Section PermutationParity.
Variable T : finType.
Implicit Types (s t u v : {perm T}) (x y z a b : T).
(* Note that pcycle s x is the orbit of x by <[s]> under the action aperm. *)
(* Hence, the pcycle lemmas below are special cases of more general lemmas *)
(* on orbits that will be stated in action.v.                              *)
(*   Defining pcycle directly here avoids a dependency of matrix.v on      *)
(* action.v and hence morphism.v.                                          *)
Definition aperm x s := s x.
Definition pcycle s x := aperm x @: <[s]>.
Definition pcycles s := pcycle s @: T.
Definition odd_perm (s : perm_type T) := odd #|T| (+) odd #|pcycles s|.

Lemma odd_permV s : odd_perm s^-1 = odd_perm s.
Definition lift_perm i j s := perm (can_inj (lift_permK i j s)).