author | huffman |
Fri, 20 Jun 2008 22:51:50 +0200 | |
changeset 27309 | c74270fd72a8 |
parent 27297 | 2c42b1505f25 |
child 27310 | d0229bc6c461 |
permissions | -rw-r--r-- |
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(* Title: HOLCF/LowerPD.thy |
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ID: $Id$ |
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Author: Brian Huffman |
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*) |
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header {* Lower powerdomain *} |
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theory LowerPD |
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imports CompactBasis |
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begin |
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subsection {* Basis preorder *} |
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definition |
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lower_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<flat>" 50) where |
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"lower_le = (\<lambda>u v. \<forall>x\<in>Rep_pd_basis u. \<exists>y\<in>Rep_pd_basis v. x \<sqsubseteq> y)" |
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lemma lower_le_refl [simp]: "t \<le>\<flat> t" |
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unfolding lower_le_def by fast |
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lemma lower_le_trans: "\<lbrakk>t \<le>\<flat> u; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> t \<le>\<flat> v" |
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unfolding lower_le_def |
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apply (rule ballI) |
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apply (drule (1) bspec, erule bexE) |
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apply (drule (1) bspec, erule bexE) |
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apply (erule rev_bexI) |
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apply (erule (1) trans_less) |
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done |
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interpretation lower_le: preorder [lower_le] |
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by (rule preorder.intro, rule lower_le_refl, rule lower_le_trans) |
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lemma lower_le_minimal [simp]: "PDUnit compact_bot \<le>\<flat> t" |
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unfolding lower_le_def Rep_PDUnit |
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by (simp, rule Rep_pd_basis_nonempty [folded ex_in_conv]) |
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lemma PDUnit_lower_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<flat> PDUnit y" |
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unfolding lower_le_def Rep_PDUnit by fast |
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lemma PDPlus_lower_mono: "\<lbrakk>s \<le>\<flat> t; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<flat> PDPlus t v" |
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unfolding lower_le_def Rep_PDPlus by fast |
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lemma PDPlus_lower_less: "t \<le>\<flat> PDPlus t u" |
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unfolding lower_le_def Rep_PDPlus by fast |
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lemma lower_le_PDUnit_PDUnit_iff [simp]: |
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"(PDUnit a \<le>\<flat> PDUnit b) = a \<sqsubseteq> b" |
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unfolding lower_le_def Rep_PDUnit by fast |
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lemma lower_le_PDUnit_PDPlus_iff: |
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"(PDUnit a \<le>\<flat> PDPlus t u) = (PDUnit a \<le>\<flat> t \<or> PDUnit a \<le>\<flat> u)" |
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unfolding lower_le_def Rep_PDPlus Rep_PDUnit by fast |
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lemma lower_le_PDPlus_iff: "(PDPlus t u \<le>\<flat> v) = (t \<le>\<flat> v \<and> u \<le>\<flat> v)" |
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unfolding lower_le_def Rep_PDPlus by fast |
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lemma lower_le_induct [induct set: lower_le]: |
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assumes le: "t \<le>\<flat> u" |
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assumes 1: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)" |
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assumes 2: "\<And>t u a. P (PDUnit a) t \<Longrightarrow> P (PDUnit a) (PDPlus t u)" |
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assumes 3: "\<And>t u v. \<lbrakk>P t v; P u v\<rbrakk> \<Longrightarrow> P (PDPlus t u) v" |
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shows "P t u" |
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using le |
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apply (induct t arbitrary: u rule: pd_basis_induct) |
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apply (erule rev_mp) |
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apply (induct_tac u rule: pd_basis_induct) |
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apply (simp add: 1) |
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apply (simp add: lower_le_PDUnit_PDPlus_iff) |
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apply (simp add: 2) |
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apply (subst PDPlus_commute) |
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apply (simp add: 2) |
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apply (simp add: lower_le_PDPlus_iff 3) |
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done |
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lemma approx_pd_lower_chain: |
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"approx_pd n t \<le>\<flat> approx_pd (Suc n) t" |
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apply (induct t rule: pd_basis_induct) |
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apply (simp add: compact_basis.take_chain) |
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apply (simp add: PDPlus_lower_mono) |
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done |
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lemma approx_pd_lower_le: "approx_pd i t \<le>\<flat> t" |
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apply (induct t rule: pd_basis_induct) |
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apply (simp add: compact_basis.take_less) |
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apply (simp add: PDPlus_lower_mono) |
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done |
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lemma approx_pd_lower_mono: |
