--- a/src/HOLCF/ConvexPD.thy Sat Nov 27 14:34:54 2010 -0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,651 +0,0 @@
-(* Title: HOLCF/ConvexPD.thy
- Author: Brian Huffman
-*)
-
-header {* Convex powerdomain *}
-
-theory ConvexPD
-imports UpperPD LowerPD
-begin
-
-subsection {* Basis preorder *}
-
-definition
- convex_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<natural>" 50) where
- "convex_le = (\<lambda>u v. u \<le>\<sharp> v \<and> u \<le>\<flat> v)"
-
-lemma convex_le_refl [simp]: "t \<le>\<natural> t"
-unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl)
-
-lemma convex_le_trans: "\<lbrakk>t \<le>\<natural> u; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> t \<le>\<natural> v"
-unfolding convex_le_def by (fast intro: upper_le_trans lower_le_trans)
-
-interpretation convex_le: preorder convex_le
-by (rule preorder.intro, rule convex_le_refl, rule convex_le_trans)
-
-lemma upper_le_minimal [simp]: "PDUnit compact_bot \<le>\<natural> t"
-unfolding convex_le_def Rep_PDUnit by simp
-
-lemma PDUnit_convex_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<natural> PDUnit y"
-unfolding convex_le_def by (fast intro: PDUnit_upper_mono PDUnit_lower_mono)
-
-lemma PDPlus_convex_mono: "\<lbrakk>s \<le>\<natural> t; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<natural> PDPlus t v"
-unfolding convex_le_def by (fast intro: PDPlus_upper_mono PDPlus_lower_mono)
-
-lemma convex_le_PDUnit_PDUnit_iff [simp]:
- "(PDUnit a \<le>\<natural> PDUnit b) = (a \<sqsubseteq> b)"
-unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit by fast
-
-lemma convex_le_PDUnit_lemma1:
- "(PDUnit a \<le>\<natural> t) = (\<forall>b\<in>Rep_pd_basis t. a \<sqsubseteq> b)"
-unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit
-using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast
-
-lemma convex_le_PDUnit_PDPlus_iff [simp]:
- "(PDUnit a \<le>\<natural> PDPlus t u) = (PDUnit a \<le>\<natural> t \<and> PDUnit a \<le>\<natural> u)"
-unfolding convex_le_PDUnit_lemma1 Rep_PDPlus by fast
-
-lemma convex_le_PDUnit_lemma2:
- "(t \<le>\<natural> PDUnit b) = (\<forall>a\<in>Rep_pd_basis t. a \<sqsubseteq> b)"
-unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit
-using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast
-
-lemma convex_le_PDPlus_PDUnit_iff [simp]:
- "(PDPlus t u \<le>\<natural> PDUnit a) = (t \<le>\<natural> PDUnit a \<and> u \<le>\<natural> PDUnit a)"
-unfolding convex_le_PDUnit_lemma2 Rep_PDPlus by fast
-
-lemma convex_le_PDPlus_lemma:
- assumes z: "PDPlus t u \<le>\<natural> z"
- shows "\<exists>v w. z = PDPlus v w \<and> t \<le>\<natural> v \<and> u \<le>\<natural> w"
-proof (intro exI conjI)
- let ?A = "{b\<in>Rep_pd_basis z. \<exists>a\<in>Rep_pd_basis t. a \<sqsubseteq> b}"
- let ?B = "{b\<in>Rep_pd_basis z. \<exists>a\<in>Rep_pd_basis u. a \<sqsubseteq> b}"
- let ?v = "Abs_pd_basis ?A"
- let ?w = "Abs_pd_basis ?B"
- have Rep_v: "Rep_pd_basis ?v = ?A"
- apply (rule Abs_pd_basis_inverse)
- apply (rule Rep_pd_basis_nonempty [of t, folded ex_in_conv, THEN exE])
- apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify)
- apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE)
- apply (simp add: pd_basis_def)
- apply fast
- done
- have Rep_w: "Rep_pd_basis ?w = ?B"
- apply (rule Abs_pd_basis_inverse)
