(*************************************************************************)
Definition corestep_fun {C M : Type} (sem : @CoreSemantics C M) :=
forall (m m' m'' : M) c c' c'',
corestep sem c m c' m' ->
corestep sem c m c'' m'' ->
c'=c'' /\ m'=m''.
(**  Multistepping *)
Section corestepN.
Context {C M E:Type} (Sem:@CoreSemantics C M).
Fixpoint corestepN (n:nat) : C -> M -> C -> M -> Prop :=
match n with
| O => fun c m c' m' => (c,m) = (c',m')
| S k => fun c1 m1 c3 m3 => exists c2, exists m2,
corestep Sem c1 m1 c2 m2 /\
corestepN k c2 m2 c3 m3
end.

(* Specifications: *)
Lemma corestepN_add : forall n m c1 m1 c3 m3,
corestepN (n+m) c1 m1 c3 m3 <->
exists c2, exists m2,
corestepN n c1 m1 c2 m2 /\
corestepN m c2 m2 c3 m3.
Lemma corestepN_add : forall n m c1 m1 c3 m3,
corestepN (n+m) c1 m1 c3 m3 <->
exists c2, exists m2,
corestepN n c1 m1 c2 m2 /\
corestepN m c2 m2 c3 m3.