Section PermutationParity.
Variable T : finType.
Implicit Types (s t u v : {perm T}) (x y z a b : T).
(* Note that porbit s x is the orbit of x by <[s]> under the action aperm. *)
(* Hence, the porbit lemmas below are special cases of more general lemmas *)
(* on orbits that will be stated in action.v.                              *)
(*   Defining porbit directly here avoids a dependency of matrix.v on      *)
(* action.v and hence morphism.v.                                          *)
Definition aperm x s := s x.
Definition porbit s x := aperm x @: <[s]>.
Definition porbits s := porbit s @: T.
Definition odd_perm (s : perm_type T) := odd #|T| (+) odd #|porbits s|.
Canonical Sym_group : {group {perm T}} := Group Sym_group_set.
Definition lift_perm_fun i j s k :=
if unlift i k is Some k' then lift j (s k') else j.

Lemma lift_permK i j s :
cancel (lift_perm_fun i j s) (lift_perm_fun j i s^-1).