- apply (rule Rep_pd_basis_nonempty [of u, folded ex_in_conv, THEN exE])
- apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify)
- apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE)
- apply (simp add: pd_basis_def)
- apply fast
- done
- show "z = PDPlus ?v ?w"
- apply (insert z)
- apply (simp add: convex_le_def, erule conjE)
- apply (simp add: Rep_pd_basis_inject [symmetric] Rep_PDPlus)
- apply (simp add: Rep_v Rep_w)
- apply (rule equalityI)
- apply (rule subsetI)
- apply (simp only: upper_le_def)
- apply (drule (1) bspec, erule bexE)
- apply (simp add: Rep_PDPlus)
- apply fast
- apply fast
- done
- show "t \<le>\<natural> ?v" "u \<le>\<natural> ?w"
- apply (insert z)
- apply (simp_all add: convex_le_def upper_le_def lower_le_def Rep_PDPlus Rep_v Rep_w)
- apply fast+
- done
-qed
-
-lemma convex_le_induct [induct set: convex_le]:
- assumes le: "t \<le>\<natural> u"
- assumes 2: "\<And>t u v. \<lbrakk>P t u; P u v\<rbrakk> \<Longrightarrow> P t v"
- assumes 3: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)"
- assumes 4: "\<And>t u v w. \<lbrakk>P t v; P u w\<rbrakk> \<Longrightarrow> P (PDPlus t u) (PDPlus v w)"
- shows "P t u"
-using le apply (induct t arbitrary: u rule: pd_basis_induct)
-apply (erule rev_mp)
-apply (induct_tac u rule: pd_basis_induct1)
-apply (simp add: 3)
-apply (simp, clarify, rename_tac a b t)
-apply (subgoal_tac "P (PDPlus (PDUnit a) (PDUnit a)) (PDPlus (PDUnit b) t)")
-apply (simp add: PDPlus_absorb)
-apply (erule (1) 4 [OF 3])
-apply (drule convex_le_PDPlus_lemma, clarify)
-apply (simp add: 4)
-done
-
-
-subsection {* Type definition *}
-
-typedef (open) 'a convex_pd =
- "{S::'a pd_basis set. convex_le.ideal S}"
-by (fast intro: convex_le.ideal_principal)
-
-instantiation convex_pd :: ("domain") below
-begin
-
-definition
- "x \<sqsubseteq> y \<longleftrightarrow> Rep_convex_pd x \<subseteq> Rep_convex_pd y"
-
-instance ..
-end
-
-instance convex_pd :: ("domain") po
-using type_definition_convex_pd below_convex_pd_def
-by (rule convex_le.typedef_ideal_po)
-
-instance convex_pd :: ("domain") cpo
-using type_definition_convex_pd below_convex_pd_def
-by (rule convex_le.typedef_ideal_cpo)
-
-definition
- convex_principal :: "'a pd_basis \<Rightarrow> 'a convex_pd" where
- "convex_principal t = Abs_convex_pd {u. u \<le>\<natural> t}"
-
-interpretation convex_pd:
- ideal_completion convex_le convex_principal Rep_convex_pd
-using type_definition_convex_pd below_convex_pd_def
-using convex_principal_def pd_basis_countable
-by (rule convex_le.typedef_ideal_completion)
-
-text {* Convex powerdomain is pointed *}
-
-lemma convex_pd_minimal: "convex_principal (PDUnit compact_bot) \<sqsubseteq> ys"
-by (induct ys rule: convex_pd.principal_induct, simp, simp)
-
-instance convex_pd :: ("domain") pcpo
-by intro_classes (fast intro: convex_pd_minimal)
-
-lemma inst_convex_pd_pcpo: "\<bottom> = convex_principal (PDUnit compact_bot)"
-by (rule convex_pd_minimal [THEN UU_I, symmetric])
-
-
-subsection {* Monadic unit and plus *}
-
-definition
- convex_unit :: "'a \<rightarrow> 'a convex_pd" where
- "convex_unit = compact_basis.basis_fun (\<lambda>a. convex_principal (PDUnit a))"
-
-definition
- convex_plus :: "'a convex_pd \<rightarrow> 'a convex_pd \<rightarrow> 'a convex_pd" where
- "convex_plus = convex_pd.basis_fun (\<lambda>t. convex_pd.basis_fun (\<lambda>u.