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"t \<le>\<flat> u \<Longrightarrow> approx_pd n t \<le>\<flat> approx_pd n u" |
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apply (erule lower_le_induct) |
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apply (simp add: compact_basis.take_mono) |
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apply (simp add: lower_le_PDUnit_PDPlus_iff) |
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apply (simp add: lower_le_PDPlus_iff) |
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done |
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subsection {* Type definition *} |
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cpodef (open) 'a lower_pd = |
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"{S::'a pd_basis cset. lower_le.ideal (Rep_cset S)}" |
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by (rule lower_le.cpodef_ideal_lemma) |
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lemma ideal_Rep_lower_pd: "lower_le.ideal (Rep_cset (Rep_lower_pd xs))" |
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by (rule Rep_lower_pd [unfolded mem_Collect_eq]) |
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definition |
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lower_principal :: "'a pd_basis \<Rightarrow> 'a lower_pd" where |
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"lower_principal t = Abs_lower_pd (Abs_cset {u. u \<le>\<flat> t})" |
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lemma Rep_lower_principal: |
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"Rep_cset (Rep_lower_pd (lower_principal t)) = {u. u \<le>\<flat> t}" |
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unfolding lower_principal_def |
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by (simp add: Abs_lower_pd_inverse lower_le.ideal_principal) |
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interpretation lower_pd: |
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ideal_completion |
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[lower_le approx_pd lower_principal "\<lambda>x. Rep_cset (Rep_lower_pd x)"] |
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apply unfold_locales |
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apply (rule approx_pd_lower_le) |
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apply (rule approx_pd_idem) |
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apply (erule approx_pd_lower_mono) |
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apply (rule approx_pd_lower_chain) |
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apply (rule finite_range_approx_pd) |
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apply (rule approx_pd_covers) |
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apply (rule ideal_Rep_lower_pd) |
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apply (simp add: cont2contlubE [OF cont_Rep_lower_pd] Rep_cset_lub) |
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apply (rule Rep_lower_principal) |
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apply (simp only: less_lower_pd_def sq_le_cset_def) |
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done |
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text {* Lower powerdomain is pointed *} |
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lemma lower_pd_minimal: "lower_principal (PDUnit compact_bot) \<sqsubseteq> ys" |
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by (induct ys rule: lower_pd.principal_induct, simp, simp) |
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instance lower_pd :: (bifinite) pcpo |
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by intro_classes (fast intro: lower_pd_minimal) |
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lemma inst_lower_pd_pcpo: "\<bottom> = lower_principal (PDUnit compact_bot)" |
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by (rule lower_pd_minimal [THEN UU_I, symmetric]) |
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text {* Lower powerdomain is profinite *} |
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instantiation lower_pd :: (profinite) profinite |
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begin |
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definition |
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approx_lower_pd_def: "approx = lower_pd.completion_approx" |
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instance |
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apply (intro_classes, unfold approx_lower_pd_def) |
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apply (simp add: lower_pd.chain_completion_approx) |
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apply (rule lower_pd.lub_completion_approx) |
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apply (rule lower_pd.completion_approx_idem) |
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apply (rule lower_pd.finite_fixes_completion_approx) |
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done |
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end |
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instance lower_pd :: (bifinite) bifinite .. |
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lemma approx_lower_principal [simp]: |
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"approx n\<cdot>(lower_principal t) = lower_principal (approx_pd n t)" |
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unfolding approx_lower_pd_def |
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by (rule lower_pd.completion_approx_principal) |
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lemma approx_eq_lower_principal: |
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"\<exists>t\<in>Rep_cset (Rep_lower_pd xs). |
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approx n\<cdot>xs = lower_principal (approx_pd n t)" |
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unfolding approx_lower_pd_def |
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by (rule lower_pd.completion_approx_eq_principal) |
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subsection {* Monadic unit and plus *} |
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definition |
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lower_unit :: "'a \<rightarrow> 'a lower_pd" where |
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"lower_unit = compact_basis.basis_fun (\<lambda>a. lower_principal (PDUnit a))" |
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definition |
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lower_plus :: "'a lower_pd \<rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd" where |
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"lower_plus = lower_pd.basis_fun (\<lambda>t. lower_pd.basis_fun (\<lambda>u. |
