theory Atoms = Main + Swap + Terms:

consts atms   :: "(string \<times> string) list \<Rightarrow> string set"

primrec

"atms [] = {}"

"atms (x#xs) = ((atms xs) \<union>  {fst(x),snd(x)})"

lemma [simp]: 

  "atms (xs@ys) = atms xs \<union> atms ys"

apply(induct_tac xs, auto)

done

lemma [simp]: 

  "atms (rev pi) = atms pi"

apply(induct_tac pi, simp_all)

done

lemma a_not_in_atms: 

  "a\<notin>atms pi \<longrightarrow> a = swapas pi a" 

apply(induct_tac pi, auto)

done

lemma swapas_pi_ineq_a: 

  "swapas pi a \<noteq> a \<longrightarrow> a\<in>atms pi"

apply(case_tac "a\<in>atms pi")

apply(simp)+

apply(drule a_not_in_atms[THEN mp])

apply(simp)

done

lemma a_ineq_swapas_pi: 

  "a \<noteq> swapas pi a \<longrightarrow> a\<in>atms pi"

apply(case_tac "a\<in>atms pi")

apply(simp_all add: a_not_in_atms)

done

lemma swapas_pi_in_atms[THEN mp]: 

  "a\<in>atms pi \<longrightarrow> swapas pi a\<in>atms pi"

apply(subgoal_tac "\<forall>pi1. (a\<in>atms pi1) \<longrightarrow> swapas pi a\<in>(atms pi1 \<union> atms pi)")

apply(force)

apply(induct_tac pi)

apply(auto)

done

end