- convex_principal (PDPlus t u)))"
-
-abbreviation
- convex_add :: "'a convex_pd \<Rightarrow> 'a convex_pd \<Rightarrow> 'a convex_pd"
- (infixl "+\<natural>" 65) where
- "xs +\<natural> ys == convex_plus\<cdot>xs\<cdot>ys"
-
-syntax
- "_convex_pd" :: "args \<Rightarrow> 'a convex_pd" ("{_}\<natural>")
-
-translations
- "{x,xs}\<natural>" == "{x}\<natural> +\<natural> {xs}\<natural>"
- "{x}\<natural>" == "CONST convex_unit\<cdot>x"
-
-lemma convex_unit_Rep_compact_basis [simp]:
- "{Rep_compact_basis a}\<natural> = convex_principal (PDUnit a)"
-unfolding convex_unit_def
-by (simp add: compact_basis.basis_fun_principal PDUnit_convex_mono)
-
-lemma convex_plus_principal [simp]:
- "convex_principal t +\<natural> convex_principal u = convex_principal (PDPlus t u)"
-unfolding convex_plus_def
-by (simp add: convex_pd.basis_fun_principal
- convex_pd.basis_fun_mono PDPlus_convex_mono)
-
-interpretation convex_add: semilattice convex_add proof
- fix xs ys zs :: "'a convex_pd"
- show "(xs +\<natural> ys) +\<natural> zs = xs +\<natural> (ys +\<natural> zs)"
- apply (induct xs ys arbitrary: zs rule: convex_pd.principal_induct2, simp, simp)
- apply (rule_tac x=zs in convex_pd.principal_induct, simp)
- apply (simp add: PDPlus_assoc)
- done
- show "xs +\<natural> ys = ys +\<natural> xs"
- apply (induct xs ys rule: convex_pd.principal_induct2, simp, simp)
- apply (simp add: PDPlus_commute)
- done
- show "xs +\<natural> xs = xs"
- apply (induct xs rule: convex_pd.principal_induct, simp)
- apply (simp add: PDPlus_absorb)
- done
-qed
-
-lemmas convex_plus_assoc = convex_add.assoc
-lemmas convex_plus_commute = convex_add.commute
-lemmas convex_plus_absorb = convex_add.idem
-lemmas convex_plus_left_commute = convex_add.left_commute
-lemmas convex_plus_left_absorb = convex_add.left_idem
-
-text {* Useful for @{text "simp add: convex_plus_ac"} *}
-lemmas convex_plus_ac =
- convex_plus_assoc convex_plus_commute convex_plus_left_commute
-
-text {* Useful for @{text "simp only: convex_plus_aci"} *}
-lemmas convex_plus_aci =
- convex_plus_ac convex_plus_absorb convex_plus_left_absorb
-
-lemma convex_unit_below_plus_iff [simp]:
- "{x}\<natural> \<sqsubseteq> ys +\<natural> zs \<longleftrightarrow> {x}\<natural> \<sqsubseteq> ys \<and> {x}\<natural> \<sqsubseteq> zs"
-apply (induct x rule: compact_basis.principal_induct, simp)
-apply (induct ys rule: convex_pd.principal_induct, simp)
-apply (induct zs rule: convex_pd.principal_induct, simp)
-apply simp
-done
-
-lemma convex_plus_below_unit_iff [simp]:
- "xs +\<natural> ys \<sqsubseteq> {z}\<natural> \<longleftrightarrow> xs \<sqsubseteq> {z}\<natural> \<and> ys \<sqsubseteq> {z}\<natural>"
-apply (induct xs rule: convex_pd.principal_induct, simp)
-apply (induct ys rule: convex_pd.principal_induct, simp)
-apply (induct z rule: compact_basis.principal_induct, simp)
-apply simp
-done
-
-lemma convex_unit_below_iff [simp]: "{x}\<natural> \<sqsubseteq> {y}\<natural> \<longleftrightarrow> x \<sqsubseteq> y"
-apply (induct x rule: compact_basis.principal_induct, simp)
