Definition lift_perm i j s := perm (can_inj (lift_permK i j s)).

(* Specifications: *)
Lemma lift_perm_id i j s : lift_perm i j s i = j.
Lemma lift_permM i j k s t : lift_perm i j s * lift_perm j k t = lift_perm i k (s * t).
Lemma lift_perm1 i : lift_perm i i 1 = 1.
Lemma lift_permV i j s : (lift_perm i j s)^-1 = lift_perm j i s^-1.