src/HOL/Isar_examples/NestedDatatype.thy
author wenzelm
Thu Dec 22 00:28:43 2005 +0100 (2005-12-22)
changeset 18460 9a1458cb2956
parent 18153 a084aa91f701
child 23373 ead82c82da9e
permissions -rw-r--r--
tuned induct proofs;
     1 
     2 (* $Id$ *)
     3 
     4 header {* Nested datatypes *}
     5 
     6 theory NestedDatatype imports Main begin
     7 
     8 subsection {* Terms and substitution *}
     9 
    10 datatype ('a, 'b) "term" =
    11     Var 'a
    12   | App 'b "('a, 'b) term list"
    13 
    14 consts
    15   subst_term :: "('a => ('a, 'b) term) => ('a, 'b) term => ('a, 'b) term"
    16   subst_term_list ::
    17     "('a => ('a, 'b) term) => ('a, 'b) term list => ('a, 'b) term list"
    18 
    19 primrec (subst)
    20   "subst_term f (Var a) = f a"
    21   "subst_term f (App b ts) = App b (subst_term_list f ts)"
    22   "subst_term_list f [] = []"
    23   "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts"
    24 
    25 
    26 text {*
    27  \medskip A simple lemma about composition of substitutions.
    28 *}
    29 
    30 lemma "subst_term (subst_term f1 o f2) t =
    31       subst_term f1 (subst_term f2 t)"
    32   and "subst_term_list (subst_term f1 o f2) ts =
    33       subst_term_list f1 (subst_term_list f2 ts)"
    34   by (induct t and ts) simp_all
    35 
    36 lemma "subst_term (subst_term f1 o f2) t =
    37   subst_term f1 (subst_term f2 t)"
    38 proof -
    39   let "?P t" = ?thesis
    40   let ?Q = "\<lambda>ts. subst_term_list (subst_term f1 o f2) ts =
    41     subst_term_list f1 (subst_term_list f2 ts)"
    42   show ?thesis
    43   proof (induct t)
    44     fix a show "?P (Var a)" by simp
    45   next
    46     fix b ts assume "?Q ts"
    47     thus "?P (App b ts)" by (simp add: o_def)
    48   next
    49     show "?Q []" by simp
    50   next
    51     fix t ts
    52     assume "?P t" "?Q ts" thus "?Q (t # ts)" by simp
    53   qed
    54 qed
    55 
    56 
    57 subsection {* Alternative induction *}
    58 
    59 theorem term_induct' [case_names Var App]:
    60   assumes var: "!!a. P (Var a)"
    61     and app: "!!b ts. list_all P ts ==> P (App b ts)"
    62   shows "P t"
    63 proof (induct t)
    64   fix a show "P (Var a)" by (rule var)
    65 next
    66   fix b t ts assume "list_all P ts"
    67   thus "P (App b ts)" by (rule app)
    68 next
    69   show "list_all P []" by simp
    70 next
    71   fix t ts assume "P t" "list_all P ts"
    72   thus "list_all P (t # ts)" by simp
    73 qed
    74 
    75 lemma
    76   "subst_term (subst_term f1 o f2) t = subst_term f1 (subst_term f2 t)"
    77 proof (induct t rule: term_induct')
    78   case (Var a)
    79   show ?case by (simp add: o_def)
    80 next
    81   case (App b ts)
    82   thus ?case by (induct ts) simp_all
    83 qed
    84 
    85 end