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(* Title: HOL/HOLCF/ConvexPD.thy |
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Author: Brian Huffman |
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*) |
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section {* Convex powerdomain *} |
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theory ConvexPD |
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imports UpperPD LowerPD |
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begin |
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subsection {* Basis preorder *} |
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definition |
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convex_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<natural>" 50) where |
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"convex_le = (\<lambda>u v. u \<le>\<sharp> v \<and> u \<le>\<flat> v)" |
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lemma convex_le_refl [simp]: "t \<le>\<natural> t" |
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unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl) |
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lemma convex_le_trans: "\<lbrakk>t \<le>\<natural> u; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> t \<le>\<natural> v" |
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unfolding convex_le_def by (fast intro: upper_le_trans lower_le_trans) |
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interpretation convex_le: preorder convex_le |
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by (rule preorder.intro, rule convex_le_refl, rule convex_le_trans) |
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lemma upper_le_minimal [simp]: "PDUnit compact_bot \<le>\<natural> t" |
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unfolding convex_le_def Rep_PDUnit by simp |
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lemma PDUnit_convex_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<natural> PDUnit y" |
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unfolding convex_le_def by (fast intro: PDUnit_upper_mono PDUnit_lower_mono) |
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lemma PDPlus_convex_mono: "\<lbrakk>s \<le>\<natural> t; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<natural> PDPlus t v" |
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unfolding convex_le_def by (fast intro: PDPlus_upper_mono PDPlus_lower_mono) |
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lemma convex_le_PDUnit_PDUnit_iff [simp]: |
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"(PDUnit a \<le>\<natural> PDUnit b) = (a \<sqsubseteq> b)" |
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unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit by fast |
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lemma convex_le_PDUnit_lemma1: |
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"(PDUnit a \<le>\<natural> t) = (\<forall>b\<in>Rep_pd_basis t. a \<sqsubseteq> b)" |
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unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit |
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using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast |
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lemma convex_le_PDUnit_PDPlus_iff [simp]: |
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"(PDUnit a \<le>\<natural> PDPlus t u) = (PDUnit a \<le>\<natural> t \<and> PDUnit a \<le>\<natural> u)" |
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unfolding convex_le_PDUnit_lemma1 Rep_PDPlus by fast |
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lemma convex_le_PDUnit_lemma2: |
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"(t \<le>\<natural> PDUnit b) = (\<forall>a\<in>Rep_pd_basis t. a \<sqsubseteq> b)" |
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unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit |
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using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast |
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lemma convex_le_PDPlus_PDUnit_iff [simp]: |
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"(PDPlus t u \<le>\<natural> PDUnit a) = (t \<le>\<natural> PDUnit a \<and> u \<le>\<natural> PDUnit a)" |
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unfolding convex_le_PDUnit_lemma2 Rep_PDPlus by fast |
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lemma convex_le_PDPlus_lemma: |
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assumes z: "PDPlus t u \<le>\<natural> z" |
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shows "\<exists>v w. z = PDPlus v w \<and> t \<le>\<natural> v \<and> u \<le>\<natural> w" |
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proof (intro exI conjI) |
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let ?A = "{b\<in>Rep_pd_basis z. \<exists>a\<in>Rep_pd_basis t. a \<sqsubseteq> b}" |
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let ?B = "{b\<in>Rep_pd_basis z. \<exists>a\<in>Rep_pd_basis u. a \<sqsubseteq> b}" |
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let ?v = "Abs_pd_basis ?A" |
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let ?w = "Abs_pd_basis ?B" |
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have Rep_v: "Rep_pd_basis ?v = ?A" |
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apply (rule Abs_pd_basis_inverse) |
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apply (rule Rep_pd_basis_nonempty [of t, folded ex_in_conv, THEN exE]) |
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apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify) |
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apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE) |
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apply (simp add: pd_basis_def) |
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apply fast |
