Section safety.
Context { G C D Z : Type }
(Hcore : CoreSemantics G C mem )
(ge : G)
(P : val -> mem -> Prop).
Fixpoint safeN (n:nat) (c:C) (m:mem) : Prop :=
match n with
| O => True
| S n' =>
match halted Hcore c with
| None =>
exists c', exists m',
corestep Hcore ge c m c' m' /\
safeN n' c' m'
| Some i => P i m
end
end.
Definition corestep_fun  :=
forall ge m q m1 q1 m2 q2 ,
corestep Hcore ge q m q1 m1 ->
corestep Hcore ge q m q2 m2 ->
(q1, m1) = (q2, m2).

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