Section closed_safety.
Variable (G C D Z M : Type).
Variable sem : CoreSemantics G C M.
Variable ge : G.
Fixpoint safeN (n : nat) (c : C) (m : M) : Prop :=
match n with
| O => True
| S n' =>
match halted sem c with
| None =>
exists c', exists m',
corestep sem ge c m c' m' /\
safeN n' c' m'
| Some i => True
end
end.

Lemma safe_downward1 n c m : safeN (S n) c m -> safeN n c m.