src/HOL/HOLCF/ConvexPD.thy
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(*  Title:      HOL/HOLCF/ConvexPD.thy
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    Author:     Brian Huffman
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*)
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header {* 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 (open) '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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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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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
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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
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lemma convex_unit_eq_iff [simp]: "{x}\<natural> = {y}\<natural> \<longleftrightarrow> x = y"
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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)
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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)
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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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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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subsection {* Induction rules *}
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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)"
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  shows "P (xs::'a convex_pd)"
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apply (induct xs rule: convex_pd.principal_induct, rule P)
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apply (induct_tac a rule: pd_basis_induct1)
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apply (simp only: convex_unit_Rep_compact_basis [symmetric])
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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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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)
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   303
apply (induct_tac a rule: pd_basis_induct)
25904
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   304
apply (simp only: convex_unit_Rep_compact_basis [symmetric] unit)
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   305
apply (simp only: convex_plus_principal [symmetric] plus)
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parents:
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   306
done
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   307
8161f137b0e9 new theory of powerdomains
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parents:
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   308
8161f137b0e9 new theory of powerdomains
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parents:
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   309
subsection {* Monadic bind *}
8161f137b0e9 new theory of powerdomains
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   310
8161f137b0e9 new theory of powerdomains
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   311
definition
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parents:
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   312
  convex_bind_basis ::
8161f137b0e9 new theory of powerdomains
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parents:
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   313
  "'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where
8161f137b0e9 new theory of powerdomains
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parents:
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   314
  "convex_bind_basis = fold_pd
8161f137b0e9 new theory of powerdomains
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parents:
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   315
    (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
41399
ad093e4638e2 changed syntax of powerdomain binary union operators
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parents: 41394
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   316
    (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<natural> y\<cdot>f)"
25904
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   317
26927
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parents: 26806
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   318
lemma ACI_convex_bind:
41399
ad093e4638e2 changed syntax of powerdomain binary union operators
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parents: 41394
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   319
  "class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<natural> y\<cdot>f)"
25904
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parents:
diff changeset
   320
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
   321
apply (simp add: convex_plus_assoc)
25904
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   322
apply (simp add: convex_plus_commute)
29990
b11793ea15a3 avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents: 29672
diff changeset
   323
apply (simp add: eta_cfun)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   324
done
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   325
8161f137b0e9 new theory of powerdomains
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parents:
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   326
lemma convex_bind_basis_simps [simp]:
8161f137b0e9 new theory of powerdomains
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parents:
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   327
  "convex_bind_basis (PDUnit a) =
8161f137b0e9 new theory of powerdomains
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parents:
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   328
    (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   329
  "convex_bind_basis (PDPlus t u) =
41399
ad093e4638e2 changed syntax of powerdomain binary union operators
huffman
parents: 41394
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   330
    (\<Lambda> f. convex_bind_basis t\<cdot>f \<union>\<natural> convex_bind_basis u\<cdot>f)"
25904
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   331
unfolding convex_bind_basis_def
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   332
apply -
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   333
apply (rule fold_pd_PDUnit [OF ACI_convex_bind])
8684b5240f11 rename locales;
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parents: 26806
diff changeset
   334
