proposition: Theorem H10p_to_PA2_satisfiability e : H10p e <-> exists M rho, (M, rho) ⊨ PA2 /\ (M, rho) ⊨ (encode_problem e).
proof: Proof. split. - intros H. exists Standard_Model, (empty_PA2_env _). split. + intros phi H1. now apply std_model_correct. + now apply H10p_to_PA2_standard_model_sat. - intros [M [rho [H1 H2]]]. eapply H10p_to_PA2_model_sat with (M := M) (rho := empty_PA2_env _). + intros phi H rho'. eapply sat_ext_closed. 4: apply H1, H. all: apply PA2_closed, H. + eapply sat_ext_closed_FO. apply encoding_first_order. apply encoding_closed. apply H2. Qed.
imports: PeanoNat Lia Vector List Undecidability.SOL(SOL PA2) Undecidability.Shared.Libs.PSL(Vectors) Undecidability.SOL.Util(Syntax Subst Tarski PA2_facts PA2_categoricity) Undecidability.Synthetic(Definitions DecidabilityFacts EnumerabilityFacts ListEnumerabilityFacts ReducibilityFacts) Undecidability.H10(H10p) Undecidability.Shared.Dec