proposition: Lemma nth_error_iP_Some {i} : i < length iP -> exists j n, nth_error iP i = Some (i, (j, n)).
proof: Proof. rewrite length_iP. move /in_iPI => [j [n /nth_error_Some_In_combineP /nth_error_combine_SomeP ?]]. by exists j, n. Qed.
imports: Relation_Operators Operators_Properties Lia PeanoNat List Undecidability.StackMachines.Reductions.CM1_HALT_to_SMNdl_UB.SMX_facts Undecidability.StackMachines.Util(Nat_facts List_facts) ssreflect ssrbool ssrfun