src/HOLCF/LowerPD.thy
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(*  Title:      HOLCF/LowerPD.thy
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    Author:     Brian Huffman
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*)
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header {* Lower powerdomain *}
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theory LowerPD
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imports CompactBasis
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begin
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subsection {* Basis preorder *}
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definition
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  lower_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<flat>" 50) where
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  "lower_le = (\<lambda>u v. \<forall>x\<in>Rep_pd_basis u. \<exists>y\<in>Rep_pd_basis v. x \<sqsubseteq> y)"
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lemma lower_le_refl [simp]: "t \<le>\<flat> t"
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unfolding lower_le_def by fast
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lemma lower_le_trans: "\<lbrakk>t \<le>\<flat> u; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> t \<le>\<flat> v"
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unfolding lower_le_def
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apply (rule ballI)
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apply (drule (1) bspec, erule bexE)
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apply (drule (1) bspec, erule bexE)
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apply (erule rev_bexI)
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apply (erule (1) below_trans)
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done
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interpretation lower_le: preorder lower_le
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by (rule preorder.intro, rule lower_le_refl, rule lower_le_trans)
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lemma lower_le_minimal [simp]: "PDUnit compact_bot \<le>\<flat> t"
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unfolding lower_le_def Rep_PDUnit
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by (simp, rule Rep_pd_basis_nonempty [folded ex_in_conv])
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lemma PDUnit_lower_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<flat> PDUnit y"
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unfolding lower_le_def Rep_PDUnit by fast
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lemma PDPlus_lower_mono: "\<lbrakk>s \<le>\<flat> t; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<flat> PDPlus t v"
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unfolding lower_le_def Rep_PDPlus by fast
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lemma PDPlus_lower_le: "t \<le>\<flat> PDPlus t u"
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unfolding lower_le_def Rep_PDPlus by fast
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lemma lower_le_PDUnit_PDUnit_iff [simp]:
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  "(PDUnit a \<le>\<flat> PDUnit b) = (a \<sqsubseteq> b)"
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lemma lower_le_PDUnit_PDPlus_iff:
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  "(PDUnit a \<le>\<flat> PDPlus t u) = (PDUnit a \<le>\<flat> t \<or> PDUnit a \<le>\<flat> u)"
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unfolding lower_le_def Rep_PDPlus Rep_PDUnit by fast
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lemma lower_le_PDPlus_iff: "(PDPlus t u \<le>\<flat> v) = (t \<le>\<flat> v \<and> u \<le>\<flat> v)"
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unfolding lower_le_def Rep_PDPlus by fast
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lemma lower_le_induct [induct set: lower_le]:
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  assumes le: "t \<le>\<flat> u"
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  assumes 1: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)"
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  assumes 2: "\<And>t u a. P (PDUnit a) t \<Longrightarrow> P (PDUnit a) (PDPlus t u)"
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  assumes 3: "\<And>t u v. \<lbrakk>P t v; P u v\<rbrakk> \<Longrightarrow> P (PDPlus t u) v"
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  shows "P t u"
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using le
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apply (induct t arbitrary: u rule: pd_basis_induct)
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apply (erule rev_mp)
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apply (induct_tac u rule: pd_basis_induct)
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apply (simp add: 1)
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apply (simp add: lower_le_PDUnit_PDPlus_iff)
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apply (simp add: 2)
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apply (subst PDPlus_commute)
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apply (simp add: 2)
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apply (simp add: lower_le_PDPlus_iff 3)
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done
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subsection {* Type definition *}
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typedef (open) 'a lower_pd =
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  "{S::'a pd_basis set. lower_le.ideal S}"
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by (fast intro: lower_le.ideal_principal)
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instantiation lower_pd :: ("domain") below
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begin
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definition
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  "x \<sqsubseteq> y \<longleftrightarrow> Rep_lower_pd x \<subseteq> Rep_lower_pd y"
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instance ..