-apply (induct y rule: compact_basis.principal_induct, simp)
-apply simp
-done
-
-lemma convex_unit_eq_iff [simp]: "{x}\<natural> = {y}\<natural> \<longleftrightarrow> x = y"
-unfolding po_eq_conv by simp
-
-lemma convex_unit_strict [simp]: "{\<bottom>}\<natural> = \<bottom>"
-using convex_unit_Rep_compact_basis [of compact_bot]
-by (simp add: inst_convex_pd_pcpo)
-
-lemma convex_unit_bottom_iff [simp]: "{x}\<natural> = \<bottom> \<longleftrightarrow> x = \<bottom>"
-unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)
-
-lemma compact_convex_unit: "compact x \<Longrightarrow> compact {x}\<natural>"
-by (auto dest!: compact_basis.compact_imp_principal)
-
-lemma compact_convex_unit_iff [simp]: "compact {x}\<natural> \<longleftrightarrow> compact x"
-apply (safe elim!: compact_convex_unit)
-apply (simp only: compact_def convex_unit_below_iff [symmetric])
-apply (erule adm_subst [OF cont_Rep_cfun2])
-done
-
-lemma compact_convex_plus [simp]:
- "\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<natural> ys)"
-by (auto dest!: convex_pd.compact_imp_principal)
-
-
-subsection {* Induction rules *}
-
-lemma convex_pd_induct1:
- assumes P: "adm P"
- assumes unit: "\<And>x. P {x}\<natural>"
- assumes insert: "\<And>x ys. \<lbrakk>P {x}\<natural>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<natural> +\<natural> ys)"
- shows "P (xs::'a convex_pd)"
-apply (induct xs rule: convex_pd.principal_induct, rule P)
-apply (induct_tac a rule: pd_basis_induct1)
-apply (simp only: convex_unit_Rep_compact_basis [symmetric])
-apply (rule unit)
-apply (simp only: convex_unit_Rep_compact_basis [symmetric]
- convex_plus_principal [symmetric])
-apply (erule insert [OF unit])
-done
-
-lemma convex_pd_induct
- [case_names adm convex_unit convex_plus, induct type: convex_pd]:
- assumes P: "adm P"
- assumes unit: "\<And>x. P {x}\<natural>"
- assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<natural> ys)"
- shows "P (xs::'a convex_pd)"
-apply (induct xs rule: convex_pd.principal_induct, rule P)
-apply (induct_tac a rule: pd_basis_induct)
-apply (simp only: convex_unit_Rep_compact_basis [symmetric] unit)
-apply (simp only: convex_plus_principal [symmetric] plus)
-done
-
-
-subsection {* Monadic bind *}
-
-definition
- convex_bind_basis ::
- "'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where
- "convex_bind_basis = fold_pd
- (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
- (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<natural> y\<cdot>f)"
-
-lemma ACI_convex_bind:
- "class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<natural> y\<cdot>f)"
-apply unfold_locales
-apply (simp add: convex_plus_assoc)
-apply (simp add: convex_plus_commute)
-apply (simp add: eta_cfun)
-done
-
-lemma convex_bind_basis_simps [simp]:
- "convex_bind_basis (PDUnit a) =
- (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
- "convex_bind_basis (PDPlus t u) =
- (\<Lambda> f. convex_bind_basis t\<cdot>f +\<natural> convex_bind_basis u\<cdot>f)"
-unfolding convex_bind_basis_def
-apply -
-apply (rule fold_pd_PDUnit [OF ACI_convex_bind])
