author | huffman |
Thu, 19 Jun 2008 22:50:58 +0200 | |
changeset 27289 | c49d427867aa |
parent 27267 | 5ebfb7f25ebb |
child 27297 | 2c42b1505f25 |
permissions | -rw-r--r-- |
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(* Title: HOLCF/UpperPD.thy |
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ID: $Id$ |
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Author: Brian Huffman |
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*) |
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header {* Upper powerdomain *} |
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theory UpperPD |
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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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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) trans_less) |
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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_less: "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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lemma approx_pd_upper_chain: |
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"approx_pd n t \<le>\<sharp> 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_upper_mono) |
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done |
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lemma approx_pd_upper_le: "approx_pd i t \<le>\<sharp> 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_upper_mono) |
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done |
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lemma approx_pd_upper_mono: |
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"t \<le>\<sharp> u \<Longrightarrow> approx_pd n t \<le>\<sharp> approx_pd n u" |
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apply (erule upper_le_induct) |
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apply (simp add: compact_basis.take_mono) |
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apply (simp add: upper_le_PDPlus_PDUnit_iff) |
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apply (simp add: upper_le_PDPlus_iff) |
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done |
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subsection {* Type definition *} |
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cpodef (open) 'a upper_pd = |
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"{S::'a::profinite pd_basis set. upper_le.ideal S}" |
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apply (simp add: upper_le.adm_ideal) |
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apply (fast intro: upper_le.ideal_principal) |
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done |
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lemma ideal_Rep_upper_pd: "upper_le.ideal (Rep_upper_pd x)" |
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by (rule Rep_upper_pd [unfolded mem_Collect_eq]) |
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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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lemma Rep_upper_principal: |
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"Rep_upper_pd (upper_principal t) = {u. u \<le>\<sharp> t}" |
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unfolding upper_principal_def |
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apply (rule Abs_upper_pd_inverse [unfolded mem_Collect_eq]) |
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apply (rule upper_le.ideal_principal) |
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done |
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interpretation upper_pd: |
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ideal_completion [upper_le approx_pd upper_principal Rep_upper_pd] |
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apply unfold_locales |
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apply (rule approx_pd_upper_le) |
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apply (rule approx_pd_idem) |
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apply (erule approx_pd_upper_mono) |
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apply (rule approx_pd_upper_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_upper_pd) |
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apply (rule cont_Rep_upper_pd) |
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apply (rule Rep_upper_principal) |
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apply (simp only: less_upper_pd_def less_set_eq) |
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done |
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text {* Upper powerdomain is pointed *} |
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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 UU_I, symmetric]) |
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text {* Upper powerdomain is profinite *} |
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instantiation upper_pd :: (profinite) profinite |
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begin |
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definition |
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approx_upper_pd_def: "approx = upper_pd.completion_approx" |
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instance |
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apply (intro_classes, unfold approx_upper_pd_def) |
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apply (simp add: upper_pd.chain_completion_approx) |
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apply (rule upper_pd.lub_completion_approx) |
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apply (rule upper_pd.completion_approx_idem) |
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apply (rule upper_pd.finite_fixes_completion_approx) |
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done |
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end |
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instance upper_pd :: (bifinite) bifinite .. |