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end
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instance lower_pd :: ("domain") po
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using type_definition_lower_pd below_lower_pd_def
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by (rule lower_le.typedef_ideal_po)
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instance lower_pd :: ("domain") cpo
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using type_definition_lower_pd below_lower_pd_def
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by (rule lower_le.typedef_ideal_cpo)
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definition
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  lower_principal :: "'a pd_basis \<Rightarrow> 'a lower_pd" where
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  "lower_principal t = Abs_lower_pd {u. u \<le>\<flat> t}"
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interpretation lower_pd:
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  ideal_completion lower_le lower_principal Rep_lower_pd
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using type_definition_lower_pd below_lower_pd_def
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using lower_principal_def pd_basis_countable
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by (rule lower_le.typedef_ideal_completion)
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text {* Lower powerdomain is pointed *}
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lemma lower_pd_minimal: "lower_principal (PDUnit compact_bot) \<sqsubseteq> ys"
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by (induct ys rule: lower_pd.principal_induct, simp, simp)
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instance lower_pd :: ("domain") pcpo
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by intro_classes (fast intro: lower_pd_minimal)
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lemma inst_lower_pd_pcpo: "\<bottom> = lower_principal (PDUnit compact_bot)"
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by (rule lower_pd_minimal [THEN UU_I, symmetric])
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subsection {* Monadic unit and plus *}
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definition
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  lower_unit :: "'a \<rightarrow> 'a lower_pd" where
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  "lower_unit = compact_basis.basis_fun (\<lambda>a. lower_principal (PDUnit a))"
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definition
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  lower_plus :: "'a lower_pd \<rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd" where
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  "lower_plus = lower_pd.basis_fun (\<lambda>t. lower_pd.basis_fun (\<lambda>u.
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      lower_principal (PDPlus t u)))"
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abbreviation
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  lower_add :: "'a lower_pd \<Rightarrow> 'a lower_pd \<Rightarrow> 'a lower_pd"
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    (infixl "+\<flat>" 65) where
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  "xs +\<flat> ys == lower_plus\<cdot>xs\<cdot>ys"
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syntax
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  "_lower_pd" :: "args \<Rightarrow> 'a lower_pd" ("{_}\<flat>")
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translations
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  "{x,xs}\<flat>" == "{x}\<flat> +\<flat> {xs}\<flat>"
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  "{x}\<flat>" == "CONST lower_unit\<cdot>x"
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lemma lower_unit_Rep_compact_basis [simp]:
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  "{Rep_compact_basis a}\<flat> = lower_principal (PDUnit a)"
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unfolding lower_unit_def
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by (simp add: compact_basis.basis_fun_principal PDUnit_lower_mono)
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lemma lower_plus_principal [simp]:
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  "lower_principal t +\<flat> lower_principal u = lower_principal (PDPlus t u)"
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unfolding lower_plus_def
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by (simp add: lower_pd.basis_fun_principal
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    lower_pd.basis_fun_mono PDPlus_lower_mono)
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interpretation lower_add: semilattice lower_add proof
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  fix xs ys zs :: "'a lower_pd"
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  show "(xs +\<flat> ys) +\<flat> zs = xs +\<flat> (ys +\<flat> zs)"
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    apply (induct xs ys arbitrary: zs rule: lower_pd.principal_induct2, simp, simp)
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    apply (rule_tac x=zs in lower_pd.principal_induct, simp)
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    apply (simp add: PDPlus_assoc)
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    done
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  show "xs +\<flat> ys = ys +\<flat> xs"
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    apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp)
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    apply (simp add: PDPlus_commute)
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    done
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  show "xs +\<flat> xs = xs"
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    apply (induct xs rule: lower_pd.principal_induct, simp)
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    apply (simp add: PDPlus_absorb)
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    done
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qed
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lemmas lower_plus_assoc = lower_add.assoc
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lemmas lower_plus_commute = lower_add.commute
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lemmas lower_plus_absorb = lower_add.idem
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lemmas lower_plus_left_commute = lower_add.left_commute
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lemmas lower_plus_left_absorb = lower_add.left_idem
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text {* Useful for @{text "simp add: lower_plus_ac"} *}
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lemmas lower_plus_ac =
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  lower_plus_assoc lower_plus_commute lower_plus_left_commute
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text {* Useful for @{text "simp only: lower_plus_aci"} *}
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lemmas lower_plus_aci =
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  lower_plus_ac lower_plus_absorb lower_plus_left_absorb
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lemma lower_plus_below1: "xs \<sqsubseteq> xs +\<flat> ys"
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apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp)
