Require Export Bmc.Core.
Definition k_ind_step (T : trans)
(P : prop) (i: nat) : Prop :=
forall ss : sseq,
path T ss 0 i -> safety_nth_offset P ss 0 i -> P ss.[i].
Definition k_induction_post (I : prop) (T : trans) (P : prop) (k: nat) : Prop :=
k_ind_step T P (S k) /\ safety_nth I T P k.
(**)
(**)

Theorem soundness_k_induction1 :
forall (I : prop) (T : trans) (P : prop),
( exists k, k_induction_post I T P k ) ->
forall (i : nat), prop_nth_init I T P i.