--- a/src/HOLCF/ConvexPD.thy Fri May 16 22:35:25 2008 +0200
+++ b/src/HOLCF/ConvexPD.thy Fri May 16 23:25:37 2008 +0200
@@ -148,14 +148,11 @@
done
lemma ideal_Rep_convex_pd: "convex_le.ideal (Rep_convex_pd xs)"
-by (rule Rep_convex_pd [simplified])
+by (rule Rep_convex_pd [unfolded mem_Collect_eq])
lemma Rep_convex_pd_mono: "xs \<sqsubseteq> ys \<Longrightarrow> Rep_convex_pd xs \<subseteq> Rep_convex_pd ys"
unfolding less_convex_pd_def less_set_eq .
-
-subsection {* Principal ideals *}
-
definition
convex_principal :: "'a pd_basis \<Rightarrow> 'a convex_pd" where
"convex_principal t = Abs_convex_pd {u. u \<le>\<natural> t}"
@@ -168,7 +165,7 @@
done
interpretation convex_pd:
- bifinite_basis [convex_le approx_pd convex_principal Rep_convex_pd]
+ ideal_completion [convex_le approx_pd convex_principal Rep_convex_pd]
apply unfold_locales
apply (rule approx_pd_convex_le)
apply (rule approx_pd_idem)
@@ -183,13 +180,16 @@
done
lemma convex_principal_less_iff [simp]:
- "(convex_principal t \<sqsubseteq> convex_principal u) = (t \<le>\<natural> u)"
-unfolding less_convex_pd_def Rep_convex_principal less_set_eq
-by (fast intro: convex_le_refl elim: convex_le_trans)
+ "convex_principal t \<sqsubseteq> convex_principal u \<longleftrightarrow> t \<le>\<natural> u"
+by (rule convex_pd.principal_less_iff)
+
+lemma convex_principal_eq_iff [simp]:
+ "convex_principal t = convex_principal u \<longleftrightarrow> t \<le>\<natural> u \<and> u \<le>\<natural> t"
+by (rule convex_pd.principal_eq_iff)
lemma convex_principal_mono:
"t \<le>\<natural> u \<Longrightarrow> convex_principal t \<sqsubseteq> convex_principal u"
-by (rule convex_principal_less_iff [THEN iffD2])
+by (rule convex_pd.principal_mono)
lemma compact_convex_principal: "compact (convex_principal t)"
by (rule convex_pd.compact_principal)
@@ -198,7 +198,7 @@
by (induct ys rule: convex_pd.principal_induct, simp, simp)
instance convex_pd :: (bifinite) pcpo
-by (intro_classes, fast intro: convex_pd_minimal)
+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])
@@ -209,51 +209,27 @@
instance convex_pd :: (profinite) approx ..
defs (overloaded)
- approx_convex_pd_def:
- "approx \<equiv> (\<lambda>n. convex_pd.basis_fun (\<lambda>t. convex_principal (approx_pd n t)))"
+ approx_convex_pd_def: "approx \<equiv> convex_pd.completion_approx"
+
+instance convex_pd :: (profinite) profinite
+apply (intro_classes, unfold approx_convex_pd_def)
+apply (simp add: convex_pd.chain_completion_approx)
+apply (rule convex_pd.lub_completion_approx)
+apply (rule convex_pd.completion_approx_idem)
+apply (rule convex_pd.finite_fixes_completion_approx)
+done
+
+instance convex_pd :: (bifinite) bifinite ..
lemma approx_convex_principal [simp]:
"approx n\<cdot>(convex_principal t) = convex_principal (approx_pd n t)"
unfolding approx_convex_pd_def
-apply (rule convex_pd.basis_fun_principal)
-apply (erule convex_principal_mono [OF approx_pd_convex_mono])
-done
-
-lemma chain_approx_convex_pd:
- "chain (approx :: nat \<Rightarrow> 'a convex_pd \<rightarrow> 'a convex_pd)"
-unfolding approx_convex_pd_def
-by (rule convex_pd.chain_basis_fun_take)
-
-lemma lub_approx_convex_pd:
- "(\<Squnion>i. approx i\<cdot>xs) = (xs::'a convex_pd)"
-unfolding approx_convex_pd_def
-by (rule convex_pd.lub_basis_fun_take)
-
-lemma approx_convex_pd_idem:
- "approx n\<cdot>(approx n\<cdot>xs) = approx n\<cdot>(xs::'a convex_pd)"
-apply (induct xs rule: convex_pd.principal_induct, simp)
-apply (simp add: approx_pd_idem)
-done
+by (rule convex_pd.completion_approx_principal)
lemma approx_eq_convex_principal:
"\<exists>t\<in>Rep_convex_pd xs. approx n\<cdot>xs = convex_principal (approx_pd n t)"
unfolding approx_convex_pd_def
-by (rule convex_pd.basis_fun_take_eq_principal)
-
-lemma finite_fixes_approx_convex_pd:
- "finite {xs::'a convex_pd. approx n\<cdot>xs = xs}"
-unfolding approx_convex_pd_def
-by (rule convex_pd.finite_fixes_basis_fun_take)
-
-instance convex_pd :: (profinite) profinite
-apply intro_classes
-apply (simp add: chain_approx_convex_pd)
-apply (rule lub_approx_convex_pd)
-apply (rule approx_convex_pd_idem)
-apply (rule finite_fixes_approx_convex_pd)
-done
-
-instance convex_pd :: (bifinite) bifinite ..
