proposition: Lemma nat_alg_is_preinitial_sec : is_preinitial_sec nat_alg_z.
proof: Proof. intro E. pose (x := pr2 E). unfold is_preinitial, algebra_section. use make_nat_alg_sec. - apply nat_rect. + exact (x (true,, fromempty) (empty_rect (pr1 E ∘ fromempty))). + intros ? fn. refine (x (false,, λ _, n) (λ b : unit, fn)). - cbn. apply maponpaths. apply funextsec; intro e; induction e. - reflexivity. Defined.
imports: UniMath.Foundations.Preamble UniMath.Foundations.PartA UniMath.Foundations.PartD UniMath.Foundations.UnivalenceAxiom UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Categories.Type.Core UniMath.Induction.FunctorAlgebras_legacy UniMath.Induction.PolynomialFunctors UniMath.Induction.W.Core UniMath.Induction.W.Fibered