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done |
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have Rep_w: "Rep_pd_basis ?w = ?B" |
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apply (rule Abs_pd_basis_inverse) |
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apply (rule Rep_pd_basis_nonempty [of u, folded ex_in_conv, THEN exE]) |
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apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify) |
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apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE) |
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apply (simp add: pd_basis_def) |
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apply fast |
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done |
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show "z = PDPlus ?v ?w" |
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apply (insert z) |
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apply (simp add: convex_le_def, erule conjE) |
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apply (simp add: Rep_pd_basis_inject [symmetric] Rep_PDPlus) |
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apply (simp add: Rep_v Rep_w) |
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apply (rule equalityI) |
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apply (rule subsetI) |
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apply (simp only: upper_le_def) |
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apply (drule (1) bspec, erule bexE) |
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apply (simp add: Rep_PDPlus) |
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apply fast |
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apply fast |
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done |
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show "t \<le>\<natural> ?v" "u \<le>\<natural> ?w" |
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apply (insert z) |
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apply (simp_all add: convex_le_def upper_le_def lower_le_def Rep_PDPlus Rep_v Rep_w) |
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apply fast+ |
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done |
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qed |
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lemma convex_le_induct [induct set: convex_le]: |
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assumes le: "t \<le>\<natural> u" |
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assumes 2: "\<And>t u v. \<lbrakk>P t u; P u v\<rbrakk> \<Longrightarrow> P t v" |
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assumes 3: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)" |
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assumes 4: "\<And>t u v w. \<lbrakk>P t v; P u w\<rbrakk> \<Longrightarrow> P (PDPlus t u) (PDPlus v w)" |
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shows "P t u" |
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using le 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_induct1) |
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apply (simp add: 3) |
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apply (simp, clarify, rename_tac a b t) |
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apply (subgoal_tac "P (PDPlus (PDUnit a) (PDUnit a)) (PDPlus (PDUnit b) t)") |
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apply (simp add: PDPlus_absorb) |
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apply (erule (1) 4 [OF 3]) |
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apply (drule convex_le_PDPlus_lemma, clarify) |
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apply (simp add: 4) |
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done |
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subsection {* Type definition *} |
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typedef 'a convex_pd = |
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"{S::'a pd_basis set. convex_le.ideal S}" |
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by (rule convex_le.ex_ideal) |
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type_notation (xsymbols) convex_pd ("('(_')\<natural>)") |
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instantiation convex_pd :: (bifinite) below |
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begin |
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definition |
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"x \<sqsubseteq> y \<longleftrightarrow> Rep_convex_pd x \<subseteq> Rep_convex_pd y" |
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instance .. |
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end |
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instance convex_pd :: (bifinite) po |
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using type_definition_convex_pd below_convex_pd_def |
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by (rule convex_le.typedef_ideal_po) |
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instance convex_pd :: (bifinite) cpo |
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using type_definition_convex_pd below_convex_pd_def |
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by (rule convex_le.typedef_ideal_cpo) |
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definition |
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convex_principal :: "'a pd_basis \<Rightarrow> 'a convex_pd" where |
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"convex_principal t = Abs_convex_pd {u. u \<le>\<natural> t}" |
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interpretation convex_pd: |
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ideal_completion convex_le convex_principal Rep_convex_pd |
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using type_definition_convex_pd below_convex_pd_def |
