proposition: Lemma inductive_sat (rho : nat -> V) x : (x .: rho) ⊨ is_inductive $0 -> M_inductive x.
proof: Proof. Proof using VIEQ M_ZF. cbn. setoid_rewrite VIEQ. split. - destruct H as [[y Hy] _]. enough (∅ = y) as -> by apply Hy. apply M_ext; trivial; intros z Hz; exfalso. + now apply M_eset in Hz. + firstorder easy. - intros y [z Hz] % H. enough (σ y = z) as -> by 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 Undecidability(Shared.ListAutomation)