_le.ideal_principal) |
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instantiation lower_pd :: (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 UU_I, 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.basis_fun (\<lambda>a. lower_principal (PDUnit a))" |
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definition |
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lower_plus :: "'a lower_pd \<rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd" where |
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"lower_plus = lower_pd.basis_fun (\<lambda>t. lower_pd.basis_fun (\<lambda>u. |
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lower_principal (PDPlus t u)))" |
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abbreviation |
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lower_add :: "'a lower_pd \<Rightarrow> 'a lower_pd \<Rightarrow> 'a lower_pd" |
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(infixl "+\<flat>" 65) where |
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"xs +\<flat> ys == lower_plus\<cdot>xs\<cdot>ys" |
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syntax |
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"_lower_pd" :: "args \<Rightarrow> 'a lower_pd" ("{_}\<flat>") |
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translations |
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"{x,xs}\<flat>" == "{x}\<flat> +\<flat> {xs}\<flat>" |
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"{x}\<flat>" == "CONST lower_unit\<cdot>x" |
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lemma lower_unit_Rep_compact_basis [simp]: |
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"{Rep_compact_basis a}\<flat> = lower_principal (PDUnit a)" |
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unfolding lower_unit_def |
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by (simp add: compact_basis.basis_fun_principal PDUnit_lower_mono) |
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lemma lower_plus_principal [simp]: |
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"lower_principal t +\<flat> lower_principal u = lower_principal (PDPlus t u)" |
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unfolding lower_plus_def |
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by (simp add: lower_pd.basis_fun_principal |
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lower_pd.basis_fun_mono PDPlus_lower_mono) |
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interpretation lower_add: semilattice lower_add proof |
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fix xs ys zs :: "'a lower_pd" |
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show "(xs +\<flat> ys) +\<flat> zs = xs +\<flat> (ys +\<flat> zs)" |
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apply (induct xs ys arbitrary: zs rule: lower_pd.principal_induct2, simp, simp) |
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apply (rule_tac x=zs in lower_pd.principal_induct, simp) |
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apply (simp add: PDPlus_assoc) |
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done |
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show "xs +\<flat> ys = ys +\<flat> xs" |
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apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) |
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apply (simp add: PDPlus_commute) |
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done |
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show "xs +\<flat> xs = xs" |
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apply (induct xs rule: lower_pd.principal_induct, simp) |
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apply (simp add: PDPlus_absorb) |
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done |
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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 +\<flat> ys" |
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apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) |
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apply (simp add: PDPlus_lower_le) |
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done |
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lemma lower_plus_below2: "ys \<sqsubseteq> xs +\<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 +\<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: |
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"xs +\<flat> ys \<sqsubseteq> zs \<longleftrightarrow> xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs" |
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apply safe |
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apply (erule 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: |
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"{x}\<flat> \<sqsubseteq> ys +\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs" |
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apply (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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|
217 |
apply simp |
26927 | 218 |
done |
219 |
||
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220 |
lemmas lower_pd_below_simps = |
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221 |
lower_unit_below_iff |
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222 |
lower_plus_below_iff |
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|
223 |
lower_unit_below_plus_iff |
25904 | 224 |
|
26927 | 225 |
lemma lower_unit_eq_iff [simp]: "{x}\<flat> = {y}\<flat> \<longleftrightarrow> x = y" |
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|
226 |
by (simp add: po_eq_conv) |
26927 | 227 |
|
228 |
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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|
230 |
by (simp add: inst_lower_pd_pcpo) |
26927 | 231 |
|
40321
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|
232 |
