Proof using.
induction xs; intros.
- auto.
- simpl in *. repeat break_let.
rewrite IHxs. break_let. find_inversion. auto.
Qed.

(* Specifications: *)
Lemma fst_execute_log' :
forall log st tr,
fst (execute_log' log st tr) = tr ++ fst (execute_log' log st []).