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lower_principal (PDPlus t u)))" |
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abbreviation |
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lower_add :: "'a lower_pd \<Rightarrow> 'a lower_pd \<Rightarrow> 'a lower_pd" |
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(infixl "+\<flat>" 65) where |
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"xs +\<flat> ys == lower_plus\<cdot>xs\<cdot>ys" |
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syntax |
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"_lower_pd" :: "args \<Rightarrow> 'a lower_pd" ("{_}\<flat>") |
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translations |
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"{x,xs}\<flat>" == "{x}\<flat> +\<flat> {xs}\<flat>" |
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"{x}\<flat>" == "CONST lower_unit\<cdot>x" |
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lemma lower_unit_Rep_compact_basis [simp]: |
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"{Rep_compact_basis a}\<flat> = lower_principal (PDUnit a)" |
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unfolding lower_unit_def |
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by (simp add: compact_basis.basis_fun_principal PDUnit_lower_mono) |
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lemma lower_plus_principal [simp]: |
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"lower_principal t +\<flat> lower_principal u = lower_principal (PDPlus t u)" |
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unfolding lower_plus_def |
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by (simp add: lower_pd.basis_fun_principal |
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lower_pd.basis_fun_mono PDPlus_lower_mono) |
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lemma approx_lower_unit [simp]: |
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"approx n\<cdot>{x}\<flat> = {approx n\<cdot>x}\<flat>" |
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apply (induct x rule: compact_basis.principal_induct, simp) |
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apply (simp add: approx_Rep_compact_basis) |
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done |
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lemma approx_lower_plus [simp]: |
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"approx n\<cdot>(xs +\<flat> ys) = (approx n\<cdot>xs) +\<flat> (approx n\<cdot>ys)" |
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by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp) |
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lemma lower_plus_assoc: "(xs +\<flat> ys) +\<flat> zs = xs +\<flat> (ys +\<flat> zs)" |
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apply (induct xs ys arbitrary: zs rule: lower_pd.principal_induct2, simp, simp) |
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apply (rule_tac x=zs in lower_pd.principal_induct, simp) |
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apply (simp add: PDPlus_assoc) |
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done |
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lemma lower_plus_commute: "xs +\<flat> ys = ys +\<flat> xs" |
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apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) |
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apply (simp add: PDPlus_commute) |
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done |
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lemma lower_plus_absorb: "xs +\<flat> xs = xs" |
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apply (induct xs rule: lower_pd.principal_induct, simp) |
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apply (simp add: PDPlus_absorb) |
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done |
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interpretation aci_lower_plus: ab_semigroup_idem_mult ["op +\<flat>"] |
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by unfold_locales |
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(rule lower_plus_assoc lower_plus_commute lower_plus_absorb)+ |
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lemma lower_plus_left_commute: "xs +\<flat> (ys +\<flat> zs) = ys +\<flat> (xs +\<flat> zs)" |
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by (rule aci_lower_plus.mult_left_commute) |
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lemma lower_plus_left_absorb: "xs +\<flat> (xs +\<flat> ys) = xs +\<flat> ys" |
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by (rule aci_lower_plus.mult_left_idem) |
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lemmas lower_plus_aci = aci_lower_plus.mult_ac_idem |
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lemma lower_plus_less1: "xs \<sqsubseteq> xs +\<flat> ys" |
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apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) |
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apply (simp add: PDPlus_lower_less) |
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done |
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lemma lower_plus_less2: "ys \<sqsubseteq> xs +\<flat> ys" |
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by (subst lower_plus_commute, rule lower_plus_less1) |
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lemma lower_plus_least: "\<lbrakk>xs \<sqsubseteq> zs; ys \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs +\<flat> ys \<sqsubseteq> zs" |
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apply (subst lower_plus_absorb [of zs, symmetric]) |
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apply (erule (1) monofun_cfun [OF monofun_cfun_arg]) |
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done |
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lemma lower_plus_less_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 trans_less [OF lower_plus_less1]) |
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apply (erule trans_less [OF lower_plus_less2]) |
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apply (erule (1) lower_plus_least) |
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done |
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lemma lower_unit_less_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 (rule iffI) |
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apply (subgoal_tac |
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"adm (\<lambda>f. f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>ys \<or> f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>zs)") |
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apply (drule admD, rule chain_approx) |
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apply (drule_tac f="approx i" in monofun_cfun_arg) |
