doc-src/TutorialI/Sets/Examples.thy

A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
the tactic has also changed and allows the abstaction over fuction occurences whose type is nat or int.

(* ID:         $Id$ *)
theory Examples = Main:

ML "reset eta_contract"
ML "Pretty.setmargin 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

constdefs
  prime   :: "nat set"
    "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