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apply (simp add: PDPlus_lower_le)
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done
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lemma lower_plus_below2: "ys \<sqsubseteq> xs +\<flat> ys"
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by (subst lower_plus_commute, rule lower_plus_below1)
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lemma lower_plus_least: "\<lbrakk>xs \<sqsubseteq> zs; ys \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs +\<flat> ys \<sqsubseteq> zs"
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apply (subst lower_plus_absorb [of zs, symmetric])
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apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
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done
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lemma lower_plus_below_iff:
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  "xs +\<flat> ys \<sqsubseteq> zs \<longleftrightarrow> xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs"
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apply safe
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apply (erule below_trans [OF lower_plus_below1])
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apply (erule below_trans [OF lower_plus_below2])
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apply (erule (1) lower_plus_least)
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done
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lemma lower_unit_below_plus_iff:
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  "{x}\<flat> \<sqsubseteq> ys +\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs"
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apply (induct x rule: compact_basis.principal_induct, simp)
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apply (induct ys rule: lower_pd.principal_induct, simp)
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apply (induct zs rule: lower_pd.principal_induct, simp)
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apply (simp add: lower_le_PDUnit_PDPlus_iff)
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done
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lemma lower_unit_below_iff [simp]: "{x}\<flat> \<sqsubseteq> {y}\<flat> \<longleftrightarrow> x \<sqsubseteq> y"
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apply (induct x rule: compact_basis.principal_induct, simp)
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apply (induct y rule: compact_basis.principal_induct, simp)
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apply simp
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done
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lemmas lower_pd_below_simps =
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  lower_unit_below_iff
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  lower_plus_below_iff
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  lower_unit_below_plus_iff
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lemma lower_unit_eq_iff [simp]: "{x}\<flat> = {y}\<flat> \<longleftrightarrow> x = y"
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by (simp add: po_eq_conv)
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lemma lower_unit_strict [simp]: "{\<bottom>}\<flat> = \<bottom>"
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using lower_unit_Rep_compact_basis [of compact_bot]
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by (simp add: inst_lower_pd_pcpo)
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lemma lower_unit_bottom_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>"
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unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff)
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lemma lower_plus_bottom_iff [simp]:
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  "xs +\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>"
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apply safe
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apply (rule UU_I, erule subst, rule lower_plus_below1)
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apply (rule UU_I, erule subst, rule lower_plus_below2)
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apply (rule lower_plus_absorb)
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done
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lemma lower_plus_strict1 [simp]: "\<bottom> +\<flat> ys = ys"
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apply (rule below_antisym [OF _ lower_plus_below2])
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apply (simp add: lower_plus_least)
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done
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lemma lower_plus_strict2 [simp]: "xs +\<flat> \<bottom> = xs"
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apply (rule below_antisym [OF _ lower_plus_below1])
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apply (simp add: lower_plus_least)
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done
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lemma compact_lower_unit: "compact x \<Longrightarrow> compact {x}\<flat>"
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by (auto dest!: compact_basis.compact_imp_principal)
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lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x"
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apply (safe elim!: compact_lower_unit)
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apply (simp only: compact_def lower_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_lower_plus [simp]:
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  "\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<flat> ys)"
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by (auto dest!: lower_pd.compact_imp_principal)
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subsection {* Induction rules *}
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lemma lower_pd_induct1:
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  assumes P: "adm P"
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  assumes unit: "\<And>x. P {x}\<flat>"
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  assumes insert:
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    "\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> +\<flat> ys)"
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  shows "P (xs::'a lower_pd)"
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apply (induct xs rule: lower_pd.principal_induct, rule P)
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apply (induct_tac a rule: pd_basis_induct1)
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apply (simp only: lower_unit_Rep_compact_basis [symmetric])
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apply (rule unit)
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apply (simp only: lower_unit_Rep_compact_basis [symmetric]
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                  lower_plus_principal [symmetric])
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apply (erule insert [OF unit])
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done
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lemma lower_pd_induct
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  [case_names adm lower_unit lower_plus, induct type: lower_pd]:
