src/Doc/Tutorial/Sets/Examples.thy

 theory Examples imports Complex_Main begin
 
 declare [[eta_contract = false]]
 
-text{*membership, intersection *}
-text{*difference and empty set*}
-text{*complement, union and universal set*}
+text\<open>membership, intersection\<close>
+text\<open>difference and empty set\<close>
+text\<open>complement, union and universal set\<close>
 
 lemma "(x \<in> A \<inter> B) = (x \<in> A \<and> x \<in> B)"
 by blast
 
-text{*
+text\<open>
 @{thm[display] IntI[no_vars]}
 \rulename{IntI}
 
 @{thm[display] IntD1[no_vars]}
 \rulename{IntD1}
 
 @{thm[display] IntD2[no_vars]}
 \rulename{IntD2}
-*}
+\<close>
 
 lemma "(x \<in> -A) = (x \<notin> A)"
 by blast
 
-text{*
+text\<open>
 @{thm[display] Compl_iff[no_vars]}
 \rulename{Compl_iff}
-*}
+\<close>
 
 lemma "- (A \<union> B) = -A \<inter> -B"
 by blast
 
-text{*
+text\<open>
 @{thm[display] Compl_Un[no_vars]}
 \rulename{Compl_Un}
-*}
+\<close>
 
 lemma "A-A = {}"
 by blast
 
-text{*
+text\<open>
 @{thm[display] Diff_disjoint[no_vars]}
 \rulename{Diff_disjoint}
-*}
+\<close>
 
 
 
 lemma "A \<union> -A = UNIV"
 by blast
 
-text{*
+text\<open>
 @{thm[display] Compl_partition[no_vars]}
 \rulename{Compl_partition}
-*}
-
-text{*subset relation*}
-
-
-text{*
+\<close>
+
+text\<open>subset relation\<close>
+
+
+text\<open>
 @{thm[display] subsetI[no_vars]}
 \rulename{subsetI}
 
 @{thm[display] subsetD[no_vars]}
 \rulename{subsetD}
-*}
+\<close>
 
 lemma "((A \<union> B) \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
 by blast
 
-text{*
+text\<open>
 @{thm[display] Un_subset_iff[no_vars]}
 \rulename{Un_subset_iff}
-*}
+\<close>
 
 lemma "(A \<subseteq> -B) = (B \<subseteq> -A)"
 by blast
 
 lemma "(A <= -B) = (B <= -A)"
   oops
 
-text{*ASCII version: blast fails because of overloading because
- it doesn't have to be sets*}
+text\<open>ASCII version: blast fails because of overloading because
+ it doesn't have to be sets\<close>
 
 lemma "((A:: 'a set) <= -B) = (B <= -A)"
 by blast
 
-text{*A type constraint lets it work*}
-
-text{*An issue here: how do we discuss the distinction between ASCII and
-symbol notation?  Here the latter disambiguates.*}
-
-
-text{*
+text\<open>A type constraint lets it work\<close>
+
+text\<open>An issue here: how do we discuss the distinction between ASCII and
+symbol notation?  Here the latter disambiguates.\<close>
+
+
+text\<open>
 set extensionality
 
 @{thm[display] set_eqI[no_vars]}
 \rulename{set_eqI}
 
 @{thm[display] equalityI[no_vars]}
 \rulename{equalityI}
 
 @{thm[display] equalityE[no_vars]}
 \rulename{equalityE}
-*}
-
-
-text{*finite sets: insertion and membership relation*}
-text{*finite set notation*}
+\<close>
+
+
+text\<open>finite sets: insertion and membership relation\<close>
+text\<open>finite set notation\<close>
 
 lemma "insert x A = {x} \<union> A"
 by blast
 
-text{*
+text\<open>
 @{thm[display] insert_is_Un[no_vars]}
 \rulename{insert_is_Un}
-*}
+\<close>
 
 lemma "{a,b} \<union> {c,d} = {a,b,c,d}"
 by blast
 
 lemma "{a,b} \<inter> {b,c} = {b}"
 apply auto
 oops
-text{*fails because it isn't valid*}
+text\<open>fails because it isn't valid\<close>
 
 lemma "{a,b} \<inter> {b,c} = (if a=c then {a,b} else {b})"
 apply simp
 by blast
 
