theory demo imports Main begin

section ‹Inductive predicates for lists›

datatype 'a list = Nil ("[]") | Cons 'a "'a list" ("_ # _")

fun length :: "'a list ⇒ nat" where
  "length [] = 0" | "length (x # xs) = 1 + length xs"

inductive ζ :: "'a list ⇒ nat ⇒ bool" where
Nil[intro!]:  "ζ [] 0" |
Cons[intro]: "ζ xs l ⟹ ζ (x # xs) (1 + l)"

(* Not the answer? *)
lemma "ζ xs 42"
  oops