+by (rule convex_pd.completion_approx_eq_principal)
lemma compact_imp_convex_principal:
"compact xs \<Longrightarrow> \<exists>t. xs = convex_principal t"
@@ -266,10 +242,7 @@
lemma convex_principal_induct:
"\<lbrakk>adm P; \<And>t. P (convex_principal t)\<rbrakk> \<Longrightarrow> P xs"
-apply (erule approx_induct, rename_tac xs)
-apply (cut_tac n=n and xs=xs in approx_eq_convex_principal)
-apply (clarify, simp)
-done
+by (rule convex_pd.principal_induct)
lemma convex_principal_induct2:
"\<lbrakk>\<And>ys. adm (\<lambda>xs. P xs ys); \<And>xs. adm (\<lambda>ys. P xs ys);
@@ -282,54 +255,12 @@
done
-subsection {* Monadic unit *}
+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))"
-lemma convex_unit_Rep_compact_basis [simp]:
- "convex_unit\<cdot>(Rep_compact_basis a) = convex_principal (PDUnit a)"
-unfolding convex_unit_def
-apply (rule compact_basis.basis_fun_principal)
-apply (rule convex_principal_mono)
-apply (erule PDUnit_convex_mono)
-done
-
-lemma convex_unit_strict [simp]: "convex_unit\<cdot>\<bottom> = \<bottom>"
-unfolding inst_convex_pd_pcpo Rep_compact_bot [symmetric] by simp
-
-lemma approx_convex_unit [simp]:
- "approx n\<cdot>(convex_unit\<cdot>x) = convex_unit\<cdot>(approx n\<cdot>x)"
-apply (induct x rule: compact_basis_induct, simp)
-apply (simp add: approx_Rep_compact_basis)
-done
-
-lemma convex_unit_less_iff [simp]:
- "(convex_unit\<cdot>x \<sqsubseteq> convex_unit\<cdot>y) = (x \<sqsubseteq> y)"
- apply (rule iffI)
- apply (rule bifinite_less_ext)
- apply (drule_tac f="approx i" in monofun_cfun_arg, simp)
- apply (cut_tac x="approx i\<cdot>x" in compact_imp_Rep_compact_basis, simp)
- apply (cut_tac x="approx i\<cdot>y" in compact_imp_Rep_compact_basis, simp)
- apply (clarify, simp add: compact_le_def)
- apply (erule monofun_cfun_arg)
-done
-
-lemma convex_unit_eq_iff [simp]:
- "(convex_unit\<cdot>x = convex_unit\<cdot>y) = (x = y)"
-unfolding po_eq_conv by simp
-
-lemma convex_unit_strict_iff [simp]: "(convex_unit\<cdot>x = \<bottom>) = (x = \<bottom>)"
-unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)
-
-lemma compact_convex_unit_iff [simp]:
- "compact (convex_unit\<cdot>x) = compact x"
-unfolding bifinite_compact_iff by simp
-
-
-subsection {* Monadic plus *}
-
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.
