theory Examples imports Main Binomial begin

 ML "eta_contract := false"

 ML "Pretty.margin_default := 64"

 text{*membership, intersection *}

 text{*difference and empty set*}

 text{*complement, union and universal set*}

 lemma "(x \<in> A \<inter> B) = (x \<in> A \<and> x \<in> B)"

 by blast

 text{*

 @{thm[display] IntI[no_vars]}

 \rulename{IntI}

 @{thm[display] IntD1[no_vars]}

 \rulename{IntD1}

 @{thm[display] IntD2[no_vars]}

 \rulename{IntD2}

 *}

 lemma "(x \<in> -A) = (x \<notin> A)"

 by blast

 text{*

 @{thm[display] Compl_iff[no_vars]}

 \rulename{Compl_iff}

 *}

 lemma "- (A \<union> B) = -A \<inter> -B"

 by blast

 text{*

 @{thm[display] Compl_Un[no_vars]}

 \rulename{Compl_Un}

 *}

 lemma "A-A = {}"

 by blast

 text{*

 @{thm[display] Diff_disjoint[no_vars]}

 \rulename{Diff_disjoint}

 *}

 lemma "A \<union> -A = UNIV"

 by blast

 text{*

 @{thm[display] Compl_partition[no_vars]}

 \rulename{Compl_partition}

 *}

 text{*subset relation*}

 text{*

 @{thm[display] subsetI[no_vars]}

 \rulename{subsetI}

 @{thm[display] subsetD[no_vars]}

 \rulename{subsetD}

 *}

 lemma "((A \<union> B) \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"

 by blast

 text{*

 @{thm[display] Un_subset_iff[no_vars]}

 \rulename{Un_subset_iff}

 *}

 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*}

 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{*

 set extensionality

 @{thm[display] set_ext[no_vars]}

 \rulename{set_ext}

 @{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*}

 lemma "insert x A = {x} \<union> A"

 by blast

 text{*

 @{thm[display] insert_is_Un[no_vars]}

 \rulename{insert_is_Un}

 *}

 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*}

 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*}

 lemma "(a \<in> {z. P z}) = P a"

 by blast

 text{*

 @{thm[display] mem_Collect_eq[no_vars]}

 \rulename{mem_Collect_eq}

 *}

 lemma "{x. x \<in> A} = A"

 by blast

 text{*

 @{thm[display] Collect_mem_eq[no_vars]}

 \rulename{Collect_mem_eq}

 *}

 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}"

 by blast

 definition prime :: "nat set" where

     "prime == {p. 1<p & (ALL m. m dvd p --> m=1 | m=p)}"

 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*}

 lemma "(\<exists>x\<in>A. P x) = (\<exists>x. x\<in>A \<and> P x)"

 by blast

 text{*

 @{thm[display] bexI[no_vars]}

 \rulename{bexI}

 *}

 text{*

 @{thm[display] bexE[no_vars]}

 \rulename{bexE}

 *}

 lemma "(\<forall>x\<in>A. P x) = (\<forall>x. x\<in>A \<longrightarrow> P x)"

 by blast

 text{*

 @{thm[display] ballI[no_vars]}

 \rulename{ballI}

 *}

 text{*

 @{thm[display] bspec[no_vars]}

 \rulename{bspec}

 *}

 text{*indexed unions and variations*}

 lemma "(\<Union>x. B x) = (\<Union>x\<in>UNIV. B x)"

 by blast

 text{*

 @{thm[display] UN_iff[no_vars]}

 \rulename{UN_iff}

 *}

 text{*

 @{thm[display] Union_iff[no_vars]}

 \rulename{Union_iff}

 *}

 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{*

 @{thm[display] UN_I[no_vars]}

 \rulename{UN_I}

 *}

 text{*

 @{thm[display] UN_E[no_vars]}

 \rulename{UN_E}

 *}

 text{*indexed intersections*}

 lemma "(\<Inter>x. B x) = {y. \<forall>x. y \<in> B x}"

 by blast

 text{*

 @{thm[display] INT_iff[no_vars]}

 \rulename{INT_iff}

 *}

 text{*

 @{thm[display] Inter_iff[no_vars]}

 \rulename{Inter_iff}

 *}

 text{*mention also card, Pow, etc.*}

 text{*

 @{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}

 *}

 end