--- a/NEWS Mon Jul 25 16:55:48 2011 +0200
+++ b/NEWS Tue Jul 26 10:03:19 2011 +0200
@@ -79,6 +79,10 @@
SUPR_apply ~> SUP_apply
INCOMPATIBILITY.
+* Theorem collections ball_simps and bex_simps do not contain theorems referring
+to UNION any longer; these have been moved to collection UN_ball_bex_simps.
+INCOMPATIBILITY.
+
* Archimedian_Field.thy:
floor now is defined as parameter of a separate type class floor_ceiling.
--- a/src/HOL/Complete_Lattice.thy Mon Jul 25 16:55:48 2011 +0200
+++ b/src/HOL/Complete_Lattice.thy Tue Jul 26 10:03:19 2011 +0200
@@ -476,6 +476,14 @@
from dual.Sup_eq_top_iff show ?thesis .
qed
+lemma INF_eq_bot_iff:
+ "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
+ unfolding INF_def Inf_eq_bot_iff by auto
+
+lemma SUP_eq_top_iff:
+ "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
+ unfolding SUP_def Sup_eq_top_iff by auto
+
end
@@ -929,7 +937,7 @@
lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
by (fact SUP_union)
-lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)" -- "FIXME generalize"
+lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
by blast
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
@@ -1067,44 +1075,13 @@
"\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
by auto
-lemma ball_simps [simp,no_atp]:
- "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
- "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
- "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
- "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
- "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
- "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
- "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
+lemma UN_ball_bex_simps [simp, no_atp]:
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
- "\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
- "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
- "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
- "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
- by auto
-
-lemma bex_simps [simp,no_atp]:
- "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
- "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
- "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
- "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
- "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
+ "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
"\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
- "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
- "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
- "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
by auto
-lemma (in complete_linorder) INF_eq_bot_iff:
- fixes f :: "'b \<Rightarrow> 'a"
- shows "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
- unfolding INF_def Inf_eq_bot_iff by auto
-
-lemma (in complete_linorder) SUP_eq_top_iff:
- fixes f :: "'b \<Rightarrow> 'a"
- shows "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
- unfolding SUP_def Sup_eq_top_iff by auto
-
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Mon Jul 25 16:55:48 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Tue Jul 26 10:03:19 2011 +0200
@@ -2539,7 +2539,7 @@
fixes s :: "'a::real_normed_vector set"
assumes "open s"
shows "open(convex hull s)"
- unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
+ unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(8)
proof(rule, rule) fix a
assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Jul 25 16:55:48 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Tue Jul 26 10:03:19 2011 +0200
@@ -2505,7 +2505,7 @@
by auto
from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
- from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
+ from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
have "f a \<notin> span (f ` b)" using tha
using "2.hyps"(2)
"2.prems"(3) by auto
--- a/src/HOL/Set.thy Mon Jul 25 16:55:48 2011 +0200
+++ b/src/HOL/Set.thy Tue Jul 26 10:03:19 2011 +0200
@@ -1540,6 +1540,30 @@
lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
by iprover
+lemma ball_simps [simp, no_atp]:
+ "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
+ "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
+ "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
+ "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
+ "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
+ "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
+ "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
+ "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
+ "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
+ "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
+ by auto
+
+lemma bex_simps [simp, no_atp]:
+ "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
+ "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
+ "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
+ "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
+ "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
+ "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
+ "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
+ "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
+ by auto
+
subsubsection {* Monotonicity of various operations *}