(* $Id$ *)
header {* Nested datatypes *}
theory NestedDatatype imports Main begin
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)"
and "subst_term_list (subst_term f1 o f2) ts =
subst_term_list f1 (subst_term_list f2 ts)"
by (induct t and ts) 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"
then show "?P (App b ts)"
by (simp only: subst.simps)
next
show "?Q []" by simp
next
fix t ts
assume "?P t" "?Q ts" then show "?Q (t # ts)"
by (simp only: subst.simps)
qed
qed
subsection {* Alternative induction *}
theorem term_induct' [case_names Var App]:
assumes var: "!!a. P (Var a)"
and app: "!!b ts. list_all P ts ==> P (App b ts)"
shows "P t"
proof (induct t)
fix a show "P (Var a)" by (rule var)
next
fix b t ts assume "list_all P ts"
then show "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"
then show "list_all P (t # ts)" by simp
qed
lemma
"subst_term (subst_term f1 o f2) t = subst_term f1 (subst_term f2 t)"
proof (induct t rule: term_induct')
case (Var a)
show ?case by (simp add: o_def)
next
case (App b ts)
then show ?case by (induct ts) simp_all
qed
end