{A B C P : Type} (f : TriJoinRecData A B C P) (a : A) (b : B) (c : C) : ap_trijoin (trijoin_rec f) a b c = (trijoin_rec_beta_join12 f a b @@ trijoin_rec_beta_join23 f b c) @ j123 f a b c @ (trijoin_rec_beta_join13 f a c)^. Proof. lhs nrapply ap_trijoin_transport. rewrite (apD_homotopic (Join_rec_beta_jglue _ _ _ _) (jglue b c)). rewrite Join_ind_beta_jglue. rewrite (concat_p_pp _ (transport_paths_Fr (jglue b c) (j12 f a b)) _). rewrite (concat_Ap (transport_paths_Fr (jglue b c))). nrapply trijoin_rec_beta_join123_helper. Qed.
Require Import Basics Types.Paths WildCat Join.Core HoTT.Tactics.