Admitted. (* OUT-OF-SCOPE *)

(* Specifications: *)
Theorem multi_trans :
forall (X:Type) (R: relation X) (x y z : X),
multi R x y  ->
multi R y z ->
multi R x z.
Theorem loop_never_stops : forall st st',
~(loop / st \\ st').
Theorem ev_plus_plus : forall n m p,
ev (n+m) -> ev (n+p) -> ev (m+p).
Theorem Sn_le_Sm__n_le_m : forall n m,
S n <= S m -> n <= m.
Theorem le_plus_l : forall a b,
a <= a + b.
Theorem lt_S : forall n m,
n < m ->
n < S m.
Theorem or_commut : forall P Q : Prop,
P \/ Q  -> Q \/ P.
Theorem exists_example_2 : forall n,
(exists m, n = 4 + m) ->
(exists o, n = 2 + o).
Theorem dist_not_exists : forall (X:Type) (P : X -> Prop),
(forall x, P x) -> ~ (exists x, ~ P x).