(* ID:         $Id$ *)

 theory Blast imports Main begin

 lemma "((\<exists>x. \<forall>y. p(x)=p(y)) = ((\<exists>x. q(x))=(\<forall>y. p(y))))   =    

        ((\<exists>x. \<forall>y. q(x)=q(y)) = ((\<exists>x. p(x))=(\<forall>y. q(y))))"

 by blast

 text{*\noindent Until now, we have proved everything using only induction and

 simplification.  Substantial proofs require more elaborate types of

 inference.*}

 lemma "(\<forall>x. honest(x) \<and> industrious(x) \<longrightarrow> healthy(x)) \<and>  

        \<not> (\<exists>x. grocer(x) \<and> healthy(x)) \<and> 

        (\<forall>x. industrious(x) \<and> grocer(x) \<longrightarrow> honest(x)) \<and> 

        (\<forall>x. cyclist(x) \<longrightarrow> industrious(x)) \<and> 

        (\<forall>x. \<not>healthy(x) \<and> cyclist(x) \<longrightarrow> \<not>honest(x))  

        \<longrightarrow> (\<forall>x. grocer(x) \<longrightarrow> \<not>cyclist(x))";

 by blast

 lemma "(\<Union>i\<in>I. A(i)) \<inter> (\<Union>j\<in>J. B(j)) =

         (\<Union>i\<in>I. \<Union>j\<in>J. A(i) \<inter> B(j))"

 by blast

 text {*

 @{thm[display] mult_is_0}

  \rulename{mult_is_0}}

 @{thm[display] finite_Un}

  \rulename{finite_Un}}

 *};

 lemma [iff]: "(xs@ys = []) = (xs=[] & ys=[])"

   apply (induct_tac xs)

   by (simp_all);

 (*ideas for uses of intro, etc.: ex/Primes/is_gcd_unique?*)

 end