theory Examples imports Main Binomial begin
declare [[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_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*}
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