@@ -340,40 +271,69 @@
(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
+ convex_principal_mono PDUnit_convex_mono)
+
lemma convex_plus_principal [simp]:
- "convex_plus\<cdot>(convex_principal t)\<cdot>(convex_principal u) =
- convex_principal (PDPlus t u)"
+ "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)
+lemma approx_convex_unit [simp]:
+ "approx n\<cdot>{x}\<natural> = {approx n\<cdot>x}\<natural>"
+apply (induct x rule: compact_basis_induct, simp)
+apply (simp add: approx_Rep_compact_basis)
+done
+
lemma approx_convex_plus [simp]:
- "approx n\<cdot>(convex_plus\<cdot>xs\<cdot>ys) = convex_plus\<cdot>(approx n\<cdot>xs)\<cdot>(approx n\<cdot>ys)"
+ "approx n\<cdot>(xs +\<natural> ys) = approx n\<cdot>xs +\<natural> approx n\<cdot>ys"
by (induct xs ys rule: convex_principal_induct2, simp, simp, simp)
-lemma convex_plus_commute: "convex_plus\<cdot>xs\<cdot>ys = convex_plus\<cdot>ys\<cdot>xs"
-apply (induct xs ys rule: convex_principal_induct2, simp, simp)
-apply (simp add: PDPlus_commute)
-done
-
lemma convex_plus_assoc:
- "convex_plus\<cdot>(convex_plus\<cdot>xs\<cdot>ys)\<cdot>zs = convex_plus\<cdot>xs\<cdot>(convex_plus\<cdot>ys\<cdot>zs)"
+ "(xs +\<natural> ys) +\<natural> zs = xs +\<natural> (ys +\<natural> zs)"
apply (induct xs ys arbitrary: zs rule: convex_principal_induct2, simp, simp)
apply (rule_tac xs=zs in convex_principal_induct, simp)
apply (simp add: PDPlus_assoc)
done
-lemma convex_plus_absorb: "convex_plus\<cdot>xs\<cdot>xs = xs"
+lemma convex_plus_commute: "xs +\<natural> ys = ys +\<natural> xs"
+apply (induct xs ys rule: convex_principal_induct2, simp, simp)
+apply (simp add: PDPlus_commute)
+done
+
+lemma convex_plus_absorb: "xs +\<natural> xs = xs"
apply (induct xs rule: convex_principal_induct, simp)
apply (simp add: PDPlus_absorb)
done
+interpretation aci_convex_plus: ab_semigroup_idem_mult ["op +\<natural>"]
+ by unfold_locales
+ (rule convex_plus_assoc convex_plus_commute convex_plus_absorb)+
+
+lemma convex_plus_left_commute: "xs +\<natural> (ys +\<natural> zs) = ys +\<natural> (xs +\<natural> zs)"
+by (rule aci_convex_plus.mult_left_commute)
+
+lemma convex_plus_left_absorb: "xs +\<natural> (xs +\<natural> ys) = xs +\<natural> ys"
+by (rule aci_convex_plus.mult_left_idem)
+
+lemmas convex_plus_aci = aci_convex_plus.mult_ac_idem
+
lemma convex_unit_less_plus_iff [simp]:
- "(convex_unit\<cdot>x \<sqsubseteq> convex_plus\<cdot>ys\<cdot>zs) =
- (convex_unit\<cdot>x \<sqsubseteq> ys \<and> convex_unit\<cdot>x \<sqsubseteq> zs)"
+ "{x}\<natural> \<sqsubseteq> ys +\<natural> zs \<longleftrightarrow> {x}\<natural> \<sqsubseteq> ys \<and> {x}\<natural> \<sqsubseteq> zs"
apply (rule iffI)
apply (subgoal_tac
- "adm (\<lambda>f. f\<cdot>(convex_unit\<cdot>x) \<sqsubseteq> f\<cdot>ys \<and> f\<cdot>(convex_unit\<cdot>x) \<sqsubseteq> f\<cdot>zs)")
+ "adm (\<lambda>f. f\<cdot>{x}\<natural> \<sqsubseteq> f\<cdot>ys \<and> f\<cdot>{x}\<natural> \<sqsubseteq> f\<cdot>zs)")
apply (drule admD, rule chain_approx)
apply (drule_tac f="approx i" in monofun_cfun_arg)
apply (cut_tac x="approx i\<cdot>x" in compact_imp_Rep_compact_basis, simp)
@@ -383,16 +343,15 @@
apply simp
apply simp
apply (erule conjE)
- apply (subst convex_plus_absorb [of "convex_unit\<cdot>x", symmetric])
+ apply (subst convex_plus_absorb [of "{x}\<natural>", symmetric])
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done
lemma convex_plus_less_unit_iff [simp]:
- "(convex_plus\<cdot>xs\<cdot>ys \<sqsubseteq> convex_unit\<cdot>z) =
- (xs \<sqsubseteq> convex_unit\<cdot>z \<and> ys \<sqsubseteq> convex_unit\<cdot>z)"
