Theory Term

theory Term
imports Main
(*  Title:      HOL/Induct/Term.thy
Author: Stefan Berghofer, TU Muenchen
*)


header {* Terms over a given alphabet *}

theory Term
imports Main
begin

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


text {* \medskip Substitution function on terms *}

primrec subst_term :: "('a => ('a, 'b) term) => ('a, 'b) term => ('a, 'b) term"
and subst_term_list :: "('a => ('a, 'b) term) => ('a, 'b) term list => ('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"


text {* \medskip A simple theorem about composition of substitutions *}

lemma subst_comp:
"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


text {* \medskip Alternative induction rule *}

lemma
assumes var: "!!v. P (Var v)"
and app: "!!f ts. (∀t ∈ set ts. P t) ==> P (App f ts)"
shows term_induct2: "P t"
and "∀t ∈ set ts. P t"
apply (induct t and ts)
apply (rule var)
apply (rule app)
apply assumption
apply simp_all
done

end