`{Funext} (n : nat) {A B : Type} (C : B -> Type) (f : A -> B) : ExtendableAlong n f C <~> PathSplit n (fun (g : forall b, C b) => g oD f). Proof. generalize dependent C; simple_induction n n IHn; intros C. 1:apply equiv_idmap. refine (_ *E _); simpl. - refine (equiv_functor_forall' 1 _); intros g; simpl. refine (equiv_functor_sigma' 1 _); intros rec. apply equiv_path_forall. - refine (equiv_functor_forall' 1 _); intros h. refine (equiv_functor_forall' 1 _); intros k; simpl. refine (_ oE IHn (fun b => h b = k b)). apply equiv_inverse. refine (equiv_functor_pathsplit n _ _ (equiv_apD10 _ _ _) (equiv_apD10 _ _ _) _). intros []; reflexivity. Defined.
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.