+ "xs +\<natural> ys \<sqsubseteq> {z}\<natural> \<longleftrightarrow> xs \<sqsubseteq> {z}\<natural> \<and> ys \<sqsubseteq> {z}\<natural>"
apply (rule iffI)
apply (subgoal_tac
- "adm (\<lambda>f. f\<cdot>xs \<sqsubseteq> f\<cdot>(convex_unit\<cdot>z) \<and> f\<cdot>ys \<sqsubseteq> f\<cdot>(convex_unit\<cdot>z))")
+ "adm (\<lambda>f. f\<cdot>xs \<sqsubseteq> f\<cdot>{z}\<natural> \<and> f\<cdot>ys \<sqsubseteq> f\<cdot>{z}\<natural>)")
apply (drule admD, rule chain_approx)
apply (drule_tac f="approx i" in monofun_cfun_arg)
apply (cut_tac xs="approx i\<cdot>xs" in compact_imp_convex_principal, simp)
@@ -402,18 +361,46 @@
apply simp
apply simp
apply (erule conjE)
- apply (subst convex_plus_absorb [of "convex_unit\<cdot>z", symmetric])
+ apply (subst convex_plus_absorb [of "{z}\<natural>", symmetric])
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done
+lemma convex_unit_less_iff [simp]: "{x}\<natural> \<sqsubseteq> {y}\<natural> \<longleftrightarrow> x \<sqsubseteq> y"
+ apply (rule iffI)
+ apply (rule bifinite_less_ext)
+ apply (drule_tac f="approx i" in monofun_cfun_arg, simp)
+ apply (cut_tac x="approx i\<cdot>x" in compact_imp_Rep_compact_basis, simp)
+ apply (cut_tac x="approx i\<cdot>y" in compact_imp_Rep_compact_basis, simp)
+ apply (clarify, simp add: compact_le_def)
+ apply (erule monofun_cfun_arg)
+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>"
+unfolding inst_convex_pd_pcpo Rep_compact_bot [symmetric] by simp
+
+lemma convex_unit_strict_iff [simp]: "{x}\<natural> = \<bottom> \<longleftrightarrow> x = \<bottom>"
+unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)
+
+lemma compact_convex_unit_iff [simp]:
+ "compact {x}\<natural> \<longleftrightarrow> compact x"
+unfolding bifinite_compact_iff by simp
+
+lemma compact_convex_plus [simp]:
+ "\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<natural> ys)"
+apply (drule compact_imp_convex_principal)+
+apply (auto simp add: compact_convex_principal)
+done
+
subsection {* Induction rules *}
lemma convex_pd_induct1:
assumes P: "adm P"
- assumes unit: "\<And>x. P (convex_unit\<cdot>x)"
- assumes insert:
- "\<And>x ys. \<lbrakk>P (convex_unit\<cdot>x); P ys\<rbrakk> \<Longrightarrow> P (convex_plus\<cdot>(convex_unit\<cdot>x)\<cdot>ys)"
+ 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_principal_induct, rule P)
apply (induct_tac t rule: pd_basis_induct1)
@@ -426,8 +413,8 @@
lemma convex_pd_induct:
assumes P: "adm P"
- assumes unit: "\<And>x. P (convex_unit\<cdot>x)"
- assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (convex_plus\<cdot>xs\<cdot>ys)"
+ 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_principal_induct, rule P)
apply (induct_tac t rule: pd_basis_induct)
@@ -443,9 +430,10 @@
"'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. convex_plus\<cdot>(x\<cdot>f)\<cdot>(y\<cdot>f))"
+ (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<natural> y\<cdot>f)"
-lemma ACI_convex_bind: "ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. convex_plus\<cdot>(x\<cdot>f)\<cdot>(y\<cdot>f))"
+lemma ACI_convex_bind:
+ "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)
@@ -456,11 +444,11 @@
"convex_bind_basis (PDUnit a) =
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
"convex_bind_basis (PDPlus t u) =
- (\<Lambda> f. convex_plus\<cdot>(convex_bind_basis t\<cdot>f)\<cdot>(convex_bind_basis u\<cdot>f))"
+ (\<Lambda> f. convex_bind_basis t\<cdot>f +\<natural> convex_bind_basis u\<cdot>f)"
unfolding convex_bind_basis_def
apply -
-apply (rule ab_semigroup_idem_mult.fold_pd_PDUnit [OF ACI_convex_bind])
-apply (rule ab_semigroup_idem_mult.fold_pd_PDPlus [OF ACI_convex_bind])
