proposition: Theorem equiv_istrunc_istrunc_loops `{Funext} n X : IsTrunc n.+2 X <~> forall (x : X), IsTrunc n.+1 (loops [X, x]).
proof: Proof. srapply equiv_iff_hprop. intro tr_loops. apply istrunc_S; intros x y. apply istrunc_S; intros p. destruct p. nrapply tr_loops. Defined.
imports: HoTT.Basics HoTT.Types HFiber Factorization Truncations.Core Truncations.Connectedness HProp Pointed.Core Pointed.pEquiv WildCat Spaces.Nat.Core