-apply (rule fold_pd_PDPlus [OF ACI_convex_bind])
-done
-
-lemma convex_bind_basis_mono:
- "t \<le>\<natural> u \<Longrightarrow> convex_bind_basis t \<sqsubseteq> convex_bind_basis u"
-apply (erule convex_le_induct)
-apply (erule (1) below_trans)
-apply (simp add: monofun_LAM monofun_cfun)
-apply (simp add: monofun_LAM monofun_cfun)
-done
-
-definition
- convex_bind :: "'a convex_pd \<rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where
- "convex_bind = convex_pd.basis_fun convex_bind_basis"
-
-lemma convex_bind_principal [simp]:
- "convex_bind\<cdot>(convex_principal t) = convex_bind_basis t"
-unfolding convex_bind_def
-apply (rule convex_pd.basis_fun_principal)
-apply (erule convex_bind_basis_mono)
-done
-
-lemma convex_bind_unit [simp]:
- "convex_bind\<cdot>{x}\<natural>\<cdot>f = f\<cdot>x"
-by (induct x rule: compact_basis.principal_induct, simp, simp)
-
-lemma convex_bind_plus [simp]:
- "convex_bind\<cdot>(xs +\<natural> ys)\<cdot>f = convex_bind\<cdot>xs\<cdot>f +\<natural> convex_bind\<cdot>ys\<cdot>f"
-by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
-
-lemma convex_bind_strict [simp]: "convex_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
-unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit)
-
-lemma convex_bind_bind:
- "convex_bind\<cdot>(convex_bind\<cdot>xs\<cdot>f)\<cdot>g =
- convex_bind\<cdot>xs\<cdot>(\<Lambda> x. convex_bind\<cdot>(f\<cdot>x)\<cdot>g)"
-by (induct xs, simp_all)
-
-
-subsection {* Map *}
-
-definition
- convex_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a convex_pd \<rightarrow> 'b convex_pd" where
- "convex_map = (\<Lambda> f xs. convex_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<natural>))"
-
-lemma convex_map_unit [simp]:
- "convex_map\<cdot>f\<cdot>{x}\<natural> = {f\<cdot>x}\<natural>"
-unfolding convex_map_def by simp
-
-lemma convex_map_plus [simp]:
- "convex_map\<cdot>f\<cdot>(xs +\<natural> ys) = convex_map\<cdot>f\<cdot>xs +\<natural> convex_map\<cdot>f\<cdot>ys"
-unfolding convex_map_def by simp
-
-lemma convex_map_bottom [simp]: "convex_map\<cdot>f\<cdot>\<bottom> = {f\<cdot>\<bottom>}\<natural>"
-unfolding convex_map_def by simp
-
-lemma convex_map_ident: "convex_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
-by (induct xs rule: convex_pd_induct, simp_all)
-
-lemma convex_map_ID: "convex_map\<cdot>ID = ID"
-by (simp add: cfun_eq_iff ID_def convex_map_ident)
-
-lemma convex_map_map:
- "convex_map\<cdot>f\<cdot>(convex_map\<cdot>g\<cdot>xs) = convex_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
-by (induct xs rule: convex_pd_induct, simp_all)
-
-lemma ep_pair_convex_map: "ep_pair e p \<Longrightarrow> ep_pair (convex_map\<cdot>e) (convex_map\<cdot>p)"
-apply default
-apply (induct_tac x rule: convex_pd_induct, simp_all add: ep_pair.e_inverse)
-apply (induct_tac y rule: convex_pd_induct)
-apply (simp_all add: ep_pair.e_p_below monofun_cfun)
-done
-
-lemma deflation_convex_map: "deflation d \<Longrightarrow> deflation (convex_map\<cdot>d)"
-apply default
-apply (induct_tac x rule: convex_pd_induct, simp_all add: deflation.idem)
