(* Title: HOL/HOLCF/ConvexPD.thy
Author: Brian Huffman
*)
section \<open>Convex powerdomain\<close>
theory ConvexPD
imports UpperPD LowerPD
begin
subsection \<open>Basis preorder\<close>
definition
convex_le :: "'a pd_basis \ 'a pd_basis \ bool" (infix "\\" 50) where
"convex_le = (\u v. u \\ v \ u \\ v)"
lemma convex_le_refl [simp]: "t \\ t"
unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl)
lemma convex_le_trans: "\t \\ u; u \\ v\ \ t \\ 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 \\ t"
unfolding convex_le_def Rep_PDUnit by simp
lemma PDUnit_convex_mono: "x \ y \ PDUnit x \\ PDUnit y"
unfolding convex_le_def by (fast intro: PDUnit_upper_mono PDUnit_lower_mono)
lemma PDPlus_convex_mono: "\s \\ t; u \\ v\ \ PDPlus s u \\ 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 \\ PDUnit b) = (a \ b)"
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit by fast
lemma convex_le_PDUnit_lemma1:
"(PDUnit a \\ t) = (\b\Rep_pd_basis t. a \ 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 \\ PDPlus t u) = (PDUnit a \\ t \ PDUnit a \\ u)"
unfolding convex_le_PDUnit_lemma1 Rep_PDPlus by fast
lemma convex_le_PDUnit_lemma2:
"(t \\ PDUnit b) = (\a\Rep_pd_basis t. a \ 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 \\ PDUnit a) = (t \\ PDUnit a \ u \\ PDUnit a)"
unfolding convex_le_PDUnit_lemma2 Rep_PDPlus by fast
lemma convex_le_PDPlus_lemma:
assumes z: "PDPlus t u \\ z"
shows "\v w. z = PDPlus v w \ t \\ v \ u \\ w"
proof (intro exI conjI)
let ?A = "{b\Rep_pd_basis z. \a\Rep_pd_basis t. a \ b}"
let ?B = "{b\Rep_pd_basis z. \a\Rep_pd_basis u. a \ 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 \\ ?v" "u \\ ?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 \\ u"
assumes 2: "\t u v. \P t u; P u v\ \ P t v"
assumes 3: "\a b. a \ b \ P (PDUnit a) (PDUnit b)"
assumes 4: "\t u v w. \P t v; P u w\ \ 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 \<open>Type definition\<close>
typedef 'a convex_pd ("('(_')\)") =
"{S::'a pd_basis set. convex_le.ideal S}"
by (rule convex_le.ex_ideal)
instantiation convex_pd :: (bifinite) below
begin
definition
"x \ y \ Rep_convex_pd x \ Rep_convex_pd y"
instance ..
end
instance convex_pd :: (bifinite) po
using type_definition_convex_pd below_convex_pd_def
by (rule convex_le.typedef_ideal_po)
instance convex_pd :: (bifinite) cpo
using type_definition_convex_pd below_convex_pd_def
by (rule convex_le.typedef_ideal_cpo)
definition
convex_principal :: "'a pd_basis \ 'a convex_pd" where
"convex_principal t = Abs_convex_pd {u. u \\ 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 \<open>Convex powerdomain is pointed\<close>
lemma convex_pd_minimal: "convex_principal (PDUnit compact_bot) \ ys"
by (induct ys rule: convex_pd.principal_induct, simp, simp)
instance convex_pd :: (bifinite) pcpo
by intro_classes (fast intro: convex_pd_minimal)
lemma inst_convex_pd_pcpo: "\ = convex_principal (PDUnit compact_bot)"
by (rule convex_pd_minimal [THEN bottomI, symmetric])
subsection \<open>Monadic unit and plus\<close>
definition
convex_unit :: "'a \ 'a convex_pd" where
"convex_unit = compact_basis.extension (\a. convex_principal (PDUnit a))"
definition
convex_plus :: "'a convex_pd \ 'a convex_pd \ 'a convex_pd" where
"convex_plus = convex_pd.extension (\t. convex_pd.extension (\u.