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apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp) |
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apply (cut_tac x="approx i\<cdot>ys" in lower_pd.compact_imp_principal, simp) |
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apply (cut_tac x="approx i\<cdot>zs" in lower_pd.compact_imp_principal, simp) |
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apply (clarify, simp add: lower_le_PDUnit_PDPlus_iff) |
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apply simp |
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apply simp |
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apply (erule disjE) |
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apply (erule trans_less [OF _ lower_plus_less1]) |
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apply (erule trans_less [OF _ lower_plus_less2]) |
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done |
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lemma lower_unit_less_iff [simp]: "{x}\<flat> \<sqsubseteq> {y}\<flat> \<longleftrightarrow> x \<sqsubseteq> y" |
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apply (rule iffI) |
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apply (rule profinite_less_ext) |
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apply (drule_tac f="approx i" in monofun_cfun_arg, simp) |
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apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp) |
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apply (cut_tac x="approx i\<cdot>y" in compact_basis.compact_imp_principal, simp) |
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apply clarsimp |
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apply (erule monofun_cfun_arg) |
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done |
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lemmas lower_pd_less_simps = |
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lower_unit_less_iff |
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lower_plus_less_iff |
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lower_unit_less_plus_iff |
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lemma fooble: |
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fixes f :: "'a::po \<Rightarrow> 'b::po" |
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assumes f: "\<And>x y. f x \<sqsubseteq> f y \<longleftrightarrow> x \<sqsubseteq> y" |
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shows "f x = f y \<longleftrightarrow> x = y" |
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unfolding po_eq_conv by (simp add: f) |
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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 (rule lower_unit_less_iff [THEN fooble]) |
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lemma lower_unit_strict [simp]: "{\<bottom>}\<flat> = \<bottom>" |
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unfolding inst_lower_pd_pcpo Rep_compact_bot [symmetric] by simp |
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||
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lemma lower_unit_strict_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>" |
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unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff) |
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lemma lower_plus_strict_iff [simp]: |
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"xs +\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>" |
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apply safe |
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apply (rule UU_I, erule subst, rule lower_plus_less1) |
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apply (rule UU_I, erule subst, rule lower_plus_less2) |
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apply (rule lower_plus_absorb) |
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done |
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lemma lower_plus_strict1 [simp]: "\<bottom> +\<flat> ys = ys" |
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apply (rule antisym_less [OF _ lower_plus_less2]) |
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apply (simp add: lower_plus_least) |
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done |
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lemma lower_plus_strict2 [simp]: "xs +\<flat> \<bottom> = xs" |
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apply (rule antisym_less [OF _ lower_plus_less1]) |
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apply (simp add: lower_plus_least) |
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done |
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lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x" |
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unfolding profinite_compact_iff by simp |
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lemma compact_lower_plus [simp]: |
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"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<flat> ys)" |
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by (auto dest!: lower_pd.compact_imp_principal) |
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subsection {* Induction rules *} |
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lemma lower_pd_induct1: |
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assumes P: "adm P" |
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assumes unit: "\<And>x. P {x}\<flat>" |
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assumes insert: |
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"\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> +\<flat> ys)" |
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shows "P (xs::'a lower_pd)" |
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apply (induct xs rule: lower_pd.principal_induct, rule P) |
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apply (induct_tac a rule: pd_basis_induct1) |
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apply (simp only: lower_unit_Rep_compact_basis [symmetric]) |
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apply (rule unit) |
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apply (simp only: lower_unit_Rep_compact_basis [symmetric] |
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lower_plus_principal [symmetric]) |
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apply (erule insert [OF unit]) |
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done |
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lemma lower_pd_induct: |
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assumes P: "adm P" |
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assumes unit: "\<And>x. P {x}\<flat>" |