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  assumes P: "adm P"
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  assumes unit: "\<And>x. P {x}\<flat>"
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  assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<flat> ys)"
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  shows "P (xs::'a lower_pd)"
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apply (induct xs rule: lower_pd.principal_induct, rule P)
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apply (induct_tac a rule: pd_basis_induct)
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apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit)
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apply (simp only: lower_plus_principal [symmetric] plus)
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done
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subsection {* Monadic bind *}
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definition
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  lower_bind_basis ::
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  "'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
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  "lower_bind_basis = fold_pd
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    (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
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    (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)"
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lemma ACI_lower_bind:
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  "class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)"
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huffman
parents:
diff changeset
   308
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
   309
apply (simp add: lower_plus_assoc)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   310
apply (simp add: lower_plus_commute)
29990
b11793ea15a3 avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents: 29672
diff changeset
   311
apply (simp add: eta_cfun)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   312
done
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   313
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   314
lemma lower_bind_basis_simps [simp]:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   315
  "lower_bind_basis (PDUnit a) =
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   316
    (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   317
  "lower_bind_basis (PDPlus t u) =
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   318
    (\<Lambda> f. lower_bind_basis t\<cdot>f +\<flat> lower_bind_basis u\<cdot>f)"
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   319
unfolding lower_bind_basis_def
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   320
apply -
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   321
apply (rule fold_pd_PDUnit [OF ACI_lower_bind])
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   322
apply (rule fold_pd_PDPlus [OF ACI_lower_bind])
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   323
done
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   324
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   325
lemma lower_bind_basis_mono:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   326
  "t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u"
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
   327
unfolding cfun_below_iff
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   328
apply (erule lower_le_induct, safe)
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   329
apply (simp add: monofun_cfun)
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
   330
apply (simp add: rev_below_trans [OF lower_plus_below1])
99fe356cbbc2 rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents: 30729
diff changeset
   331
apply (simp add: lower_plus_below_iff)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   332
done
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   333
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   334
definition
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   335
  lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   336
  "lower_bind = lower_pd.basis_fun lower_bind_basis"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   337
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   338
lemma lower_bind_principal [simp]:
8161f137b0e9 new theory of powerdomains
huffman
parents:
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   339
  "lower_bind\<cdot>(lower_principal t) = lower_bind_basis t"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   340
unfolding lower_bind_def
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   341
apply (rule lower_pd.basis_fun_principal)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   342
apply (erule lower_bind_basis_mono)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   343
done
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   344
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   345
lemma lower_bind_unit [simp]:
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   346
  "lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x"
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   347
by (induct x rule: compact_basis.principal_induct, simp, simp)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   348
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   349
lemma lower_bind_plus [simp]:
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   350
  "lower_bind\<cdot>(xs +\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f +\<flat> lower_bind\<cdot>ys\<cdot>f"
27289
c49d427867aa move lemmas into locales;
huffman
parents: 27267
diff changeset
   351
by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp)
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   352
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   353
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   354
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   355
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   356
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   357
subsection {* Map *}
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   358
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   359
definition
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   360
  lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   361
  "lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))"
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   362
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   363
lemma lower_map_unit [simp]:
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   364
  "lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>"
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   365
unfolding lower_map_def by simp
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   366
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   367
lemma lower_map_plus [simp]:
26927
8684b5240f11 rename locales;
huffman
parents: 26806
diff changeset
   368
  "lower_map\<cdot>f\<cdot>(xs +\<flat> ys) = lower_map\<cdot>f\<cdot>xs +\<flat> lower_map\<cdot>f\<cdot>ys"
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   369
unfolding lower_map_def by simp
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   370
40577
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   371
