End NTransitive.
Arguments dtuple_on {sT} n%N S%g.
Arguments ntransitive {gT sT} n%N A%g S%g to%act.
Arguments n_act {gT sT} to {n} t a.
Notation "n .-dtuple ( S )" := (dtuple_on n S)
(at level 8, format "n .-dtuple ( S )") : set_scope.
Notation "[ 'transitive' ^ n A , 'on' S | to ]" := (ntransitive n A S to)
(at level 0, n at level 8,
format "[ 'transitive' ^ n  A ,  'on'  S  |  to ]") : form_scope.
Section NTransitveProp.
Variables (gT : finGroupType) (sT : finType).
Variables (to : {action gT &-> sT}) (G : {group gT}) (S : {set sT}).

(* Specifications: *)
Lemma card_uniq_tuple n (t : n.-tuple sT) : uniq t -> #|t| = n.
Lemma dtuple_on_subset n (S1 S2 : {set sT}) t :
S1 \subset S2 -> t \in n.-dtuple(S1) -> t \in n.-dtuple(S2).
Lemma ntransitive_weak k m :
k <= m -> [transitive^m G, on S | to] -> [transitive^k G, on S | to].
Lemma ntransitive1 m :
0 < m -> [transitive^m G, on S | to] -> [transitive G, on S | to].
Lemma ntransitive_primitive m :
1 < m -> [transitive^m G, on S | to] -> [primitive G, on S | to].