Global Instance Injection_equiv_hlist_cons ls i a b (c : hlist ls) d
: Injective (equiv_hlist (Hcons a c) (@Hcons i ls b d)) :=
{ result := @eqv i a b /\ equiv_hlist c d
; injection := @equiv_hlist_Hcons _ _ _ _ _ _ }.
Global Instance Injection_equiv_hlist_app a b (c c' : hlist a) (d d' : hlist b)
: Injective (equiv_hlist (hlist_app c d) (hlist_app c' d')) :=
{ result := equiv_hlist c c' /\ equiv_hlist d d'
; injection := fun x => proj2 (@equiv_hlist_app _ _ _ _ _ _) x }.
End equiv.

(* Specifications: *)
Lemma hlist_nil_eta : forall (h : hlist nil), h = Hnil.
Lemma hlist_cons_eta : forall a b (h : hlist (a :: b)),
h = Hcons (hlist_hd h) (hlist_tl h).
Lemma Hcons_inv
: forall l ls a b c d,
@eq (hlist (l :: ls)) (Hcons a b) (Hcons c d) ->
a = c /\ b = d.