1 

     2 theory Swap = Main:

     3 

     4 consts 

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

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

     7 primrec

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

     9 

    10 primrec

    11   "swapas []     a = a"

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

    13 

    14 lemma [simp]: 

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

    16 apply(auto)

    17 done

    18 

    19 lemma swapas_append: 

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

    21 apply(induct_tac pi1, auto)

    22 done 

    23 

    24 lemma [simp]: 

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

    26 apply(induct_tac pi)

    27 apply(simp_all add: swapas_append, auto)

    28 done

    29 

    30 lemma swapas_rev_pi_a: 

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

    32 apply(auto)

    33 done

    34 

    35 lemma [simp]: 

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

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

    38 apply(simp)

    39 done

    40 

    41 lemma swapas_rev_pi_b: 

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

    43 apply(auto)

    44 done

    45 

    46 lemma swapas_comm: 

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

    48 apply(auto)

    49 apply(simp_all only: swapas_append)

    50 apply(auto)

    51 apply(drule swapas_rev_pi_a, simp)

    52 apply(drule swapas_rev_pi_a, simp)

    53 done

    54 

    55 end