src/HOL/Induct/Term.thy
 author wenzelm Mon, 01 Oct 2001 15:46:35 +0200 changeset 11649 dfb59b9954a6 parent 11549 e7265e70fd7c child 11809 c9ffdd63dd93 permissions -rw-r--r--
tuned;
```
(*  Title:      HOL/Induct/Term.thy
ID:         \$Id\$
Author:     Stefan Berghofer,  TU Muenchen
*)

header {* Terms over a given alphabet *}

theory Term = Main:

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

text {* \medskip Substitution function on terms *}

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_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 theorem about composition of substitutions *}

lemma subst_comp:
"(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))"
apply (induct t and ts rule: term.induct)
apply simp_all
done

text {* \medskip Alternative induction rule *}

lemma term_induct2:
"(!!v. P (Var v)) ==>
(!!f ts. list_all P ts ==> P (App f ts))
==> P t"
proof -
case rule_context
have "P t \<and> list_all P ts"
apply (induct t and ts rule: term.induct)
apply (rule rule_context)
apply (rule rule_context)
apply assumption
apply simp_all
done
thus ?thesis ..
qed

end
```