proposition: Lemma solutions_derivations B f n k : M_solutions B f (numeral n) -> k <= n -> M_opair (numeral k) (M_enc_stack (derivations B k)) ∈ f.
proof: Proof. Proof using M_ZF. intros H Hk; induction k; cbn. - apply H. - assert (Hk' : k <= n) by lia. specialize (IHk Hk'). destruct H as [_ H]. eapply H in IHk; eauto. + now apply numeral_lt. + apply M_combinations_step. Qed.
imports: Undecidability.FOL.ZF Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Sets(ZF) Undecidability.FOL.Sets.ZF Undecidability.FOL.Reductions.PCPb_to_ZF Lia Undecidability.PCP(PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb) Undecidability(Shared.ListAutomation) Morphisms