-text{*or just force or auto.  blast alone can't handle the if-then-else*}
-
-text{*next: some comprehension examples*}
+text\<open>or just force or auto.  blast alone can't handle the if-then-else\<close>
+
+text\<open>next: some comprehension examples\<close>
 
 lemma "(a \<in> {z. P z}) = P a"
 by blast
 
-text{*
+text\<open>
 @{thm[display] mem_Collect_eq[no_vars]}
 \rulename{mem_Collect_eq}
-*}
+\<close>
 
 lemma "{x. x \<in> A} = A"
 by blast
 
-text{*
+text\<open>
 @{thm[display] Collect_mem_eq[no_vars]}
 \rulename{Collect_mem_eq}
-*}
+\<close>
 
 lemma "{x. P x \<or> x \<in> A} = {x. P x} \<union> A"
 by blast
 
 lemma "{x. P x \<longrightarrow> Q x} = -{x. P x} \<union> {x. Q x}"

 
 lemma "{p*q | p q. p\<in>prime \<and> q\<in>prime} =
        {z. \<exists>p q. z = p*q \<and> p\<in>prime \<and> q\<in>prime}"
 by (rule refl)
 
-text{*binders*}
-
-text{*bounded quantifiers*}
+text\<open>binders\<close>
+
+text\<open>bounded quantifiers\<close>
 
 lemma "(\<exists>x\<in>A. P x) = (\<exists>x. x\<in>A \<and> P x)"
 by blast
 
-text{*
+text\<open>
 @{thm[display] bexI[no_vars]}
 \rulename{bexI}
-*}
-
-text{*
+\<close>
+
+text\<open>
 @{thm[display] bexE[no_vars]}
 \rulename{bexE}
-*}
+\<close>
 
 lemma "(\<forall>x\<in>A. P x) = (\<forall>x. x\<in>A \<longrightarrow> P x)"
 by blast
 
-text{*
+text\<open>
 @{thm[display] ballI[no_vars]}
 \rulename{ballI}
-*}
-
-text{*
+\<close>
+
+text\<open>
 @{thm[display] bspec[no_vars]}
 \rulename{bspec}
-*}
-
-text{*indexed unions and variations*}
+\<close>
+
+text\<open>indexed unions and variations\<close>
 
 lemma "(\<Union>x. B x) = (\<Union>x\<in>UNIV. B x)"
 by blast
 
-text{*
+text\<open>
 @{thm[display] UN_iff[no_vars]}
 \rulename{UN_iff}
-*}
-
-text{*
+\<close>
+
+text\<open>
 @{thm[display] Union_iff[no_vars]}
 \rulename{Union_iff}
-*}
+\<close>
 
 lemma "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
 by blast
 
 lemma "\<Union>S = (\<Union>x\<in>S. x)"
 by blast
 
-text{*
+text\<open>
 @{thm[display] UN_I[no_vars]}
 \rulename{UN_I}
-*}
-
-text{*
+\<close>
+
+text\<open>
 @{thm[display] UN_E[no_vars]}
 \rulename{UN_E}
-*}
-
-text{*indexed intersections*}
+\<close>
+
+text\<open>indexed intersections\<close>
 
 lemma "(\<Inter>x. B x) = {y. \<forall>x. y \<in> B x}"
 by blast
 
-text{*
+text\<open>
 @{thm[display] INT_iff[no_vars]}
 \rulename{INT_iff}
-*}
-
-text{*
+\<close>
+
+text\<open>
 @{thm[display] Inter_iff[no_vars]}
 \rulename{Inter_iff}
-*}
-
-text{*mention also card, Pow, etc.*}
-
-
-text{*
+\<close>
+
+text\<open>mention also card, Pow, etc.\<close>
+
+
+text\<open>
 @{thm[display] card_Un_Int[no_vars]}
 \rulename{card_Un_Int}
 
 @{thm[display] card_Pow[no_vars]}
 \rulename{card_Pow}
 
 @{thm[display] n_subsets[no_vars]}
 \rulename{n_subsets}
-*}
+\<close>
 
 end