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

Lemma snd_execute_log' :
forall log st tr,
snd (execute_log' log st tr) = snd (execute_log' log st []).