1.1 --- a/NEWS Mon Jul 25 14:10:12 2011 +0200
1.2 +++ b/NEWS Mon Jul 25 23:27:20 2011 +0200
1.3 @@ -79,6 +79,10 @@
1.4 SUPR_apply ~> SUP_apply
1.5 INCOMPATIBILITY.
1.6
1.7 +* Theorem collections ball_simps and bex_simps do not contain theorems referring
1.8 +to UNION any longer; these have been moved to collection UN_ball_bex_simps.
1.9 +INCOMPATIBILITY.
1.10 +
1.11 * Archimedian_Field.thy:
1.12 floor now is defined as parameter of a separate type class floor_ceiling.
1.13
2.1 --- a/src/HOL/Complete_Lattice.thy Mon Jul 25 14:10:12 2011 +0200
2.2 +++ b/src/HOL/Complete_Lattice.thy Mon Jul 25 23:27:20 2011 +0200
2.3 @@ -476,6 +476,14 @@
2.4 from dual.Sup_eq_top_iff show ?thesis .
2.5 qed
2.6
2.7 +lemma INF_eq_bot_iff:
2.8 + "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
2.9 + unfolding INF_def Inf_eq_bot_iff by auto
2.10 +
2.11 +lemma SUP_eq_top_iff:
2.12 + "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
2.13 + unfolding SUP_def Sup_eq_top_iff by auto
2.14 +
2.15 end
2.16
2.17
2.18 @@ -929,7 +937,7 @@
2.19 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)"
2.20 by (fact SUP_union)
2.21
2.22 -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"
2.23 +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)"
2.24 by blast
2.25
2.26 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
2.27 @@ -1067,44 +1075,13 @@
2.28 "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
2.29 by auto
2.30
2.31 -lemma ball_simps [simp,no_atp]:
2.32 - "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
2.33 - "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
2.34 - "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
2.35 - "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
2.36 - "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
2.37 - "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
2.38 - "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
2.39 +lemma UN_ball_bex_simps [simp, no_atp]:
2.40 "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
2.41 - "\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
2.42 - "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
2.43 - "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
2.44 - "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
2.45 - by auto
2.46 -
2.47 -lemma bex_simps [simp,no_atp]:
2.48 - "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
2.49 - "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
2.50 - "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
2.51 - "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
2.52 - "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
2.53 + "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
2.54 "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
2.55 "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
2.56 - "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
2.57 - "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
2.58 - "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
2.59 by auto
2.60
2.61 -lemma (in complete_linorder) INF_eq_bot_iff:
2.62 - fixes f :: "'b \<Rightarrow> 'a"
2.63 - shows "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
2.64 - unfolding INF_def Inf_eq_bot_iff by auto
2.65 -
2.66 -lemma (in complete_linorder) SUP_eq_top_iff:
2.67 - fixes f :: "'b \<Rightarrow> 'a"
2.68 - shows "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
2.69 - unfolding SUP_def Sup_eq_top_iff by auto
2.70 -
2.71
2.72 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
2.73
3.1 --- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Mon Jul 25 14:10:12 2011 +0200
3.2 +++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Mon Jul 25 23:27:20 2011 +0200
3.3 @@ -2539,7 +2539,7 @@
3.4 fixes s :: "'a::real_normed_vector set"
3.5 assumes "open s"
3.6 shows "open(convex hull s)"
3.7 - unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
3.8 + unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(8)
3.9 proof(rule, rule) fix a
3.10 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"
3.11 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
4.1 --- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Jul 25 14:10:12 2011 +0200
4.2 +++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Jul 25 23:27:20 2011 +0200
4.3 @@ -2505,7 +2505,7 @@
4.4 by auto
4.5 from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
4.6 have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
4.7 - from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
4.8 + from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
4.9 have "f a \<notin> span (f ` b)" using tha
4.10 using "2.hyps"(2)
4.11 "2.prems"(3) by auto
5.1 --- a/src/HOL/Set.thy Mon Jul 25 14:10:12 2011 +0200
5.2 +++ b/src/HOL/Set.thy Mon Jul 25 23:27:20 2011 +0200
5.3 @@ -1540,6 +1540,30 @@
5.4 lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
5.5 by iprover
5.6
5.7 +lemma ball_simps [simp, no_atp]:
5.8 + "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
5.9 + "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
5.10 + "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
5.11 + "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
5.12 + "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
5.13 + "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
5.14 + "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
5.15 + "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
5.16 + "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
5.17 + "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
5.18 + by auto
5.19 +
5.20 +lemma bex_simps [simp, no_atp]:
5.21 + "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
5.22 + "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
5.23 + "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
5.24 + "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
5.25 + "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
5.26 + "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
5.27 + "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
5.28 + "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
5.29 + by auto
5.30 +
5.31
5.32 subsubsection {* Monotonicity of various operations *}
5.33