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|.

Lemma porbits_mul_tperm s x y : let t := tperm x y in
#|porbits (t * s)| + (x \notin porbit s y).*2 = #|porbits s| + (x != y).