convex_principal (PDPlus t u)))"
abbreviation
convex_add :: "'a convex_pd \ 'a convex_pd \ 'a convex_pd"
(infixl "\\" 65) where
"xs \\ ys == convex_plus\xs\ys"
syntax
"_convex_pd" :: "args \ logic" ("{_}\")
translations
"{x,xs}\" == "{x}\ \\ {xs}\"
"{x}\" == "CONST convex_unit\x"
lemma convex_unit_Rep_compact_basis [simp]:
"{Rep_compact_basis a}\ = convex_principal (PDUnit a)"
unfolding convex_unit_def
by (simp add: compact_basis.extension_principal PDUnit_convex_mono)
lemma convex_plus_principal [simp]:
"convex_principal t \\ convex_principal u = convex_principal (PDPlus t u)"
unfolding convex_plus_def
by (simp add: convex_pd.extension_principal
convex_pd.extension_mono PDPlus_convex_mono)
interpretation convex_add: semilattice convex_add proof
fix xs ys zs :: "'a convex_pd"
show "(xs \\ ys) \\ zs = xs \\ (ys \\ zs)"
apply (induct xs rule: convex_pd.principal_induct, simp)
apply (induct ys rule: convex_pd.principal_induct, simp)
apply (induct zs rule: convex_pd.principal_induct, simp)
apply (simp add: PDPlus_assoc)
done
show "xs \\ ys = ys \\ xs"
apply (induct xs rule: convex_pd.principal_induct, simp)
apply (induct ys rule: convex_pd.principal_induct, simp)
apply (simp add: PDPlus_commute)
done
show "xs \\ 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 \<open>Useful for \<open>simp add: convex_plus_ac\<close>\<close>
lemmas convex_plus_ac =
convex_plus_assoc convex_plus_commute convex_plus_left_commute
text \<open>Useful for \<open>simp only: convex_plus_aci\<close>\<close>
lemmas convex_plus_aci =
convex_plus_ac convex_plus_absorb convex_plus_left_absorb
lemma convex_unit_below_plus_iff [simp]:
"{x}\ \ ys \\ zs \ {x}\ \ ys \ {x}\ \ 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 \\ ys \ {z}\ \ xs \ {z}\ \ ys \ {z}\"
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}\ \ {y}\ \ x \ 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}\ = {y}\ \ x = y"
unfolding po_eq_conv by simp
lemma convex_unit_strict [simp]: "{\}\ = \"
using convex_unit_Rep_compact_basis [of compact_bot]
by (simp add: inst_convex_pd_pcpo)
lemma convex_unit_bottom_iff [simp]: "{x}\ = \ \ x = \"
unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)
lemma compact_convex_unit: "compact x \ compact {x}\"
by (auto dest!: compact_basis.compact_imp_principal)
lemma compact_convex_unit_iff [simp]: "compact {x}\ \ 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]:
"\compact xs; compact ys\ \ compact (xs \\ ys)"
by (auto dest!: convex_pd.compact_imp_principal)
subsection \<open>Induction rules\<close>
lemma convex_pd_induct1:
assumes P: "adm P"
assumes unit: "\x. P {x}\"
assumes insert: "\x ys. \P {x}\; P ys\ \ P ({x}\ \\ 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: "\x. P {x}\"
assumes plus: "\xs ys. \P xs; P ys\ \ P (xs \\ 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 \<open>Monadic bind\<close>
definition
convex_bind_basis ::
"'a pd_basis \ ('a \ 'b convex_pd) \ '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 \<union>\<natural> y\<cdot>f)"
lemma ACI_convex_bind:
"semilattice (\x y. \ f. x\f \\ y\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 \<union>\<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 \\ u \ convex_bind_basis t \ 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 \ ('a \ 'b convex_pd) \ 'b convex_pd" where
"convex_bind = convex_pd.extension convex_bind_basis"
syntax
"_convex_bind" :: "[logic, logic, logic] \ logic"
("(3\\_\_./ _)" [0, 0, 10] 10)
translations
