src/HOL/Isar_examples/NestedDatatype.thy

theorems [cases type: bool] = case_split;

header {* Nested datatypes *};

theory NestedDatatype = Main:;

subsection {* Terms and substitution *};

datatype ('a, 'b) "term" =
    Var 'a
  | App 'b "('a, 'b) term list";

consts
  subst_term :: "('a => ('a, 'b) term) => ('a, 'b) term => ('a, 'b) term"
  subst_term_list ::
    "('a => ('a, 'b) term) => ('a, 'b) term list => ('a, 'b) term list";

primrec (subst)
  "subst_term f (Var a) = f a"
  "subst_term f (App b ts) = App b (subst_term_list f ts)"
  "subst_term_list f [] = []"
  "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts";


text {*
 \medskip A simple lemma about composition of substitutions.
*};

lemma
   "subst_term (subst_term f1 o f2) t =
      subst_term f1 (subst_term f2 t) &
    subst_term_list (subst_term f1 o f2) ts =
      subst_term_list f1 (subst_term_list f2 ts)";
  by (induct t and ts rule: term.induct) simp_all;

lemma "subst_term (subst_term f1 o f2) t = subst_term f1 (subst_term f2 t)";
proof -;
  let "?P t" = ?thesis;
  let ?Q = "\\<lambda>ts. subst_term_list (subst_term f1 o f2) ts =
    subst_term_list f1 (subst_term_list f2 ts)";
  show ?thesis;
  proof (induct t);
    fix a; show "?P (Var a)"; by simp;
  next;
    fix b ts; assume "?Q ts";
    thus "?P (App b ts)"; by (simp add: o_def);
  next;
    show "?Q []"; by simp;
  next;
    fix t ts;
    assume "?P t" "?Q ts"; thus "?Q (t # ts)"; by simp;
  qed;
qed;


subsection {* Alternative induction *};

theorem term_induct' [case_names Var App]:
 "(!!a. P (Var a)) ==> (!!b ts. list_all P ts ==> P (App b ts)) ==> P t";
proof -;
  assume var: "!!a. P (Var a)";
  assume app: "!!b ts. list_all P ts ==> P (App b ts)";
  show ?thesis;
  proof (induct P t);
    fix a; show "P (Var a)"; by (rule var);
  next;
    fix b t ts; assume "list_all P ts";
    thus "P (App b ts)"; by (rule app);
  next;
    show "list_all P []"; by simp;
  next;
    fix t ts; assume "P t" "list_all P ts";
    thus "list_all P (t # ts)"; by simp;
  qed;
qed;

lemma
  "subst_term (subst_term f1 o f2) t = subst_term f1 (subst_term f2 t)"
  (is "?P t");
proof (induct ?P t rule: term_induct');
  case Var;
  show "?P (Var a)"; by (simp add: o_def);
next;
  case App;
  have "?this --> ?P (App b ts)";
    by (induct ts) simp_all;
  thus "..."; ..;
qed;

end;