apply (rule fold_pd_PDPlus [OF ACI_convex_bind])
25904
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parents:
diff changeset
   335
done
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   336
8161f137b0e9 new theory of powerdomains
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parents:
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   337
lemma convex_bind_basis_mono:
8161f137b0e9 new theory of powerdomains
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parents:
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   338
  "t \<le>\<natural> u \<Longrightarrow> convex_bind_basis t \<sqsubseteq> convex_bind_basis u"
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   339
apply (erule convex_le_induct)
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
   340
apply (erule (1) below_trans)
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
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   341
apply (simp add: monofun_LAM monofun_cfun)
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   342
apply (simp add: monofun_LAM monofun_cfun)
25904
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huffman
parents:
diff changeset
   343
done
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   344
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   345
definition
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   346
  convex_bind :: "'a convex_pd \<rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41289
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   347
  "convex_bind = convex_pd.extension convex_bind_basis"
25904
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parents:
diff changeset
   348
41036
4acbacd6c5bc add set-union-like syntax for powerdomain bind operators
huffman
parents: 40888
diff changeset
   349
syntax
4acbacd6c5bc add set-union-like syntax for powerdomain bind operators
huffman
parents: 40888
diff changeset
   350
  "_convex_bind" :: "[logic, logic, logic] \<Rightarrow> logic"
4acbacd6c5bc add set-union-like syntax for powerdomain bind operators
huffman
parents: 40888
diff changeset
   351
    ("(3\<Union>\<natural>_\<in>_./ _)" [0, 0, 10] 10)
4acbacd6c5bc add set-union-like syntax for powerdomain bind operators
huffman
parents: 40888
diff changeset
   352
4acbacd6c5bc add set-union-like syntax for powerdomain bind operators
huffman
parents: 40888
diff changeset
   353
translations
4acbacd6c5bc add set-union-like syntax for powerdomain bind operators
huffman
parents: 40888
diff changeset
   354
  "\<Union>\<natural>x\<in>xs. e" == "CONST convex_bind\<cdot>xs\<cdot>(\<Lambda> x. e)"
4acbacd6c5bc add set-union-like syntax for powerdomain bind operators
huffman
parents: 40888
diff changeset
   355
25904
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huffman
parents:
diff changeset
   356
lemma convex_bind_principal [simp]:
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parents:
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   357
  "convex_bind\<cdot>(convex_principal t) = convex_bind_basis t"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   358
unfolding convex_bind_def
41394
51c866d1b53b rename function ideal_completion.basis_fun to ideal_completion.extension
huffman
parents: 41289
diff changeset
   359
apply (rule convex_pd.extension_principal)
25904
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huffman
parents:
diff changeset
   360
apply (erule convex_bind_basis_mono)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   361
done
8161f137b0e9 new theory of powerdomains
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parents:
diff changeset
   362
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   363
lemma convex_bind_unit [simp]:
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   364
  "convex_bind\<cdot>{x}\<natural>\<cdot>f = f\<cdot>x"
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   365
by (induct x rule: compact_basis.principal_induct, simp, simp)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   366
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   367
lemma convex_bind_plus [simp]:
41399
ad093e4638e2 changed syntax of powerdomain binary union operators
huffman
parents: 41394
diff changeset
   368
  "convex_bind\<cdot>(xs \<union>\<natural> ys)\<cdot>f = convex_bind\<cdot>xs\<cdot>f \<union>\<natural> convex_bind\<cdot>ys\<cdot>f"
41402
b647212cee03 remove lemma ideal_completion.principal_induct2, use principal_induct twice instead
huffman
parents: 41399
diff changeset
   369
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
   370
    induct ys rule: convex_pd.principal_induct, simp, simp)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   371
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   372
lemma convex_bind_strict [simp]: "convex_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   373
unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   374
40589
0e77e45d2ffc add bind_bind rules for powerdomains
huffman
parents: 40577
diff changeset
   375
lemma convex_bind_bind:
0e77e45d2ffc add bind_bind rules for powerdomains
huffman
parents: 40577
diff changeset
   376
  "convex_bind\<cdot>(convex_bind\<cdot>xs\<cdot>f)\<cdot>g =
0e77e45d2ffc add bind_bind rules for powerdomains
huffman
parents: 40577
diff changeset
   377
    convex_bind\<cdot>xs\<cdot>(\<Lambda> x. convex_bind\<cdot>(f\<cdot>x)\<cdot>g)"
0e77e45d2ffc add bind_bind rules for powerdomains
huffman
parents: 40577
diff changeset
   378
by (induct xs, simp_all)
0e77e45d2ffc add bind_bind rules for powerdomains
huffman
parents: 40577
diff changeset
   379
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   380
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   381
subsection {* Map *}
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   382
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   383
definition
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   384
  convex_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a convex_pd \<rightarrow> 'b convex_pd" where
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   385
  "convex_map = (\<Lambda> f xs. convex_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<natural>))"
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   386
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   387
lemma convex_map_unit [simp]:
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   388
  "convex_map\<cdot>f\<cdot>{x}\<natural> = {f\<cdot>x}\<natural>"
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   389
unfolding convex_map_def by simp
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   390
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   391
