(n : nat) {A B} (f : A -> B) (C : B -> Type) : (forall b, ooExtendableAlong (const_tt (hfiber f b)) (fun _ => C b)) -> ExtendableAlong n f C. Proof. revert C. simple_induction n n IHn; intros C null; [exact tt | split]. - intros g. exists (fun b => (fst (null b 1%nat) (fun x => x.2 # g x.1)).1 tt). intros a. rewrite (path_unit tt (const_tt _ a)). exact ((fst (null (f a) 1%nat) _).2 (a ; 1)). - intros h k. apply IHn; intros b. apply ooextendable_homotopy, null. Defined.
Require Import HoTT.Basics HoTT.Types. Require Import Extensions NullHomotopy. Require Import Modality. Module Import IsLocal_Internal. Module Import IsNull_Internal.