lemma lower_unit_bottom_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>" |
26927 | 233 |
unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff) |
234 |
||
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|
235 |
lemma lower_plus_bottom_iff [simp]: |
26927 | 236 |
"xs +\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>" |
237 |
apply safe |
|
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|
238 |
apply (rule UU_I, erule subst, rule lower_plus_below1) |
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|
239 |
apply (rule UU_I, erule subst, rule lower_plus_below2) |
26927 | 240 |
apply (rule lower_plus_absorb) |
241 |
done |
|
242 |
||
243 |
lemma lower_plus_strict1 [simp]: "\<bottom> +\<flat> ys = ys" |
|
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|
244 |
apply (rule below_antisym [OF _ lower_plus_below2]) |
26927 | 245 |
apply (simp add: lower_plus_least) |
246 |
done |
|
247 |
||
248 |
lemma lower_plus_strict2 [simp]: "xs +\<flat> \<bottom> = xs" |
|
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|
249 |
apply (rule below_antisym [OF _ lower_plus_below1]) |
26927 | 250 |
apply (simp add: lower_plus_least) |
251 |
done |
|
252 |
||
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253 |
lemma compact_lower_unit: "compact x \<Longrightarrow> compact {x}\<flat>" |
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254 |
by (auto dest!: compact_basis.compact_imp_principal) |
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255 |
|
26927 | 256 |
lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x" |
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257 |
apply (safe elim!: compact_lower_unit) |
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258 |
apply (simp only: compact_def lower_unit_below_iff [symmetric]) |
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|
259 |
apply (erule adm_subst [OF cont_Rep_CFun2]) |
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260 |
done |
26927 | 261 |
|
262 |
lemma compact_lower_plus [simp]: |
|
263 |
"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<flat> ys)" |
|
27289 | 264 |
by (auto dest!: lower_pd.compact_imp_principal) |
26927 | 265 |
|
25904 | 266 |
|
267 |
subsection {* Induction rules *} |
|
268 |
||
269 |
lemma lower_pd_induct1: |
|
270 |
assumes P: "adm P" |
|
26927 | 271 |
assumes unit: "\<And>x. P {x}\<flat>" |
25904 | 272 |
assumes insert: |
26927 | 273 |
"\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> +\<flat> ys)" |
25904 | 274 |
shows "P (xs::'a lower_pd)" |
27289 | 275 |
apply (induct xs rule: lower_pd.principal_induct, rule P) |
276 |
apply (induct_tac a rule: pd_basis_induct1) |
|
25904 | 277 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric]) |
278 |
apply (rule unit) |
|
279 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] |
|
280 |
lower_plus_principal [symmetric]) |
|
281 |
apply (erule insert [OF unit]) |
|
282 |
done |
|
283 |
||
284 |
lemma lower_pd_induct: |
|
285 |
assumes P: "adm P" |
|
26927 | 286 |
assumes unit: "\<And>x. P {x}\<flat>" |
287 |
assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<flat> ys)" |
|
25904 | 288 |
shows "P (xs::'a lower_pd)" |
27289 | 289 |
apply (induct xs rule: lower_pd.principal_induct, rule P) |
290 |
apply (induct_tac a rule: pd_basis_induct) |
|
25904 | 291 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit) |
292 |
apply (simp only: lower_plus_principal [symmetric] plus) |
|
293 |
done |
|
294 |
||
295 |
||
296 |
subsection {* Monadic bind *} |
|
297 |
||
298 |
definition |
|
299 |
lower_bind_basis :: |
|
300 |
"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
|
301 |
"lower_bind_basis = fold_pd |
|
302 |
(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a)) |
|
26927 | 303 |
(\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" |
25904 | 304 |
|
26927 | 305 |
lemma ACI_lower_bind: |
36635
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|
306 |
"class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" |
25904 | 307 |
apply unfold_locales |
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locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
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|
308 |
apply (simp add: lower_plus_assoc) |
25904 | 309 |
apply (simp add: lower_plus_commute) |
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|
310 |
apply (simp add: eta_cfun) |
25904 | 311 |
done |
312 |
||
313 |
lemma lower_bind_basis_simps [simp]: |
|
314 |
"lower_bind_basis (PDUnit a) = |
|
315 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" |
|
316 |
"lower_bind_basis (PDPlus t u) = |
|
26927 | 317 |
(\<Lambda> f. lower_bind_basis t\<cdot>f +\<flat> lower_bind_basis u\<cdot>f)" |
25904 | 318 |
unfolding lower_bind_basis_def |
319 |
apply - |
|
26927 | 320 |
apply (rule fold_pd_PDUnit [OF ACI_lower_bind]) |
321 |
apply (rule fold_pd_PDPlus [OF ACI_lower_bind]) |
|
25904 | 322 |
done |
323 |
||
324 |
lemma lower_bind_basis_mono: |
|
325 |
"t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u" |
|
40002
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|
326 |
unfolding cfun_below_iff |
25904 | 327 |
apply (erule lower_le_induct, safe) |
27289 | 328 |
apply (simp add: monofun_cfun) |
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|
329 |
apply (simp add: rev_below_trans [OF lower_plus_below1]) |
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|
330 |
apply (simp add: lower_plus_below_iff) |
25904 | 331 |
done |
332 |
||
333 |
definition |
|
334 |
lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
|
335 |
"lower_bind = lower_pd.basis_fun lower_bind_basis" |
|
336 |
||
337 |
lemma lower_bind_principal [simp]: |
|
338 |
"lower_bind\<cdot>(lower_principal t) = lower_bind_basis t" |
|
339 |
unfolding lower_bind_def |
|
340 |
apply (rule lower_pd.basis_fun_principal) |
|
341 |
apply (erule lower_bind_basis_mono) |
|
342 |
done |
|
343 |
||
344 |
lemma lower_bind_unit [simp]: |
|
26927 | 345 |
"lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x" |