lemma convex_map_plus [simp]:
41399
ad093e4638e2 changed syntax of powerdomain binary union operators
huffman
parents: 41394
diff changeset
   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
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   393
unfolding convex_map_def by simp
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   394
40577
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   395
lemma convex_map_bottom [simp]: "convex_map\<cdot>f\<cdot>\<bottom> = {f\<cdot>\<bottom>}\<natural>"
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   396
unfolding convex_map_def by simp
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   397
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   398
lemma convex_map_ident: "convex_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   399
by (induct xs rule: convex_pd_induct, simp_all)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   400
33808
31169fdc5ae7 add map_ID lemmas
huffman
parents: 33585
diff changeset
   401
lemma convex_map_ID: "convex_map\<cdot>ID = ID"
40002
c5b5f7a3a3b1 new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents: 39989
diff changeset
   402
by (simp add: cfun_eq_iff ID_def convex_map_ident)
33808
31169fdc5ae7 add map_ID lemmas
huffman
parents: 33585
diff changeset
   403
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   404
lemma convex_map_map:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   405
  "convex_map\<cdot>f\<cdot>(convex_map\<cdot>g\<cdot>xs) = convex_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   406
by (induct xs rule: convex_pd_induct, simp_all)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   407
41110
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   408
lemma convex_bind_map:
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   409
  "convex_bind\<cdot>(convex_map\<cdot>f\<cdot>xs)\<cdot>g = convex_bind\<cdot>xs\<cdot>(\<Lambda> x. g\<cdot>(f\<cdot>x))"
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   410
by (simp add: convex_map_def convex_bind_bind)
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   411
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   412
lemma convex_map_bind:
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   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))"
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   414
by (simp add: convex_map_def convex_bind_bind)
32099ee71a2f new powerdomain lemmas
huffman
parents: 41036
diff changeset
   415
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   416
lemma ep_pair_convex_map: "ep_pair e p \<Longrightarrow> ep_pair (convex_map\<cdot>e) (convex_map\<cdot>p)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   417
apply default
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   418
apply (induct_tac x rule: convex_pd_induct, simp_all add: ep_pair.e_inverse)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   419
apply (induct_tac y rule: convex_pd_induct)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   420
apply (simp_all add: ep_pair.e_p_below monofun_cfun)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   421
done
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   422
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   423
lemma deflation_convex_map: "deflation d \<Longrightarrow> deflation (convex_map\<cdot>d)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   424
apply default
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
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
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   501
lemma convex_join_bottom [simp]: "convex_join\<cdot>\<bottom> = \<bottom>"
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   502
unfolding convex_join_def by simp
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   503
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   504
lemma convex_join_map_unit:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   505
  "convex_join\<cdot>(convex_map\<cdot>convex_unit\<cdot>xs) = xs"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   506
by (induct xs rule: convex_pd_induct, simp_all)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   507
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   508
lemma convex_join_map_join:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   509
  "convex_join\<cdot>(convex_map\<cdot>convex_join\<cdot>xsss) = convex_join\<cdot>(convex_join\<cdot>xsss)"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   510
by (induct xsss rule: convex_pd_induct, simp_all)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   511
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   512
lemma convex_join_map_map:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   513
  "convex_join\<cdot>(convex_map\<cdot>(convex_map\<cdot>f)\<cdot>xss) =
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   514
   convex_map\<cdot>f\<cdot>(convex_join\<cdot>xss)"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   515
by (induct xss rule: convex_pd_induct, simp_all)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   516
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   517
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   518
subsection {* Conversions to other powerdomains *}
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   519
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   520
text {* Convex to upper *}
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   521
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   522
lemma convex_le_imp_upper_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<sharp> u"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   523
unfolding convex_le_def by simp
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   524
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   525
definition
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   528
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   529
lemma convex_to_upper_principal [simp]:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   530
  "convex_to_upper\<cdot>(convex_principal t) = upper_principal t"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   533
apply (rule upper_pd.principal_mono)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   534
apply (erule convex_le_imp_upper_le)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   535
done
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   536
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   537
lemma convex_to_upper_unit [simp]:
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   538
