Section thread_stepN.
Context {G TID SCH TR C M E res thread_type:Type}
(Sem:@ConcurSemantics G TID SCH TR C M res (*thread_type*)) (ge:G).
Fixpoint thread_stepN (n:nat) : SCH -> C -> M -> C -> M -> Prop :=
match n with
| O => fun U c m c' m' => (c,m) = ( c',m')
| S k => fun U c1 m1 c3 m3 => exists c2, exists m2,
@thread_step _ _ _ _ _ _ _ (*_*) Sem ge U c1 m1 c2 m2 /\
thread_stepN k U c2 m2 c3 m3
end.

Lemma thread_stepN_add : forall n m U c1 m1 c3 m3,
thread_stepN (n+m) U c1 m1 c3 m3 <->
exists c2, exists m2,
thread_stepN n U c1 m1 c2 m2 /\
thread_stepN m U c2 m2 c3 m3.