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using convex_principal_def pd_basis_countable |
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by (rule convex_le.typedef_ideal_completion) |
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text {* Convex powerdomain is pointed *} |
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lemma convex_pd_minimal: "convex_principal (PDUnit compact_bot) \<sqsubseteq> ys" |
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by (induct ys rule: convex_pd.principal_induct, simp, simp) |
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instance convex_pd :: (bifinite) pcpo |
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by intro_classes (fast intro: convex_pd_minimal) |
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lemma inst_convex_pd_pcpo: "\<bottom> = convex_principal (PDUnit compact_bot)" |
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by (rule convex_pd_minimal [THEN bottomI, symmetric]) |
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subsection {* Monadic unit and plus *} |
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definition |
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convex_unit :: "'a \<rightarrow> 'a convex_pd" where |
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"convex_unit = compact_basis.extension (\<lambda>a. convex_principal (PDUnit a))" |
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definition |
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convex_plus :: "'a convex_pd \<rightarrow> 'a convex_pd \<rightarrow> 'a convex_pd" where |
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"convex_plus = convex_pd.extension (\<lambda>t. convex_pd.extension (\<lambda>u. |
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convex_principal (PDPlus t u)))" |
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abbreviation |
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convex_add :: "'a convex_pd \<Rightarrow> 'a convex_pd \<Rightarrow> 'a convex_pd" |
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(infixl "\<union>\<natural>" 65) where |
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"xs \<union>\<natural> ys == convex_plus\<cdot>xs\<cdot>ys" |
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syntax |
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"_convex_pd" :: "args \<Rightarrow> logic" ("{_}\<natural>") |
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translations |
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"{x,xs}\<natural>" == "{x}\<natural> \<union>\<natural> {xs}\<natural>" |
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"{x}\<natural>" == "CONST convex_unit\<cdot>x" |
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lemma convex_unit_Rep_compact_basis [simp]: |
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"{Rep_compact_basis a}\<natural> = convex_principal (PDUnit a)" |
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unfolding convex_unit_def |
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by (simp add: compact_basis.extension_principal PDUnit_convex_mono) |
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lemma convex_plus_principal [simp]: |
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"convex_principal t \<union>\<natural> convex_principal u = convex_principal (PDPlus t u)" |
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unfolding convex_plus_def |
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by (simp add: convex_pd.extension_principal |
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convex_pd.extension_mono PDPlus_convex_mono) |
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interpretation convex_add: semilattice convex_add proof |
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fix xs ys zs :: "'a convex_pd" |
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show "(xs \<union>\<natural> ys) \<union>\<natural> zs = xs \<union>\<natural> (ys \<union>\<natural> zs)" |
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apply (induct xs rule: convex_pd.principal_induct, simp) |
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apply (induct ys rule: convex_pd.principal_induct, simp) |
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apply (induct zs rule: convex_pd.principal_induct, simp) |
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apply (simp add: PDPlus_assoc) |
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done |
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show "xs \<union>\<natural> ys = ys \<union>\<natural> xs" |
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apply (induct xs rule: convex_pd.principal_induct, simp) |
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apply (induct ys rule: convex_pd.principal_induct, simp) |
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apply (simp add: PDPlus_commute) |
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done |
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show "xs \<union>\<natural> xs = xs" |
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apply (induct xs rule: convex_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 convex_plus_assoc = convex_add.assoc |
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lemmas convex_plus_commute = convex_add.commute |
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lemmas convex_plus_absorb = convex_add.idem |
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lemmas convex_plus_left_commute = convex_add.left_commute |
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lemmas convex_plus_left_absorb = convex_add.left_idem |
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text {* Useful for @{text "simp add: convex_plus_ac"} *} |
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lemmas convex_plus_ac = |
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convex_plus_assoc convex_plus_commute convex_plus_left_commute |
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|
