(* local injectivity properties *)

(* Specifications: *)
Lemma injmP : reflect {in D &, injective f} ('injm f).
Lemma card_im_injm : (#|f @* D| == #|D|) = 'injm f.
Lemma order_injm x : x \in D -> #[f x] = #[x].
Lemma injm1 x : x \in D -> f x = 1 -> x = 1.
Lemma injmD1 A : f @* A^# = (f @* A)^#.
Lemma injm_norm A : A \subset D -> f @* 'N(A) = 'N_(f @* D)(f @* A).
Lemma injm_norms A B :
A \subset D -> B \subset D -> (f @* A \subset 'N(f @* B)) = (A \subset 'N(B)).
Lemma injm_cents A B :
A \subset D -> B \subset D -> (f @* A \subset 'C(f @* B)) = (A \subset 'C(B)).