proposition: Lemma inductive_sat (rho : nat -> V) x : (x .: rho) ⊨ is_inductive $0 -> M_inductive x.
proof: Proof. Proof using M_ZF. cbn. split. - destruct H as [[y Hy] _]. enough (H : ∅ ≡ y). { eapply set_equiv_elem; eauto. now apply set_equiv_equiv. apply Hy. } apply M_ext; trivial; intros z Hz; exfalso. + now apply M_eset in Hz. + firstorder easy. - intros y [z Hz] % H. enough (Hx : σ y ≡ z). { eapply set_equiv_elem; eauto. now apply set_equiv_equiv. apply Hz. } apply M_ext; trivial. + intros a Ha % sigma_el; trivial. now apply Hz. + intros a Ha % Hz. now apply sigma_el. Qed.
imports: Undecidability.FOL.ZF Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Sets(minZF ZF) Undecidability.FOL.Sets.ZF Undecidability.FOL.Reductions(PCPb_to_ZF PCPb_to_ZFeq PCPb_to_minZF) Undecidability(Shared.ListAutomation) Morphisms