{A B A' B'} (f : A -> A') (g : B -> B') : join_sym A' B' o functor_join f g == functor_join g f o join_sym A B. Proof. snrapply Join_ind_FlFr. 1, 2: reflexivity. intros a b; cbn beta. apply equiv_p1_1q. lhs nrefine (ap_compose' (functor_join f g) _ (jglue a b)). lhs nrefine (ap _ (functor_join_beta_jglue _ _ _ _)). lhs nrapply join_sym_beta_jglue. symmetry. lhs nrefine (ap_compose' (join_sym A B) _ (jglue a b)). lhs nrefine (ap _ (join_sym_beta_jglue a b)). refine (ap_V _ (jglue b a) @ ap inverse _). apply functor_join_beta_jglue. Defined.
Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare. Require Import NullHomotopy. Require Import Extensions. Require Import Colimits.Pushout. Require Import Truncations.Core Truncations.Connectedness. Require Import Pointed.Core. Require Import WildCat. Require Import Spaces.Nat.Core.