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text {* Useful for @{text "simp only: convex_plus_aci"} *} |
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lemmas convex_plus_aci = |
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convex_plus_ac convex_plus_absorb convex_plus_left_absorb |
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233 |
|
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lemma convex_unit_below_plus_iff [simp]: |
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"{x}\<natural> \<sqsubseteq> ys \<union>\<natural> zs \<longleftrightarrow> {x}\<natural> \<sqsubseteq> ys \<and> {x}\<natural> \<sqsubseteq> zs" |
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apply (induct x rule: compact_basis.principal_induct, simp) |
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apply (induct ys rule: convex_pd.principal_induct, simp) |
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apply (induct zs rule: convex_pd.principal_induct, simp) |
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apply simp |
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done |
241 |
||
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lemma convex_plus_below_unit_iff [simp]: |
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"xs \<union>\<natural> ys \<sqsubseteq> {z}\<natural> \<longleftrightarrow> xs \<sqsubseteq> {z}\<natural> \<and> ys \<sqsubseteq> {z}\<natural>" |
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apply (induct xs rule: convex_pd.principal_induct, simp) |
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apply (induct ys rule: convex_pd.principal_induct, simp) |
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apply (induct z rule: compact_basis.principal_induct, simp) |
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apply simp |
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done |
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lemma convex_unit_below_iff [simp]: "{x}\<natural> \<sqsubseteq> {y}\<natural> \<longleftrightarrow> x \<sqsubseteq> y" |
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apply (induct x rule: compact_basis.principal_induct, simp) |
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apply (induct y rule: compact_basis.principal_induct, simp) |
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apply simp |
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done |
255 |
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256 |
lemma convex_unit_eq_iff [simp]: "{x}\<natural> = {y}\<natural> \<longleftrightarrow> x = y" |
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257 |
unfolding po_eq_conv by simp |
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lemma convex_unit_strict [simp]: "{\<bottom>}\<natural> = \<bottom>" |
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using convex_unit_Rep_compact_basis [of compact_bot] |
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by (simp add: inst_convex_pd_pcpo) |
26927 | 262 |
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lemma convex_unit_bottom_iff [simp]: "{x}\<natural> = \<bottom> \<longleftrightarrow> x = \<bottom>" |
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unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff) |
265 |
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lemma compact_convex_unit: "compact x \<Longrightarrow> compact {x}\<natural>" |
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by (auto dest!: compact_basis.compact_imp_principal) |
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|
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lemma compact_convex_unit_iff [simp]: "compact {x}\<natural> \<longleftrightarrow> compact x" |
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apply (safe elim!: compact_convex_unit) |
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apply (simp only: compact_def convex_unit_below_iff [symmetric]) |
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apply (erule adm_subst [OF cont_Rep_cfun2]) |
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done |
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lemma compact_convex_plus [simp]: |
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"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs \<union>\<natural> ys)" |
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by (auto dest!: convex_pd.compact_imp_principal) |
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|
25904 | 279 |
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subsection {* Induction rules *} |
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||
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lemma convex_pd_induct1: |
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assumes P: "adm P" |
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assumes unit: "\<And>x. P {x}\<natural>" |
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assumes insert: "\<And>x ys. \<lbrakk>P {x}\<natural>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<natural> \<union>\<natural> ys)" |
25904 | 286 |
shows "P (xs::'a convex_pd)" |
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apply (induct xs rule: convex_pd.principal_induct, rule P) |
288 |
apply (induct_tac a rule: pd_basis_induct1) |
|
25904 | 289 |
apply (simp only: convex_unit_Rep_compact_basis [symmetric]) |
290 |
apply (rule unit) |
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apply (simp only: convex_unit_Rep_compact_basis [symmetric] |
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convex_plus_principal [symmetric]) |
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apply (erule insert [OF unit]) |
|
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done |
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||
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lemma convex_pd_induct |
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[case_names adm convex_unit convex_plus, induct type: convex_pd]: |
