1 theory Example

     2 imports Main

     3 begin

     4 

     5 text {* 

     6   A small example theory proving a few facts

     7   about list append and list reverse.

     8 *}

     9 

    10 datatype 'a list2 =

    11     nil2                   ("[]")

    12   | cons2 "'a"  "'a list2" ("_ ## _")

    13 

    14 abbreviation

    15   singleton :: "'a \<Rightarrow> 'a list2" ("[_]")

    16 where

    17   "[x] \<equiv> x##[]"

    18 

    19 fun

    20   append2 :: "'a list2 \<Rightarrow> 'a list2 \<Rightarrow> 'a list2" ("_ @@ _")

    21 where

    22   append2_nil2:  "[] @@ ys = ys"

    23 | append2_cons2: "(x##xs) @@ ys = x##(xs @@ ys)"

    24 

    25 fun

    26   rev2 :: "'a list2 \<Rightarrow> 'a list2"

    27 where

    28   "rev2 [] = []"

    29 | "rev2 (x##xs) = (rev2 xs) @@ (x##[])"

    30 

    31 lemma append2_nil2R[simp]:

    32   shows "xs @@ [] = xs"

    33 by (induct xs) (auto)

    34 

    35 lemma append2_assoc[simp]:

    36   shows "(xs @@ ys) @@ zs = xs @@ (ys @@ zs)"

    37 by (induct xs) (auto)

    38 

    39 lemma rev2_append2[simp]: 

    40   shows "rev2 (xs @@ ys) = (rev2 ys) @@ (rev2 xs)"

    41 by (induct xs) (auto)

    42 

    43 lemma rev2_rev2:

    44   shows "rev2 (rev2 xs) = xs"

    45 by (induct xs) (auto)

    46 

    47 end