"\\x\xs. e" == "CONST convex_bind\xs\(\ x. e)"
lemma convex_bind_principal [simp]:
"convex_bind\(convex_principal t) = convex_bind_basis t"
unfolding convex_bind_def
apply (rule convex_pd.extension_principal)
apply (erule convex_bind_basis_mono)
done
lemma convex_bind_unit [simp]:
"convex_bind\{x}\\f = f\x"
by (induct x rule: compact_basis.principal_induct, simp, simp)
lemma convex_bind_plus [simp]:
"convex_bind\(xs \\ ys)\f = convex_bind\xs\f \\ convex_bind\ys\f"
by (induct xs rule: convex_pd.principal_induct, simp,
induct ys rule: convex_pd.principal_induct, simp, simp)
lemma convex_bind_strict [simp]: "convex_bind\\\f = f\\"
unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit)
lemma convex_bind_bind:
"convex_bind\(convex_bind\xs\f)\g =
convex_bind\<cdot>xs\<cdot>(\<Lambda> x. convex_bind\<cdot>(f\<cdot>x)\<cdot>g)"
by (induct xs, simp_all)
subsection \<open>Map\<close>
definition
convex_map :: "('a \ 'b) \ 'a convex_pd \ 'b convex_pd" where
"convex_map = (\ f xs. convex_bind\xs\(\ x. {f\x}\))"
lemma convex_map_unit [simp]:
"convex_map\f\{x}\ = {f\x}\"
unfolding convex_map_def by simp
lemma convex_map_plus [simp]:
"convex_map\f\(xs \\ ys) = convex_map\f\xs \\ convex_map\f\ys"
unfolding convex_map_def by simp
lemma convex_map_bottom [simp]: "convex_map\f\\ = {f\\}\"
unfolding convex_map_def by simp
lemma convex_map_ident: "convex_map\(\ x. x)\xs = xs"
by (induct xs rule: convex_pd_induct, simp_all)
lemma convex_map_ID: "convex_map\ID = ID"
by (simp add: cfun_eq_iff ID_def convex_map_ident)
lemma convex_map_map:
"convex_map\f\(convex_map\g\xs) = convex_map\(\ x. f\(g\x))\xs"
by (induct xs rule: convex_pd_induct, simp_all)
lemma convex_bind_map:
"convex_bind\(convex_map\f\xs)\g = convex_bind\xs\(\ x. g\(f\x))"
by (simp add: convex_map_def convex_bind_bind)
lemma convex_map_bind:
"convex_map\f\(convex_bind\xs\g) = convex_bind\xs\(\ x. convex_map\f\(g\x))"
by (simp add: convex_map_def convex_bind_bind)
lemma ep_pair_convex_map: "ep_pair e p \ ep_pair (convex_map\e) (convex_map\p)"
apply standard
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 \ deflation (convex_map\d)"
apply standard
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\d)"
proof (rule finite_deflation_intro)
interpret d: finite_deflation d by fact
from d.deflation_axioms show "deflation (convex_map\d)"
by (rule deflation_convex_map)
have "finite (range (\x. d\x))" by (rule d.finite_range)
hence "finite (Rep_compact_basis -` range (\x. d\x))"
by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
hence "finite (Pow (Rep_compact_basis -` range (\x. d\x)))" by simp
hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\x. d\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 (\x. d\x))))" by simp
hence "finite (range (\xs. convex_map\d\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 "\b. d\(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\d\xs = xs}"
by (rule finite_range_imp_finite_fixes)
qed
subsection \<open>Convex powerdomain is bifinite\<close>
lemma approx_chain_convex_map:
assumes "approx_chain a"
shows "approx_chain (\i. convex_map\(a i))"
using assms unfolding approx_chain_def
by (simp add: lub_APP convex_map_ID finite_deflation_convex_map)
instance convex_pd :: (bifinite) bifinite
proof
show "\(a::nat \ 'a convex_pd \ 'a convex_pd). approx_chain a"
using bifinite [where 'a='a]
by (fast intro!: approx_chain_convex_map)
qed
subsection \<open>Join\<close>
definition
convex_join :: "'a convex_pd convex_pd \ 'a convex_pd" where