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assumes P: "adm P" |
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assumes unit: "\<And>x. P {x}\<natural>" |
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assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs \<union>\<natural> ys)" |
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shows "P (xs::'a convex_pd)" |
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apply (induct xs rule: convex_pd.principal_induct, rule P) |
303 |
apply (induct_tac a rule: pd_basis_induct) |
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apply (simp only: convex_unit_Rep_compact_basis [symmetric] unit) |
305 |
apply (simp only: convex_plus_principal [symmetric] plus) |
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306 |
done |
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307 |
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308 |
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309 |
subsection {* Monadic bind *} |
|
310 |
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311 |
definition |
|
312 |
convex_bind_basis :: |
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"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where |
|
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"convex_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 \<union>\<natural> y\<cdot>f)" |
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|
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lemma ACI_convex_bind: |
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"semilattice (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<natural> y\<cdot>f)" |
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apply unfold_locales |
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apply (simp add: convex_plus_assoc) |
25904 | 322 |
apply (simp add: convex_plus_commute) |
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apply (simp add: eta_cfun) |
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done |
325 |
||
326 |
lemma convex_bind_basis_simps [simp]: |
|
327 |
"convex_bind_basis (PDUnit a) = |
|
328 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" |
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329 |
"convex_bind_basis (PDPlus t u) = |
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330 |
(\<Lambda> f. convex_bind_basis t\<cdot>f \<union>\<natural> convex_bind_basis u\<cdot>f)" |
25904 | 331 |
unfolding convex_bind_basis_def |
332 |
apply - |
|
26927 | 333 |
apply (rule fold_pd_PDUnit [OF ACI_convex_bind]) |
334 |
apply (rule fold_pd_PDPlus [OF ACI_convex_bind]) |
|
25904 | 335 |
done |
336 |
||
337 |
lemma convex_bind_basis_mono: |
|
338 |
"t \<le>\<natural> u \<Longrightarrow> convex_bind_basis t \<sqsubseteq> convex_bind_basis u" |
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apply (erule convex_le_induct) |
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apply (erule (1) below_trans) |
27289 | 341 |
apply (simp add: monofun_LAM monofun_cfun) |
342 |
apply (simp add: monofun_LAM monofun_cfun) |
|
25904 | 343 |
done |
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||
345 |
definition |
|
346 |
convex_bind :: "'a convex_pd \<rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where |
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347 |
"convex_bind = convex_pd.extension convex_bind_basis" |
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syntax |
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350 |
"_convex_bind" :: "[logic, logic, logic] \<Rightarrow> logic" |
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351 |
("(3\<Union>\<natural>_\<in>_./ _)" [0, 0, 10] 10) |
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352 |
|
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353 |
translations |
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"\<Union>\<natural>x\<in>xs. e" == "CONST convex_bind\<cdot>xs\<cdot>(\<Lambda> x. e)" |
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355 |
|
25904 | 356 |
lemma convex_bind_principal [simp]: |
357 |
"convex_bind\<cdot>(convex_principal t) = convex_bind_basis t" |
|
358 |
unfolding convex_bind_def |
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|
359 |
apply (rule convex_pd.extension_principal) |
25904 | 360 |
apply (erule convex_bind_basis_mono) |
361 |
done |
|
362 |
||
363 |
lemma convex_bind_unit [simp]: |
|
26927 | 364 |
"convex_bind\<cdot>{x}\<natural>\<cdot>f = f\<cdot>x" |
27289 | 365 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 366 |
|
367 |
lemma convex_bind_plus [simp]: |
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"convex_bind\<cdot>(xs \<union>\<natural> ys)\<cdot>f = convex_bind\<cdot>xs\<cdot>f \<union>\<natural> convex_bind\<cdot>ys\<cdot>f" |
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by (induct xs rule: convex_pd.principal_induct, simp, |
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370 |
induct ys rule: convex_pd.principal_induct, simp, simp) |
25904 | 371 |
|
372 |
lemma convex_bind_strict [simp]: "convex_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" |
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373 |
unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit) |
|
374 |
||
40589 | 375 |
lemma convex_bind_bind: |
376 |
"convex_bind\<cdot>(convex_bind\<cdot>xs\<cdot>f)\<cdot>g = |
|
377 |
convex_bind\<cdot>xs\<cdot>(\<Lambda> x. convex_bind\<cdot>(f\<cdot>x)\<cdot>g)" |
|
378 |
by (induct xs, simp_all) |
|
379 |
||
25904 | 380 |
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381 |
subsection {* Map *} |
25904 | 382 |
|
383 |
definition |
|
384 |
convex_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a convex_pd \<rightarrow> 'b convex_pd" where |
|
26927 | 385 |
"convex_map = (\<Lambda> f xs. convex_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<natural>))" |
25904 | 386 |
|
387 |
lemma convex_map_unit [simp]: |
|
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"convex_map\<cdot>f\<cdot>{x}\<natural> = {f\<cdot>x}\<natural>" |
