Definition explicit_safety (U:schedule)  (tr:event_trace) (st:machine_state) (m:mem): Prop:=
    exp_safety _ _ (fun U stm => is_true (ssrbool.isSome (halted_machine (U, fst(fst stm), snd(fst stm)))))
                   (fun U stm stm' => internal_step U (snd(fst stm)) (snd stm) (snd(fst stm')) (snd stm'))
                   (fun U stm U' stm' => external_step U (fst(fst stm)) (snd(fst stm)) (snd stm) U' (fst(fst stm')) (snd(fst stm')) (snd stm'))
                   (fun U stm => @valid (fst(fst stm),snd(fst stm), snd stm) U) U (tr,st,m).Definition explicit_safety_bounded (U:schedule)  (tr:event_trace)(st:machine_state) (m:mem): Prop:=
    exp_safety _ _ (fun U stm => is_true (ssrbool.isSome (halted_machine (U, fst(fst stm), snd(fst stm)))))
                   (fun U stm stm' => internal_step U (snd(fst stm)) (snd stm) (snd(fst stm')) (snd stm'))
                   (fun U stm U' stm' => external_step U (fst(fst stm))  (snd(fst stm)) (snd stm) U' (fst(fst stm')) (snd(fst stm')) (snd stm'))
                   (fun U stm => @valid_bound (fst(fst stm),snd(fst stm), snd stm) U) U (tr,st,m).

Lemma safe_kstep_trace_irr: forall U tr tr' tp m, safe_kstep (U,tr,tp) m -> safe_kstep (U,tr',tp) m.