"convex_join = (\ xss. convex_bind\xss\(\ xs. xs))"
lemma convex_join_unit [simp]:
"convex_join\{xs}\ = xs"
unfolding convex_join_def by simp
lemma convex_join_plus [simp]:
"convex_join\(xss \\ yss) = convex_join\xss \\ convex_join\yss"
unfolding convex_join_def by simp
lemma convex_join_bottom [simp]: "convex_join\\ = \"
unfolding convex_join_def by simp
lemma convex_join_map_unit:
"convex_join\(convex_map\convex_unit\xs) = xs"
by (induct xs rule: convex_pd_induct, simp_all)
lemma convex_join_map_join:
"convex_join\(convex_map\convex_join\xsss) = convex_join\(convex_join\xsss)"
by (induct xsss rule: convex_pd_induct, simp_all)
lemma convex_join_map_map:
"convex_join\(convex_map\(convex_map\f)\xss) =
convex_map\<cdot>f\<cdot>(convex_join\<cdot>xss)"
by (induct xss rule: convex_pd_induct, simp_all)
subsection \<open>Conversions to other powerdomains\<close>
text \<open>Convex to upper\<close>
lemma convex_le_imp_upper_le: "t \\ u \ t \\ u"
unfolding convex_le_def by simp
definition
convex_to_upper :: "'a convex_pd \ 'a upper_pd" where
"convex_to_upper = convex_pd.extension upper_principal"
lemma convex_to_upper_principal [simp]:
"convex_to_upper\(convex_principal t) = upper_principal t"
unfolding convex_to_upper_def
apply (rule convex_pd.extension_principal)
apply (rule upper_pd.principal_mono)
apply (erule convex_le_imp_upper_le)
done
lemma convex_to_upper_unit [simp]:
"convex_to_upper\{x}\ = {x}\"
by (induct x rule: compact_basis.principal_induct, simp, simp)
lemma convex_to_upper_plus [simp]:
"convex_to_upper\(xs \\ ys) = convex_to_upper\xs \\ convex_to_upper\ys"
by (induct xs rule: convex_pd.principal_induct, simp,
induct ys rule: convex_pd.principal_induct, simp, simp)
lemma convex_to_upper_bind [simp]:
"convex_to_upper\(convex_bind\xs\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\(convex_map\f\xs) = upper_map\f\(convex_to_upper\xs)"
by (simp add: convex_map_def upper_map_def cfcomp_LAM)
lemma convex_to_upper_join [simp]:
"convex_to_upper\(convex_join\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 \<open>Convex to lower\<close>
lemma convex_le_imp_lower_le: "t \\ u \ t \\ u"
unfolding convex_le_def by simp
definition
convex_to_lower :: "'a convex_pd \ 'a lower_pd" where
"convex_to_lower = convex_pd.extension lower_principal"
lemma convex_to_lower_principal [simp]:
"convex_to_lower\(convex_principal t) = lower_principal t"
unfolding convex_to_lower_def
apply (rule convex_pd.extension_principal)
apply (rule lower_pd.principal_mono)
apply (erule convex_le_imp_lower_le)
done
lemma convex_to_lower_unit [simp]:
"convex_to_lower\{x}\ = {x}\"
by (induct x rule: compact_basis.principal_induct, simp, simp)
lemma convex_to_lower_plus [simp]:
"convex_to_lower\(xs \\ ys) = convex_to_lower\xs \\ convex_to_lower\ys"
by (induct xs rule: convex_pd.principal_induct, simp,
induct ys rule: convex_pd.principal_induct, simp, simp)
lemma convex_to_lower_bind [simp]:
"convex_to_lower\(convex_bind\xs\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\(convex_map\f\xs) = lower_map\f\(convex_to_lower\xs)"
by (simp add: convex_map_def lower_map_def cfcomp_LAM)
lemma convex_to_lower_join [simp]:
"convex_to_lower\(convex_join\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 \<open>Ordering property\<close>
lemma convex_pd_below_iff:
"(xs \ 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 \\ ys" and ys="zs \\ ws"]
for xs ys zs ws
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
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