27289 | 346 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 347 |
|
348 |
lemma lower_bind_plus [simp]: |
|
26927 | 349 |
"lower_bind\<cdot>(xs +\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f +\<flat> lower_bind\<cdot>ys\<cdot>f" |
27289 | 350 |
by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp) |
25904 | 351 |
|
352 |
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" |
|
353 |
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit) |
|
354 |
||
355 |
||
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|
356 |
subsection {* Map *} |
25904 | 357 |
|
358 |
definition |
|
359 |
lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where |
|
26927 | 360 |
"lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))" |
25904 | 361 |
|
362 |
lemma lower_map_unit [simp]: |
|
26927 | 363 |
"lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>" |
25904 | 364 |
unfolding lower_map_def by simp |
365 |
||
366 |
lemma lower_map_plus [simp]: |
|
26927 | 367 |
"lower_map\<cdot>f\<cdot>(xs +\<flat> ys) = lower_map\<cdot>f\<cdot>xs +\<flat> lower_map\<cdot>f\<cdot>ys" |
25904 | 368 |
unfolding lower_map_def by simp |
369 |
||
370 |
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" |
|
371 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
372 |
||
33808 | 373 |
lemma lower_map_ID: "lower_map\<cdot>ID = ID" |
40002
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|
374 |
by (simp add: cfun_eq_iff ID_def lower_map_ident) |
33808 | 375 |
|
25904 | 376 |
lemma lower_map_map: |
377 |
"lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" |
|
378 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
379 |
||
33585
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changeset
|
380 |
lemma ep_pair_lower_map: "ep_pair e p \<Longrightarrow> ep_pair (lower_map\<cdot>e) (lower_map\<cdot>p)" |
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changeset
|
381 |
apply default |
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changeset
|
382 |
apply (induct_tac x rule: lower_pd_induct, simp_all add: ep_pair.e_inverse) |
35901
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huffman
parents:
34973
diff
changeset
|
383 |
apply (induct_tac y rule: lower_pd_induct) |
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huffman
parents:
34973
diff
changeset
|
384 |
apply (simp_all add: ep_pair.e_p_below monofun_cfun) |
33585
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changeset
|
385 |
done |
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huffman
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31076
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changeset
|
386 |
|
8d39394fe5cf
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huffman
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31076
diff
changeset
|
387 |
lemma deflation_lower_map: "deflation d \<Longrightarrow> deflation (lower_map\<cdot>d)" |
8d39394fe5cf
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huffman
parents:
31076
diff
changeset
|
388 |
apply default |
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huffman
parents:
31076
diff
changeset
|
389 |
apply (induct_tac x rule: lower_pd_induct, simp_all add: deflation.idem) |
35901
12f09bf2c77f
fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents:
34973
diff
changeset
|
390 |
apply (induct_tac x rule: lower_pd_induct) |
12f09bf2c77f
fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents:
34973
diff
changeset
|
391 |
apply (simp_all add: deflation.below monofun_cfun) |
33585
8d39394fe5cf
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huffman
parents:
31076
diff
changeset
|
392 |
done |
8d39394fe5cf
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huffman
parents:
31076
diff
changeset
|
393 |
|
39974
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|
394 |
(* FIXME: long proof! *) |
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|
395 |
lemma finite_deflation_lower_map: |
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changeset
|
396 |
assumes "finite_deflation d" shows "finite_deflation (lower_map\<cdot>d)" |
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changeset
|
397 |
proof (rule finite_deflation_intro) |
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changeset
|
398 |
interpret d: finite_deflation d by fact |
b525988432e9
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diff
changeset
|
399 |
have "deflation d" by fact |
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changeset
|
400 |
thus "deflation (lower_map\<cdot>d)" by (rule deflation_lower_map) |
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major reorganization/simplification of HOLCF type classes:
huffman
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diff
changeset
|
401 |
have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range) |
b525988432e9
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diff
changeset
|
402 |
hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))" |
b525988432e9
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huffman
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diff
changeset
|
403 |
by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
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39970
diff
changeset
|
404 |
hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
405 |
hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
406 |
by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
407 |
hence *: "finite (lower_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
408 |
hence "finite (range (\<lambda>xs. lower_map\<cdot>d\<cdot>xs))" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
409 |
apply (rule rev_finite_subset) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
410 |
apply clarsimp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
411 |
apply (induct_tac xs rule: lower_pd.principal_induct) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
412 |
apply (simp add: adm_mem_finite *) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
413 |