-apply (induct_tac x rule: convex_pd_induct)
-apply (simp_all add: deflation.below monofun_cfun)
-done
-
-(* FIXME: long proof! *)
-lemma finite_deflation_convex_map:
- assumes "finite_deflation d" shows "finite_deflation (convex_map\<cdot>d)"
-proof (rule finite_deflation_intro)
- interpret d: finite_deflation d by fact
- have "deflation d" by fact
- thus "deflation (convex_map\<cdot>d)" by (rule deflation_convex_map)
- have "finite (range (\<lambda>x. d\<cdot>x))" by (rule d.finite_range)
- hence "finite (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))"
- by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
- hence "finite (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x)))" by simp
- hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))"
- by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject)
- hence *: "finite (convex_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\<lambda>x. d\<cdot>x))))" by simp
- hence "finite (range (\<lambda>xs. convex_map\<cdot>d\<cdot>xs))"
- apply (rule rev_finite_subset)
- apply clarsimp
- apply (induct_tac xs rule: convex_pd.principal_induct)
- apply (simp add: adm_mem_finite *)
- apply (rename_tac t, induct_tac t rule: pd_basis_induct)
- apply (simp only: convex_unit_Rep_compact_basis [symmetric] convex_map_unit)
- apply simp
- apply (subgoal_tac "\<exists>b. d\<cdot>(Rep_compact_basis a) = Rep_compact_basis b")
- apply clarsimp
- apply (rule imageI)
- apply (rule vimageI2)
- apply (simp add: Rep_PDUnit)
- apply (rule range_eqI)
- apply (erule sym)
- apply (rule exI)
- apply (rule Abs_compact_basis_inverse [symmetric])
- apply (simp add: d.compact)
- apply (simp only: convex_plus_principal [symmetric] convex_map_plus)
- apply clarsimp
- apply (rule imageI)
- apply (rule vimageI2)
- apply (simp add: Rep_PDPlus)
- done
- thus "finite {xs. convex_map\<cdot>d\<cdot>xs = xs}"
- by (rule finite_range_imp_finite_fixes)
-qed
-
-subsection {* Convex powerdomain is a domain *}
-
-definition
- convex_approx :: "nat \<Rightarrow> udom convex_pd \<rightarrow> udom convex_pd"
-where
- "convex_approx = (\<lambda>i. convex_map\<cdot>(udom_approx i))"
-
-lemma convex_approx: "approx_chain convex_approx"
-using convex_map_ID finite_deflation_convex_map
-unfolding convex_approx_def by (rule approx_chain_lemma1)
-
-definition convex_defl :: "defl \<rightarrow> defl"
-where "convex_defl = defl_fun1 convex_approx convex_map"
-
-lemma cast_convex_defl:
- "cast\<cdot>(convex_defl\<cdot>A) =
- udom_emb convex_approx oo convex_map\<cdot>(cast\<cdot>A) oo udom_prj convex_approx"
-using convex_approx finite_deflation_convex_map
-unfolding convex_defl_def by (rule cast_defl_fun1)
-
-instantiation convex_pd :: ("domain") liftdomain
-begin
-
-definition
- "emb = udom_emb convex_approx oo convex_map\<cdot>emb"
-
-definition
- "prj = convex_map\<cdot>prj oo udom_prj convex_approx"
-
-definition
- "defl (t::'a convex_pd itself) = convex_defl\<cdot>DEFL('a)"
-
-definition
- "(liftemb :: 'a convex_pd u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
