theory Swap = Main:

consts 

  swapa   :: "(string \<times> string) \<Rightarrow> string \<Rightarrow> string"

  swapas  :: "(string \<times> string)list \<Rightarrow> string \<Rightarrow> string"

primrec

  "swapa (b,c) a = (if b = a then c else (if c = a then b else a))"

primrec

  "swapas []     a = a"

  "swapas (x#pi) a = swapa x (swapas pi a)"

lemma [simp]: 

  "swapas [(a,a)] b=b"

apply(auto)

done

lemma swapas_append: 

  "swapas (pi1@pi2) a = swapas pi1 (swapas pi2 a)"

apply(induct_tac pi1, auto)

done 

lemma [simp]: 

  "swapas (rev pi) (swapas pi a) = a"

apply(induct_tac pi)

apply(simp_all add: swapas_append, auto)

done

lemma swapas_rev_pi_a: 

  "swapas pi a = b \<Longrightarrow> swapas (rev pi) b = a"

apply(auto)

done

lemma [simp]: 

  "swapas pi (swapas (rev pi) a) = a"

apply(rule swapas_rev_pi_a[of "(rev pi)" _ _,simplified])

apply(simp)

done

lemma swapas_rev_pi_b: 

  "swapas (rev pi) a = b \<Longrightarrow> swapas pi b = a"

apply(auto)

done

lemma swapas_comm: 

  "(swapas (pi@[(a,b)]) c) = (swapas ([(swapas pi a,swapas pi b)]@pi) c)"

apply(auto)

apply(simp_all only: swapas_append)

apply(auto)

apply(drule swapas_rev_pi_a, simp)

apply(drule swapas_rev_pi_a, simp)

done

end