(n : nat) {X Y : pType} (f : X ->* Y) : pequiv_ppi_ptr_iterated_loops n Y o* fmap (pPi n) f ==* fmap (pTr 0) (fmap (iterated_loops n) f) o* pequiv_ppi_ptr_iterated_loops n X. Proof. destruct n; unfold pequiv_ppi_ptr_iterated_loops. 1: refine (pmap_postcompose_idmap _ @* (pmap_precompose_idmap _)^*). refine (pmap_postcompose_idmap _ @* _ @* (pmap_precompose_idmap _)^*). srapply phomotopy_homotopy_hset; reflexivity. Defined.
Require Import Basics Types Pointed HSet. Require Import Modalities.Modality. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Algebra.AbGroups. Require Import WildCat.