-
-definition
- "(liftprj :: udom \<rightarrow> 'a convex_pd u) = u_map\<cdot>prj oo udom_prj u_approx"
-
-definition
- "liftdefl (t::'a convex_pd itself) = u_defl\<cdot>DEFL('a convex_pd)"
-
-instance
-using liftemb_convex_pd_def liftprj_convex_pd_def liftdefl_convex_pd_def
-proof (rule liftdomain_class_intro)
- show "ep_pair emb (prj :: udom \<rightarrow> 'a convex_pd)"
- unfolding emb_convex_pd_def prj_convex_pd_def
- using ep_pair_udom [OF convex_approx]
- by (intro ep_pair_comp ep_pair_convex_map ep_pair_emb_prj)
-next
- show "cast\<cdot>DEFL('a convex_pd) = emb oo (prj :: udom \<rightarrow> 'a convex_pd)"
- unfolding emb_convex_pd_def prj_convex_pd_def defl_convex_pd_def cast_convex_defl
- by (simp add: cast_DEFL oo_def cfun_eq_iff convex_map_map)
-qed
-
-end
-
-text {* DEFL of type constructor = type combinator *}
-
-lemma DEFL_convex: "DEFL('a convex_pd) = convex_defl\<cdot>DEFL('a)"
-by (rule defl_convex_pd_def)
-
-
-subsection {* Join *}
-
-definition
- convex_join :: "'a convex_pd convex_pd \<rightarrow> 'a convex_pd" where
- "convex_join = (\<Lambda> xss. convex_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"
-
-lemma convex_join_unit [simp]:
- "convex_join\<cdot>{xs}\<natural> = xs"
-unfolding convex_join_def by simp
-
-lemma convex_join_plus [simp]:
- "convex_join\<cdot>(xss +\<natural> yss) = convex_join\<cdot>xss +\<natural> convex_join\<cdot>yss"
-unfolding convex_join_def by simp
-
-lemma convex_join_bottom [simp]: "convex_join\<cdot>\<bottom> = \<bottom>"
-unfolding convex_join_def by simp
-
-lemma convex_join_map_unit:
- "convex_join\<cdot>(convex_map\<cdot>convex_unit\<cdot>xs) = xs"
-by (induct xs rule: convex_pd_induct, simp_all)
-
-lemma convex_join_map_join:
- "convex_join\<cdot>(convex_map\<cdot>convex_join\<cdot>xsss) = convex_join\<cdot>(convex_join\<cdot>xsss)"
-by (induct xsss rule: convex_pd_induct, simp_all)
-
-lemma convex_join_map_map:
- "convex_join\<cdot>(convex_map\<cdot>(convex_map\<cdot>f)\<cdot>xss) =
- convex_map\<cdot>f\<cdot>(convex_join\<cdot>xss)"
-by (induct xss rule: convex_pd_induct, simp_all)
-
-
-subsection {* Conversions to other powerdomains *}
-
-text {* Convex to upper *}
-
-lemma convex_le_imp_upper_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<sharp> u"
-unfolding convex_le_def by simp
-
-definition
- convex_to_upper :: "'a convex_pd \<rightarrow> 'a upper_pd" where
- "convex_to_upper = convex_pd.basis_fun upper_principal"
-
-lemma convex_to_upper_principal [simp]:
- "convex_to_upper\<cdot>(convex_principal t) = upper_principal t"
-unfolding convex_to_upper_def
-apply (rule convex_pd.basis_fun_principal)
-apply (rule upper_pd.principal_mono)
-apply (erule convex_le_imp_upper_le)
-done
-
-lemma convex_to_upper_unit [simp]:
- "convex_to_upper\<cdot>{x}\<natural> = {x}\<sharp>"
-by (induct x rule: compact_basis.principal_induct, simp, simp)
-
-lemma convex_to_upper_plus [simp]:
- "convex_to_upper\<cdot>(xs +\<natural> ys) = convex_to_upper\<cdot>xs +\<sharp> convex_to_upper\<cdot>ys"
-by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
