open nat
open eq.ops

inductive even : nat → Prop :=
| even_zero        : even zero
| even_succ_of_odd : ∀ {a}, odd a  → even (succ a)
with odd : nat → Prop :=
| odd_succ_of_even : ∀ {a}, even a → odd (succ a)

example : even 1 → false :=
begin
  intro He1,
  cases He1 with (a, Ho0),
  cases Ho0
end

example : even 3 → false :=
begin
  intro He3,
  cases He3 with (a, Ho2),
  cases Ho2 with (a, He1),
  cases He1 with (a, Ho0),
  cases Ho0
end