`{Univalence} {X : Type@{u}} : IsEmbedding (@paths X). Proof. snrapply isembedding_isequiv_ap. intros x1 x2. snrapply (isequiv_isequiv_compose_embedding (ap_paths_inverse x1 x2)). - unfold ap_paths_inverse. nrefine (mapinO_compose (O:=Tr (-1)) _ (equiv_path_inverse x2 x1 oE _)). 2: exact _. nrefine (mapinO_compose _ (functor_forall_id _)). 1: exact _. rapply mapinO_functor_forall_id. intro y. apply isembedding_equiv_fun. - simpl. srapply (isequiv_homotopic idmap). intros []. reflexivity. Defined.
Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.