+apply (rule fold_pd_PDUnit [OF ACI_convex_bind])
+apply (rule fold_pd_PDPlus [OF ACI_convex_bind])
done
lemma monofun_LAM:
@@ -487,12 +475,11 @@
done
lemma convex_bind_unit [simp]:
- "convex_bind\<cdot>(convex_unit\<cdot>x)\<cdot>f = f\<cdot>x"
+ "convex_bind\<cdot>{x}\<natural>\<cdot>f = f\<cdot>x"
by (induct x rule: compact_basis_induct, simp, simp)
lemma convex_bind_plus [simp]:
- "convex_bind\<cdot>(convex_plus\<cdot>xs\<cdot>ys)\<cdot>f =
- convex_plus\<cdot>(convex_bind\<cdot>xs\<cdot>f)\<cdot>(convex_bind\<cdot>ys\<cdot>f)"
+ "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_principal_induct2, simp, simp, simp)
lemma convex_bind_strict [simp]: "convex_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
@@ -503,7 +490,7 @@
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. convex_unit\<cdot>(f\<cdot>x)))"
+ "convex_map = (\<Lambda> f xs. convex_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<natural>))"
definition
convex_join :: "'a convex_pd convex_pd \<rightarrow> 'a convex_pd" where
@@ -514,17 +501,15 @@
unfolding convex_map_def by simp
lemma convex_map_plus [simp]:
- "convex_map\<cdot>f\<cdot>(convex_plus\<cdot>xs\<cdot>ys) =
- convex_plus\<cdot>(convex_map\<cdot>f\<cdot>xs)\<cdot>(convex_map\<cdot>f\<cdot>ys)"
+ "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_join_unit [simp]:
- "convex_join\<cdot>(convex_unit\<cdot>xs) = xs"
+ "convex_join\<cdot>{xs}\<natural> = xs"
unfolding convex_join_def by simp
lemma convex_join_plus [simp]:
- "convex_join\<cdot>(convex_plus\<cdot>xss\<cdot>yss) =
- convex_plus\<cdot>(convex_join\<cdot>xss)\<cdot>(convex_join\<cdot>yss)"
+ "convex_join\<cdot>(xss +\<natural> yss) = convex_join\<cdot>xss +\<natural> convex_join\<cdot>yss"
unfolding convex_join_def by simp
lemma convex_map_ident: "convex_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
@@ -571,12 +556,11 @@
done
lemma convex_to_upper_unit [simp]:
- "convex_to_upper\<cdot>(convex_unit\<cdot>x) = upper_unit\<cdot>x"
+ "convex_to_upper\<cdot>{x}\<natural> = {x}\<sharp>"
by (induct x rule: compact_basis_induct, simp, simp)
lemma convex_to_upper_plus [simp]:
- "convex_to_upper\<cdot>(convex_plus\<cdot>xs\<cdot>ys) =
- upper_plus\<cdot>(convex_to_upper\<cdot>xs)\<cdot>(convex_to_upper\<cdot>ys)"
+ "convex_to_upper\<cdot>(xs +\<natural> ys) = convex_to_upper\<cdot>xs +\<sharp> convex_to_upper\<cdot>ys"
by (induct xs ys rule: convex_principal_induct2, simp, simp, simp)
lemma approx_convex_to_upper:
@@ -601,12 +585,11 @@
done
lemma convex_to_lower_unit [simp]:
- "convex_to_lower\<cdot>(convex_unit\<cdot>x) = lower_unit\<cdot>x"
+ "convex_to_lower\<cdot>{x}\<natural> = {x}\<flat>"
by (induct x rule: compact_basis_induct, simp, simp)
lemma convex_to_lower_plus [simp]:
- "convex_to_lower\<cdot>(convex_plus\<cdot>xs\<cdot>ys) =
- lower_plus\<cdot>(convex_to_lower\<cdot>xs)\<cdot>(convex_to_lower\<cdot>ys)"
+ "convex_to_lower\<cdot>(xs +\<natural> ys) = convex_to_lower\<cdot>xs +\<flat> convex_to_lower\<cdot>ys"
by (induct xs ys rule: convex_principal_induct2, simp, simp, simp)
lemma approx_convex_to_lower:
@@ -629,4 +612,19 @@
apply (simp add: approx_convex_to_upper approx_convex_to_lower convex_le_def)
done
+lemmas convex_plus_less_plus_iff =
+ convex_pd_less_iff [where xs="xs +\<natural> ys" and ys="zs +\<natural> ws", standard]
+
+lemmas convex_pd_less_simps =
+ convex_unit_less_plus_iff
+ convex_plus_less_unit_iff
+ convex_plus_less_plus_iff
+ convex_unit_less_iff
+ convex_to_upper_unit
+ convex_to_upper_plus
+ convex_to_lower_unit
+ convex_to_lower_plus
+ upper_pd_less_simps
+ lower_pd_less_simps
+
end