lemma lower_map_bottom [simp]: "lower_map\<cdot>f\<cdot>\<bottom> = {f\<cdot>\<bottom>}\<flat>"
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   372
unfolding lower_map_def by simp
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   373
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   374
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   375
by (induct xs rule: lower_pd_induct, simp_all)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   376
33808
31169fdc5ae7 add map_ID lemmas
huffman
parents: 33585
diff changeset
   377
lemma lower_map_ID: "lower_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
   378
by (simp add: cfun_eq_iff ID_def lower_map_ident)
33808
31169fdc5ae7 add map_ID lemmas
huffman
parents: 33585
diff changeset
   379
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   380
lemma lower_map_map:
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   381
  "lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   382
by (induct xs rule: lower_pd_induct, simp_all)
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   383
33585
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   384
lemma ep_pair_lower_map: "ep_pair e p \<Longrightarrow> ep_pair (lower_map\<cdot>e) (lower_map\<cdot>p)"
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   385
apply default
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   386
apply (induct_tac x rule: lower_pd_induct, simp_all add: ep_pair.e_inverse)
35901
12f09bf2c77f fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents: 34973
diff changeset
   387
apply (induct_tac y rule: lower_pd_induct)
12f09bf2c77f fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents: 34973
diff changeset
   388
apply (simp_all add: ep_pair.e_p_below monofun_cfun)
33585
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   389
done
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   390
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   391
lemma deflation_lower_map: "deflation d \<Longrightarrow> deflation (lower_map\<cdot>d)"
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   392
apply default
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   393
apply (induct_tac x rule: lower_pd_induct, simp_all add: deflation.idem)
35901
12f09bf2c77f fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents: 34973
diff changeset
   394
apply (induct_tac x rule: lower_pd_induct)
12f09bf2c77f fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents: 34973
diff changeset
   395
apply (simp_all add: deflation.below monofun_cfun)
33585
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   396
done
8d39394fe5cf ep_pair and deflation lemmas for powerdomain map functions
huffman
parents: 31076
diff changeset
   397
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   398
(* FIXME: long proof! *)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   399
lemma finite_deflation_lower_map:
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   400
  assumes "finite_deflation d" shows "finite_deflation (lower_map\<cdot>d)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   401
proof (rule finite_deflation_intro)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   402
  interpret d: finite_deflation d by fact
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   403
  have "deflation d" by fact
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   404
  thus "deflation (lower_map\<cdot>d)" by (rule deflation_lower_map)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   405
  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
   406
  hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   407
    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
   408
  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
   409
  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
   410
    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
   411
  hence *: "finite (lower_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   412
  hence "finite (range (\<lambda>xs. lower_map\<cdot>d\<cdot>xs))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   413
    apply (rule rev_finite_subset)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   414
    apply clarsimp
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   415
    apply (induct_tac xs rule: lower_pd.principal_induct)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   416
    apply (simp add: adm_mem_finite *)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   417
    apply (rename_tac t, induct_tac t rule: pd_basis_induct)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   418
    apply (simp only: lower_unit_Rep_compact_basis [symmetric] lower_map_unit)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   419
    apply simp
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   420
    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
   421
    apply clarsimp
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   422
    apply (rule imageI)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   423
    apply (rule vimageI2)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   424
    apply (simp add: Rep_PDUnit)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   425
    apply (rule range_eqI)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   426
    apply (erule sym)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   427
    apply (rule exI)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   428
    apply (rule Abs_compact_basis_inverse [symmetric])
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   429
    apply (simp add: d.compact)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   430
    apply (simp only: lower_plus_principal [symmetric] lower_map_plus)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   431
    apply clarsimp
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   432
    apply (rule imageI)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   433
    apply (rule vimageI2)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   434
    apply (simp add: Rep_PDPlus)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   435
    done
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   436
  thus "finite {xs. lower_map\<cdot>d\<cdot>xs = xs}"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   437
    by (rule finite_range_imp_finite_fixes)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   438
qed
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   439
40497
d2e876d6da8c rename class 'bifinite' to 'domain'
huffman
parents: 40494
diff changeset
   440
subsection {* Lower powerdomain is a domain *}
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   441
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   442
definition
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   443
  lower_approx :: "nat \<Rightarrow> udom lower_pd \<rightarrow> udom lower_pd"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   444
where
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   445
  "lower_approx = (\<lambda>i. lower_map\<cdot>(udom_approx i))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   446
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   447
lemma lower_approx: "approx_chain lower_approx"
40484
768f7e264e2b reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents: 40436
diff changeset
   448
using lower_map_ID finite_deflation_lower_map
768f7e264e2b reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents: 40436
diff changeset