-
-lemma convex_to_upper_bind [simp]:
- "convex_to_upper\<cdot>(convex_bind\<cdot>xs\<cdot>f) =
- upper_bind\<cdot>(convex_to_upper\<cdot>xs)\<cdot>(convex_to_upper oo f)"
-by (induct xs rule: convex_pd_induct, simp, simp, simp)
-
-lemma convex_to_upper_map [simp]:
- "convex_to_upper\<cdot>(convex_map\<cdot>f\<cdot>xs) = upper_map\<cdot>f\<cdot>(convex_to_upper\<cdot>xs)"
-by (simp add: convex_map_def upper_map_def cfcomp_LAM)
-
-lemma convex_to_upper_join [simp]:
- "convex_to_upper\<cdot>(convex_join\<cdot>xss) =
- upper_bind\<cdot>(convex_to_upper\<cdot>xss)\<cdot>convex_to_upper"
-by (simp add: convex_join_def upper_join_def cfcomp_LAM eta_cfun)
-
-text {* Convex to lower *}
-
-lemma convex_le_imp_lower_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<flat> u"
-unfolding convex_le_def by simp
-
-definition
- convex_to_lower :: "'a convex_pd \<rightarrow> 'a lower_pd" where
- "convex_to_lower = convex_pd.basis_fun lower_principal"
-
-lemma convex_to_lower_principal [simp]:
- "convex_to_lower\<cdot>(convex_principal t) = lower_principal t"
-unfolding convex_to_lower_def
-apply (rule convex_pd.basis_fun_principal)
-apply (rule lower_pd.principal_mono)
-apply (erule convex_le_imp_lower_le)
-done
-
-lemma convex_to_lower_unit [simp]:
- "convex_to_lower\<cdot>{x}\<natural> = {x}\<flat>"
-by (induct x rule: compact_basis.principal_induct, simp, simp)
-
-lemma convex_to_lower_plus [simp]:
- "convex_to_lower\<cdot>(xs +\<natural> ys) = convex_to_lower\<cdot>xs +\<flat> convex_to_lower\<cdot>ys"
-by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
-
-lemma convex_to_lower_bind [simp]:
- "convex_to_lower\<cdot>(convex_bind\<cdot>xs\<cdot>f) =
- lower_bind\<cdot>(convex_to_lower\<cdot>xs)\<cdot>(convex_to_lower oo f)"
-by (induct xs rule: convex_pd_induct, simp, simp, simp)
-
-lemma convex_to_lower_map [simp]:
- "convex_to_lower\<cdot>(convex_map\<cdot>f\<cdot>xs) = lower_map\<cdot>f\<cdot>(convex_to_lower\<cdot>xs)"
-by (simp add: convex_map_def lower_map_def cfcomp_LAM)
-
-lemma convex_to_lower_join [simp]:
- "convex_to_lower\<cdot>(convex_join\<cdot>xss) =
- lower_bind\<cdot>(convex_to_lower\<cdot>xss)\<cdot>convex_to_lower"
-by (simp add: convex_join_def lower_join_def cfcomp_LAM eta_cfun)
-
-text {* Ordering property *}
-
-lemma convex_pd_below_iff:
- "(xs \<sqsubseteq> ys) =
- (convex_to_upper\<cdot>xs \<sqsubseteq> convex_to_upper\<cdot>ys \<and>
- convex_to_lower\<cdot>xs \<sqsubseteq> convex_to_lower\<cdot>ys)"
-apply (induct xs rule: convex_pd.principal_induct, simp)
-apply (induct ys rule: convex_pd.principal_induct, simp)
-apply (simp add: convex_le_def)
-done
-
-lemmas convex_plus_below_plus_iff =
- convex_pd_below_iff [where xs="xs +\<natural> ys" and ys="zs +\<natural> ws", standard]
-
-lemmas convex_pd_below_simps =
- convex_unit_below_plus_iff
- convex_plus_below_unit_iff
- convex_plus_below_plus_iff
- convex_unit_below_iff
- convex_to_upper_unit
- convex_to_upper_plus
- convex_to_lower_unit
- convex_to_lower_plus
- upper_pd_below_simps
- lower_pd_below_simps
-
-end