apply (rename_tac t, induct_tac t rule: pd_basis_induct) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
414 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] lower_map_unit) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
415 |
apply simp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
416 |
apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b") |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
417 |
apply clarsimp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
418 |
apply (rule imageI) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
419 |
apply (rule vimageI2) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
420 |
apply (simp add: Rep_PDUnit) |
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parents:
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|
421 |
apply (rule range_eqI) |
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|
422 |
apply (erule sym) |
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diff
changeset
|
423 |
apply (rule exI) |
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|
424 |
apply (rule Abs_compact_basis_inverse [symmetric]) |
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changeset
|
425 |
apply (simp add: d.compact) |
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changeset
|
426 |
apply (simp only: lower_plus_principal [symmetric] lower_map_plus) |
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changeset
|
427 |
apply clarsimp |
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changeset
|
428 |
apply (rule imageI) |
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changeset
|
429 |
apply (rule vimageI2) |
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changeset
|
430 |
apply (simp add: Rep_PDPlus) |
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changeset
|
431 |
done |
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|
432 |
thus "finite {xs. lower_map\<cdot>d\<cdot>xs = xs}" |
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changeset
|
433 |
by (rule finite_range_imp_finite_fixes) |
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|
434 |
qed |
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|
435 |
|
39986 | 436 |
subsection {* Lower powerdomain is a bifinite domain *} |
39974
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|
437 |
|
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|
438 |
definition |
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|
439 |
lower_approx :: "nat \<Rightarrow> udom lower_pd \<rightarrow> udom lower_pd" |
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|
440 |
where |
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|
441 |
"lower_approx = (\<lambda>i. lower_map\<cdot>(udom_approx i))" |
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|
442 |
|
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|
443 |
lemma lower_approx: "approx_chain lower_approx" |
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|
444 |
proof (rule approx_chain.intro) |
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|
445 |
show "chain (\<lambda>i. lower_approx i)" |
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changeset
|
446 |
unfolding lower_approx_def by simp |
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changeset
|
447 |
show "(\<Squnion>i. lower_approx i) = ID" |
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changeset
|
448 |
unfolding lower_approx_def |
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changeset
|
449 |
by (simp add: lub_distribs lower_map_ID) |
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changeset
|
450 |
show "\<And>i. finite_deflation (lower_approx i)" |
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changeset
|
451 |
unfolding lower_approx_def |
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changeset
|
452 |
by (intro finite_deflation_lower_map finite_deflation_udom_approx) |
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changeset
|
453 |
qed |
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changeset
|
454 |
|
39989
ad60d7311f43
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huffman
parents:
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changeset
|
455 |
definition lower_defl :: "defl \<rightarrow> defl" |
ad60d7311f43
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huffman
parents:
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changeset
|
456 |
where "lower_defl = defl_fun1 lower_approx lower_map" |
39974
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|
457 |
|
39989
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huffman
parents:
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changeset
|
458 |
lemma cast_lower_defl: |
ad60d7311f43
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huffman
parents:
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diff
changeset
|
459 |
"cast\<cdot>(lower_defl\<cdot>A) = |
39974
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|
460 |
udom_emb lower_approx oo lower_map\<cdot>(cast\<cdot>A) oo udom_prj lower_approx" |
39989
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huffman
parents:
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diff
changeset
|
461 |
unfolding lower_defl_def |
ad60d7311f43
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huffman
parents:
39986
diff
changeset
|
462 |
apply (rule cast_defl_fun1 [OF lower_approx]) |
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changeset
|
463 |
apply (erule finite_deflation_lower_map) |
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changeset
|
464 |
done |
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|
465 |
|
39986 | 466 |
instantiation lower_pd :: (bifinite) bifinite |
39974
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|
467 |
begin |
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|
468 |
|
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changeset
|
469 |
definition |
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|
470 |