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lemma approx_upper_principal [simp]: |
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"approx n\<cdot>(upper_principal t) = upper_principal (approx_pd n t)" |
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unfolding approx_upper_pd_def |
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by (rule upper_pd.completion_approx_principal) |
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lemma approx_eq_upper_principal: |
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"\<exists>t\<in>Rep_upper_pd xs. approx n\<cdot>xs = upper_principal (approx_pd n t)" |
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unfolding approx_upper_pd_def |
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by (rule upper_pd.completion_approx_eq_principal) |
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subsection {* Monadic unit and plus *} |
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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.basis_fun (\<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.basis_fun (\<lambda>t. upper_pd.basis_fun (\<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 "+\<sharp>" 65) where |
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"xs +\<sharp> ys == upper_plus\<cdot>xs\<cdot>ys" |
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syntax |
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"_upper_pd" :: "args \<Rightarrow> 'a upper_pd" ("{_}\<sharp>") |
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translations |
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"{x,xs}\<sharp>" == "{x}\<sharp> +\<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.basis_fun_principal PDUnit_upper_mono) |
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lemma upper_plus_principal [simp]: |
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"upper_principal t +\<sharp> upper_principal u = upper_principal (PDPlus t u)" |
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unfolding upper_plus_def |
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by (simp add: upper_pd.basis_fun_principal |
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upper_pd.basis_fun_mono PDPlus_upper_mono) |
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lemma approx_upper_unit [simp]: |
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"approx n\<cdot>{x}\<sharp> = {approx n\<cdot>x}\<sharp>" |
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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_upper_plus [simp]: |
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"approx n\<cdot>(xs +\<sharp> ys) = (approx n\<cdot>xs) +\<sharp> (approx n\<cdot>ys)" |
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by (induct xs ys rule: upper_pd.principal_induct2, simp, simp, simp) |
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lemma upper_plus_assoc: "(xs +\<sharp> ys) +\<sharp> zs = xs +\<sharp> (ys +\<sharp> zs)" |
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apply (induct xs ys arbitrary: zs rule: upper_pd.principal_induct2, simp, simp) |
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apply (rule_tac x=zs in upper_pd.principal_induct, simp) |
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apply (simp add: PDPlus_assoc) |
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done |
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lemma upper_plus_commute: "xs +\<sharp> ys = ys +\<sharp> xs" |
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apply (induct xs ys rule: upper_pd.principal_induct2, simp, simp) |
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apply (simp add: PDPlus_commute) |
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done |
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lemma upper_plus_absorb: "xs +\<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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interpretation aci_upper_plus: ab_semigroup_idem_mult ["op +\<sharp>"] |
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by unfold_locales |
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(rule upper_plus_assoc upper_plus_commute upper_plus_absorb)+ |
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lemma upper_plus_left_commute: "xs +\<sharp> (ys +\<sharp> zs) = ys +\<sharp> (xs +\<sharp> zs)" |
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by (rule aci_upper_plus.mult_left_commute) |
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lemma upper_plus_left_absorb: "xs +\<sharp> (xs +\<sharp> ys) = xs +\<sharp> ys" |
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by (rule aci_upper_plus.mult_left_idem) |
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lemmas upper_plus_aci = aci_upper_plus.mult_ac_idem |
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lemma upper_plus_less1: "xs +\<sharp> ys \<sqsubseteq> xs" |
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apply (induct xs ys rule: upper_pd.principal_induct2, simp, simp) |
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apply (simp add: PDPlus_upper_less) |
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done |
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lemma upper_plus_less2: "xs +\<sharp> ys \<sqsubseteq> ys" |
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by (subst upper_plus_commute, rule upper_plus_less1) |
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lemma upper_plus_greatest: "\<lbrakk>xs \<sqsubseteq> ys; xs \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs \<sqsubseteq> ys +\<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]) |
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done |
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lemma upper_less_plus_iff: |
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"xs \<sqsubseteq> ys +\<sharp> zs \<longleftrightarrow> xs \<sqsubseteq> ys \<and> xs \<sqsubseteq> zs" |
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apply safe |
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apply (erule trans_less [OF _ upper_plus_less1]) |
