Theory Term

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

section ‹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 ∘ f2) t =
subst_term f1 (subst_term f2 t)"
and "subst_term_list (subst_term f1 ∘ f2) ts =
subst_term_list f1 (subst_term_list f2 ts)"
by (induct t and ts rule: subst_term.induct subst_term_list.induct) 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 rule: subst_term.induct subst_term_list.induct)
apply (rule var)
apply (rule app)
apply assumption
apply simp_all
done

end
```