1 

     2 theory Atoms = Main + Swap + Terms:

     3 

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

     5 primrec

     6 "atms [] = {}"

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

     8 

     9 lemma [simp]: 

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

    11 apply(induct_tac xs, auto)

    12 done

    13 

    14 lemma [simp]: 

    15   "atms (rev pi) = atms pi"

    16 apply(induct_tac pi, simp_all)

    17 done

    18 

    19 lemma a_not_in_atms: 

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

    21 apply(induct_tac pi, auto)

    22 done

    23 

    24 lemma swapas_pi_ineq_a: 

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

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

    27 apply(simp)+

    28 apply(drule a_not_in_atms[THEN mp])

    29 apply(simp)

    30 done

    31 

    32 lemma a_ineq_swapas_pi: 

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

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

    35 apply(simp_all add: a_not_in_atms)

    36 done

    37 

    38 lemma swapas_pi_in_atms[THEN mp]: 

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

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

    41 apply(force)

    42 apply(induct_tac pi)

    43 apply(auto)

    44 done

    45 

    46 end