proposition: Theorem H10p_to_FA_ext_sat E : H10p E <-> ext_entailment_PA (embed E).
proof: Proof. split. - intros [sigma HE]. intros D I rho Hext H. eapply subst_exist_sat. apply problem_to_ext_model. + apply Hext. + intros. apply H. now constructor. + apply HE. + rewrite <-exists_close_form; apply embed_is_closed. - intros H. specialize (H nat interp_nat id nat_extensional). unfold embed in *. apply subst_exist_sat2 in H. now apply nat_sat'. apply nat_is_PA_model. Qed.
imports: Undecidability.Synthetic.Definitions Undecidability.FOL(Syntax.Facts Deduction.FullNDFacts Semantics.Tarski.FullFacts Semantics.Tarski.FullSoundness) Undecidability.FOL.PA Undecidability.FOL.Arithmetics(TarskiFacts DeductionFacts NatModel) Undecidability.H10(H10p) List Lia