25904 | 389 |
unfolding convex_map_def by simp |
390 |
||
391 |
lemma convex_map_plus [simp]: |
|
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|
392 |
"convex_map\<cdot>f\<cdot>(xs \<union>\<natural> ys) = convex_map\<cdot>f\<cdot>xs \<union>\<natural> convex_map\<cdot>f\<cdot>ys" |
25904 | 393 |
unfolding convex_map_def by simp |
394 |
||
40577 | 395 |
lemma convex_map_bottom [simp]: "convex_map\<cdot>f\<cdot>\<bottom> = {f\<cdot>\<bottom>}\<natural>" |
396 |
unfolding convex_map_def by simp |
|
397 |
||
25904 | 398 |
lemma convex_map_ident: "convex_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" |
399 |
by (induct xs rule: convex_pd_induct, simp_all) |
|
400 |
||
33808 | 401 |
lemma convex_map_ID: "convex_map\<cdot>ID = ID" |
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|
402 |
by (simp add: cfun_eq_iff ID_def convex_map_ident) |
33808 | 403 |
|
25904 | 404 |
lemma convex_map_map: |
405 |
"convex_map\<cdot>f\<cdot>(convex_map\<cdot>g\<cdot>xs) = convex_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" |
|
406 |
by (induct xs rule: convex_pd_induct, simp_all) |
|
407 |
||
41110 | 408 |
lemma convex_bind_map: |
409 |
"convex_bind\<cdot>(convex_map\<cdot>f\<cdot>xs)\<cdot>g = convex_bind\<cdot>xs\<cdot>(\<Lambda> x. g\<cdot>(f\<cdot>x))" |
|
410 |
by (simp add: convex_map_def convex_bind_bind) |
|
411 |
||
412 |
lemma convex_map_bind: |
|
413 |
"convex_map\<cdot>f\<cdot>(convex_bind\<cdot>xs\<cdot>g) = convex_bind\<cdot>xs\<cdot>(\<Lambda> x. convex_map\<cdot>f\<cdot>(g\<cdot>x))" |
|
414 |
by (simp add: convex_map_def convex_bind_bind) |
|
415 |
||
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416 |
lemma ep_pair_convex_map: "ep_pair e p \<Longrightarrow> ep_pair (convex_map\<cdot>e) (convex_map\<cdot>p)" |
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|
417 |
apply default |
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|
418 |
apply (induct_tac x rule: convex_pd_induct, simp_all add: ep_pair.e_inverse) |
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|
419 |
apply (induct_tac y rule: convex_pd_induct) |
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|
420 |
apply (simp_all add: ep_pair.e_p_below monofun_cfun) |
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|
421 |
done |
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|
422 |
|
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|
423 |
lemma deflation_convex_map: "deflation d \<Longrightarrow> deflation (convex_map\<cdot>d)" |
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424 |
apply default |
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diff
changeset
|
425 |
apply (induct_tac x rule: convex_pd_induct, simp_all add: deflation.idem) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
426 |
apply (induct_tac x rule: convex_pd_induct) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
427 |
apply (simp_all add: deflation.below monofun_cfun) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
428 |
done |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
429 |
|
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
430 |
(* FIXME: long proof! *) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
431 |
lemma finite_deflation_convex_map: |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
432 |
assumes "finite_deflation d" shows "finite_deflation (convex_map\<cdot>d)" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
433 |
proof (rule finite_deflation_intro) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
434 |
interpret d: finite_deflation d by fact |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
435 |
have "deflation d" by fact |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
436 |
thus "deflation (convex_map\<cdot>d)" by (rule deflation_convex_map) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
437 |
have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
438 |
hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
439 |
by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
440 |
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
|
441 |
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
|
442 |
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
|
443 |
hence *: "finite (convex_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
|
444 |
hence "finite (range (\<lambda>xs. convex_map\<cdot>d\<cdot>xs))" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
445 |
apply (rule rev_finite_subset) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
446 |
apply clarsimp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
447 |
apply (induct_tac xs rule: convex_pd.principal_induct) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
448 |
apply (simp add: adm_mem_finite *) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
449 |
apply (rename_tac t, induct_tac t rule: pd_basis_induct) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
450 |
apply (simp only: convex_unit_Rep_compact_basis [symmetric] convex_map_unit) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
451 |
apply simp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
452 |
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
|
453 |
apply clarsimp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
454 |
apply (rule imageI) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
455 |
apply (rule vimageI2) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
456 |
apply (simp add: Rep_PDUnit) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
457 |
apply (rule range_eqI) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
458 |
apply (erule sym) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
459 |
apply (rule exI) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
460 |
apply (rule Abs_compact_basis_inverse [symmetric]) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
461 |
apply (simp add: d.compact) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
462 |
