Proof using.
induction log; intros.
- auto.
- simpl. break_let. rewrite IHlog.
rewrite IHlog with (tr := [(eInput a, o)]).
auto.
Qed.

(* Specifications: *)
Lemma execute_log_correct' :
forall log st,
step_1_star st (snd (execute_log' log st []))
(fst (execute_log' log st [])).