Smash_ind_beta_gluer' {P : Smash X Y -> Type} {Psm : forall a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr} (Pgl : forall a, DPath P (gluel a) (Psm a pt) Pl) (Pgr : forall b, DPath P (gluer b) (Psm pt b) Pr) (a b : Y) : apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluer' a b) = (Pgr a) @Dp ((Pgr b)^D). Proof. lhs nrapply dp_apD_pp. apply ap011. 1: apply Smash_ind_beta_gluer. lhs nrapply dp_apD_V. apply ap. apply Smash_ind_beta_gluer. Defined.
Require Import Basics.Overture Basics.PathGroupoids Basics.Tactics Basics.Equivalences. Require Import Types.Sum Types.Bool Types.Paths Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Equiv. Require Import Colimits.Pushout. Require Import Cubical.DPath. Require Import Pointed.Core.