  "convex_to_upper\<cdot>{x}\<natural> = {x}\<sharp>"
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   539
by (induct x rule: compact_basis.principal_induct, simp, simp)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   540
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   545
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   546
lemma convex_to_upper_bind [simp]:
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   547
  "convex_to_upper\<cdot>(convex_bind\<cdot>xs\<cdot>f) =
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   548
    upper_bind\<cdot>(convex_to_upper\<cdot>xs)\<cdot>(convex_to_upper oo f)"
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   549
by (induct xs rule: convex_pd_induct, simp, simp, simp)
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   550
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   551
lemma convex_to_upper_map [simp]:
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   552
  "convex_to_upper\<cdot>(convex_map\<cdot>f\<cdot>xs) = upper_map\<cdot>f\<cdot>(convex_to_upper\<cdot>xs)"
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   553
by (simp add: convex_map_def upper_map_def cfcomp_LAM)
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   554
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   555
lemma convex_to_upper_join [simp]:
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   556
  "convex_to_upper\<cdot>(convex_join\<cdot>xss) =
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   557
    upper_bind\<cdot>(convex_to_upper\<cdot>xss)\<cdot>convex_to_upper"
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   558
by (simp add: convex_join_def upper_join_def cfcomp_LAM eta_cfun)
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   559
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   560
text {* Convex to lower *}
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   561
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   562
lemma convex_le_imp_lower_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<flat> u"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   563
unfolding convex_le_def by simp
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   564
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   565
definition
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   568
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   569
lemma convex_to_lower_principal [simp]:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   570
  "convex_to_lower\<cdot>(convex_principal t) = lower_principal t"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   573
apply (rule lower_pd.principal_mono)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   574
apply (erule convex_le_imp_lower_le)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   575
done
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   576
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   577
lemma convex_to_lower_unit [simp]:
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   578
  "convex_to_lower\<cdot>{x}\<natural> = {x}\<flat>"
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   579
by (induct x rule: compact_basis.principal_induct, simp, simp)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   580
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   585
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   586
lemma convex_to_lower_bind [simp]:
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   587
  "convex_to_lower\<cdot>(convex_bind\<cdot>xs\<cdot>f) =
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   588
    lower_bind\<cdot>(convex_to_lower\<cdot>xs)\<cdot>(convex_to_lower oo f)"
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   589
by (induct xs rule: convex_pd_induct, simp, simp, simp)
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   590
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   591
lemma convex_to_lower_map [simp]:
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   592
  "convex_to_lower\<cdot>(convex_map\<cdot>f\<cdot>xs) = lower_map\<cdot>f\<cdot>(convex_to_lower\<cdot>xs)"
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   593
by (simp add: convex_map_def lower_map_def cfcomp_LAM)
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   594
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   595
lemma convex_to_lower_join [simp]:
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   596
  "convex_to_lower\<cdot>(convex_join\<cdot>xss) =
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   597
    lower_bind\<cdot>(convex_to_lower\<cdot>xss)\<cdot>convex_to_lower"
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   598
by (simp add: convex_join_def lower_join_def cfcomp_LAM eta_cfun)
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   599
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   600
text {* Ordering property *}
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   601
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
   602
lemma convex_pd_below_iff:
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   603
  "(xs \<sqsubseteq> ys) =
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   604
    (convex_to_upper\<cdot>xs \<sqsubseteq> convex_to_upper\<cdot>ys \<and>
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   609
done
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   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
b1e1508643b1 eliminated obsolete "standard";
wenzelm
parents: 42151
diff changeset
   612
  convex_pd_below_iff [where xs="xs \<union>\<natural> ys" and ys="zs \<union>\<natural> ws"]
b1e1508643b1 eliminated obsolete "standard";
wenzelm
parents: 42151
diff changeset
   613
  for xs ys zs ws
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   614
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
   615
lemmas convex_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
   616
  convex_unit_below_plus_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
   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 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 30729
diff changeset
   619
  convex_unit_below_iff
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   620
  convex_to_upper_unit
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   621
  convex_to_upper_plus
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   622
  convex_to_lower_unit
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   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
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   626
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   627
end