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apply (erule trans_less [OF _ upper_plus_less2]) |
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apply (erule (1) upper_plus_greatest) |
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done |
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lemma upper_plus_less_unit_iff: |
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"xs +\<sharp> ys \<sqsubseteq> {z}\<sharp> \<longleftrightarrow> xs \<sqsubseteq> {z}\<sharp> \<or> ys \<sqsubseteq> {z}\<sharp>" |
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apply (rule iffI) |
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apply (subgoal_tac |
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"adm (\<lambda>f. f\<cdot>xs \<sqsubseteq> f\<cdot>{z}\<sharp> \<or> f\<cdot>ys \<sqsubseteq> f\<cdot>{z}\<sharp>)") |
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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>xs" in upper_pd.compact_imp_principal, simp) |
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apply (cut_tac x="approx i\<cdot>ys" in upper_pd.compact_imp_principal, simp) |
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apply (cut_tac x="approx i\<cdot>z" in compact_basis.compact_imp_principal, simp) |
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apply (clarify, simp add: upper_le_PDPlus_PDUnit_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 upper_plus_less1]) |
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apply (erule trans_less [OF upper_plus_less2]) |
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done |
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lemma upper_unit_less_iff [simp]: "{x}\<sharp> \<sqsubseteq> {y}\<sharp> \<longleftrightarrow> x \<sqsubseteq> y" |
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apply (rule iffI) |
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apply (rule bifinite_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 upper_pd_less_simps = |
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upper_unit_less_iff |
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upper_less_plus_iff |
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upper_plus_less_unit_iff |
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lemma upper_unit_eq_iff [simp]: "{x}\<sharp> = {y}\<sharp> \<longleftrightarrow> x = y" |
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unfolding po_eq_conv by simp |
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lemma upper_unit_strict [simp]: "{\<bottom>}\<sharp> = \<bottom>" |
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unfolding inst_upper_pd_pcpo Rep_compact_bot [symmetric] by simp |
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lemma upper_plus_strict1 [simp]: "\<bottom> +\<sharp> ys = \<bottom>" |
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by (rule UU_I, rule upper_plus_less1) |
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lemma upper_plus_strict2 [simp]: "xs +\<sharp> \<bottom> = \<bottom>" |
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by (rule UU_I, rule upper_plus_less2) |
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lemma upper_unit_strict_iff [simp]: "{x}\<sharp> = \<bottom> \<longleftrightarrow> x = \<bottom>" |
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unfolding upper_unit_strict [symmetric] by (rule upper_unit_eq_iff) |
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lemma upper_plus_strict_iff [simp]: |
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"xs +\<sharp> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<or> ys = \<bottom>" |
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apply (rule iffI) |
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apply (erule rev_mp) |
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apply (rule upper_pd.principal_induct2 [where x=xs and y=ys], simp, simp) |
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apply (simp add: inst_upper_pd_pcpo upper_pd.principal_eq_iff |
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upper_le_PDPlus_PDUnit_iff) |
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apply auto |
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done |
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lemma compact_upper_unit_iff [simp]: "compact {x}\<sharp> \<longleftrightarrow> compact x" |
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unfolding bifinite_compact_iff by simp |
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327 |
||
328 |
lemma compact_upper_plus [simp]: |
|
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"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<sharp> ys)" |
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by (auto dest!: upper_pd.compact_imp_principal) |
26927 | 331 |
|
25904 | 332 |
|
333 |
subsection {* Induction rules *} |
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334 |
||
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lemma upper_pd_induct1: |
|
336 |
assumes P: "adm P" |
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assumes unit: "\<And>x. P {x}\<sharp>" |
338 |
assumes insert: "\<And>x ys. \<lbrakk>P {x}\<sharp>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<sharp> +\<sharp> ys)" |
|
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shows "P (xs::'a upper_pd)" |
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apply (induct xs rule: upper_pd.principal_induct, rule P) |
341 |
apply (induct_tac a rule: pd_basis_induct1) |
|
25904 | 342 |
apply (simp only: upper_unit_Rep_compact_basis [symmetric]) |
343 |
apply (rule unit) |
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344 |
apply (simp only: upper_unit_Rep_compact_basis [symmetric] |
|
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upper_plus_principal [symmetric]) |
|
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apply (erule insert [OF unit]) |
|
347 |
done |
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348 |
||
349 |
lemma upper_pd_induct: |
|
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assumes P: "adm P" |
