proposition: Lemma path_bool_prop : HProp = Bool.
proof: Proof. apply path_universe_uncurried. srapply equiv_adjointify. - exact (fun P => if LEM P _ then true else false). - exact (fun b : Bool => if b then merely Unit else merely Empty). - intros []; destruct LEM as [H|H]; auto. + destruct (H (tr tt)). + apply (@merely_destruct Empty); try easy. exact _. - intros P. destruct LEM as [H|H]; apply equiv_path_iff_hprop. + split; auto. intros _. apply tr. exact tt. + split; try easy. intros HE. apply (@merely_destruct Empty); try easy. exact _. Qed.
imports: HoTT(TruncType ExcludedMiddle abstract_algebra) HoTT(PropResizing.PropResizing) HoTT(Spaces.Nat.Core Spaces.Card) HoTT(Equiv.BiInv) HoTT(HIT.unique_choice) HoTT.Sets(Ordinals Hartogs Powers GCH AC)