"emb = udom_emb lower_approx oo lower_map\<cdot>emb" |
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changeset
|
471 |
|
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changeset
|
472 |
definition |
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|
473 |
"prj = lower_map\<cdot>prj oo udom_prj lower_approx" |
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changeset
|
474 |
|
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changeset
|
475 |
definition |
39989
ad60d7311f43
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huffman
parents:
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diff
changeset
|
476 |
"defl (t::'a lower_pd itself) = lower_defl\<cdot>DEFL('a)" |
39974
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|
477 |
|
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|
478 |
instance proof |
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changeset
|
479 |
show "ep_pair emb (prj :: udom \<rightarrow> 'a lower_pd)" |
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changeset
|
480 |
unfolding emb_lower_pd_def prj_lower_pd_def |
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changeset
|
481 |
using ep_pair_udom [OF lower_approx] |
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changeset
|
482 |
by (intro ep_pair_comp ep_pair_lower_map ep_pair_emb_prj) |
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diff
changeset
|
483 |
next |
39989
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huffman
parents:
39986
diff
changeset
|
484 |
show "cast\<cdot>DEFL('a lower_pd) = emb oo (prj :: udom \<rightarrow> 'a lower_pd)" |
ad60d7311f43
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huffman
parents:
39986
diff
changeset
|
485 |
unfolding emb_lower_pd_def prj_lower_pd_def defl_lower_pd_def cast_lower_defl |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
39989
diff
changeset
|
486 |
by (simp add: cast_DEFL oo_def cfun_eq_iff lower_map_map) |
39974
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|
487 |
qed |
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changeset
|
488 |
|
25904 | 489 |
end |
39974
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changeset
|
490 |
|
39989
ad60d7311f43
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huffman
parents:
39986
diff
changeset
|
491 |
lemma DEFL_lower: "DEFL('a lower_pd) = lower_defl\<cdot>DEFL('a)" |
ad60d7311f43
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huffman
parents:
39986
diff
changeset
|
492 |
by (rule defl_lower_pd_def) |
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changeset
|
493 |
|
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changeset
|
494 |
|
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|
495 |
subsection {* Join *} |
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|
496 |
|
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changeset
|
497 |
definition |
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changeset
|
498 |
lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where |
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changeset
|
499 |
"lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" |
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changeset
|
500 |
|
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changeset
|
501 |
lemma lower_join_unit [simp]: |
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changeset
|
502 |
"lower_join\<cdot>{xs}\<flat> = xs" |
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diff
changeset
|
503 |
unfolding lower_join_def by simp |
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parents:
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diff
changeset
|
504 |
|
b525988432e9
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changeset
|
505 |
lemma lower_join_plus [simp]: |
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changeset
|
506 |
"lower_join\<cdot>(xss +\<flat> yss) = lower_join\<cdot>xss +\<flat> lower_join\<cdot>yss" |
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huffman
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diff
changeset
|
507 |
unfolding lower_join_def by simp |
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huffman
parents:
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diff
changeset
|
508 |
|
b525988432e9
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changeset
|
509 |
lemma lower_join_map_unit: |
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diff
changeset
|
510 |
"lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs" |
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changeset
|
511 |
by (induct xs rule: lower_pd_induct, simp_all) |
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diff
changeset
|
512 |
|
b525988432e9
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diff
changeset
|
513 |
lemma lower_join_map_join: |
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diff
changeset
|
514 |
"lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)" |
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huffman
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changeset
|
515 |
by (induct xsss rule: lower_pd_induct, simp_all) |
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parents:
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diff
changeset
|
516 |
|
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diff
changeset
|
517 |
lemma lower_join_map_map: |
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diff
changeset
|
518 |
"lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) = |
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changeset
|
519 |
lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)" |
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changeset
|
520 |
by (induct xss rule: lower_pd_induct, simp_all) |
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parents:
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diff
changeset
|
521 |
|
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changeset
|
522 |
end |