apply (simp only: convex_plus_principal [symmetric] convex_map_plus) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
463 |
apply clarsimp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
464 |
apply (rule imageI) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
465 |
apply (rule vimageI2) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
466 |
apply (simp add: Rep_PDPlus) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
467 |
done |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
468 |
thus "finite {xs. convex_map\<cdot>d\<cdot>xs = xs}" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
469 |
by (rule finite_range_imp_finite_fixes) |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
470 |
qed |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
471 |
|
41289
f655912ac235
minimize imports; move domain class instances for powerdomain types into Powerdomains.thy
huffman
parents:
41288
diff
changeset
|
472 |
subsection {* Convex powerdomain is bifinite *} |
39974
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
473 |
|
41286
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
474 |
lemma approx_chain_convex_map: |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
475 |
assumes "approx_chain a" |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
476 |
shows "approx_chain (\<lambda>i. convex_map\<cdot>(a i))" |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
477 |
using assms unfolding approx_chain_def |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
478 |
by (simp add: lub_APP convex_map_ID finite_deflation_convex_map) |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
479 |
|
41288
a19edebad961
powerdomain theories require class 'bifinite' instead of 'domain'
huffman
parents:
41287
diff
changeset
|
480 |
instance convex_pd :: (bifinite) bifinite |
41286
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
481 |
proof |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
482 |
show "\<exists>(a::nat \<Rightarrow> 'a convex_pd \<rightarrow> 'a convex_pd). approx_chain a" |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
483 |
using bifinite [where 'a='a] |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
484 |
by (fast intro!: approx_chain_convex_map) |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
485 |
qed |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41111
diff
changeset
|
486 |
|
39974
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
487 |
subsection {* Join *} |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
488 |
|
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
489 |
definition |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
490 |
convex_join :: "'a convex_pd convex_pd \<rightarrow> 'a convex_pd" where |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
491 |
"convex_join = (\<Lambda> xss. convex_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
492 |
|
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
493 |
lemma convex_join_unit [simp]: |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
494 |
"convex_join\<cdot>{xs}\<natural> = xs" |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
495 |
unfolding convex_join_def by simp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
496 |
|
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
497 |
lemma convex_join_plus [simp]: |
41399
ad093e4638e2
changed syntax of powerdomain binary union operators
huffman
parents:
41394
diff
changeset
|
498 |
"convex_join\<cdot>(xss \<union>\<natural> yss) = convex_join\<cdot>xss \<union>\<natural> convex_join\<cdot>yss" |
39974
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
499 |
unfolding convex_join_def by simp |
b525988432e9
major reorganization/simplification of HOLCF type classes:
huffman
parents:
39970
diff
changeset
|
500 |
|
40577 | 501 |
lemma convex_join_bottom [simp]: "convex_join\<cdot>\<bottom> = \<bottom>" |
502 |
unfolding convex_join_def by simp |
|
503 |
||
25904 | 504 |
lemma convex_join_map_unit: |
505 |
"convex_join\<cdot>(convex_map\<cdot>convex_unit\<cdot>xs) = xs" |
|
506 |
by (induct xs rule: convex_pd_induct, simp_all) |
|
507 |
||
508 |
lemma convex_join_map_join: |
|
509 |
"convex_join\<cdot>(convex_map\<cdot>convex_join\<cdot>xsss) = convex_join\<cdot>(convex_join\<cdot>xsss)" |
|
510 |
by (induct xsss rule: convex_pd_induct, simp_all) |
|
511 |
||
512 |
lemma convex_join_map_map: |
|
513 |
"convex_join\<cdot>(convex_map\<cdot>(convex_map\<cdot>f)\<cdot>xss) = |
|
514 |
convex_map\<cdot>f\<cdot>(convex_join\<cdot>xss)" |
|
515 |
by (induct xss rule: convex_pd_induct, simp_all) |
|
516 |
||
517 |
||
518 |
subsection {* Conversions to other powerdomains *} |
|
519 |
||
520 |
text {* Convex to upper *} |
|
521 |
||
522 |
lemma convex_le_imp_upper_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<sharp> u" |
|
523 |
unfolding convex_le_def by simp |
|
524 |
||
525 |
definition |
|
526 |
convex_to_upper :: "'a convex_pd \<rightarrow> 'a upper_pd" where |
|
41394
51c866d1b53b
rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents:
41289
diff
changeset
|
527 |
"convex_to_upper = convex_pd.extension upper_principal" |
25904 | 528 |
|
529 |
lemma convex_to_upper_principal [simp]: |
|
530 |
"convex_to_upper\<cdot>(convex_principal t) = upper_principal t" |
|
531 |
unfolding convex_to_upper_def |
|
41394
51c866d1b53b
rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents:
41289
diff
changeset
|
532 |
apply (rule convex_pd.extension_principal) |
27289 | 533 |
apply (rule upper_pd.principal_mono) |
25904 | 534 |
apply (erule convex_le_imp_upper_le) |
535 |
done |
|
536 |
||
537 |
lemma convex_to_upper_unit [simp]: |
|
26927 | 538 |
"convex_to_upper\<cdot>{x}\<natural> = {x}\<sharp>" |
27289 | 539 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 540 |
|
541 |
lemma convex_to_upper_plus [simp]: |
|
41399
ad093e4638e2
changed syntax of powerdomain binary union operators
huffman
parents:
41394
diff
changeset
|
542 |
"convex_to_upper\<cdot>(xs \<union>\<natural> ys) = convex_to_upper\<cdot>xs \<union>\<sharp> convex_to_upper\<cdot>ys" |
