--- a/src/HOL/Finite_Set.thy Sun Jul 12 10:14:51 2009 +0200
+++ b/src/HOL/Finite_Set.thy Sun Jul 12 11:25:56 2009 +0200
@@ -832,16 +832,16 @@
begin
lemma ab_semigroup_idem_mult_sup: "ab_semigroup_idem_mult sup"
-by (rule lower_semilattice.ab_semigroup_idem_mult_inf)(rule dual_lattice)
+by (rule lower_semilattice.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)"
-by(rule lower_semilattice.fold_inf_insert)(rule dual_lattice)
+by(rule lower_semilattice.fold_inf_insert)(rule dual_semlattice)
lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c"
-by(rule lower_semilattice.inf_le_fold_inf)(rule dual_lattice)
+by(rule lower_semilattice.inf_le_fold_inf)(rule dual_semilattice)
lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A"
-by(rule lower_semilattice.fold_inf_le_inf)(rule dual_lattice)
+by(rule lower_semilattice.fold_inf_le_inf)(rule dual_semilattice)
end
@@ -2785,7 +2785,7 @@
apply(erule exE)
apply(rule order_trans)
apply(erule (1) fold1_belowI)
-apply(erule (1) lower_semilattice.fold1_belowI [OF dual_lattice])
+apply(erule (1) lower_semilattice.fold1_belowI [OF dual_semilattice])
done
lemma sup_Inf_absorb [simp]:
@@ -2797,7 +2797,7 @@
lemma inf_Sup_absorb [simp]:
"finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
by (simp add: Sup_fin_def inf_absorb1
- lower_semilattice.fold1_belowI [OF dual_lattice])
+ lower_semilattice.fold1_belowI [OF dual_semilattice])
end
--- a/src/HOL/Lattices.thy Sun Jul 12 10:14:51 2009 +0200
+++ b/src/HOL/Lattices.thy Sun Jul 12 11:25:56 2009 +0200
@@ -29,7 +29,7 @@
text {* Dual lattice *}
-lemma dual_lattice:
+lemma dual_semilattice:
"lower_semilattice (op \<ge>) (op >) sup"
by (rule lower_semilattice.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)
@@ -180,6 +180,11 @@
context lattice
begin
+lemma dual_lattice:
+ "lattice (op \<ge>) (op >) sup inf"
+ by (rule lattice.intro, rule dual_semilattice, rule upper_semilattice.intro, rule dual_order)
+ (unfold_locales, auto)
+
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
@@ -252,12 +257,148 @@
"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
by(simp add:ACI inf_sup_distrib1)
+lemma dual_distrib_lattice:
+ "distrib_lattice (op \<ge>) (op >) sup inf"
+ by (rule distrib_lattice.intro, rule dual_lattice)
+ (unfold_locales, fact inf_sup_distrib1)
+
lemmas distrib =
sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
end
+subsection {* Boolean algebras *}
+
+class boolean_algebra = distrib_lattice + top + bot + minus + uminus +
+ assumes inf_compl_bot: "x \<sqinter> - x = bot"
+ and sup_compl_top: "x \<squnion> - x = top"
+ assumes diff_eq: "x - y = x \<sqinter> - y"
+begin
+
+lemma dual_boolean_algebra:
+ "boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) top bot"
+ by (rule boolean_algebra.intro, rule dual_distrib_lattice)
+ (unfold_locales,
+ auto simp add: inf_compl_bot sup_compl_top diff_eq less_le_not_le)
+
+lemma compl_inf_bot:
+ "- x \<sqinter> x = bot"
+ by (simp add: inf_commute inf_compl_bot)
+
+lemma compl_sup_top:
+ "- x \<squnion> x = top"
+ by (simp add: sup_commute sup_compl_top)
+
+lemma inf_bot_left [simp]:
+ "bot \<sqinter> x = bot"
+ by (rule inf_absorb1) simp
+
+lemma inf_bot_right [simp]:
+ "x \<sqinter> bot = bot"
+ by (rule inf_absorb2) simp
+
+lemma sup_top_left [simp]:
+ "top \<squnion> x = top"
+ by (rule sup_absorb1) simp
+
+lemma sup_top_right [simp]:
+ "x \<squnion> top = top"
+ by (rule sup_absorb2) simp
+
+lemma inf_top_left [simp]:
+ "top \<sqinter> x = x"
+ by (rule inf_absorb2) simp
+
+lemma inf_top_right [simp]:
+ "x \<sqinter> top = x"
+ by (rule inf_absorb1) simp
+
+lemma sup_bot_left [simp]:
+ "bot \<squnion> x = x"
+ by (rule sup_absorb2) simp
+
+lemma sup_bot_right [simp]:
+ "x \<squnion> bot = x"
+ by (rule sup_absorb1) simp
+
+lemma compl_unique:
+ assumes "x \<sqinter> y = bot"
+ and "x \<squnion> y = top"
+ shows "- x = y"
+proof -
+ have "(x \<sqinter> - x) \<squnion> (- x \<sqinter> y) = (x \<sqinter> y) \<squnion> (- x \<sqinter> y)"
+ using inf_compl_bot assms(1) by simp
+ then have "(- x \<sqinter> x) \<squnion> (- x \<sqinter> y) = (y \<sqinter> x) \<squnion> (y \<sqinter> - x)"
+ by (simp add: inf_commute)
+ then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)"
+ by (simp add: inf_sup_distrib1)
+ then have "- x \<sqinter> top = y \<sqinter> top"
+ using sup_compl_top assms(2) by simp
+ then show "- x = y" by (simp add: inf_top_right)
+qed
+
+lemma double_compl [simp]:
+ "- (- x) = x"
+ using compl_inf_bot compl_sup_top by (rule compl_unique)
+
+lemma compl_eq_compl_iff [simp]:
+ "- x = - y \<longleftrightarrow> x = y"
+proof
+ assume "- x = - y"
+ then have "- x \<sqinter> y = bot"
+ and "- x \<squnion> y = top"
+ by (simp_all add: compl_inf_bot compl_sup_top)
+ then have "- (- x) = y" by (rule compl_unique)
+ then show "x = y" by simp
+next
+ assume "x = y"
+ then show "- x = - y" by simp
+qed
+
+lemma compl_bot_eq [simp]:
+ "- bot = top"
+proof -
+ from sup_compl_top have "bot \<squnion> - bot = top" .