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assumes unit: "\<And>x. P {x}\<sharp>" |
352 |
assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<sharp> ys)" |
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shows "P (xs::'a upper_pd)" |
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apply (induct xs rule: upper_pd.principal_induct, rule P) |
355 |
apply (induct_tac a rule: pd_basis_induct) |
|
25904 | 356 |
apply (simp only: upper_unit_Rep_compact_basis [symmetric] unit) |
357 |
apply (simp only: upper_plus_principal [symmetric] plus) |
|
358 |
done |
|
359 |
||
360 |
||
361 |
subsection {* Monadic bind *} |
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362 |
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definition |
|
364 |
upper_bind_basis :: |
|
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"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b upper_pd) \<rightarrow> 'b upper_pd" where |
|
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"upper_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 +\<sharp> y\<cdot>f)" |
25904 | 369 |
|
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lemma ACI_upper_bind: |
371 |
"ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<sharp> 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
|
373 |
apply (simp add: upper_plus_assoc) |
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apply (simp add: upper_plus_commute) |
375 |
apply (simp add: upper_plus_absorb eta_cfun) |
|
376 |
done |
|
377 |
||
378 |
lemma upper_bind_basis_simps [simp]: |
|
379 |
"upper_bind_basis (PDUnit a) = |
|
380 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" |
|
381 |
"upper_bind_basis (PDPlus t u) = |
|
26927 | 382 |
(\<Lambda> f. upper_bind_basis t\<cdot>f +\<sharp> upper_bind_basis u\<cdot>f)" |
25904 | 383 |
unfolding upper_bind_basis_def |
384 |
apply - |
|
26927 | 385 |
apply (rule fold_pd_PDUnit [OF ACI_upper_bind]) |
386 |
apply (rule fold_pd_PDPlus [OF ACI_upper_bind]) |
|
25904 | 387 |
done |
388 |
||
389 |
lemma upper_bind_basis_mono: |
|
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"t \<le>\<sharp> u \<Longrightarrow> upper_bind_basis t \<sqsubseteq> upper_bind_basis u" |
|
391 |
unfolding expand_cfun_less |
|
392 |
apply (erule upper_le_induct, safe) |
|
27289 | 393 |
apply (simp add: monofun_cfun) |
25904 | 394 |
apply (simp add: trans_less [OF upper_plus_less1]) |
395 |
apply (simp add: upper_less_plus_iff) |
|
396 |
done |
|
397 |
||
398 |
definition |
|
399 |
upper_bind :: "'a upper_pd \<rightarrow> ('a \<rightarrow> 'b upper_pd) \<rightarrow> 'b upper_pd" where |
|
400 |
"upper_bind = upper_pd.basis_fun upper_bind_basis" |
|
401 |
||
402 |
lemma upper_bind_principal [simp]: |
|
403 |
"upper_bind\<cdot>(upper_principal t) = upper_bind_basis t" |
|
404 |
unfolding upper_bind_def |
|
405 |
apply (rule upper_pd.basis_fun_principal) |
|
406 |
apply (erule upper_bind_basis_mono) |
|
407 |
done |
|
408 |
||
409 |
lemma upper_bind_unit [simp]: |
|
26927 | 410 |
"upper_bind\<cdot>{x}\<sharp>\<cdot>f = f\<cdot>x" |
27289 | 411 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 412 |
|
413 |
lemma upper_bind_plus [simp]: |
|
26927 | 414 |
"upper_bind\<cdot>(xs +\<sharp> ys)\<cdot>f = upper_bind\<cdot>xs\<cdot>f +\<sharp> upper_bind\<cdot>ys\<cdot>f" |
27289 | 415 |
by (induct xs ys rule: upper_pd.principal_induct2, simp, simp, simp) |
25904 | 416 |
|
417 |
lemma upper_bind_strict [simp]: "upper_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" |
|
418 |
unfolding upper_unit_strict [symmetric] by (rule upper_bind_unit) |
|
419 |
||
420 |
||
421 |
subsection {* Map and join *} |
|
422 |
||
423 |
definition |
|
424 |
upper_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a upper_pd \<rightarrow> 'b upper_pd" where |
|
26927 | 425 |
"upper_map = (\<Lambda> f xs. upper_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<sharp>))" |
25904 | 426 |
|
427 |
definition |
|
428 |
upper_join :: "'a upper_pd upper_pd \<rightarrow> 'a upper_pd" where |
|
429 |
"upper_join = (\<Lambda> xss. upper_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" |
|
430 |
||
431 |
lemma upper_map_unit [simp]: |
|
26927 | 432 |
"upper_map\<cdot>f\<cdot>{x}\<sharp> = {f\<cdot>x}\<sharp>" |
25904 | 433 |
unfolding upper_map_def by simp |
434 |
||
435 |
lemma upper_map_plus [simp]: |
|
26927 | 436 |
"upper_map\<cdot>f\<cdot>(xs +\<sharp> ys) = upper_map\<cdot>f\<cdot>xs +\<sharp> upper_map\<cdot>f\<cdot>ys" |
25904 | 437 |
unfolding upper_map_def by simp |
438 |
||
439 |
lemma upper_join_unit [simp]: |
|
26927 | 440 |
"upper_join\<cdot>{xs}\<sharp> = xs" |
25904 | 441 |
unfolding upper_join_def by simp |
442 |
||
443 |
lemma upper_join_plus [simp]: |
|
26927 | 444 |
"upper_join\<cdot>(xss +\<sharp> yss) = upper_join\<cdot>xss +\<sharp> upper_join\<cdot>yss" |
25904 | 445 |
unfolding upper_join_def by simp |
446 |
||
447 |
lemma upper_map_ident: "upper_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" |
|
448 |
by (induct xs rule: upper_pd_induct, simp_all) |
|
449 |
||
450 |
lemma upper_map_map: |
|
451 |
"upper_map\<cdot>f\<cdot>(upper_map\<cdot>g\<cdot>xs) = upper_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" |
|
452 |
by (induct xs rule: upper_pd_induct, simp_all) |
|
453 |
||
454 |
lemma upper_join_map_unit: |
|
455 |
"upper_join\<cdot>(upper_map\<cdot>upper_unit\<cdot>xs) = xs" |
|
456 |
by (induct xs rule: upper_pd_induct, simp_all) |
|
457 |
||
458 |
lemma upper_join_map_join: |
|
459 |
"upper_join\<cdot>(upper_map\<cdot>upper_join\<cdot>xsss) = upper_join\<cdot>(upper_join\<cdot>xsss)" |
|
460 |
by (induct xsss rule: upper_pd_induct, simp_all) |
|
461 |
||
462 |
lemma upper_join_map_map: |
|
463 |
"upper_join\<cdot>(upper_map\<cdot>(upper_map\<cdot>f)\<cdot>xss) = |
|
464 |
upper_map\<cdot>f\<cdot>(upper_join\<cdot>xss)" |
|
465 |
by (induct xss rule: upper_pd_induct, simp_all) |
|
466 |
||
467 |
lemma upper_map_approx: "upper_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs" |
|
468 |
by (induct xs rule: upper_pd_induct, simp_all) |
|
469 |
||
470 |
end |