   449
unfolding lower_approx_def by (rule approx_chain_lemma1)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   450
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   451
definition lower_defl :: "defl \<rightarrow> defl"
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   452
where "lower_defl = defl_fun1 lower_approx lower_map"
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   453
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   454
lemma cast_lower_defl:
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   455
  "cast\<cdot>(lower_defl\<cdot>A) =
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   456
    udom_emb lower_approx oo lower_map\<cdot>(cast\<cdot>A) oo udom_prj lower_approx"
40484
768f7e264e2b reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents: 40436
diff changeset
   457
using lower_approx finite_deflation_lower_map
768f7e264e2b reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents: 40436
diff changeset
   458
unfolding lower_defl_def by (rule cast_defl_fun1)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   459
40497
d2e876d6da8c rename class 'bifinite' to 'domain'
huffman
parents: 40494
diff changeset
   460
instantiation lower_pd :: ("domain") liftdomain
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   461
begin
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   462
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   463
definition
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   464
  "emb = udom_emb lower_approx oo lower_map\<cdot>emb"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   465
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   466
definition
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   467
  "prj = lower_map\<cdot>prj oo udom_prj lower_approx"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   468
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   469
definition
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   470
  "defl (t::'a lower_pd itself) = lower_defl\<cdot>DEFL('a)"
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   471
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   472
definition
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   473
  "(liftemb :: 'a lower_pd u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   474
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   475
definition
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   476
  "(liftprj :: udom \<rightarrow> 'a lower_pd u) = u_map\<cdot>prj oo udom_prj u_approx"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   477
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   478
definition
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   479
  "liftdefl (t::'a lower_pd itself) = u_defl\<cdot>DEFL('a lower_pd)"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   480
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   481
instance
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   482
using liftemb_lower_pd_def liftprj_lower_pd_def liftdefl_lower_pd_def
40494
db8a09daba7b add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents: 40491
diff changeset
   483
proof (rule liftdomain_class_intro)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   484
  show "ep_pair emb (prj :: udom \<rightarrow> 'a lower_pd)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   485
    unfolding emb_lower_pd_def prj_lower_pd_def
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   486
    using ep_pair_udom [OF lower_approx]
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   487
    by (intro ep_pair_comp ep_pair_lower_map ep_pair_emb_prj)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   488
next
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   489
  show "cast\<cdot>DEFL('a lower_pd) = emb oo (prj :: udom \<rightarrow> 'a lower_pd)"
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   490
    unfolding emb_lower_pd_def prj_lower_pd_def defl_lower_pd_def cast_lower_defl
40002
c5b5f7a3a3b1 new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents: 39989
diff changeset
   491
    by (simp add: cast_DEFL oo_def cfun_eq_iff lower_map_map)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   492
qed
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   493
25904
8161f137b0e9 new theory of powerdomains
huffman
parents:
diff changeset
   494
end
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   495
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   496
lemma DEFL_lower: "DEFL('a lower_pd) = lower_defl\<cdot>DEFL('a)"
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39986
diff changeset
   497
by (rule defl_lower_pd_def)
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   498
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   499
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   500
subsection {* Join *}
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   501
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   502
definition
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   503
  lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   504
  "lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   505
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   506
lemma lower_join_unit [simp]:
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   507
  "lower_join\<cdot>{xs}\<flat> = xs"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   508
unfolding lower_join_def by simp
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   509
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   510
lemma lower_join_plus [simp]:
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   511
  "lower_join\<cdot>(xss +\<flat> yss) = lower_join\<cdot>xss +\<flat> lower_join\<cdot>yss"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   512
unfolding lower_join_def by simp
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   513
40577
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   514
lemma lower_join_bottom [simp]: "lower_join\<cdot>\<bottom> = \<bottom>"
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   515
unfolding lower_join_def by simp
5c6225a1c2c0 add lemmas about powerdomains
huffman
parents: 40576
diff changeset
   516
39974
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   517
lemma lower_join_map_unit:
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   518
  "lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   519
by (induct xs rule: lower_pd_induct, simp_all)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   520
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   521
lemma lower_join_map_join:
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   522
  "lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   523
by (induct xsss rule: lower_pd_induct, simp_all)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   524
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   525
lemma lower_join_map_map:
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   526
  "lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) =
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   527
   lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)"
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   528
by (induct xss rule: lower_pd_induct, simp_all)
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   529
b525988432e9 major reorganization/simplification of HOLCF type classes:
huffman
parents: 39970
diff changeset
   530
end