+ then show ?thesis by simp
+qed
+
+lemma compl_top_eq [simp]:
+ "- top = bot"
+proof -
+ from inf_compl_bot have "top \<sqinter> - top = bot" .
+ then show ?thesis by simp
+qed
+
+lemma compl_inf [simp]:
+ "- (x \<sqinter> y) = - x \<squnion> - y"
+proof (rule compl_unique)
+ have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = ((x \<sqinter> y) \<sqinter> - x) \<squnion> ((x \<sqinter> y) \<sqinter> - y)"
+ by (rule inf_sup_distrib1)
+ also have "... = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
+ by (simp only: inf_commute inf_assoc inf_left_commute)
+ finally show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = bot"
+ by (simp add: inf_compl_bot)
+next
+ have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (x \<squnion> (- x \<squnion> - y)) \<sqinter> (y \<squnion> (- x \<squnion> - y))"
+ by (rule sup_inf_distrib2)
+ also have "... = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
+ by (simp only: sup_commute sup_assoc sup_left_commute)
+ finally show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = top"
+ by (simp add: sup_compl_top)
+qed
+
+lemma compl_sup [simp]:
+ "- (x \<squnion> y) = - x \<sqinter> - y"
+proof -
+ interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" top bot
+ by (rule dual_boolean_algebra)
+ then show ?thesis by simp
+qed
+
+end
+
+
subsection {* Uniqueness of inf and sup *}
lemma (in lower_semilattice) inf_unique:
@@ -330,17 +471,24 @@
subsection {* Bool as lattice *}
-instantiation bool :: distrib_lattice
+instantiation bool :: boolean_algebra
begin
definition
+ bool_Compl_def: "uminus = Not"
+
+definition
+ bool_diff_def: "A - B \<longleftrightarrow> A \<and> \<not> B"
+
+definition
inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
definition
sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
-instance
- by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
+instance proof
+qed (simp_all add: inf_bool_eq sup_bool_eq le_bool_def
+ bot_bool_eq top_bool_eq bool_Compl_def bool_diff_def, auto)
end
@@ -369,7 +517,33 @@
end
instance "fun" :: (type, distrib_lattice) distrib_lattice
- by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
+proof
+qed (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
+
+instantiation "fun" :: (type, uminus) uminus
+begin
+
+definition
+ fun_Compl_def: "- A = (\<lambda>x. - A x)"
+
+instance ..
+
+end
+
+instantiation "fun" :: (type, minus) minus
+begin
+
+definition
+ fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
+
+instance ..
+
+end
+
+instance "fun" :: (type, boolean_algebra) boolean_algebra
+proof
+qed (simp_all add: inf_fun_eq sup_fun_eq bot_fun_eq top_fun_eq fun_Compl_def fun_diff_def
+ inf_compl_bot sup_compl_top diff_eq)
text {* redundant bindings *}
--- a/src/HOL/Set.thy Sun Jul 12 10:14:51 2009 +0200
+++ b/src/HOL/Set.thy Sun Jul 12 11:25:56 2009 +0200
@@ -10,7 +10,6 @@
text {* A set in HOL is simply a predicate. *}
-
subsection {* Basic syntax *}
global
@@ -363,46 +362,6 @@
Bex_def: "Bex A P == EX x. x:A & P(x)"
Bex1_def: "Bex1 A P == EX! x. x:A & P(x)"
-instantiation "fun" :: (type, minus) minus
-begin
-
-definition
- fun_diff_def: "A - B = (%x. A x - B x)"
-
-instance ..
-
-end
-
-instantiation bool :: minus
-begin
-
-definition
- bool_diff_def: "A - B = (A & ~ B)"
-
-instance ..
-
-end
-
-instantiation "fun" :: (type, uminus) uminus
-begin
-
-definition
- fun_Compl_def: "- A = (%x. - A x)"
-
-instance ..
-
-end
-
-instantiation bool :: uminus
-begin
-
-definition
- bool_Compl_def: "- A = (~ A)"
-
-instance ..
-
-end
-
definition Pow :: "'a set => 'a set set" where
Pow_def: "Pow A = {B. B \<le> A}"