header {* Nested datatypes *}
theory Nested_Datatype
imports Main
begin
subsection {* Terms and substitution *}
datatype ('a, 'b) "term" =
Var 'a
| App 'b "('a, 'b) term list"
primrec subst_term :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term \<Rightarrow> ('a, 'b) term"
and subst_term_list :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term list \<Rightarrow> ('a, 'b) term list"
where
"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"
lemmas subst_simps = subst_term_subst_term_list.simps
text {* \medskip A simple lemma about composition of substitutions. *}
lemma
"subst_term (subst_term f1 \<circ> f2) t =
subst_term f1 (subst_term f2 t)"
and
"subst_term_list (subst_term f1 \<circ> f2) ts =
subst_term_list f1 (subst_term_list f2 ts)"
by (induct t and ts) simp_all
lemma "subst_term (subst_term f1 \<circ> f2) t =
subst_term f1 (subst_term f2 t)"
proof -
let "?P t" = ?thesis
let ?Q = "\<lambda>ts. subst_term_list (subst_term f1 \<circ> 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: "\<And>a. P (Var a)"
and app: "\<And>b ts. (\<forall>t \<in> set ts. P t) \<Longrightarrow> 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 "\<forall>t \<in> set ts. P t"
then show "P (App b ts)" by (rule app)
next
show "\<forall>t \<in> set []. P t" by simp
next
fix t ts assume "P t" "\<forall>t' \<in> set ts. P t'"
then show "\<forall>t' \<in> set (t # ts). P t'" by simp
qed
lemma "subst_term (subst_term f1 \<circ> 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