41402
b647212cee03
remove lemma ideal_completion.principal_induct2, use principal_induct twice instead
huffman
parents:
41399
diff
changeset
|
543 |
by (induct xs rule: convex_pd.principal_induct, simp, |
b647212cee03
remove lemma ideal_completion.principal_induct2, use principal_induct twice instead
huffman
parents:
41399
diff
changeset
|
544 |
induct ys rule: convex_pd.principal_induct, simp, simp) |
25904 | 545 |
|
27289 | 546 |
lemma convex_to_upper_bind [simp]: |
547 |
"convex_to_upper\<cdot>(convex_bind\<cdot>xs\<cdot>f) = |
|
548 |
upper_bind\<cdot>(convex_to_upper\<cdot>xs)\<cdot>(convex_to_upper oo f)" |
|
549 |
by (induct xs rule: convex_pd_induct, simp, simp, simp) |
|
550 |
||
551 |
lemma convex_to_upper_map [simp]: |
|
552 |
"convex_to_upper\<cdot>(convex_map\<cdot>f\<cdot>xs) = upper_map\<cdot>f\<cdot>(convex_to_upper\<cdot>xs)" |
|
553 |
by (simp add: convex_map_def upper_map_def cfcomp_LAM) |
|
554 |
||
555 |
lemma convex_to_upper_join [simp]: |
|
556 |
"convex_to_upper\<cdot>(convex_join\<cdot>xss) = |
|
557 |
upper_bind\<cdot>(convex_to_upper\<cdot>xss)\<cdot>convex_to_upper" |
|
558 |
by (simp add: convex_join_def upper_join_def cfcomp_LAM eta_cfun) |
|
559 |
||
25904 | 560 |
text {* Convex to lower *} |
561 |
||
562 |
lemma convex_le_imp_lower_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<flat> u" |
|
563 |
unfolding convex_le_def by simp |
|
564 |
||
565 |
definition |
|
566 |
convex_to_lower :: "'a convex_pd \<rightarrow> 'a lower_pd" where |
|
41394
51c866d1b53b
rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents:
41289
diff
changeset
|
567 |
"convex_to_lower = convex_pd.extension lower_principal" |
25904 | 568 |
|
569 |
lemma convex_to_lower_principal [simp]: |
|
570 |
"convex_to_lower\<cdot>(convex_principal t) = lower_principal t" |
|
571 |
unfolding convex_to_lower_def |
|
41394
51c866d1b53b
rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents:
41289
diff
changeset
|
572 |
apply (rule convex_pd.extension_principal) |
27289 | 573 |
apply (rule lower_pd.principal_mono) |
25904 | 574 |
apply (erule convex_le_imp_lower_le) |
575 |
done |
|
576 |
||
577 |
lemma convex_to_lower_unit [simp]: |
|
26927 | 578 |
"convex_to_lower\<cdot>{x}\<natural> = {x}\<flat>" |
27289 | 579 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 580 |
|
581 |
lemma convex_to_lower_plus [simp]: |
|
41399
ad093e4638e2
changed syntax of powerdomain binary union operators
huffman
parents:
41394
diff
changeset
|
582 |
"convex_to_lower\<cdot>(xs \<union>\<natural> ys) = convex_to_lower\<cdot>xs \<union>\<flat> convex_to_lower\<cdot>ys" |
41402
b647212cee03
remove lemma ideal_completion.principal_induct2, use principal_induct twice instead
huffman
parents:
41399
diff
changeset
|
583 |
by (induct xs rule: convex_pd.principal_induct, simp, |
b647212cee03
remove lemma ideal_completion.principal_induct2, use principal_induct twice instead
huffman
parents:
41399
diff
changeset
|
584 |
induct ys rule: convex_pd.principal_induct, simp, simp) |
25904 | 585 |
|
27289 | 586 |
lemma convex_to_lower_bind [simp]: |
587 |
"convex_to_lower\<cdot>(convex_bind\<cdot>xs\<cdot>f) = |
|
588 |
lower_bind\<cdot>(convex_to_lower\<cdot>xs)\<cdot>(convex_to_lower oo f)" |
|
589 |
by (induct xs rule: convex_pd_induct, simp, simp, simp) |
|
590 |
||
591 |
lemma convex_to_lower_map [simp]: |
|
592 |
"convex_to_lower\<cdot>(convex_map\<cdot>f\<cdot>xs) = lower_map\<cdot>f\<cdot>(convex_to_lower\<cdot>xs)" |
|
593 |
by (simp add: convex_map_def lower_map_def cfcomp_LAM) |
|
594 |
||
595 |
lemma convex_to_lower_join [simp]: |
|
596 |
"convex_to_lower\<cdot>(convex_join\<cdot>xss) = |
|
597 |
lower_bind\<cdot>(convex_to_lower\<cdot>xss)\<cdot>convex_to_lower" |
|
598 |
by (simp add: convex_join_def lower_join_def cfcomp_LAM eta_cfun) |
|
599 |
||
25904 | 600 |
text {* Ordering property *} |
601 |
||
31076
99fe356cbbc2
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huffman
parents:
30729
diff
changeset
|
602 |
lemma convex_pd_below_iff: |
25904 | 603 |
"(xs \<sqsubseteq> ys) = |
604 |
(convex_to_upper\<cdot>xs \<sqsubseteq> convex_to_upper\<cdot>ys \<and> |
|
605 |
convex_to_lower\<cdot>xs \<sqsubseteq> convex_to_lower\<cdot>ys)" |
|
39970
9023b897e67a
simplify proofs of powerdomain inequalities
Brian Huffman <brianh@cs.pdx.edu>
parents:
37770
diff
changeset
|
606 |
apply (induct xs rule: convex_pd.principal_induct, simp) |
9023b897e67a
simplify proofs of powerdomain inequalities
Brian Huffman <brianh@cs.pdx.edu>
parents:
37770
diff
changeset
|
607 |
apply (induct ys rule: convex_pd.principal_induct, simp) |
9023b897e67a
simplify proofs of powerdomain inequalities
Brian Huffman <brianh@cs.pdx.edu>
parents:
37770
diff
changeset
|
608 |
apply (simp add: convex_le_def) |
25904 | 609 |
done |
610 |
||
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
611 |
lemmas convex_plus_below_plus_iff = |
45606 | 612 |
convex_pd_below_iff [where xs="xs \<union>\<natural> ys" and ys="zs \<union>\<natural> ws"] |
613 |
for xs ys zs ws |
|
26927 | 614 |
|
31076
99fe356cbbc2
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huffman
parents:
30729
diff
changeset
|
615 |
lemmas convex_pd_below_simps = |
99fe356cbbc2
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huffman
parents:
30729
diff
changeset
|
616 |
convex_unit_below_plus_iff |
99fe356cbbc2
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huffman
parents:
30729
diff
changeset
|
617 |
convex_plus_below_unit_iff |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
618 |
convex_plus_below_plus_iff |
99fe356cbbc2
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huffman
parents:
30729
diff
changeset
|
619 |
convex_unit_below_iff |
26927 | 620 |
convex_to_upper_unit |
621 |
convex_to_upper_plus |
|
622 |
convex_to_lower_unit |
|
623 |
convex_to_lower_plus |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
624 |
upper_pd_below_simps |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
625 |
lower_pd_below_simps |
26927 | 626 |
|
25904 | 627 |
end |