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assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<flat> ys)" |
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shows "P (xs::'a lower_pd)" |
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apply (induct xs rule: lower_pd.principal_induct, rule P) |
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apply (induct_tac a rule: pd_basis_induct) |
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apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit) |
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apply (simp only: lower_plus_principal [symmetric] plus) |
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367 |
done |
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368 |
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369 |
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subsection {* Monadic bind *} |
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definition |
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lower_bind_basis :: |
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"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
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"lower_bind_basis = fold_pd |
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(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a)) |
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(\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" |
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lemma ACI_lower_bind: |
380 |
"ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" |
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apply unfold_locales |
26041
c2e15e65165f
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
parents:
25925
diff
changeset
|
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apply (simp add: lower_plus_assoc) |
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apply (simp add: lower_plus_commute) |
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apply (simp add: lower_plus_absorb eta_cfun) |
|
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done |
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lemma lower_bind_basis_simps [simp]: |
|
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"lower_bind_basis (PDUnit a) = |
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(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" |
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"lower_bind_basis (PDPlus t u) = |
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(\<Lambda> f. lower_bind_basis t\<cdot>f +\<flat> lower_bind_basis u\<cdot>f)" |
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unfolding lower_bind_basis_def |
393 |
apply - |
|
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apply (rule fold_pd_PDUnit [OF ACI_lower_bind]) |
395 |
apply (rule fold_pd_PDPlus [OF ACI_lower_bind]) |
|
25904 | 396 |
done |
397 |
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lemma lower_bind_basis_mono: |
|
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"t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u" |
|
400 |
unfolding expand_cfun_less |
|
401 |
apply (erule lower_le_induct, safe) |
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apply (simp add: monofun_cfun) |
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apply (simp add: rev_trans_less [OF lower_plus_less1]) |
404 |
apply (simp add: lower_plus_less_iff) |
|
405 |
done |
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406 |
||
407 |
definition |
|
408 |
lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
|
409 |
"lower_bind = lower_pd.basis_fun lower_bind_basis" |
|
410 |
||
411 |
lemma lower_bind_principal [simp]: |
|
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"lower_bind\<cdot>(lower_principal t) = lower_bind_basis t" |
|
413 |
unfolding lower_bind_def |
|
414 |
apply (rule lower_pd.basis_fun_principal) |
|
415 |
apply (erule lower_bind_basis_mono) |
|
416 |
done |
|
417 |
||
418 |
lemma lower_bind_unit [simp]: |
|
26927 | 419 |
"lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x" |
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by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 421 |
|
422 |
lemma lower_bind_plus [simp]: |
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26927 | 423 |
"lower_bind\<cdot>(xs +\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f +\<flat> lower_bind\<cdot>ys\<cdot>f" |
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by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp) |
25904 | 425 |
|
426 |
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" |
|
427 |
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit) |
|
428 |
||
429 |
||
430 |
subsection {* Map and join *} |
|
431 |
||
432 |
definition |
|
433 |
lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where |
|
26927 | 434 |
"lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))" |
25904 | 435 |
|
436 |
definition |
|
437 |
lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where |
|
438 |
"lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" |
|
439 |
||
440 |
lemma lower_map_unit [simp]: |
|
26927 | 441 |
"lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>" |
25904 | 442 |
unfolding lower_map_def by simp |
443 |
||
444 |
lemma lower_map_plus [simp]: |
|
26927 | 445 |
"lower_map\<cdot>f\<cdot>(xs +\<flat> ys) = lower_map\<cdot>f\<cdot>xs +\<flat> lower_map\<cdot>f\<cdot>ys" |
25904 | 446 |
unfolding lower_map_def by simp |
447 |
||
448 |
lemma lower_join_unit [simp]: |
|
26927 | 449 |
"lower_join\<cdot>{xs}\<flat> = xs" |
25904 | 450 |
unfolding lower_join_def by simp |
451 |
||
452 |
lemma lower_join_plus [simp]: |
|
26927 | 453 |
"lower_join\<cdot>(xss +\<flat> yss) = lower_join\<cdot>xss +\<flat> lower_join\<cdot>yss" |
25904 | 454 |
unfolding lower_join_def by simp |
455 |
||
456 |
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" |
|
457 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
458 |
||
459 |
lemma lower_map_map: |
|
460 |
"lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" |
|
461 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
462 |
||
463 |
lemma lower_join_map_unit: |
|
464 |
"lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs" |
|
465 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
466 |
||
467 |
lemma lower_join_map_join: |
|
468 |
"lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)" |
|
469 |
by (induct xsss rule: lower_pd_induct, simp_all) |
|
470 |
||
471 |
lemma lower_join_map_map: |
|
472 |
"lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) = |
|
473 |
lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)" |
|
474 |
by (induct xss rule: lower_pd_induct, simp_all) |
|
475 |
||
476 |
lemma lower_map_approx: "lower_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs" |
|
477 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
478 |
||
479 |
end |